PLACE N RETURN BOX to remove this checkout from your record. TO AVOID FINES mun on Of baton 6‘. duo. DATE DUE DATE DUE DATE DUE [rm—[f MSU I. An Milan-five Action/Equal Opportunity Indltution A STUDY OF CALCIUM OXALAIE CRYSTALLIZAIION IN KIDNEY STONE FDRHAIION BY Carol ShuhuiLin Hsu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1989 gdeOBK ABSTRACT A STUDY OF CALCIUH.OXALATE CRYSTALLIZATION IN KIDNEY STONE FORMATION By Carol ShuhuiLin Hsu This work was undertaken to gain a fundamental understanding of the physical and chemical mechanisms involved in the formation of kidney stones. Calcium oxalate is the major component of kidney stones and was chosen as the representative of stone minerals. Experiments were performed in a mixed-suspension, mixed-product-removal (MSMPR) crystallizer and in a flow chamber crystallizer. Previous work has shown the significant influence of urinary macromolecules on calcium oxalate crystallization. Both inhibition and promotion of crystal growth and crystal aggregation by these biopolymers has been reported; however, crystal growth and crystal aggregation mechanisms were not isolated in these studies. In work conducted with the MSMPR crystallizer, the linear population density model has been used to describe nonlinear data; crystal growth rates were obtained but crystal aggregation was neglected. Crystal aggregation has been suspected as being important in stone formation. In this work, experiments with the MSMPR crystallizer were conducted to characterize calcium oxalate crystallization. The entire MSMPR crystallization process was characterized by unsteady state, quasi-steady state, and steady state periods. At unsteady state, nucleation, crystal growth, and crystal aggregation all occurred in the system. In passing from the unsteady to the steady state, the crystallization process progressed with reduced nucleation and crystal aggregation . At steady state, only crystal growth was important. The unsteady state operation of the MSMPR crystallizer was of interest due to the presence of crystal aggregation; however, a clear understanding of the crystal aggregation mechanism depends upon knowing crystal growth kinetics. An approach was developed to obtain crystal growth rate for the entire process, including unsteady and steady states, by identifying the shifts of particle size distributions along the x-axis with time. The growth rates obtained from this method correlated well with the results of the linear population density model and the experiments with a flow chamber crystallizer. With this method, a model urinary biopolymer, poly-L-glutamate (PCA), was identified as a growth rate inhibitor. The flow chamber crystallizer experiments were conducted to investigate the effects of healthy and injured cultured epithelium (Maden Darby Canine Kidney (MDCK) cells) on the growth of single calcium oxalate crystals in the presence of flow shear. Healthy MDCK cells were found to inhibit crystal nucleation. The crystal growth rate of calcium oxalate monohydrate (COM) decreased in the order of crystal growth on glass > MDCK cells treated with HCl > MDCK cells treated with papain > healthy MDCK cells. The biopolymer additive, heparin, inhibited COM crystal growth on healthy MDCK cells. Copyright by CAROL SHUHUILIN HSU 1989 DEDICATION This work is dedicated to the author's parents, Mr. and Mrs. Lin-Kou Lin, the author's parents-in-law, Mr. and Mrs. Chin-Ann Hsu, Cheazone, Victoria, and Eleanor. ACKNOWLEDGMENTS I am greatly grateful to my research advisor, Dr. Daina M. Briedis, for her patience, encouragement, and financial support. Her guidance in the preparation of this dissertation and her friendship and encouragement make this work possible. Many thanks are due to Dr. William S. Spielman in the Department of Physiology for his kind help in the preparation of MDCK cells, to Dr. Mackenzie L. Davis in the Department of Civil and Environmental Engineering for lending the AA spectrophotometer, to Dr. Kalinath Mukherjee in the Department of Metallurgy, Mechanics, and Materials Science and Dr. Duncan F. Sibley in the Department of Geological Science for their lending of X-ray diffractometers. I also want to thank Dr. Donald K. Anderson and Dr. Charles A. Petty in the Department of Chemical Engineering and Dr. Paul Kindel in the Department of Biochemistry for their work on my guidance committee. Appreciation is also due the other faculty and staff of the Chemical Engineering Department. Special thanks are due to my parents and my parents-in-law for their care of my children and constant support, to my sisters and brothers (-in-law) for their love, and to many friends for their great concerns and their prayers. I thank my husband, Cheazone, for his understanding and encouragement. Finally, I thank the Creator for His everlasting care and love, especially for allowing me to know His great name, who is also my Father through Jesus Christ. vi TABLE OF CONTENTS CHAPTER 1 INTRODUCTION AND BACKGROUND 1-1. Introduction 1-2. Causes of Renal Stone Disease - 1-2-1. §£gn§_gompositions 1-2-2. Theories of Stone Formation - 1-3. Stone Formation and Crystallization Mechanisms 1-3-1. §upersaturation - 1-3-2. Nucleation 1-3-3. t owt 1-3-4. C ta re ation - 1-3-4-1. Diffuse Double Layer of A Single Particle in Solution - l~3-4-2. Repulsive Force of Two Approaching Particles 1-3-4-3. Attractive Force Between Two Particles 1-3-4-4. Total Energy of Two Interacting Particles and Stability of Particle Aggregation and Dispersion 1-3-4-5. Kinetics of Aggregation - l-h. Review of Experimental Methods l-A-l. Batch Experiments - 1-4-2. Constant Composition Method - 1-4-3. MSMPR Crystallize; 1-4-4. Sin e Crystal Growth - 1-5. Scope of Research - vii 10 13 16 l8 19 21 23 26 26 29 30 37 37 CHAPTER 2 SUPERSATURATION IN ELECTROLTTE SOLUTIONS 2-1. 2-2. 2-3. 2-4. 2-5. CHAPTER 3 PARTICLE SIZE DISTRIBUTIONS, POPULATION BALANCE EQUATIONS 3-1. 3-2. 3-3. 3-3-1. 3-3-2. 3-3-3. 3-3-3- 3-3-3- 3-3-3- 3-3-3- 3-4. 3-4-1. 3-4-2. 3-4-3. 3-5. 3-5-1. 3-5-2. Saturation Definition of Supersaturation - Fbrmation of Complexes Calculation of Supersaturations - Accuracy of the Calculated Supersaturations - AND DETERMINATION OF PARTICLE SIZE DISTRIBUTIONS Introduction Particle Size Distributions and Population Balance Equations - Determination of Particle Size Distributions a e at in e o the LZON (R) ste Co at' a icle ' Di t ' ut’o s w‘th the Instrument Counted Datg - Comments on the ELZONE(R) Analysis Program 1. "Histogram" Display - 2. Scaling Factor in Volume Data - 3. Blending of Volume Data - 4. Sampling Volume - Calibration Procedure and the 'Normalize' Function Original Calibration Procedurg Modified Calibration Procedure Calibrations with Known-Size Standard Particles - Modification of ELZONE's Data Analysis An Alternate Procedure for Sampling Volume Data Mani ulatio - viii 40 41 44 47 51 54 54 S8 60 60 62 63 63 65 67 68 69 71 73 74 75 76 CHAPTER 5 MIXED-SUSPENSION, MIXED-PRODUCT-REMOVAL CRYSTALLIZATION SYSTEM 4-1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2, Experimental Methods . . . . . . . . . . . . . . . . . . . . 30 4-2-1, Apparatus 33d Materials . . . . . . . . . . . . . . . . . . 30 4-2-2. Experimental Schemg . . . . . . . . . . . . . . . . . . . . 34 4-2-3. Measurement of Particle Size Distributions - - - - - - - - 85 4-2-4. ea u m nt 0 Ca cium oncentratio s - ~ - - . - - - - - - 86 4-2-5. Determination of Crystal Habigfiand Crystal Hydrate Ehéégfi . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4-3. Characterization of Calcium Oxalate Crystallization in.MSMPR.System,- . . . . . . . . . . . . . . . . . . . . . . 37 4-3-1. Caicium Oxalate Hydrates and Crystal Habit - - - - . - - - 87 4.3.2_ a 1; tr u o . . . . . . . . . . . . . . . . 33 4-3-3. oncentrat n the sta er - - - - - - ~ - 95 4-3-4. Crystallization Mechanisms - - - - - - - - . - - . - ~ - - 98 4-3-5. Eiigpr ofi PGA Additivg on Calcium Oxalate Crystallization . . . . . . . . . . . . . . . . . . . . . . 100 4_4. Crystal Growth Rate . . . . . . . . . . . . . . . . . . . . . 104 4-4-1. Apprppcp A; Growth Rate by Comparison of PSD Curves - - - - 105 4-4-2. Apprppch B: Growth Rate at Quasi-Steady State - - - - - - - 117 4-4-3. Co a o G owth e - - - . - - - - - - - - - - - - 120 4-5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 123 CHAPTER 5 THE EFFECT OF MDCK CELLS ON CALCIUM OXALATE GROWTH INCA FTOU’CHAMBER 5 - 1 . Inumtion . . . . . . . . . . . . . . . . . . . . . . . . 1 2 5 5-2. Precipitator Design and.Assemb1y - - - - ~ ' ~ - - - ' - - - 128 ix 5-3. 5-3-1. 5-3-2. 5-3-3. 5-3-4. 5-4. 5-4-1. 5-4-2. 5-4-3. 5-4-4. 5-4-4- 5-4-4- 5-4-4- 5-5. 5-5-1. 5-5-2. 5-5-2- 5-5-2- 5-5-2- 5-5-2- 5-5-3. 5-5-4. 5-5-4- 5-5-4- 5-5-5. Fluid Mechanics Properties in.Upper Chamber - Reypplds Number - ran tio Len Velocit o i e Shear_§£res§ Experimental Materials and Methods Maden Darby Canine Kidney Cells - Solution Preparation Experimental Set-Up - Experimental Scheme - 1. Viability Tests on MDCK Cells 2. Growth Rate Experiments: Effects of Additives and Tissues Surfaces 3. Induction Period in Closed Vessels Results and Discussion e V a i t Observations on Formation of Single Crystals 1. Nucleation 2. Heterogeneous Nucleation 3. Nucleation on Glass and on MDCK Cells - 4. Crystal Habit - Supersaturation Level in Flow Chamber - Growth Rate under Effects of Additivg and Iissue Surfaces - 1. Effects of Tissue Surfaces 2. Effects of Heparin Experimental Results and Stone Formation Hypothesis - 134 - 134 - 136 - 136 - 139 141 . 141 . 142 - 146 - 149 - 149 150 o 154 - 154 - 154 - 155 155 158 158 163 - 164 - 167 - 167 167 174 CHAPTER 6 SUMMARY, CONCLUSIONS, AND RECOMMENDATION 6-1. Sun-nary and Conclusion 6-2. Recommendations for Future Research - APPENDICES APPENDIX A MIL COMPUTER PROGRAMS - APPENDIX B CALIBRATION DATA OF THE ELZONEUO SYSTEM AND THEIR APPLICATIONS APPENDIX C COMPUTER PROGRAM 80XYABCV.BAS - APPENDIX D PHYSICAL PROPERTIES OF SOLUTIONS APPENDIX E COMPUTER PROGRAM N'DIBJ‘OR - APPENDIX 1“ SHII-TS OF THE v(L,t) CURVES - APPENDIX G COMPUTER PROGRAM GMSMI’R- BIBLIOGRAPHY xi 178 182 183 191 199 202 203 ' 209 216 221 1-1 1-2 2-1 2-2 2-3 3-1 4-1 4-2 4-3 4-4 4-5 4-6 4-7 5-1 5-2 5-3 5-4 5-5 LIST OF TABLES Growth rate correlations of calcium oxalate crystals at 30’C . . . . . . . . . . . . . . . . . . . . . Values for a - Thermodynamic solubility products of calcium oxalate hydrates at 37° C . . . . . . . . . . . . Stability constants SCOM i calculated with different stability constants for solutions containing CaClz, K2C20‘, and KCl (ionic strength buffer); ionic strength - 0.15 M F factors calculated from comparison of reported and calculated differential volume data. (abbreviated version) - - Geometrical parameters of the crystallizer ~ Experimental scheme Crystal hydrates - Correlations of growth rates with supersaturations Growth rates at quasi-steady state Comparison of growth rates with previous work Growth rates at steady state - Reynolds number, transition length, and maximum shear stress for flow chamber precipitator - - ~ - - - Original Earle's medium recipe (1X) A typical solution composition for 2 2- [Ca +1 - [C20, 1 - 0.001 M MDCK cell viability with shear stress Calculated induction periods at the reactant concentrations used in single crystal experiments - - - - - - - . . xii 14 25 41 48 52 64 83 85 88 ~ 116 120 121 - 122 - 135 ° 143 - 145 - 155 - 156 5-6 Calcium concentrations of some effuent samples - - - - - - - - 165 5-7 Rate constants of COM growth - - - - - - - - - . - - ~ ~ - - - 168 xiii 1-1 1-2 1-3 1-4 3-1 (a) (b) (e) (d) 4.1 4-2 4-3 4-4 4-6 4-7 4-3 LIST OF FIGURES Screw dislocation on crystal surface and incorporation of solute molecules into crystal lattices - - . - The diffuse- double layer and potential profile around a particle - . . Interaction energy of particles at distance H - Semi- log population density plot; linear and curved distributions - . . . . . . . . . Blending of a simulated vch and v(L) data; linear population density plot, volume density, differential volume per channel, v h by blending routine. The thicker dash lines represent the data from the 76 pm aperture tube; the thinner solid lines represent the data from the 300 pm aperture tube. Sampling volume is 1 ml for both data sets - - - - - - - - - - - Repeating unit of polyglutamate - MSMPR experimental set-up - 400 ml hybrid tank Photomicrograph of a sample taken from the experiment with 3 mM calcium and oxalate concentrations . 2 2, Population density plot A; [Ca +1 - [020, 1 - 3 mM, without PGA, r - 8 min, sampling time - 74 min 2+ 2- Population density plot B; [Ca ] - [C20, ] - 2 mM, PGA - 100 ppm, 1 - 8 min, sampling time - 68 min v(L,t) particle size distribution; 2 2- [Ca +1 - [0,0, 1 - 2 mM, PGA - 100 ppm, r - 8 min, sampling time - 68 min - v(L,t) with extrapolated small size range - xiv 12 17 22 36 66 80 81 82 89 90 91 92 93 4-9 (a) 0)) 4-10 4-11 (a) (b) (C) 4-14 4-15 4-16 4-17 4-18 4-19 4-20 4-21 5-1 5-2 5-3 5-4 5-5 Total particle numbers and the third moments for the base run; total particle numbers, NT’ the third moments, M3 . . . . . . . . . . . . . . . . . . . . 96 Calcium concentrations in the filtrates and in the suspensions; 2 2- [Ca +1 - [020‘ ] n 2 mM, PGA - 10 ppm . . . . . . . . . . . 97 Effects of PGA additive on total particle numbers, the third moments, and the maximum particle sizes; effect of PGA on total particle numbers, NT’ effect of PGA on the third moments, M3, effect of PGA on the maximum particle sizes, Lmax - - - . - - 101 Effect of PGA additive on total calcium concentrations in 2 2- suspensions; [Ca +] - [CQO‘ ] - 2 mM, 1 - 8 min - - - - - - 102 Effect of PGA additive on calcium concentrations in 2 2- filtrates; [Ca +1 - [0,01 1 - 2 mM, 1 - 8 min - - - . - - . 103 Shift of v(L, t) curve and calculation of growth rate; PGA - 100 ppm . . . . . . . . . . . . . . . . . . . . . . 107 SCOT c in the crystallizer in experiments with varied residence time . . . . . . . . . . . . . . . . . . . . . . . 108 Growth rates and supersaturations for the experiment with 1 mM calcium and oxalate concentrations - - - . - - - - . - - 110 Growth rates and supersaturations for the base run - ~ - - - 111 Growth rates in experiments with varied residence time - - - 112 Growth rates in experiments with various levels of PGA additive . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Growth rates in experiments with varied calcium and oxalate concentrations - - - - - - - - . - - - - - - - - - - 114 Langmuir isotherm of PGA adsorption on calcium oxalate crystals . . . . . . . . . . . . . . . . . . . 113 Repeating unit of heparin - - - - - - - ~ - . - - - - . - - - 126 Top view of growth chamber - - - - - - - - - - - - - - - - - 129 Exploded side view of the growth chamber - - - - - - - - - - 130 Upper chamber holder . . . . . . . . . . . . . . . . . . . . 131 Lower chamber holder . . . . . . . . . . . . . . . . . . . . 132 5-6 5-7 5-8 5-9 5-10 5-11 5-12 5-13 5-14 5-15 5-16 5-17 5-18 (a) (b) (C) 5-19 (3) (b) (C) 5-20 5-21 5-22 F-l Modeling coordinates of the flow chamber precipitator - Illustrative velocity profiles; flow rate at 40 mI/min Shear stress at interface surface; flow rate at 40 mI/min - Crystal growth flow system Induction period for the formation of CaC20‘- COM crystals on MDCK cells; Run 1102087, T - T - 1.5 mM, taken after experiment Ca Ox COM crystals on glass in an experiment with MDCK cells, a spot with cells detached; Run 1102087, TCa - TOx - 1.5 mM, taken after exper1ment COM crystals on MDCK cells treated with papain; Run 2110487, T - T - 1.25 mM, t - 15 min Ca 0x COM crystals on MDCK cells treated with HCl; Run 2102087, T - T - 1.5 mM, t - 27 min - Ca Ox COM crystals on glass; Run 2092387, TCa - Tox - 1.25 mM, t - 35 min - COM crystals on MDCK cells with 100 mg/1 heparin; Run 1082587, TCa - TOx - 1.0 mM, t - 40 min - Effluent calcium concentrations - Growth rate versus (SCOM,c)b; crystal growth on glass and on healthy MDCK cells, crystal growth on HCl treated MDCK cells, crystal growth on papain treated MDCK cells Growth rate versus (SCOM,i)b; crystal growth on glass and on healthy MDCK cells, crystal growth on HCl treated MDCK cells, crystal growth on papain treated MDCK cells Comparison of growth rates at any constant (SCOM,c)b° Inhibition of COM crystal growth on MDCK cells by heparin; TCa - T0x - 1.0m Inhibitory effect of heparin on COM crystal growth follows Langmuir adsorption isotherm - Shifts of v(L,t) for the experiment with PGA - 100 ppm; xvi 133 138 140 147 157 160 160 161 161 162 162 166 169 170 172 173 175 F-2 F-3 F-4 F-5 F-6 sampling time: 3 to 51. 6 min. Arrows and "*" identify points compared among the PSD curves - - - - - Shifts of v(L, t) for the experiment with PGA - 100 ppm; sampling time. 51. 6 to 131 min - Shifts of v(L, t) for the experiment with PGA - 100 ppm; sampling time: 131 to 206 min Shifts of v(L,t) for the base run; sampling time: 2.5 to 42.5 min - Shifts of v(L,t) for the base run; sampling time: 42.5 to 98.5 min Shifts of v(L,t) for the base run; sampling time: 98.5 to 194.5 min - xvii 210 211 212 213 214 215 CHAPTER 1 INTRODUCTION AND BACKGROUND l-l. Introduction The objective of the work described in this dissertation was to gain a fundamental understanding of the physical and chemical mechanisms involved in the formation of kidney or renal stones (calculi). Renal calculus formation is one of the oldest diseases known to man. A bladder stone found in an Egyptian mummy was estimated to have been formed five thousand years ago (Kraljevich, 1981). While the vesical calculus (bladder stones, mostly uratic in composition and found in boys) is more common in undeveloped and developing countries, renal calculus (mainly calcium oxalate, with or without apatite, found in adults) is more common in developed countries (Pyrah, 1979). In the U.S., hospital admissions for the treatment of kidney stones were about every one to four persons per 2,000 of the population (ibid.); $47 million was spent annually for the treatment of these patients (Finlayson, 1974). Besides the economic consumption, the patients also suffer emotional and physical trauma due to the extreme pain experienced when a sharp stone passes through the urinary tract. Research in nephrolithiasis (kidney stones) has often focused on metabolic and occupational causes of the disease. It has been suggested that a richer diet-~a higher intake of calcium-containing foods and animal protein--might be one of the factors which causes renal calculus formation. It was also found statistically that people with relatively sedentary occupations or individual working in high temperature environments have higher incidence of stones (Pyrah, 1979). Yet, consensus on the mechanisms of the formation of kidney stone has 1 not been achieved. Because kidney stone formation is a crystallization-precipitation process as considered from the chemical engineer’s point of view, this research is motivated by the desire for an understanding the physical and chemical mechanisms of kidney stone formation and crystallization phenomena. 1-2. Causes of Renal Stone Disease Formation of renal stone must be initiated from the molecular level at which nuclei of calculi are formed in unine. The small nuclei may attach to the surfaces of the urinary tract and grow to larger stones or more than one nuclei may combine to form larger stones that threaten a person's health. Composition of stones will be reviewed to gain priliminary understanding of the disease since the composition and morphology of kidney stones gives some information on the possible etiology of stone formation. 1-2-1. Stone Compositions Most urinary stones usually contain about 97.5% of one or more crystalline substances and 2.5% colloidal matrix (Kraljevich, 1981; Pyrah, 1979). Matrix calculi, a separate class of stones, contain 65- 85% matrix (macromolecular compounds) by weight. Sixty-three to 88 percent of stone-forming patients pass stones composed entirely or predominantly of calcium oxalate (Pyrah, 1979; Pak, 1987). It was for this reason that calcium oxalate was chosen for this study. Calcium oxalate itself is an interesting crystal for study. There are three hydrates of calcium oxalate crystals: calcium oxalate monohydrate (COM), or whewellite, CaC204-H20, which forms as monoclinic crystals; calcium oxalate dihydrate (COD), or weddellite, CaC204o(2+x)H20, x s 0.5, forming as tetragonal crystals; and calcium oxalate trihydrate (COT); CaCQO‘-(2+x)H20, 0.5 s x s 1, which appears as triclinic crystals. Because COM is thermodynamically the most stable phase, COT and COD may precipitate first but transform to COM gradually. COT has been suggested as the precursor in stone formation (Gardner, 1976; Tomazic and Nancollas, 1976), but is a rare component in urinary stones (Pyrah, 1979). Heijnen et a1. (1985), however, found that COT was common in the outermost layer (up to a few hundred microns thick) of one third of the largely whewellite stones they examined. Phosphate is another component subordinate to calcium oxalate. The phosphates are identified as apatite (hydroxyapatite, Ca1°(PO‘)6(OH)2, carbonate-apatite, Ca1°(PO‘-C030H)6-OH2), struvite (MgNH‘PO‘-6H20), brushite (CaHPO‘~2H20), whitlockite (fi-Ca3(PO4)2), and octacalcium phosphate (Ca3H2(PO‘),~6H20). Other urinary stone components are uric acid, uric acid dihydrate, cystine, and xanthine (Pyrah, 1979; Pak, 1987). These minor components were not considered in the work. Although the matrix content in stones is small, it is believed to have a complicated involvement in stone formation and composition. Cross-sections of the stones or of the decalcified stones (stones immersed in EDTA solution for about 40 days) show a concentrically laminated sturcture of the organic matrix, giving an indication of the possible "glue-like" involvement of the matrix in stone formation. The organic matrix is composed of two thirds mucoproteins and one third mucopolysaccharides by weight-~mainly high molecular weight substances found in urine. Some of the proteins in the mucoprotein fraction are hexosamine, hexuronic acid, sialic acid, and glucosamine. About half of the mucoprotein contains aspartic and glutamic acids (Kraljevich, 1981). The mucopolysaccharides may be chondroitin sulphates or may contain sugars, such as glucose and mannose. A urinary mucoprotein discovered by Tamm and Horsfall was found at elevated levels in patients with renal stone disease. Another mucoprotein, matrix substance A, has been present in the matrix of infective stones (Pyrah, 1979). 1-2-2. Theories of Stone Formpripp Several theories concerning stone formation have been prevalent-- the matrix theory, supersaturation theory, and inhibitor or promoter theory. These will be discussed in the following paragraphs. Because of the special structure of the colloid matrix in urinary stones (described above), the "matrix theory" of stone formation has been suggested. Boyce (1969, 1973) studied the relationship of the crystals to the laminar organic matrix and concluded that stones are formed by inclusion of the crystalline matter in a preformed organic matrix. Others suggested that aggregation of the small crystalline particles is the initial step and urinary macromolecules are then adsorbed onto the particle aggregates (Leal and finlayson, 1977). With the exception of matrix stones (stones without crystalline matter), most stones contain crystalline material. Calcification of the crystalline material requires the urine to be supersaturated with the crystalline mineral. This forms the basis of supersaturation theory of stone formation. For example, calcium oxalate stones are found in patients with hypercalciuria or with hyperoxaluria, the symptoms of excess calcium or oxalate in urine. Since the urine from both the stone-formers and non-stone-formers are usually supersaturated with calcium oxalate and calcium phosphate (Coe, 1978), supersaturation of urine with respect to crystalline minerals is a necessary but insufficient condition for stone formation. Either the matrix theory or the supersaturation theory alone cannot explain the stone formation. Yet, these two theories are not contradictory to each other when considered in the context of the inhibitor (promoter) theory. The inhibitor theory suggests that some substances in the urine of non-stone-formers inhibit stone formation even though the urine is supersaturated with respect to calcium oxalate or calcium phosphate. These inhibitors may be absent or present in smaller amounts in the urine of stone-formers. The deficiency in the concentration of inhibitors results in the formation of stones. The converse to this may be that the presence of some promoter materials in the urine of stone-former results in stones. Both the inhibitor and the promoter theories always emphasize the relationship of the urinary colloid (organic matrix) with the calcifying process. Although inorganic substances such as pyrophosphate, citrate, and magnesium have been reported as being inhibitors of calcium oxalate crystallization, only five to 20 percent of the inhibition by urine is due to the presence of these inorganic substances (Gardner and Doremus, 1978). Studies of urinary mucopolysaccharide and mucoprotein inhibitors have been more prevalent. Among the substances studied are hyaluronic acid, chondroitin sulfate, heparin, ribonucleotides, and the acidic peptides containing aspartic or glutamic acid residues (Vermooten, 1956; Ito and Coe, 1977; Gjaldbaek, 1982; Drach et a1., 1982). The Tamm-Horsfall mucoprotein and matrix substance A which were suggested as possible promotors to the formation of urinary stone were found to behave as inhibitors of the growth of calcium oxalate crystals (Robertson et a1., 1981). The "free particle" theory describes one possible physical mechanism of stone formation. This theory suggests that small crystals are aggregated in the urinary tract to become large particles that become kidney stones. In studies of the urine of stone-formers and of normal persons both supersaturated with calcium oxalate, the crystals found in the urine of stone-formers were greater in number, larger in size, and included a larger fraction of aggregates (Rose, 1982; Pyrah, 1979). Contrary to the free particle theory, evidence of the possibility of a nidus for stone formation--a fixed nuclei on the walls of renal system--supports the "fixed particle theory". In the fixed particle theory, a nidus may be formed on the tissue surface through heterogeneous nucleation or may be formed in the urinary system as free particles and then adsorded or "glued" to the tissue possibly through the interaction of crystals and high molecular weight urinary materials (possibly the matrix type material or biopolymers) (Robertson, Scurr, and Bridge, 1981). in yirrp studies of the effects of these macromolecular substances on the chemical and physical mechanisms of the mineralization process may provide a clear understanding of the involvement of biopolymers in the etiology of kidney stone formation. 1-3. Stone Formation and Crystallization.Mechanisms The physicochemical study of urinary calculi formation, a biological mineralization process, is best described by crystallization theories. Supersaturation of the calculus mineral in the renal fluid is required for the process to take place. In this work the process is envisioned as one in which nuclei are formed, and they grow and/or aggregate to form particles of the sizes which cause physiological problems. In crystallization processes, supersaturation of the solute substances may be generated by cooling the saturated or undersaturated solution, by evaporating the solvent, or by mixing of two reactants to increase the solute concentration in solution. In this study, supersaturation of calcium oxalate was generated by mixing of calcium and oxalate solutions. By convention, such a process is called a precipitation process; however, the term crystallization has been used commonly in calcium oxalate urolithiasis studies. Crystallization and precipitation will be used interchangeably in this study. Supersaturation is the necessary pre-condition for the crystallization process; therefore, a broad overview of supersaturation will be introduced followed by a review of nucleation, crystal growth, and crystal aggregation, the three major mechanisms of crystallization potentially important in kidney stone formation. A definition and calculation of supersaturation suitable for the calcium oxalate system will be discussed in Chapter 2. 1-3-1. u sa tio A saturated solution is the solution in equilibrium with the solid phase of the solute, which is the sparingly soluble salt, calcium oxalate, in this study. Solutions may be undersaturated or supersaturated with the solute depending on the amount of solute contained in the liquid phase. The supersaturated solution contains more solute than the amount of solute at equilibrium, and the undersaturated solution contains less solute. Thermodynamically, this is described in terms of the activity and chemical potential. The chemical potential of the solute in the liquid phase, p, is related to the activity of the solute, a, by p - p* + RT 1n a (1‘1) where R is the universal gas constant, T is the absolute temperature, and p* is the chemical potential at the reference state where the activity is 1. At saturation, the chemical potential of the solute, go, is also related to its activity, a., such that * 1.4,, - p + RT 1n a° (1-2) The activity of a solute in a supersaturated solution is higher than the activity at saturation, which is higher than the activity of the solute in a undersaturated solution. The same relationships pertain to the chemical potential. The chemical potential of the solute in a undersaturated solution will be increased by dissolving the solute from the solid phase. On the other hand, the higher value of the chemical potential in the supersaturated solution may be lowered by transferring solute from liquid phase to solid phase. In the latter process, solid phase is formed through either nucleation or chemical deposition, i.e., crystal growth. 1-3-2. Nu ea 0 Nucleation is the formation of new small particles from the solution phase. The chemical potential of a solute in a supersaturated solution provides the driving force for the formation of such a solid phase. The nucleation process results in a negative chemical potential change. Because of the looser and weaker bonding of the ions or molecules at the surface of a solid phase compared to ions in the bulk solid volume, formation of a solid surface requires a positive free energy change; this excess energy is the surface tension of a macroscopic body. As a result, the total free energy change of a system due to formation of a solid phase is comprised of two parts: the free energy change resulting from transfer of solute from the liquid to solid phase and the surface free energy change. This may be written as (Nielsen, 1964) AG - (,u° - p)An + 0A (1-3) where AG is the total free energy change of the system, An is the number of solute molecules or ion pairs transferred from the liquid phase to the solid phase, a is the surface energy per unit surface area, and A is the surface area of the newly formed particle. The process will occur spontaneously if the total free energy change is negative. Because of the positive surface energy term on the right hand side of Equation (1-3), nucleation may not necessarily occur in a supersaturated solution, i.e., the solution may be metastable. The region between saturation and the supersaturation where nucleation finally does occur is termed the metastable region (Finlayson, 1977). When considered on a microscopic scale, ion or molecular clusters are formed and may be redissolved in a supersaturated solution under the following conditions. The smaller the cluster, the larger is the surface energy per unit mass. For the clusters having a size that possesses a surface energy resulting in a negative total free energy in Equation (1-3), the cluster will survive and grow in the supersaturated 10 solution. The size of the surviving clusters is called critical size, and such clusters are called nuclei. Formation of nuclei may be termed as primary or secondary. Homogeneous and heterogeneous nucleation are primary. Formation of nuclei in a solution as the result of supersaturation alone is defined as homogeneous, and formation of nuclei in the presence of foreign particles is heterogeneous nucleation. Secondary nucleation requires the existence of crystals and is induced by small embryos generated by attrition from existing crystals either under the influence of fluid ‘shear or from collision with other crystals, reactor wall, or propeller (McCabe, Smith and Harriott, 1985). In the consideration of the energetics of the nucleation process, formation of nuclei from homogeneous nucleation requires a high supersaturation to compensate for the free energy required for the formation of a new surface. In heterogeneous or secondary nucleation, because of existing surfaces provided by the foreign particles or small embryos from attrition from crystals, nuclei may be formed in a solution of a relatively lower supersaturation, usually in the metastable region. 1-3-3- W Supersaturation in a solution may also be relieved by the addition of solute to crystals existing in solution, a surface reaction that incorporates solute molecules or ion pairs into the crystal lattice. This is commonly refered to as crystal growth. The consumption of the solute material at the crystal surface due to the surface reaction results in a concentration difference between the bulk solution and the solid-liquid interface. This concentration difference provides a 11 driving force for mass transfer across the concentration gradient towards the crystal surface. This description of crystal growth is modelled as a two-step mechanism, a mass diffusion step and a surface reaction step. The process is diffusion-controlled if the growth rate is limited by the diffusion step; the process is surface-reaction controlled if the growth rate is limited by the lattice integration step (McCabe et a1., 1985). When the growth process is surface-reaction controlled, mass transfer resistance in the solution is negligible. Several theories describe the surface-reaction controlled crystal growth as reviewed by Ohara and Reid (1973) include the mononuclear, polynuclear, and birth and spread two-dimensional nucleation theories and the Burton-Cabrera- Frank (BCF) surface diffusion and dislocation model. Two-dimensional mucleation theories suggest integration of crystals through surface formation of critical-sized embryos (surface nucleation) with assumption of zero, infinite, or intermediate growth velocity on the subsequent lateral spreading across crystal surfaces. However, growth rates predicted from these theories have been much too low in comparison to experimental data (ibid). The widely used BCF surface diffusion and dislocation model suggests formation of crystals through growth spirals which originate at a screw or edge dislocation as shown in Figure 1-1. First, solute molecules are adsorbed on the flats between steps of the growth spirals. After migration to the steps under a surface concentration gradient, the molecules are then incorporated into kinks on the spiral steps (lattice incorporation). The physical reasoning and mathematical derivation of this theory has been carefully examined by Ohara and 12 Figure 1-1 Screw dislocation on crystal surface and incorporation of solute molecules into crystal lattices. Reid (ibid), and the linear growth rate normal to the crystal surface is expressed as: G - CBT SBln(l+SB)tanh[C7/T1n(l+SB)] (1-4) where C6 - 2DSCSEkfi70/l9xsa (1-5) and C7 - 19an/2kxs (1-6) The symbols used in the above equations are summarized below: T SB CBE Ds CSE k l3 70 absolute temperature; bulk supersaturation, - (CB-CBE)/CBE; equilibrium concentration in the bulk solution; surface diffusivity; equilibrium concentration of surface adsorbed molecules if the bulk supersaturation were unity; Boltzmann constant; a retardation factor relating bulk supersaturation and surface supersaturation; kink retardation factor, related to ledge diffusion rates; 13 x3 mean distance traveled by an adsorbed molecule before it desorbs; a interfacial tension; Vm effective volume of a molecule. When SB is very low so that tanh(C7/TSB)zl, the growth rate model of Equation (1-4) predicts a second power dependence of growth rate to the bulk supersaturation, S At larger SB where tanh(C7/TSB)=C7/TSB, B' the growth rate is linearly dependent on the bulk supersaturation. In calcium oxalate crystallization studies, growth rate results have been reported in the form growth rate - kg sb (1-7) where k3 is a proportional constant, and the growth rate and the independent variable, 5, have several diverse definitions as shown in Table 1-1. Table 1-1 also shows the b values for each individual growth rate correlation and the growth rate predictions by the BOP theory. 1-3-4. a A e tio Because of the excess energy confined within surface tension, a suspension containing many fine crystals is not thermodynamically stable. Particle size redistribution may take place by Ostwald ripening or by crystal aggregation. In the ripening process, smaller crystals dissolve to provide the solute concentration for larger ones to grow; in aggregation, smaller crystals aggregate to form larger ones. Both of these mechanisms result in changes of size distributions of crystal suspensions with respect to time (Nielsen, 1964). Crystal aggregation is more prevalent in the calcium oxalate crystallization process and is the focus of this section. 14 Table 1-1 Growth rate correlations of calcium oxalate crystals at 37°C. [Growth rate, 5, b] BCE mpdei (Ohara and Reid, 1973) [G, SB’ Equation (1-4)] for high SB [G, 88’ 2] for tanh(C7/TSB)z1 [G, SB’ 1] for tanh(C7/TSB)zC7/TSB G, SB: described in context. M e nd Sm'th 9 5 [iv—“3:“ ]' [p.2+1-,..2+],], 2] 2+ 2+ [Ca 10: [Ca ] at equilibrium. * Drach pt ai,, 1978 [GL' C-03, 2.55] G : linear growth rate from MSMPR experiments (Section 1-4-3), z2G; C : COD solubility concentration in the MSMPR process at steady state operation; C : COD solubility concentration at equilibrium. heeh and Nanc as 1980 _ _dm_ _ [ dt ]9 (“1 m0): 2] m - [CaCzO‘]; mo: m at equilibrium. omaz and Na 0 as 1980 [[ (1:531 [8032+][Cioiz'lygzlk'l‘spcomj' 2] yzz activity coefficient of [Ca +] and [020‘ ']; Ksp,COM: thermodynamic solubility product of COM. Garside 9 2+ 2- 2 ([Ca ][C2O4 1Y2 -Ksp) G 2 [ L’ K ’ ] 5? GL: linear growth rate from MSMPR experiments (Section 1-4-3), =2G; Ksp: thermodynamic solubility product of COM or of COT. (cont’d.) 15 Table 1-1 (cont'd.). 2- 2 K H ’ sp,COM The three b values are for COM growth on the respective crystal faces, 110, 010, and i01. (b-llO, b-010, b-iOl) (2.15, 2.20, 2.47) 2+] *: Temperature unknown. In the discussion of biological crystallization, the terms aggregation and agglomeration are usually used interchangeably. A standardized nomenclature, however, was proposed by Randolph and Drach (1980) as follows: agglomerates are particles held by weak cohesive forces, such as electrostatic forces, and aggregates are held by strong intermolecular bondings. Agglomerates can be broken up by flow shear whereas aggregates cannot. The characteristics of aggregates were deemed to be more appropriate for this study. According to the International Union of Pure and Applied Chemisty, colloids are defined as particles having sizes ranging from 1 to 1000 nanometers (1 pm) (Hirtzel and Rajagopalan, 1985). Although the primary calcium oxalate particles may be about 10 pm in size and the aggregates may be over 50 pm, the theories underlying both aggregation and colloid coagulation phenomena are fundamentally and physically the same and have been applied in this work. The Deryaguin-Landau-Verwey-Overbeek (DLVO) theory describes coagulation of particles by considering repulsive and attractive forces of two interacting particles and explains the stability of particle aggregation or dispersion by the resultant total energy of the repulsive and attractive forces. The repulsive force of two approaching particles is due to the electrical charges surrounding each 16 single particle. The electrical environment of a particle in solution is discussed in the next section. 1-3-4-1. Diffuse Double-Layer of A Single Particle in Solution The electrical environment of particles in electrolytic media has been described by the theory of diffuse double-layer (Sonntag and Strenge, 1972; Hirtzel and Rajagopalan, 1985). In an electrolytic medium, the crystal surface may be positively or negatively charged depending on the concentration of potential determining ions, the ions that make up the crystal lattice. The "double-layer" is envisioned as two layers of charge localization-- one on the surface of the crystal and the other in a diffuse region spreading into the solution. For a negatively charged particle surface, the diffuse double layer is pictured in Figure 1-2. Potential determining ions and other ions may be adsorbed on the particle surface to form an "inner Helmholtz layer". These specifically adsorbed ions in the inner Helmholtz layer are dehydrated or dehydrated at least in the direction closest to the particle. Counter ions, ions of charges opposite to the specifically adsorbed ions, surround the inner Helmholtz layer to form the Stern layer or outer Helmholtz layer. Outside the Stern layer is the Gouy layer, a diffuse electrical layer containing counter ions, specifically adsorbed ions, and indifferent ions. The density of counter ions is higher close to the Stern layer and decreases gradually until electroneutrality is established at some distance from the particle surface. The potential energy profile of the diffuse double-layer is also shown in Figure 1-2, where p. is the surface potential and pd is the Stern potential, the potential at the Stern layer (Shaw, 1980). The 17 Bulk of solution Diffuse layer Plane of shear C) \ C) lo 9 \ Inner Helmholtz layer ()6) —~- Particle Stern layer (d)-d" Potential E Q --..---.._ Distance Figure l-2 The diffuse-double layer and potential profile around a particle. 18 electrokinetic g (zeta) potential, the potential at the surface of shear, is experimentally obtainable from electrophoresis measurement. The Stern potential may be estimated by the f potential with small errors (ibid; Adair, 1981). 1-3-4-2. Repulsive Force of Two Approaching Particles Because the crystal particles are surrounded by electrical charges, repulsive forces arise as a consequence of overlapping of the double layers when two particles approach each other. The interaction energy for two approaching particles may be described by the Poisson equation which relates net volume charge density to electrostatic potential and by the Boltzmann distribution which relates point charge density to electrical potential. The combined Poisson-Boltzmann equation has no exact analytical solutidn. Approximate solutions have been developed assuming two extreme cases for this interaction, maintenance of constant surface potential during the interaction and constant surface charge density. For two spherical particles with radii r1 and r2, nrl and nrz (5 defined in Equation (1-11)) much greater than one, and pd less than 50 mV, the repulsion potential for the constant charge case, V a, and for the constant potential case, VRw, are obtained (Shaw, 1980; Harnby et a1., 1985; Hiemenz, 1986) from the following equations: a 4 ———-—r1r2 21 1 VR - - we r1 + r2 pd n[ - exp(-KH)] (1-8) ¢ 4 r1 r2 2 VR - we r1 + r2 ¢d 1n[1 + exp(-KH)] (1-9) where e is the permittivity of the dispersion medium and H is the shortest distance between Stern layers. Since the dielectric constant 19 of a material, er, is the ratio of its permittivity to the permittivity of vacuum, 5., the permittivity 5 may be expressed as e - er 6. (1-10) The parameter n may be related to the concentration of ions in the solution according to (Hiemenz, 1986): 2 1000e N .-[ A 2 h ekT 2 21 C1 ] (1-11) i where e is the unit electric charge, 1.6 x 10.19 Coulombs, NA is the Avogadro’s number, 6.02 x 1023, and zi and Ci are the charge and concentration of the ionic species 1, respectively. The summation term in Equation (1-11) is twice the ionic strength, I, and may be replaced with 21. The parameter 5 in Equations (1-8) and (1-9) has the unit of reciprocal length; therefore, 5'1 has units of length and is a measure of the thickness of the double layer. Since 5 is proportional to JI, the thickness of double layer decreases with increase of ionic strength in solution. 1-3-4-3. Attractive Force Between Two Particles Particles are attracted to each other under the influence of van der Waals force. The van der Waals force, or London dispersion force, is the interaction of the induced dipoles from the molecules of each particle. At large particle separations, due to the propagation of the electromagnetic interaction by the speed of light, the van der Waals attraction is retarded. For spherical particles at the separation, H, smaller than ca. 10 nm, the potential energy of the non-retarded van der Waals force are (Vold and Vold, 1983; Hiemenz, 1986) 20 2 r1 r2 2 r1 r2 ' ' 2 2 + 2 2 A 6 R - (r1 + r2) R - (r1 - r2) 2 2 + 1n 2 2 (1-12) R - (r1 - r2) where A is the van der Waals-Hamaker constant and R is the separation distance between the centers of the spheres. For two slab-shaped particles of the same size, the energy of van der Waals forces (Vold, 1954) shows a prefered approach between particles--with the wide sides facing each other and the long axes parallel rather than perpendicular to each other. For the particles with width w, thickness c w, and length czw and the long axes parallel t to each other, this attraction energy is (ibid) 2 2 3/2 2 2 3/2 Aczw (w + H ) (w + (H + 2c w) ) t V - ' 2 2 + 2 2 A 12" w H w (a + 2ctw) 2 2 3/2 2(w + (H + ctw) ) 2 2 w (H + ctw) Ac w 2 _i_ 1 2 8n H + H + 2ctw - H + ctw (1-13) This equation is valid for the distances H smaller than the length of particles. H is the shortest distance between the Stern layers of two approaching particles. For particles of the same volume, particles of slab shapes exhibit greater attraction energy than spherical particles. At large distances of separation, there is no difference between Equations (1-12) and (l- 13), and both equations approach the attraction law for atoms, which is 21 proportional to the inverse of particle separation to the sixth power (ibid). 1-3-4-4. Total Energy of Two Interacting Particles and Stability of Particle Aggregation and Dispersion The total energy of interaction of two approaching particles is the sum of the repulsion and attraction energies: vT - vA + vR (1-14) Figure 1-3 illustrates the energy distribution as a function of H. The total energy curve as shown in Figure 1-3 helps explain the formation and stability of aggregates or the stability of a dispersed system. A positive potential energy is a repulsive force, and a negative potential energy is an attractive force. A system tends to lower its total energy. From the illustration in Figure 1-3, two particles may form strong aggregates at a small interparticle distance, the "primary minimum", after overcoming the energy barrier of the positive maximum of the total energy. The higher the energy maximum, the greater the energy barrier. On the other hand, reversible flocculation may occur at a larger interparticle distance at the secondary minimum as seen on the total energy curve. In a system without shear, aggregates or flocs are formed under the influence of the thermal energy, kT. If the positive energy maximum is higher than the thermal energy, the dispersed system stays stable, i.e., no aggregation takes place, or vice versa. The stability of the floc structure also depends on the depth of the secondary minimum compared to the thermal energy. Since the van der Waals forces are greater for larger particles, flocculation may be dominant as a consequence of a deeper secondary minimum for the larger particles. Potential energy Figure l-3 Energy Maximum 22 Secondary minimum ‘ r Primary minimum Distance‘between particle surfaces, H Interaction energy of particles at distance H. 23 However, in systems under shear, floc structures are not likely to persist against the flow shear. Formation of strongly bonded aggregates is the main concern in the crystallization system of this study where flow shear is present. 1-3-4-5. Kinetics of Aggregation The energy status of the particles determines thermodynamically the possibility of forming aggregates; the kinetic aspect--how quickly aggregates are formed--is determined by the collision rate of particles. By assuming that aggregates are formed by binary collisions, the kinetics of particle aggregation may be modelled similarly to the bimolecular reaction (Hansen, 1975). The rate of aggregation is proportional to the frequency of particle collisions which is proportional to the concentration of particle numbers. The aggregation rate B(v1,v2), is therefore expressed as B(V1.V2) ' 5(V1.V2) n(V1) n(V2) (1'15) where fi(vl,v2) is the collision frequency factor and n(v1) and n(v2) are the number concentrations of particles of volumes v1 and v2, respectively. The particles may be brought together by Brownian diffusion, known as perikinetic (non-agitated) aggregation, or by orthokinetic (agitated) aggregation. For Brownian motion, the collision frequency factor, the kernel, has been derived by von Smoluchowski for spherical particles as (Drake, 1972): 5301M) - -§%L [ml/3 + V21/3] [v.‘l/a + vg'l/a] (1-15) where p is the fluid viscosity. 24 In a turbulent flow, collision of particles due to the dissipation rate of turbulence and particle inertia causes orthokinetic aggregation. For colloids aggregating in an agitated tank reactor, it may be assumed that particles follow the fluid flow fairly well. As a result, aggregation due to particle inertia is not significant (Hartel and Randolph, 1986). For particles smaller than the length scale of the small turbulent eddies, the collision frequency factor for particles aggregating in isotropic turbulence is given as a _ BT(r1,r2) - a (r1 + r2) G (1-17) where G is the mean velocity gradient in the turbulent fluid and is related to the average turbulent energy dissipation rate per unit mass, :, and the kinematic viscosity of the fluid, v, according to - 1: '0' - [4] (1-18) V The kinematic viscosity is u - (p/p)8, and p is the density of the fluid. The turbulent collision frequency factor in terms of particle volume becomes __1_ 1/3 1/3 1/3 3 _2;'8 fiT(v1, v2) - a [ 4n ] [v1 + v2 ] [ y ] (1-19) Several different values of the leading coefficient a in Equations (1- 17) and (1-19) found in the literature are summarized in Table 1-2. Equations (1-17) and (1-19) are valid for isotropic turbulent flow with particles smaller than the small eddies of the turbulence. In an agitated tank, the flow becomes isotropically turbulent when a modified 4 Reynolds number is higher than 10 (Shinnar, 1961; Perry and Chilton, 25 Table 1-2 Values for a. a value Reference -§- (Camp and Stein, 1943) 1.67 (Saffman and Turner, 1956) 1.25 (Beal, 1972) a [‘15—] x (- 0 311) (Delichatsios and Probstein, 1975) 1.67 (Yuu. 1985) 1973). The dimensionless modified Reynolds number for an agitated tank is defined as 2 NDa p Re]: - —p——- (1-20) where N is the propeller speed in revolutions per second and Da is the diameter of the propeller. The length scale of small eddies is described by the Kolmogorov microscale, A, given as (Delichatsios and Probstein, 1975; Shinnar, 1961) 3 a 1- {44—} (1-21) Considering Brownian diffusion and turbulent diffusion in addition to the total potential energy of two interacting particles, the rate constant for aggregation is expressed as (Shaw, 1980; Voigt 1986; Hunter 1987) flB(V1:V2) + fiT(V1»V2) fl(V1,V2) ' W(r1,r2) (1°22) where W(r1,r2), the stability factor, is 26 exp(VT/kT) W(r1,r2) - 2 I” 2 ds (1-23) 2 s2 and s - 2 [l + H/(r1+r2)]. The aggregation rate constant described by Equation (1-22) can be estimated numerically. With the particle size distributions experimentally observed in a crystallization system, changes (increases or decreases) in number density of particles at certain sizes may be characterized by Equation (1-15). These may be used with the population balance equation to simulate the particle size distribution in a crystallization system. The relevant population balance equation and functions describing particle density will be discussed in Chapter 3. 1-4. Review of Experimental Methods This section will review briefly several experimental methods used in the study of stone formation and the effects of inhibitor substances (mostly organic macromolecules) on calcium oxalate crystallization. In these methods, studies were carried out with either batch or continuous experiments. Crystallization mechanisms were characterized by measuring turbidity, calcium or oxalate depletion (concentration), or particle size distribution. 1-4-1. fiarch Experiments Robertson and Peacock (1972) studied calcium oxalate crystal growth and aggregation by observing the changes of particle size distributions of calcium oxalate crystals with time. The crystals were incubated in metastable calcium oxalate solutions with or without the 27 addition of 5% urine. The metastable solution was prepared to allow the seeded COM crystals to grow but to avoid spontaneous nucleation. During 4 hours of incubation, the seed particles grew and aggregated. This was seen in an increase in particle numbers at a larger particle size, e.g., 20 pm. A fractional rate of growth and aggregation was calculated from the increase of numbers of particles at 20 pm in the solution containing 5% urine compared to the increase in solution containing 5% NaCl, such that fractional N20 m 4 ° N20 m 0 h urine rate of growth - N _ N and aggregation 20 pm, 4 hr. 20 pm, 0 hr. 5% NaCl sol'n (1-24) The tests with urine from stone-formers showed higher fractional rates of growth and aggregation than with normal urine. Felix and his coworkers (1977) produced calcium oxalate aggregates following the method of Robertson and Peacock except that the crystals were incubated for 2 hours to evaluate the inhibiting activity of urine. Aggregation was tested by filtering suspensions through an "agglometer", a filter of 20 pm pore size operated under a constant pressure 100 mm Hg. The volumes of suspension filtered in 2 seconds were recorded to evaluate the percentage of aggregation relative to the inhibiting activity of urine according to Vo‘vu % aggregation - -§:_T—V;— (1-25) where V0 is the volume filtered in 2 seconds for a suspension before incubation, Vu is the volume filtered for the suspension incubated in the solution with inhibiting substances, and Vc is the volume filtered for a suspension incubated in the solution absent of inhibiting compounds (control). Their results show that the inhibition of 28 aggregation by 10% urine was close to the inhibition by 100% urine. Orthophosphate and magnesium at concentrations found in urine, citrate at 10J M, pentanemonophosphonate, and uromucoid had no effect in inhibiting aggregation. Through a series of ultrafiltration, fractionation, and gel-filtration steps, the inhibitor in urine was isolated and tested; the inhibitory activity of urine was attributed to the macromolecules in urine larger than 10,000 daltons. Bowyer, Brockis, and McCulloch (1979) also used the same technique as Robertson and Peacock (1972) to measure the rate of crystal growth and aggregation. A mixture of glycosaminoglycans was isolated from urine and showed the same inhibitory effect as a 1% urine solution. The glycosaminoglycans were identified as chondroitin 6-sulphate and chondroitin 4-sulphate by infra-red spectroscopy studies. Further tests of the crystal growth and aggregation rate with the commercially available chondroitin 6-sulphate and chondroitin 4-sulphate gave inhibition identical to the glycosaminoglycans and 1% urine solutions. This batch reactor scheme with seeded COM crystals was also adopted by Nakagawa et a1. (1983) to measure the growth rate of COM crystals by the depletion rate of 1‘C-labeled oxalate. The oxalate depletion rate was found to be proportional to the second power of the concentration difference of oxalate at the reaction time and at equilibrium. A highly acidic glycoprotein with molecular weight 1.4 x 104 was found to inhibit the growth rate of COM crystals. The change in the rate constant for crystal growth with the inhibitor concentration followed a Langmuir adsorption isotherm. This suggests that the inhibition effect is due to the adsorption of inhibitor on crystal surfaces which prevents further crystal growth. An assay of amino acid content of the isolated glycoprotein inhibitor was found to 29 be rich in aspartic acid, threonine, serine, glutamic acid, and glycine and contained about two 1-carboxyglutamic acid residues. 1-4-2. 0 e d Sheehan and Nancollas (1980) suggested a constant composition method for studying the kinetics of crystal growth by measuring the amounts of calcium and oxalate solutions added to a slurry of growing calcium oxalate crystals. The calcium and oxalate solutions were added to maintain constant calcium and oxalate concentration levels in a seeded reaction vessel. Calcium concentrations in the reaction vessel were monitored by a calcium selective electrode in the reaction vessel and an automatic titrator was activated to add calcium and oxalate solutions to return the calcium concentration to its set point. The crystal growth rate, determined from the amount of solution added, was found to be proportional to the second power of the excess calcium oxalate concentration in the solution. With the constant composition method, Lanzalaco et al. (1988) investigated the effect of urinary macromolecules on COM crystal growth. The macromolecular additives tested include pre-bladder urine, normal urine, the macromolecular fractions of these urines, and macromolecules extracted from the matrix of both kidney stones and bladder stones. The inhibitory activity of the additives was defined in terms of growth rate in the absence of additive, R0, and the growth rate in the presence of additive, R according to i, % inhibition - 100 (R0 - Ri)/Ro (1-26) Their results show that pre-bladder urine and normal urine have equal inhibitory effects on COM growth. The macromolecular components 30 fractionated from urine and from stone matrix also inhibited the growth of COM. 1-4-3. MSW One of the major areas of study in understanding the physicochemical etiology of urolithiasis is the use of the continuous mixed-suspension mixed-product removal (MSMPR) crystallizer for simulating the kidney. Finlayson (1977) initiated this approach due to the similarity between the operation of kidney and the continuous crystallizer system. Since the MSMPR crystallizer has been widely used in crystallization industry, the theories of crystallization mechanisms and crystal characterization as related to this system have been well documented (Randolph and Larson, 1971). For the study of calcium oxalate crystallization, the MSMPR crystallizer system is a continuously stirred crystallization vessel with two feed streams, one stream for calcium and the other for oxalate. The concentrations in the feed streams are prepared such that the mixed solution is supersaturated with calcium oxalate. The two streams of reactants are fed continuously to the reactor vessel where calcium oxalate crystals are precipitated. The crystals and mother liquor are well mixed and withdrawn continuously from the reactor to maintain a constant suspension volume in the reactor. It is assumed that calcium oxalate crystallization does not change the total volume of suspension; therefore, the total flow rate of the inlet streams is equal to the flow rate of outlet stream. The suspension has an constant average retention time in the reactor, or residence time (T), which is calculated as the reactor volume divided by the flow rate (in or out). In steady-state operation, temperature, flow rates, volume of 31 suspension in the crystallizer, chemical composition, pH, and crystal size distribution are all constant with respect to time and position in the reactor. The crystal size distribution (CSD) or particle size distribution (PSD) of the crystals generated in the MSMPR crystallizer system may be described by the population balance equation (Randolph and Larson, 1971): 6(G n(L,t)) mirth. + LaL , £11.11. _ 0 (1.27) where n(L,t) is the particle (crystal) population density as a function _d_L_ of particle size, L, and time t. GL is the crystal growth rate, dt . The growth rate, G is the increase of particle diameter, or L' equivalent spherical diameter, with respect to time. This should be distinguished from the growth rate defined as the increase of total precipitate mass or volume with time or the increase in moles of calcium oxalate precipitate per unit area of seed surface with time. The latter definitions are usually used in the studies in which crystal "growth rate" is determined from the depletion of reactants in solution with time. If the growth rate, G is independent of crystal size, L, the L! population balance equation describing the steady state MSMPR operation becomes _n.(ltLd _r1._ GL dL + T 0 (1-28) After integration, this becomes 1n n(L) - 1n n(O) - 721:- (1-29) L where n(0) is the population density at zero size. The population density, n(L), is usually obtained experimentally from counting 32 crystals or particles with an electronic sizing instrument. Typical sizing instruments allow acquisition of n(L) data within a limited size range. Linear regression of the acquired n(L) data on a natural log __1_ scale results in a straight line with slope [- G r ] and intercept L [1n n(0)]. With known 7, the growth rate, G may be calculated from L’ the slope. This type of representation, 1n n(L) or log n(L) data versus L, is called the population density plot. It should be noted, however, that the growth rate obtained in this analysis is the actual growth rate of crystals only if no other competing phenomena (e.g. aggregation, crystal breakage) exist in the crystallizer. The intercept of the population density plot or the population density at zero size, is the nucleation rate, 8.. It may be calculated according to (Randolph and Larson, 1971): B° - n(O) GL (1-30) This technique of using the MSMPR crystallizer to estimate linear crystal growth rate and nucleation rate has been adopted by many researchers in studying modifiers of calcium oxalate crystallization. Drach and his coworkers (1978) were among the pioneers of this technique in their work with COD crystallization. COD crystals were formed in the synthetic urine supersaturated with calcium oxalate (the control), and two modifiers, pyrophosphate and methylene blue, were tested for their effect on nucleation rate, growth rate, and total crystal mass generated. The total crystal mass is the total calcium oxalate precipitated per time, including the crystals in suspension and the crystals fouling the reactor walls. Both modifiers were found to have significant inhibitory effect on nucleation rate and total crystal mass; COD growth rates with the modifiers were higher than the control. 33 It was also suggested that the inhibition of nucleation rate might be correlated to the inhibition of crystal aggregation, although aggregation was not measured. Drach, Thorson, and Randolph (1980) then investigated COD crystallization under the effects of urinary macromolecules, including urine from calcium oxalate stone-formers and from healthy persons. For these experiments, the MSMPR crystallizer system was run with 95% synthetic urine (a solution prepared to mimic urine) plus 5% natural urine. Their results show that addition of both normal (N1) and stone- former (SF) urines inhibited linear growth rate and enhanced nucleation rate of COD. Both urines had equally significant inhibition on the total crystal mass. Crystal growth rate was lower with SF urine, but nucleation rate was higher. The addition of uromucoid, a high molecular weight mucopolysaccharide, also inhibited growth rate and enhanced nucleation rate, though there was no effect on the total crystal mass. It was hypothesized that uromucoid may contribute to the difference between N1 urine and SF urine. To focus on the differences between the urine of N1 and of SF, Drach, Kraljevich, and Randolph (1982) separated urinary macromolecules into low molecular weight macromolecules (LMWMM, <30,000 daltons) and high molecular weight macromolecules (HMWMM, >50,000 daltons). Macromolecues betweem 30,000 and 50,000 daltons were not recovered from either SF or Nl urine. The urine of SF contained much more HMWMM than N1 urine, but SF urine had no LMWMM. Experiments with COD crystallization in synthetic urine showed that the LMWMM from N1 had no effect on COD nucleation rate and growth rate. Both SF and N1 HMWMM had equal effects on nucleation rate and growth rate, but SF HMWMM reduced significantly the total solid concentration produced in the 34 crystallizer (total crystal mass). It was decided that the higher concentration of HMWMM in the SF urine must play an important role in decreasing the linear crystal growth rate and increasing the nucleation rate when compared to the control for which urinary macromolecules were not added. Randolph and Drach (1981) extended the MSMPR technique to observe the effects of osmolarity, calcium and oxalate concentrations, urea, and magnesium on COD crystallization in synthetic urine. The crystal growth rate, nucleation rate, and the total solids produced in the crystallizer were found to be reduced by decreases in calcium and oxalate concentrations. However, changes in osmolarity and urea and magnesium additives did not affect COD crystallization significantly as had been observed with the addition of urinary macromolecules. In further investigations of the urinary inhibitors using an MSMPR crystallizer, Drach, Sarig, and their coworkers (1982) found that the addition of polylysine did not have a significant effect on COD crystallization. The addition of both heparin and polyglutamate, however, increased nucleation rate and decreased linear growth rate, average crystal size, and the total mass of the COD crystals formed in synthetic urine; polyglutamate had a more marked effect. Addition of polyglutamate also decreased the deposition of calcium oxalate precipitates on the wall of the crystallizer, which is a common problem in calcium oxalate crystallization in an MSMPR crystallizer (Drach, Randolph, and Miller, 1978; Randolph and Drach, 1981; Drach, Sarig, et a1., 1982; Garside et a1., 1982). In the MSMPR crystallizer studies described above, linear population density distributions were assumed in analyzing steady state particle size distributions, where crystal aggregation or agglomeration 35 was neglected. However, curved population density distributions obtained from steady state calcium oxalate crystallization (illustrated in Figure 1-4) were observed (Garside et a1., 1982; Brecevic and Garside, 1982; Robertson and Scurr, 1986). Robertson and Scurr considered the curved population density plots to be an indication of particle agglomeration. Based on the binary collision theory, a linear population density distribution of the predicted nonagglomerated crystals, was calculated from the curved distribution, nobs’ by npred' computer simulation. The degree of agglomeration, A, was defined as: - 2 nobs) -1) (2 npred A - (E n (1-31) pred The inhibition of agglomeration, IN (A), was then calculated from AC and At’ the degree of agglomeration for the control and for the tested additive, respectively, such that (Ac - At) A c IN (A) - (1-32) For the evaluation of nucleation rate, linear growth rate, and total crystal mass, Robertson and Scurr followed the analyses for the steady state linear population distribution. Several modifiers of calcium oxalate crystallization were tested with synthetic urine. For additives within their urinary concentration range, the results are summarized as follows: magnesium and citrate slightly inhibited linear growth rate and crystal mass. Pyrophosphate only inhibited crystal growth rate but to an insignificant degree. Chondroitin sulphate, ribonucleic acid, and heparin inhibited crystal growth, degree of agglomeration, and crystal mass production. Among these modifiers, ribonucleic acid acted most strongly. ln [n(L,t) #/(pm°ml)] 36 8p 6 4 2 o o : curve distribution -—-: linear distribution 0 L 1 1 1 1 5 l0 l5 20 25 L (m) Figure l-4 Semi-log population density plot; linear and curved distributions. 37 1-4-4. t G ow Special attempts have been successfully made to study the growth of calcium oxalate crystals without interference from other types of crystallization phenomena. DeLong (1988) used a growth cell to measure the facial linear growth rates of single calcium oxalate crystals. The growth cell is a small (:5 ml) isothermal chamber in which single crystals can be observed and photographed with a light microscope. Calcium and oxalate solutions were mixed in a T-junction and then introduced into the cell chamber. The mixed solution was supersaturated with calcium oxalate, so that crystal nuclei formed on the observation platform (a glass cover slip) installed in the chamber. The nuclei then grew in the supersaturated solution. Photographs were taken in a time sequence, and the growth rates of crystals were calculated from changes in linear dimensions of the photographed crystal sizes with respect to time. COM was the major crystal phase studied. The results showed that presence of the macromolecular additives, polyglutamate and heparin, inhibit single COM crystal growth. The advantage of this technique is that it allows the effects of macromolecular modifiers on crystal growth to be evaluated separately from their effects on nucleation or on aggregation. 1-5. Scope of Research. The research conducted in this study had the ultimate goal of understanding the physical and chemical formation of kidney stones. Since urinary macromolecules have been shown to be important in modifying calcium oxalate crystallization, the purpose of the work was to elucidate the effects of these macromolecules on each step of the stone formation mechanisms. To this end, the MSMPR crystallization 38 system was chosen for its similarity to crystallization in the human kidney--i.e. that the stone constituents are supplied continuously to the reacting volume, that crystals are formed through nucleation, crystal growth, and/or crystal aggregation, and that the mixture of crystals and mother liquor are continuously withdrawn from the reacting volume. The MSMPR crystallizer system also has some advantages over the batch system. Because of the continuous supply of calcium and oxalate solutions from the feed streams, supersaturation and, subsequently, growth rate in the crystallizer do not decrease as rapidly as in a batch reactor. Besides, the prolonged process of MSMPR operation makes possible the observation of crystallization mechanisms with several samplings, while sampling from batch reactors is limited by the total reacting volume and by the relatively brief reaction time for calcium oxalate precipitation. Because of the problem of crystal fouling on the calcium electrode, the constant composition method was not considered the method of choice either. The MSMPR crystallization system was adopted to characterize calcium oxalate crystallization. A complete population balance equation including an aggregation term was chosen to model the crystal size distributions instead of the simple steady state model leading to the linear population density distribution. The crystal growth rate was isolated from the particle distribution data in which both crystal growth and crystal aggregation were superimposed. The effects of tissue surface on crystal growth were also investigated due to the attention drawn to this interaction by the fixed particle theory of stone formation. It was hypothesized that injured kidney cells may provide the attachment site for a stone nidus 39 or that injured cells or healthy cells may have promotive or inhibitory effects on nucleation and crystal growth. To provide insight to these questions, experiments were conducted using Maden Darby Canine Kidney (MDCK) cells and the photomicroscopic technique for observation of single crystal growth. The growth cell from the photomicroscopic technique was modified to allow observation of calcium oxalate crystals and the kidney cells under a fully developed laminar flow. Several chemical methods of injury to MDCK cells were evaluated for their resultant effects on nucleation and growth of calcium oxalate crystals. Because of the importance of supersaturation in the crystallization process, calculation of supersaturation for the systems used in this work is addressed in Chapter 2. The MSMPR crystallization is best described by the population balance equation which requires particle size distribution data, accurate measurement of particle size is needed. The particle size distributions and population balance equation pertinent to describing the MSMPR crystallizer system and accurate determination of particle size distributions will be discussed in Chapter 3. Experiments using the MSMPR crystallizer system are the main issue of Chapter 4; the effects of MDCK cells on calcium oxalate growth is addressed in Chapter 5. The summary of this work and recommendations for future research are described in Chapter 6. CHAPTER 2 SUPERSAIURAIION IN ELECTROEYTE SOLUTIONS Supersaturation is a quantity of importance in the study of calcium oxalate crystallization. It is the driving force for nucleation and for crystal growth. Calcium oxalate supersaturations must be known to quantify and understand the precipitation phenomena observed in this study. Because solution saturation is a reference state for supersaturation, saturation of calcium oxalate will be discussed first followed by a discussion of supersaturation and ion complexation. 2-1. Saturation Calcium oxalate, the sparingly soluble salt of interest in this work, forms three hydrates depending on solution concentrations, temperature, and pH. The three hydrates are calcium oxalate monohydrate (COM), calcium oxalate dihydrate (COD), and calcium oxalate trihydrate (COT) (Garside et a1, 1982; Tomazic and Nancollas, 1979; Gardner, 1975). When any of these calcium oxalate hydrates is in equilibrium in solution, the solution is considered to be saturated with that hydrate. Thermodynamic solubility products of the phases in the saturated solutions are defined as 2+ 2, Ksp,hydrates - (Ca )eq(C20‘ ) (2-1) 2 2- where (Ca +)eq and (C20, )eq are the activities of calcium and oxalate free ions in the saturated solutions, respectively. The Ksp values at human body temperature (37°C) are listed in Table 2-1. 40 41 Table 2-1 Thermodynamic solubility products of calcium oxalate hydrates at 37°C. o Hydrate Kspxlo Reference CaCzo‘oflzo (COM) 2.51 Garside et a1., 1982 Nancollas and Gardner, 1974 CaC,O‘-2H20 (COD) 5.01 Gardner, 1975 CaCzo‘o3H20 (COT) . 7.88 Tomazic and Nancollas, 1982 (8.2) (Garside et al., 1982) 2-2. Definition of Supersaturation For solutions having higher calcium oxalate activity products than Ksp’ the solutions are supersaturated with respect to calcium oxalate. Formation of calcium oxalate crystals occurs through nucleation and crystal growth to reduce the solution activity product to the equilibrium value. To quantify the driving force for nucleation and crystal growth in the solutions supersaturated with the electrolytic calcium oxalate solute, conventional definitiOns of supersaturation will be reviewed and re-evaluated for the specific case of calcium oxalate crystallization. The conventional supersaturation ratio for molecular solutes is expressed in terms of the activities of solute in the supersaturated solution and at equilibrium, a and a., which is supersaturation ratio - -g—- (2-2) Finlayson (1977) defined a relative supersaturation as the ratio of activities of the associated calcium oxalate ion pairs similar to Equation (2-2), i.e., 42 (Cacgo‘) (CaCQO‘)eq relative supersaturation - (2-3) where (CaCQO‘) and (CaCQO‘)eq are the activities of calcium oxalate ion pair in supersaturated and in saturated solution, respectively. A value of 6.16xlO'6 M for (CaCQO‘)eq was used by Finlayson. Garside et a1. (1982) defined the supersaturation in terms of the activity products of calcium oxalate as +)(c,o,2’) - x K 2 (Ca sp,hydrate (2_4) sp,hydrate supersaturation - Gardner (1978) and Gardner and Doremus (1978) defined a relative supersaturation according to H K ) 2 2- ( <[Ca *itczo. n” - —§P*—14-——hy rate relative _ 2 (2-5) supersaturation (K )8 sp,hydrate Y2 where [Ca2+] and [C20,2'] are the concentrations of free calcium and free oxalate ions, respectively, and y2 is the activity coefficient for divalent ions in the solution. While the activity of complexed calcium oxalate in Equation (2-3) or the activity product of calcium and oxalate in Equation (2-4) stresses the contribution of either the complex species or the resultant contribution of calcium and oxalate ions to the driving force of crystallization, Equation (2-5) stresses the contribution of the individual calcium and oxalate ions. The mechanisms of electrolyte crystal growth and nucleation are not understood well enough to distinguish whether crystals grow and nucleate by integration of calcium oxalate ion pairs or individual calcium and oxalate ions. 43 Therefore, two definitions of supersaturation have been adopted for this study. The first definition is the relative supersaturation in terms of the activity of the calcium oxalate complex ion, i.e., Shydrate,c - (Cacgo‘) (CaCZO‘)eq - 1 (2-6) so The relative supersaturation in terms of activities of free calcium and free oxalate ions is defined as h -(K ) spiny—L. )8 drat (2_7) [(Ca2+) - + 2 + 2 [c1 1 - T61 / [1 + KNaCl[Na 1y, + KKCI[K 1y, ] (2-48) As will be discussed in Chapters 4 and 5, COM and COT were the most frequently encountered hydrates in this work. Supersaturations of these two hydrates may be calculated according to Equations (2-6) and (2-7). The calculations were implemented with computer programs written in BASICA for the IBM AT, IBM XT, or IBM compatible systems. The program was developed according to the following logic: define stability constants and Ksps; T T T and T in input solution pH and values of TCa’ Ox’ Na’ K’ C1 solution; 3. calculate [H+] and [OH'] from the input pH value; 4. guess concentrations for free ions, [Ca2+], [C2042-], [Na+], [K+], and [Cl']; guess activity coefficients, y1 and y2; 5. calculate new values of free-ion concentrations by Equations (2-44) to (2-48); 6. compare the guessed concentrations and the calculated values to see whether the relative difference of the concentrations of each free ions is within a preset tolerance; 51 a) if it is not, use the new calculated concentrations for the guessed values to calculate ionic strength, I, use this I value to calculate y1 and y: by Equation (2-38), and then repeat calculations from step 5; b) if tolerance satisfied, go to step 7; 7. calculate supersaturations according to Equations (2-6) and (2-7). Listings of programs are shown in Appendix A. The supersaturations thus calculated are for solutions without solid calcium oxalate precipitate. In that sense they are the initial driving forces for nucleation or for crystal growth before any precipitation takes place. 2-5. Accuracy of the Calculated Supersaturations In calculating supersaturations as described above, some unavoidable uncertainties in the parameters were noticed. The stability constants used in the modified EQUIL programs were obtained from the literature. It was assumed that stability constants corrected for infinitely dilute solutions or for the asymptotic values at zero concentrations are independent of composition but depend only on temperature. In order to obtain stability constants for infinite dilute solutions at 37°C, extrapolation or interpolation was used whenever necessary. For the stability constants without information on the dependence on temperature, values at 25°C or at 38°C were also used. For some data, different references reported different values of stability constants for similar solution conditions. The effect of using different values for stability constants on calculated supersaturations may be seen in Table 2-3, where supersaturations of COM, (3 and (S are calculated from a modified EQUIL COM,i)A COM,i)B' 2 2- program for systems containing Ca +, Ox , and KCl using different 52 Table 2-3 SCOM i calculated with different stability constants for solutions containing CaClg, K2C20‘, and KCl (ionic strength buffer); ionic strength - 0.15 M. TCa - TOx Supersaturation Difference in (mM) (SCOM,i)A (Scou,1)3 (SCOM,i)A and (SCOM,i)B (%) 0.5 1.484 1.365 8.62 0.75 2.616 2.452 6.66 1.0 3.685 3.482 5.80 1.25 4.699 4.462 5.28 1.5 5.661 5.396 4.90 Stability constants and KSp COM used in calculation f°r (SCOM,i)A (SCOM,i)B KHO 21141.6 21008.7 x KKOx 10.0 13.3 KCan 1871.03 1869 2 KCaQOx + 71.4 71.6 1.0e-14 2.57e-l4 20 K 2.493e-9 2.514e-9 53 stability constants. The stability constants used in the calculations for (S and for (S are also listed. The differences COM,i)A' cou,1)a between these two values are listed in the last column of Table 2-3. As an example, the difference between these two supersaturations is 6.66% for the lowest experimental concentrations, TCa - TOx - 0.75 mM, used in this work. Another source of uncertainty in calculating supersaturation is from the activity coefficients from Equation (2-38). This equation has the advantage of representing activity coefficients of several types of single e1ectrolytes-- uni-univalent, bi-univalent, or uni-bivalent—- in a single equation. The average deviation of values predicted by the equation was 1.6-2.2% from the true activity coefficient values for the solutions containing the single electrolytes (Davies, ibid). When the equation is applied to systems containing multiple components, the accuracy is unknown. Because of the minor deviations for the single electrolyte solutions, the equation is used in dilute systems containing multiple components. It offers reasonable approximations for the activity coefficients, but with unavoidable uncertainty. The methods described in this chapter allow calculation of supersaturation driving force for calcium oxalate crystallization for a variety of solution conditions. ~They are important for the quantification of crystal growth in this work. CHAPTER 3 PARTICLE SIEE DISTRIBUTIONS, POPULATION BALANCE EQUATIONS AND DETERMINATION OF PARTICLE SIZE DISTRIBUTIONS 3-1. Introduction Particle size distributions (PSD) or crystal size distributions (CSD) are of value in crystallization studies; they offer both qualitative and quantitative information which characterize crystallizing and precipitating systems. Randolph and Larson (1971) did some of the first work in the use of PSD to characterize crystallizing systems through the use of population balance equations. As a preamble to Chapter 4, this chapter will discuss the theoretical aspects of particle size distributions and population balance equations, as well as the practical aspects of acquisition of particle size distributions. Portions of this chapter are excerpted from two papers published as part of this work (Lin and Briedis, 1988a, 1988b). 3-2. Particle Size Distributions and Population Balance Equations There are several ways of presenting particle size distributions. One of the most popular particle size distributions used is differential population density, n(L,t) or n(L), representing number of particles per differential length per sample volume. This PSD representation uses a characteristic length of the particles, L, as the independent variable and may or may not be a function of time t. For particles of irregular shapes, the equivalent spherical diameter is usually used for the characteristic length. Assuming Lc is the smallest particles in the sample, the total particle number, NT (no./m1), may be calculated according to (Randolph S4 55 and Larson, 1971): NT(t) - I” n(L,t) dL (3-1) L c The upper limit of this integration can also be the largest particle size of the sample instead of m. The differential volume density, v(L,t), may be obtained from the number density by: 3 v(L,t) - 41%- n(L,t) (3-2) The third moment of the number density, M3, is expressed as (ibid): M3(t) - r n(L,t) I.3 dL (3-3) L C This moment is also related to the total particle volume in the sample, 3 vT(t) (pm /ml), according to: _6_. Mact) - ,, vT (3-4) The total suspension mass expressed in terms of the third moment is we) - —§-"- Mam <3-5) where p is the density of the suspended particles. For the MSMPR continuous crystallization system with nucleation, crystal growth, and aggregation involved, the unsteady state population balance equation may be written as (ibid): 8(GLn(L,t)) figgL,t) n L t _ _ _ at + 6L + 1 Ba Da (3 6) where CL is the particle growth rate, or it , in units of pm/min, and r is the residence time in the MSMPR crystallizer. The first term on the left hand side of the equation accounts for changes in particle 56 numbers in the system with time, the second term accounts for particle growth by transfer of material from the solution phase to the particles, also called the condensation term, and the third represents removal of particles by constant outlet withdrawal. Since L is the equivalent spherical diameter of particles, GL is twice the value of the growth rate defined as the procession of a surface with time as used in Chapter 5. Ba and Da are particle birth and death into and from a given size resulting from particle collisions and aggregations. Aggregation of particles may result in conservation of particle volume instead of particle diameter; that is, the volume of aggregates is the sum of the volume of the aggregate-forming particles. The population balance equation written in terms of n(x,t) is more suited for representing the aggregation terms, where x is the volume of particles of size L. The PSD n(x,t) is related to n(L,t) by n(x,t) dx - n(L,t) dL (3-7) and 3 x - -§— L (3-8) If it is assumed that aggregates are formed by collision of only two particles at a time (Hansen, 1975), the whole population balance can be written as (Swift and Friedlander, 1964; Drake, 1972) .29121s1. + °(Gv“(x'°)) + _31;,;1_ at 6x r x - -%rI n(y,x-y)n(y,t)n(X'Y.t) dy X C - n(x,t)l n(X.y)n(y.t) dy (3-9) X C 57 dx dt ' xc is the particle volume at size Lc’ and n(x,y) is the aggregation where CV is the increase of particle volume with respect to time, kernel for particles of volumes x and y, similar to the kinetic rate constant in bimolecular reactions. Because volume is proportional to the third power of L, the n(x,t) distribution tends to be weighted more heavily to larger sizes than a n(L,t) distribution. A compromise between length and volume-based distributions is to define another size distribution, m(w,t), such that (Gelbard and Seinfeld, 1978) "bJ m(w,t) - x n(x,t) ln[ x (3-10) c and 1n[—3—-] Xe w - (3-11) 1n[—fh—] x c where xc and xb are the lower and upper limits on particle volume. Let m(wx,t)- x n(x,t) ln[‘§h—] (3-12) c m(w ,t)- y n(y,t) ln[ xb ] (3-13) Y Xe and ln[‘§:x—] c x-y 1n[ xb X ] (3-14) c Applying the chain rule in changing the coordinate from x to w, the population balance equation becomes 58 at 8w 1 w - —l— XOXC x n(y x- )m(w t)m(w t) dw 2 ' y y! X'y’ w x-y Y c wb - m(w,t)! n(x,y)m(w ,t) dw (3-15) w Y Y c where Gw - g: , and, according to Equation (3-11), wb and wc are equal to l and 0, respectively. For an unsteady state continuous crystallization process in which the total particle number is small, there may be no aggregation taking place in the system. For such a case, the population balance equations describing the system may be rewritten by omitting the aggregation terms. Using the PSD n(L,t) as an example, the population balance equation for such a process may be described as: as 1124915.). W4l_ , at + GL aL + 6L + T 0 (3 16) If the growth rate G is independent of particle sizes, Equation (3-16) L may be rearranged to result in an equation to be used in Chapter 4 to calculate the growth rate, i.e., Q13 n(L,t) + _l_ 6t 1 CL - - 21 “:1 t: (3-17) 6L For the time-independent case, this formulation reduces to Equation (1-28). 3-3. Determination of Particle Size Distributions Particle size distributions modelled with the use of population balance equations are of great importance in characterizing 59 crystallization system; however, an accurate and time-economic particle sizing method is required to obtain representative CSD or PSD data. Several techniques such as sieving, sedimentation, and electronic sizing may provide particle size distribution data. In this work, because the particles generated from the MSMPR system were below 100 pm in sizes, they were below the sieve-sizing range. On the other hand, sedimentation methods require long times for particle size differentiation due to long particle settling times. Sedimentation method is not suited for the rapid analysis required for the time- dependent study of the MSMPR crystallizer system. The electronic sizing method was chosen for its capability of analyzing data with high resolution (100 or 128 sizing channels), good reproducibility (Allen, 1981), and its ability to size without alterating the physical state of the suspension samples generated from the MSMPR crystallizer. The particle sizing instruments operating on the electrical sensing zone principle typically used to determine PSDs are the Coulter Counter (Coulter Electronics, Inc., Hialeah, Florida) and the ELZONE Particle Data Analyzer (model 112 LSGD/ADC/SSA-SGS/80XY/VM5/TP50) was adapted for sizing the crystals formed from the MSMPR continuous crystallization system to be discussed 60 in Chapter 4. The practical aspects concerning the use of this instrument will be discussed in the sections that follow and were published as two papers (Lin and Briedis, 1988a, 1988b). 3-3-1. Basic Operating Principles of the ELZON§(R) System The electronic sizing instruments count particles of suspension samples in solutions with electrical conductance. The sensor parts are an orifice tube and two electrodes, one inside the orifice tube, the other outside the tube. The instrument generates vacuum to draw a sample of known volume into the orifice tube for particle counting. The electrodes sense the resistance changes of the solution caused by the passing particles when suspension solution is drawn through the orifice. With constant electric current passing through the orifice, the resistance change due to a passing particle generates a voltage pulse proportional to the particle volume (Kachel, 1979). The instrument collects the signals of the voltage pulses, discriminates the signals into size channels, and then reports the particle counts in each channel and/or other particle counting information based on these particle count data. The representative sizes of these sizing channels are obtained by calibrating the instrument with particles of known sizes called standard particles. 3-3-2. 0 e a n Part c Size D s ibu ions w th t e I strument W Based on the operating principles stated above, the population density, n(L), is related to the counted particle data, nch’ or particle number in the sampling volume in each channel, by 61 n n(L) - —-93— DF (3-18) ALch Vs where ALch is the channel width in units of pm/channel, V8 is the sampling volume for which the particle data are counted, and DF is the dilution factor used in sample preparation for particle counting. Since time is an independent variable with respect to data processing, the time independent form of PSDs or counted particle data will be used here for simplicity. A quantity useful in data manipulation, “ch,c’ the corrected count in a given channel ch, in units of no./(channel-unit sample volume), may be defined as n n - —93- or (3-19) ch,c V s The volume density, v(L), may be related to differential volume data, vch’ according to V v(L) - ———9L DF (3-20) ALCh Vs where vch is the total volume of particles in each channel in the sampling volume analyzed and may be calculated from differential population data according to 3 v - k L n ch v ch (3-21) ch where kv is the geometric shape factor of particles (Randolph and Larson, 1971, McCabe and Smith, 1976) and Lch’ the representative length of channel ch, is equivalent to particle size L. In the case where L is the equivalent spherical diameter, the geometric shape . 1? factor is “g“. 62 3-3-3. 0 e th L N (R) a s s ro ra During the course of this work with the computerized ELZONE(R) system, several limitations were discovered in its "firmware" (a program stored permanently in Read Only Memory) for quantitative volume and numbers distribution analysis. The ELZONE system used in this work consists of the primary particle sensor and counter which is interfaced to the minicomputer (Model ADC-80-XY-VMS) of the secondary instrumentation for data acquisition and processing. The data processing program stored in the ROM offers several optional functions which include reporting of data on either a population (numbers) or volume basis and in differential or cumulative display, blending of multi-orifice data for particulate systems of broad size distributions, and extrapolation of the blended or single orifice data. In analyses where only relative volumes and relative population distributions are necessary, as with the ELZONESR) "histogram” display or for percentile presentation, the existing firmware calculations are adequate. However, some characteristics of the ELZONE(R) firmware limit its application for quantitative volume or numbers distribution analyses which are needed for modeling of number-dependent phenomena (Halfon and Kaliaguine, 1976; Swift and Friedlander, 1964; Gelbard and Seinfeld, 1978; Delichatsios and Probstein, 1975; Hartel and Randolph, 1986). Such details have been neglected in descriptions of the operation and data analysis package of the ELZONE system (Particle Data, (b); Karuhn and Berg, 1984). It is the purpose of this section t35address these limitations. The comments and discussion herein are specific to the software version described in 0XY007.R01 80XY User's 63 Manual - REX 710, and may not apply to other versions distributed by Particle Data, Inc. 3-3-3-1. "Histogram" Display The population or volume data reported from the ELZONE system are given in a "histogram" representation (Particle Data, (b)); i.e., the data presented are in some proportion to the quantitatively correct data. For example, the differential population data, nch’ are given in a computer printout of the differential population display from the 80XY program. In normal instrument operation, the channel widths of each set of data counted vary with the aperture tube, size span, and current and gain settings chosen. Therefore, the reported "histogram" cannot be quantitatively compared for data sets that have been acquired at different size span, current, and gain settings or with different aperture tubes, sampling volumes, and dilution factors. For further quantitative analyses, these data sets require modifications that will be discussed in Section 3-5. 3-3-3-2. Scaling Factor in Volume Data It was noticed that the differential volume histogram data, vrep’ reported by the ELZONE firmware are not equivalent to the differential volume data, vch' Table 3-1 presents a channel-by-channel comparison of the vrep data with the volume data, v calculated from cal’ Equation (3-21) using kv - “£5, The results show that the actual conversion equation implemented by the firmware must include an additional factor, P, such that vcal and v are related according to: rep vcal - vrep F (3-22) where F is approximately constant for each individual sample. However, 64 Table 3-1 F factors calculated from comparison of reported and calculated differential volume data. (abbreviated version) Sample Parameters: 76 micron Orifice Tube, serial # 179 Volumetric: 498.4 p1 Current: 5.5 True Log: 10.0 (10) Gain: 3.5 Dia. Ch 1: 1.902 (pm) Preset Time: 21.7 (sec) Dia. Ch 128: 19.22 (pm) Elapsed Time: 21.7 (sec) Low Cal.: ch.l (1.902 pm) High Cal.: ch.128 (19.252 pm) Total Count - 10870 Total Volume - 47373 Ch Lch nch vrep vcal F Ch Lch nch vrep vcal F 24 2.892 50 3 633.23 211.1 57 5.278 77 31 5927.9 191.2 25 2.946 52 4 696.15 174.0 58 5.375 77 33 6260.7 189.7 26 3.000 51 4 721.00 180.2 59 5.474 78 35 6699.0 191.4 27 3.055 52 4 776.31 194.1 -- 28 3.111 52 4 819.79 204.9 -- 29 3.168 49 4 815.74 203.9 -- 30 3.227 48 4 844.57 211.1 -- -- 123 17.55 94 1413 2.660e5 188.3 -- 124 17.88 88 1397 2.634e5 188.5 -- 125 18.28 84 1408 2.687e5 190.8 54 4.997 66 23 4311.9 187.5 126 18.53 81 1434 2.698e5 188.2 55 5.089 70 26 4830.5 185.8 127 18.94 74 1384 2.633e5 190.2 56 5.183 72 28 5249.0 187.5 128 19.22 324 6400 1.204e6 188.2 Average value of F: 189.7 65 between several sample data sets analyzed, F has been found to vary anywhere from 189 to 1516. Information concerning this factor provided upon request from Particle Data, Inc., shows that the scaling factor F of the 80XY program is a relative value. The scaling process is designed to avoid overflows in the microprocessor memory. As a result, the volume data in every channel are normalized to a peak height of 6400 counts by the scaling factor and are then rounded off to integer values. This methodology is valid when relative volume distribution are required, but precludes quantitative comparisons of volume distributions from one sample to another. 3-3-3-3. Blending of Volume Data The blending routine in the ELZONE firmware manipulates multiple orifice data to give a complete histogram across the size range of importance. It uses data represented on a volume basis (Karuhn and Berg, 1984; Particle data, (b)). It is herein that questions about the correctness of the routine arise as the volume representation generated by the ELZONE(R) firmware as described in paragraphs above is not quantitatively correct. It is important to take into account differences between vch data and v(L) data. As an example, suppose a suspension sample is observed to have a linear population density given by the expression, 1n n(L) - -0.2L + 12.5 (Figure 3-1(a)), and is to be analyzed with two aperture tubes of 76 pm and 300 pm. These sample data are then to be blended. Identical dilutions in sample preparation and equal sampling volumes (1 ml) in data acquisition are assumed. Figures 3-1(b) and 3-1(c) show the simulated quantitatively correct v(L) data and vch data, respectively. The discontinuity seen in the vCh distribution curve 66 T V Y Y I A T l I 7: 12 ~\ (0) .l '- T_ \ E E 7' 6 . i 8 E 8 - .c 1 l v =2: " 4 q u, E 3 '1 E ‘ t - 'o .d \ :; 2 . E \ .c )0 0 0 TE 2 (b) (a) 7 6+ . E 1. "E 5 4 i- 1 1 a > 1 V: IL 2- . g / > I oo 10 20 1 1 ° 1 L ‘ ‘ L Figure 3-1 Blending of a simulated vc equivalent spherical diameter (pm) (a) (b) (e) (d) 4O SO 10 20 30 40 50 equivalent spherical diameter (pm) h and v(L) data; linear population density plot, volume density , differential volume per channel, vch by blending routine. The thicker dash lines represent the data from the 76 pm aperture tube; the thinner solid lines represent the data from the 300 pm aperture tube. Sampling volume is 1 ml for both data sets. 67 arises due to the inherently smaller channel widths for the 76 pm orifice tube than for the 300 pm orifice tube. The program performs its routine simply by adjusting the two vch curves based on matching "the same or most nearly same" (Particle Data, (b)) slopes of the curves enclosed by the low and high markers in the region where data of several channels overlap. In order to simulate the procedure, curve A in Figure 3-1(c) must be scaled down by a factor of 0.8006 (based on most nearly the same slopes between 9 to 12 pm) to yield a continuous curve as shown in Figure 3-l(d). Additionally, the blending routine in the ELZONE(R) system rescales the curve to yield a peak value of 6400. The results of this routine cannot be used quantitatively with confidence. The above assumption of equal sampling volumes in data acquisition is usually not realizable. In the analysis of populations of broad size distributions, two mercury volumetric sections are often used in acquiring data with different size aperture tubes. Such data sets are necessarily each on a different volume basis and should not be blended without correction. 3-3-3-4. Sampling Volume It has also been discovered that in its present design the instrument has no way of directly coordinating the start of sample analysis (counting of particles in each channel) with the start of intake of the calibrated sample volume, Vv’ in the system's ANALYZE or TIMED ANALYZE data acquisition mode. Therefore, even when the same aperture tube and mercury section are used for different samples, the lack of automatic coordination between intake and counting causes uncertainty in the values of vch obtained. For generating quantitative 68 number or volume distribution data, n(L) and v(L), respectively, a true sampling volume is required. To estimate the true sampling volume for each data set, Particle Data, Inc., offers an approach based on the total counts in two successive sample analyses. Analysis of a sample in the ANALYZE mode yields a distribution of particle numbers and the total count, NT,s' For the same suspension sample, another analysis is performed for a sample volume equal to the calibrated volume of mercury section, Vv’ by moving the vacuum/flush control toggle from VACUUM to COUNT. This is done in the ANALYZE mode and so does not generate the particle numbers distribution in the system's memory. The total count thus obtained is NT v' The true sampling volume, V3, for the analysis of known particle number distribution is then calculated as .EIis. V - V (3-23) 5 v N T,v However, this method is not valid in systems having particle size distributions varying with time. In order to obtain reliable data for the crystallization system as will be described in Chapter 4, an alternate approach for sampling volume as well as modification of data manipulation was used and will be described in Section 3-5. 3-4. Ca1ibration.Procedure and the “Normalize“ Function The "normalize" function, one of the unique features of the ELZONE system, is available on all the ELZONE instruments manufactured since 1970 (Particle Data, (a)). It provides a null check of calibration against any property changes in electrolyte and circuits (Particle Data, (b)). As described by Particle Data, Inc., the normalizing procedure serves to adjust the calibration for either a) 69 any electrolyte conductivity changes due to changes in electrolyte composition or temperature, or b) any circuit drift caused by aging or temperature changes. Questions have arisen as to the correctness of the normalize function and the calibration procedure. Because the operating procedure of the instrument relies greatly on the application of the normalize function for accurate data and because the calibration procedure is required for correlating the sizes of counting channels with the switch settings of the instrument, efforts were taken to verify the normalize function and the calibration technique in question. A modified calibration procedure was developed and verified to be analogous to the original calibration procedure, but it offers a much clearer understanding of the basis for the calibration procedure and a more straightforward determination of calibration constants than the original technique. In this section, the original calibration procedure will be reviewed, and the modified calibration procedure will be discussed and verified with calibration experiments. 3-4-1. W (Particle Data. (8)) The-logarithmic span mode in the ELZONE system provides a relatively broad sizing span for particle counting. In the logarithmic mode, for a given orifice tube and a given log span setting, the calibration equation as provided by the manufacturer is L (Tr-1) 2 s vTr - Kc (I,G) (3-24) where VTr is the particle volume at trigger level Tr, Kc is the calibration constant, L8 is the true log span, I and G are the current and gain switch settings on the instrument respectively, and (100) is 2(I+G). the 10 product, or' The trigger level Tr, which has a value 70 from 0 to 1, controls the discrimination between the electric pulses, which are related to particle sizes as described in Equation (3-24). The log span settings on the front panel of the instrument are only nominal values of the true log span, Ls“ The original graphical calibration method is summarized as follows. For a given orifice tube and a log span switch setting, the instrument is calibrated with one size of standard particles at several current and gain switch settings. The trigger levels corresponding to the median (half count) or modal (peak) size are recorded. The technique then requires plotting of the logarithm of a pseudo-volume versus the trigger level or "instrument scale" (dial setting, channel number, ...) at several current and gain switch settings to generate a linear calibration line. From this line, a ratio, R, is calculated from the volume at "full scale", Vf, and at "zero reading", Vz: R - (3-25) V 2 The "full scale" and "zero reading" correspond to channel 128 and channel 1, respectively (see Equation (3-29)). The calibration constants, Kc and L8 in Equation (3-24), are given as Kc - I-G-Vf (3-26) and __lsz_B._ Ls log 2 (3-27) Kc and LS are used in Equation (3-24) to calculate particle volume versus trigger level or channel. 71 Based on the understanding of the instrument’s function and design, some modifications to the original graphical procedure have been proposed. These are described in the following sections. 3-4-2. Modified Calibration Erogedurg For instruments manufactured after January 1978, the IG product is internally set as 2 v (3-28) f A modified approach to calibration is suggested as follows. The channel numbers, ch, are related to trigger levels by ch - 127 Tr + 1 (3-29) This equation is based on the understanding that as Tr is varied from 0 to 1 the instrument analyzes data from channel 1 to channel 128. As described in the Operator's Manual (Particle Data, (a)) the ELZONE(R) system is designed such that the particle-pulse amplitude for a given orifice is directly proportional to current, gain, and particle volume. In other words, if the particle-pulse amplitude, which is 2 10 ) T in the reactor. The fluid viscosity and fluid density used for the calculation of Reynolds number are shown in Appendix D. Flow rate in each stream of the inlet solutions was monitored with flowmeters (Cole-Farmer) to ensure the required residence time of the 84 solution in reactor. The on-line filters used were 0.2 pm ULTIPOR disposable filter assemblies (Pall Corporation). Reaction temperature was kept at 37°C by a water bath with an immersion circulator (Haake, model E3). Solutions were prepared using reagent grade chemicals and doubly distilled deionized water. Ca(N03)2o4H20 (from J. T. Baker) and K2C20‘-H20 (from Sigma) were used as the sources of calcium and oxalate respectively. The biopolymer additive was polyglutamate (poly-L glutamic acid, sodium salt, (PGA), from Sigma) of molecular weight 26,500. pH values of the feed solutions were adjusted to 6.00i.01 with HN03 and KOH; KN03 (J. T. Baker Chemical) was added to maintain ionic strength at 0.15 molal. Equal molar concentrations of calcium and oxalate ranging from 1 to 4 mM (based on the total volume of the mixed solution) were used. Supersaturation of calcium oxalate, pH, and ionic strength were chosen to simulate the physiological conditions of urine. 4-2-2. Experimental Scheme The experiment with calcium and oxalate concentration at 2 mM without polyglutamate additive and a residence time of eight minutes was chosen as the base case to compare directly with physiological conditions. Other experimental runs were carried out by varying calcium and oxalate concentration level, residence time, and PGA concentration. Residence time, r, was varied from 6 to 16 minutes, and the PGA concentration levels used were 1 ppm, 10 ppm, and 100 ppm. The experimental scheme is summarized in Table 4-2. 85 Table 4-2 Experimental scheme. Experimental conditions 2+ 2- Run r [Ca ]-[C20, ] PGA number (min) (mM) (ppm) 4 8 1 0 8 8 2 0 7 8 3 0 6 8 4 0 9 6 2 0 10 16 2 0 l4 8 2 1 12 8 2 10 13 8 2 100 Each experiment was initiated with 400 m2 of a saturated calcium oxalate solution in the reactor. Throughout the run, suspension solution was sampled from the crystallizer every 1 to 2 residence times for the determination of particle size distributions, total calcium concentration in the suspensions, and calcium concentration in the filtrates of suspensions. Each run was carried out for a period of about 24 residence times. 4-2-3. Mpasurement of Partiple Size Distributions The ELZONE Particle Data Analyzer (Particle Data, Inc.) described in Chapter 3 was used to determine the size distributions of crystals and aggregates. The ELZONE(R) was calibrated with standard microspheres (10.00 and 41.1 pm) and latex particles (20.27 pm). Diameters of the aperture tubes used were 76 um, 95 pm, 150 pm, or 300 um depending on the particle sizes of the samples. Generally, the 86 higher the calcium and oxalate concentrations or the longer the residence time, the larger the particles formed. A 0.15 molal KN03 solution saturated with calcium oxalate crystals was prepared as the electrolytic solution used for particle counting with the ELZONE(R). Immediately before use, the electrolytic solution was filtered through a 0.45 pm membrane filter and a 0.2 pm ULTIPOR disposable filter assembly (Pall Corporation) in series. Samples were diluted one to three or one to six by volume with the prepared electrolytic solution before particles were counted with the ELZONE(R) system. Dilution was necessary because too many particles in the samples could cause clogging at the orifice during particle counting. The particle sizing data were transmitted to an IBM PC and stored on floppy diskettes as described in Chapter 3. For further data manipulation, particle data were imported to SuperCalc 4 (tm) or Lotus 1-2-3 spreadsheets. Particle size distributions, n(L,t) or v(L,t) versus particle size, L, or m(w,t) versus w (defined in Chapter 3) were calculated and presented as figures. The distributions in the form of tabulated data or figures were printed out as needed. 4-2-4. Measurement of Calcium Concentrations Calcium concentrations in the filtrates and in the suspensions (10 m1 of each) were determined using atomic absorption (AA) spectrophotometry. The AA spectrophotometer (Varian AA-375) was made available by Dr. Mackenzie L. Davis in the Department of Civil and Environmental Engineering. The filtrates were prepared from suspension samples drawn with a syringe and filtered with 0.2 pm membrane filters assembled in the syringe filter holder. For the determination of total calcium concentration in the suspension samples, the calcium oxalate 87 crystals or precipitates in the solution were dissolved by the addition of concentrated sulphuric acid for AA analyses. 4-2-5. Determ'nation o C stal Habit and C stal H drate Phases Crystal hydrates were identified from microscopy (One-Ten (R) Microstar , AO Scientific Instruments) and X-ray diffractometry. Powder samples were collected at between 6 to 12 r and filtered through 0.8 pm membrane filters for X-ray identification. Two X-ray diffractometers were used in this work-- one made available by Dr. Kalinath Mukherjee in the Department of Metallurgy, Mechanics, and Materials Science, and the other by Dr. Duncan F. Sibley in the Department of Geological Sciences. The radiation used in both diffreactometers was CuKa filtered with Ni filters. 4-3. Characterization of Calcium Oxalate Crystallization in.HSHPR System 4-3-1. Calcium Oxalate Hydrates and Crystal Habit Aggregates of calcium oxalate crystals were formed in the precipitation processes as confirmed with the optical microscope. Figure 4-4 shows a photograph taken from the experiment with calcium and oxalate concentrations at 3 mM.. A COM (DeLong, 1988) crystal was seen as marked. The other crystals are most likely COT (Gardner, 1975). Crystal hydrates were identified by comparison of the X-ray diffractograms to the crystallography data for COM, COD (Sutor and Scheidt, 1968), and COT (Brindley, 1957). Table 4-3 summarizes the major hydrates formed for each set of experimental conditions. Because of the limits of sensitivity of the X-ray diffractometer, minor crystal 88 hydrates present in less than 5% were not shown in the diffractograms. For example, the COM seen in Figure 4-4 was not observed by X-ray diffraction. Table 4-3 Crystal hydrates. Experiment with Hydrates 2+ 2- F [Ca ]‘[C204 1 j 1 mM COM 3 2 mM COT 3 mM COT 4 mM COT Residence time: 6 min COT 16 min. COT PGA additive: 1 ppm COM and COT 10 ppm COM and COD 100 ppm COM and COD 4-3-2. Particle Size Distributions The time-dependent particle size distributions of each experimental run may be presented in the form of n(L,t), v(L,t), or w(m,t) as per the equations shown in Chapter 3. Figures 4-5 and 4-6 are typical population density plots of the samples taken between eight to ten residence times. Figure 4-5 data are without PGA additive, and Figure 4-6 data are with PGA additive. With PGA additive, the particle size distribution exhibits a wave-like shape. The wave shape of population distribution is more pronounced in the v(L,t) PSD representation (Figure 4-7). In [n(L,t) (#/,u.m.ml)] 90 a 5.. 4. 2. O o 10 2'0 3? 4'0 50 so L (Mm) Figure 4—5 Population density plot A; 2 2- [Ca +1 - [0,0, ] - 3 mM, without PGA, r - 8 min, sampling time - 74 min. 91 In [n(L,t) (#/pm.mi)1 L (Mm) Figure 4-6 Population density plot 8‘ 2- [Ca +1 - [0,0, 1 - 2 mM, PGA - 100 ppm, 1 - 8 min, sampling time - 68 min. 92 o +————-—~~.— , 0 5 10 L (,um) Figure 4-7 v(L,t) particle size distribution; 2+ 2_ [Ca ] - [C20‘ ] - 2 mM, PGA - 100 ppm, 1 - 8 min, sampling time - 68 min. 93 oo .mmcea m~_m __msm umumpoaeauxm new: Au.nv> mic drama; A830 1_ om ow om ow 0.. o e . e \\ 1. db . as x r w— r a? r 390 i e y 1 I. mamas . L _ s 4. m. an a”. r . . 1 .1 k .1 “a . as _ 1. _ _ . _ 1... e 3.: a i a i 38 ewe $533 a (1w-Lu71/cuml) q‘mA 94 Figure 4-8 also illustrates a v(L,t) PSD. The original PSD is shown with the square symbol. Because of the nature of the particle sizing insturment, particles below certain size limits were not counted; Figure 4-8 shows the size distribution above 5.5 pm. In this v(L,t) data, the size distribution follows a general trend of decreasing volume with decreasing particle sizes ranging from 5.5 pm to 23 pm. Because of this general trend over a relatively wide range of the existing particle sizes, it may be assumed that the size distribution is a monotonically smooth curve below the lower sizing limit. If a curve of the second power of particle sizes is assumed for this small size range, the whole particle size distribution is shown as the dashed curve in Figure 4-8, where the distribution above the lower counting limit is presented by straight lines connecting the data points. This fitted complete size distribution provides a feasible way to calculate the total particle number, N the third moment, M3, the T, total mass, MT, or total volume, VT’ in suspension. In this work, the computer program NTM3.FOR was written in FORTRAN to calculate NT and M3 according to Equations (3-1) and (3-3): NT(t) - I” n(L,t) dL (3-1) L c 3 M3(t) - [a n(L,t) L dL (3-3) L c . The n(L,t)'s used in the calculations were calculated from the fitted v(L,t) distributions input to the program. The theoretically calculated critical cluster size, Lc’ from homogeneous nucleation data is 0.0013 pm for calcium oxalate crystals (Walton, 1967). This value 95 was the lower bound of the integration used in calculation. IMSL spline and integration routines were used as shown in the flow chart and in the listing of the NTM3.FOR program (see Appendix E). For the base case, the N and M3 results are shown in Figure 4-9. T Figure 4-9(a) shows a maximum in N at 36 min (- 4.5 r) and Figure 4- T 9(b) shows a maximum in M3 at 24 min (z 8.1 1). Since the third moment is proportional to the total crystal mass (Equation (3-5)), both the total crystal number and total crystal mass decrease gradually after reaching their maximum values. After eight to ten residence times (64 to 80 min), the rates of decrease of NT and MT slowed. 4-3-3. Calcium Concentration in the prstallizer Figure 4-10 presents typical results from the calcium concentration measurements. The calcium concentration in the filtrate measured by AA spectrophotometry is the total calcium content in the liquid phase and is related to the solution supersaturation in the crystallizer. The total calcium concentration of the suspension is the sum of the calcium content in the crystal phase and in the liquid phase. The difference between these two concentrations is the calcium content of the crystal mass and should be proportional to M or M3 (the T shaded region). In Figure 4-10, both the filtrate calcium concentration and the suspension calcium concentration show the same trend as the N and M3 T results-- an initial increase and a subsequent decrease with respect to time. These concentrations leveled off after eight to ten residence times. The solid line in Figure 4-10 (marked theoretical) is the maximum NT(t) (#lml) "31:1 (l-umJ/ml) 96 S.OE+O7 4.0E+O7d iOE+O71 ZOE+O74 LOE+O74 01) LOE+08 200 1 8.0E+O7 .. d 6.0E+O7 4 1 t0£+074 4 ZOE+07d d CLO 0 Figure 4-9 Total particle numbers and the third moments for the base run; f T V fi r 7 f f 50 7100 Time (min) (a) total particle numbers, NT’ (b) the third moments, M3. 150 (b) 200 [00} (mM) 97 2.5 K t . 2 2 ‘.___ Unsteady ___. :.__Qua51-steady _.:._ Steadys state : state : state 2.01 : i l 2 I 4 : : ‘ 2 Calcium oxalate : 1.5d 2 scaling : 1 $ : . ‘ ~b \ : [\B 1.0- : ‘ 1 2 : 095 E : i . \ : — theoretical : t H suspension 5 : ‘ x—x filtrate : : 0.0 . . . . . e e s , 1 I , r 1 o 50 ‘100 ' ' '150 ' 200 Time (min) Figure 4—10 Calcium concentrations irrthe filtrates and in the suspensmns; 2+ 2- [C3 l ' [C201 ] - 2 mM, PGA - 10 ppm. 98 calcium concentration calculated to show the upper limit of the total calcium concentration in the suspension. It was calculated from species balance for a CSTR system (Levenspiel, 1972): [a], - 1...], [1 . {—1—}..- 1] ..p[ - +1] (.-.) where [Ca]t is the calcium concentration in the crystallizer as a function of time t, [Ca]i is the inlet calcium concentration, and [Ca]° is the initial calcium concentration in the crystallizer. Since saturated calcium oxalate solution was used in the crystallizer to initiate the operation of the MSMPR crystallizer, [Ca],‘was calculated from the COM solubility product assuming equal calcium and oxalate concentrations; an approximated value of 0.31 for the activity coefficients of calcium and of oxalate as calculated from the EQUIL- NA.BAS program (Appendix A) was used. The deviations from the theoretically calculated and the observed total calcium concentrations are due to the severe scaling effect. Because of the scale formed on the walls of the reactor, baffles, and the propeller, particle numbers and total crystal mass in suspension decreased during a run (Figures 4-9). 4-3-4. C sta l' t'o Mechanisms The distributions of calcium concentrations, the total number, and the third moment as presented in Figures 4-9 and 4-10 give an indication of the mechanisms involved in the crystallization process. The literature generally reports that steady state may be achieved in a calcium oxalate MSMPR crystallizer after eight to ten residence times (Garside, Brecevic, and Mullin 1982; Drach, Thorson and Randolph, 1980). In Figures 4-9 and 4-10, the unsteady-state nature of the 99 process is seen before 8 to 10 r as expected. After about 10 r, the concentrations level off, and the process begins to approach a steady state after at least 12 r at which point there are fewer particles in the solution (in Figure 4-9(a) the peak NT - 4.48E7 at t - 4.3 7, NT - 2.38E6 at t - 12.3 r , and N - 1.35E6 at 22.3 r). T The process may be divided into three regions, unsteady state, quasi-steady state, and steady state as shown in Figure 4-10. In this figure, three regions are divided at 8 r and at 20 1. However, this delimitation is conceptual especially at the division between the quasi-steady state and steady state. The quasi-steady state is characterized by the continuous but gradual decrease of particle numbers with time. At unsteady state, because of the presence of a high supersaturation and a large amount of crystal mass, formation of crystal nuclei may be via heterogeneous nucleation and secondary nucleation. In the early stage of the MSMPR crystallization process when there were no crystals in the system, the initial nucleation in the system is most likely heterogenous nucleation, one of the primary nucleation mechanisms, since homogeneous nucleation is rare. After a large number of nuclei formed, the presence of existing crystals suggests the occurrence of secondary nucleation. High supersaturations also ensured crystal growth, and a large number of crystals provided a high collision frequency for the formation of aggregates. Therefore, nucleation, crystal growth, and aggregation were all involved in the precipitation process. At quasi-steady to steady state, the total particle numbers decreased. Low total particle numbers may greatly reduce both secondary nucleation and particle aggregation with time. Eventually, 100 crystal growth is the only significant crystallization phenomenon in the system. 4-3-5. Efifpct of PGA Additive on Calcium Oxalate Crystallization It was observed that addition of PGA in the experimental solutions decreased the scaling effect. The total particle numbers and the third moments were higher but the maximum particle sizes observed from the PSDs were lower with the addition of PGA (Figure 4-11). Adding PGA also increased the total calcium concentrations in suspension and the calcium concentrations in filtrates (Figures 4-12 and 4-13). Because the third moment is proportional to the total crystal mass, the higher values of the third moment indicate higher total crystal mass due to addition of PGA to the solution. The higher the calcium concentration in the filtrate, the higher the supersaturation in solution. Therefore, addition of PGA increases the total crystal number, crystal retention (crystal mass), and supersaturation in the reactor, but decreases the crystal size formed in the crystallization process. These results suggest that addition of PGA may inhibit crystal growth by reducing the generation of active sites for crystal growth. As a result, a smaller number of the growth sites on the wall of the reactor reduced scaling effect. The lower growth rate also indicates a lower calcium oxalate consumption due to crystallization, which in turn results in higher supersaturation. The higher supersaturations are relieved by the formation of more but smaller crystals. PGA may inhibit crystal nucleation by providing a protective layer on the crystal debris which generate new particles since secondary nucleation is the most likely nucleation mechanism in the system. But high supersaturation results in a burst of nucleation. This may be the ”1(1) (l/mll us“) “film‘s/ml) x—-x 100 ppm H 0W“ BO 100 150 300 350 (c) l i l ”l J l 1 l 8 haul!) (gin) 20+ l .ol x—-x 100 ppm 1 a-a 0 ppm 0% . f f r i O 60 100 160 200 260 Time (min) Figure 4-11 Effects of PGA additive on total particle numbers, the third moments, and maximum particle sizes; (a) effect of PGA on total particle numbers, NT’ (b) effect of PGA on the third moments, M3, (c) effect of PGA on maximum particle sizes, Lmax' [00} (mM) 102 2i) (15- ‘ 0—9 100 ppni «e—e HJppm x—x‘ippnl O 0.0375..p‘?m,....,..-.,.... 0 50 100 150 200 Time (min) Figure 4—12 Effect of PGA additive on total calcuum concentrations In suspenSIons; 2+ 2_ [Ca ] - [C20, ] - 2 mM, 1 - 8 min. [CO] (mM) 2.0 103 1.5d 1.0a 0.5 0.0 Time (min) Figure 4—13 Effect of PGA additive on calcium concentrations in filtrates; 2+ 2- . [Ca ] - [020‘ ] - 2 mM, r - 8 min. 104 reason for the wave curve observed in the PSDs with PGA additive. When crystal growth is size independent and crystal aggregation does not significantly affect the PSDs, the wave curve is sustained to fairly large sizes. The filtrate calcium concentrations for the experiment with 100 ppm PGA addition (Figure 4-13) were not accurate because filtration of the suspensions for this experiment was difficult. It is probable that the large molecules of PGA clogged the pores of the filters. 4-4. Growth rate The discussion in Section 4-3-4 shows that crystal aggregation occurs predominantly during the unsteady state operation of the MSMPR crystallization system. Since an ultimate goal of this work was to gain an understanding of the role of crystal aggregation in kidney stone formation and the effects of biopolymers on crystal aggregation, the unsteady state period in MSMPR operation is of interest. Because crystal growth and crystal aggregation occurred simultaneously in the unsteady-state MSMPR crystallizer operation, understanding of aggregation phenomenon relies upon knowing growth rate kinetics of crystals. Two approaches were developed to find the growth rate of calcium oxalate crystals in the MSMPR crystallizer: Approach A is for the growth rate of the whole process, unsteady state through steady state, and Approach B is for the growth rate at the quasi-steady state. In these approaches, growth rate dispersion and size dependent growth were not considered based on results of single crystal growth experiments conducted in this lab (DeLong and Briedis, 1985) that showed neither phenomenon occurring. 105 4-4-1. Approach A: Growth Rppp bv Compprison of PSD Curvgp From the experiments described earlier, time dependent particle size distributions were obtained. Careful observation of time- sequenced PSDs from the experiment with 100 ppm PGA additive showed that the wave curves as seen in Figure 4-7 appeared to shift along the L axis when one PSD was compared to the next. This observation led to an understanding of the changes in PSDs for the MSMPR crystallizer as described in the following paragraphs. For an MSMPR crystallizer operating at unsteady state, the particle size distribution in the crystallizer changes with time due to the constant withdrawal of suspension from the reactor and due to crystal growth and aggregation. The constant withdrawal of suspension from the reaction vessel results in an exponential decay bath of particle number of each size and total particle number. In this case, the shape (the characteristic peaks and valleys) of the PSD curves remain the same. Adding the effect of crystal growth to the exponentially decaying PSD results in a shift of the PSD to right on the size axis, L. The crystal growth rate may be calculated from the shifted length, AL, and the time interval of data samplings, At, according to AL GL ' At (4'2) For the case of simultaneous crystal growth and crystal aggregation superimposed on the constant particle withdrawal, this growth rate equation also holds true when the changes of particle size distribution due to crystal aggregation are described by a smooth curve superimposed on the changes of PSD due to crystal growth. 106 Based on this argument, every two v(L,t) data sampled at consecutive times were compared. The In n(L,t) or m(L,t) (m(w(L),t) as a function of L) curves were also compared to confirm the shifting of PSD based on matching of the characteristic shape of the PSD curves. The crystal growth rates between every two consecutive PSDs were calculated from the shifted x-axis and the time interval of the two samplings according to Equation (4-2). Figure 4-14 shows the shift of the v(L,t) curve between two sampling times, 43 min and 51.6 min, and the calculation of growth rate from the curve shifting. In order to correlate growth rate and supersaturation, supersaturations for each run were calculated from the measured calcium concentrations of the filtrates by EQUIL-NA.BAS, a modified EQUIL program (described in Chapter 2). As an example, Figure 4-15 shows the supersaturations of COT in terms of the calcium oxalate complex, SCOT,c’ for the experiments with varied residence times. Since the experiments were all conducted with the same stirring rate, identical correlation between supersaturation and growth rate may be assumed for the growth conditions under which the same calcium oxalate phases were generated. This was applied to the experiments in which residence time was varied for the same inlet concentration. Therefore, growth rates were determined by measuring the shifts of curves for several types of PSDs, correlating growth rate with supersaturation, assuming consistent growth rate and supersaturation correlations for varied residence times, and by observing a consistent trend of the growth rate and supersaturation correlations for varied inlet concentrations and for various PGA additives. These procedures for finding growth rates are described in the following. First, growth rates were obtained by matching the characteristic 107 5e6~ E. E s . . m Sampling time: 43 min E 0 3 :i p 5e6- Sampling time: o 51.6 mig_ o 5 25 30 G = AL, .um L (5l.6-43) min Figure 4-l4 Shift of v(L,t) curve and calculation of growth rate; PGA = lOO ppm. 108 12a0 .s—e 1'== 6 nfin. ae-x 1'== 8 nun. 4 :3-3 1'== 16 nun. 10J3- 813« l 0.. [.— CD BJD- 1 o I 00 1” 4to- 2.0 ,. oppress, .-,-. - O 50 100 150 200 250 Time (min) Figure 4-l5 SCOT c in the crystallizer in experiments with varied residence times. 109 shapes of the PSD curves. These growth rate versus time data were used to find a growth rate-supersaturation correlation for each experiment. For some of the growth rate data showing large errors in the correlations, suggested growth rates for better correlation were given. The shifting of curves corresponding to the suggested growth rate was checked again for matches of the PSD curves. The growth rates with reasonable matches of the characteristic shapes of curves were accepted. Otherwise, comparisons of the curves for the growth rates were repeated until the growth rate data correlated well with the supersaturations. The next step was checking the growth rate-supersaturation correlations for a consistent trend with respect to the varied experimental conditions. For example, the growth rate-supersaturation correlations for various PGA additions should consistently decrease or increase parallel to PGA concentrations. This checking was conducted similarly for the growth rate data with varied residence time and with varied calcium and oxalate inletlconcentrations. The shifts of the v(L,t) curves from the procedures described above for the run with 100 ppm PGA and for the base run are shown in Appendix F. Figures 4-16 and 4-17 show the correlations between the growth rates and supersaturations for the experiments with calcium and oxalate concentrations at 1 mM and 2 mM. The growth rates thus obtained for the experiments with varied residence times, PGA additives, and calcium and oxalate concentrations are presented in Figures 4-18, 4-19, and 4-20. The growth rates were correlated linearly with supersaturations for several experimental conditions according to: CL - k x supersaturation (4-3) GL (pm/ min) 110 0.6 ~15 0.5 - I' r 0.4 .. r ..e O r 0434 If 0.2 - U supersaturation SCOM,c V 0.1 x supersaturation 0 GL from comparison of v(L,t) -- 6L fitted with spline approximation oxarreflsmnew...sse...o o 50 100 150 200 Time(min) Figure 4-l6 Growth rates and supersaturations for the experiment with 1 mM calcium and oxalate concentrations. lll 2.0 L-10 x supersaturation : a GL from comparison of v(L,t) -—— GL fitted with spline approximation 0.04,—.—.—.~.,---1fll,.r,,..10 0 50 100 150 200 Time (min) ALL 4 Figure 4-l7 Growth rates and supersaturations for the base run. supersaturation SCOT,c GL (pm / min) 112 2 a—a ‘r a 6 min. H 1' =- 8 min. H 1’ a 16 min. 15.. 11 .54 0 ‘r i *r i 0 50 100 150 200 Time (min) Figure 4-18 Growth rates in experiments with varied residence time. 250 113 1.5 -B— 0 ppm B -ae- 1 ppm ll + 10 ppm + 100 ppm 11 A .5 u E \ \ e u 3 . _J c (D a .l "K 5 = . ' - fi-h‘~= = a: 7 ’ i§f=f=?-3 0 ‘r r i i O 50 100 150 200 lime (min) Figure 4-19 Growth rates in experiments with various levels of PGA additive. 250 GL (pm/min) 114 +lmM +2mM +3mM +4mll 1. .5. k 0 i . r 0 50 100 150 lime (min) Figure 4-20 Growth rates in experiments with varied calcium and oxalate concentrations. 200 115 where k is the growth rate constant. The correlations are summarized in Table 4-4. As shown in this table, for the experiments generating the same cystal phase , i.e. for the runs with calcium and oxalate concentrations equal to 2 mM, 3 mM, and 4 mM and for the runs with varied residence times, the growth rate constants are the same. These consistent growth rate correlations are important in verifying the validity of this growth rate determination method. The results from measurement of calcium concentrations in filtrates are unsatisfactory in the experiments marked with an asterisk as shown in Table 4-5. Supersaturations for these runs were obtained by extrapolation from the supersaturations for the series of runs in which the same experimental parameters were varied. For example, the supersaturations for the run with 100 ppm PGA were extrapolated from the supersaturations for the runs with the addition of zero (base run), one, and 10 ppm PGA additive. Therefore, growth rates for these experiments were obtained only by comparing the PSD curves for the shift of curves between two samplings; these supersaturation-growth rate correlations are estimated only. Correlations of growth rates with the supersaturations in terms of ionic species, SCOT,i and SCOM,i’ were also implemented similarly to the correlations with the supersaturations in terms of complex species, SCOT,c and SCOM,c' It may be more plausible that growth rate is a linear function of the square of supersaturation in terms of ionic species, since both calcium and oxalate ions contribute to crystal growth. It was found that growth rates fit better linearly with the supersaturations, SCOT,c and SCOM,c' The growth rates obtained from the best fit of the linear relationship with each supersaturation 116 Table 4-4 Correlations of growth rates with supersaturations. Cpppelationp Experiment with (GL - k x supersaturation) 2+ 2- [Ca l-[C204 l: 1 mM 0.037 x SCOM,c 0.15 x SCOM,i 2 mM 0.19 x SCOT,c 0.57 x SCOT,i * 3 mM 0.19 x sCOT,c 0.57 x SCOT,i * . 4 mM 0.19 x SCOT,c 0.57 x SCOT,i Residence time: 6 min 0.19 x SCOT,c 0.57 x SCOT,i 16 min 0.19 x SCOT,c 0.57 x SCOT,i PGA additive: 1 ppm 0.12 x SCOT,c 0.392 x SCOT,i 10 ppm 0.065 x SCOT,c 0.25 x SCOT,i * 100 ppm 0.029 x SCOT,c 0.15 x SCOT,i *: estimated correlations for these experimental conditions. 117 definition are slightly different. However, due to the accuracy of growth rates obtained from this approach, one correlation cannot be differentiated from the other. Although COT was not the main hydrate formed in the experiments with PGA additive, growth rates were correlated with the supersaturation of COT, the main hydrate formed in the base run experiment, for the purpose of illustrating the effect of the addition of PGA. Inhibition of PGA on the growth rate was seen from the decrease of the correlation constants with increase in PGA concentration as shown in Table 4-5. If the inhibition by PGA of calcium oxalate crystal growth is due to PGA adsorption on the crystal surface, the growth rate constants at various PGA concentrations, [PGA], should follow the Langmuir isotherm. That is, for the growth rate constant without PGA additive, k., and the rate constants with PGA additive, kPGA' the plot of k./(k.-kPGA) versus l/[PGA] should be linear. Figure 4-21 shows the Langmuir isotherm behavior for PGA adsorption and the linear regression line with a 0.992 correlation coefficient. 4-4-2. ' ow ua - tead ate In the operation of the MSMPR crystallizer at quasi-steady state, crystal growth was the predominant crystallization mechanism. Because of the constant withdrawal of the suspension, the total particle numbers in the reactor decayed with time. Crystal growth rates at the quasi-steady state were calculated according to Equation (3-17); W“? G _ _ (3-17) L le n(L,t) 8L k0 ' kPGA 3.1 118 2.8 .1 2.1 ~ 1.6 d 1.1 'Figure 4-21 .1 u . - .1 l [PGA] Langmuir isotherm of PGA adsorption on calcium oxalate crystals. 119 The semi-log population density plots of the PSDs at quasi-steady state were printed out from the spreadsheet program and were smoothed by hand-fitting. Coordinates of points on the smoothed curves were digitized using the digitizer in the Case Center of the College of Engineering. A FORTRAN program, GMSMPR, was written using Equation (3- 17) to calculate growth rates, G from the digitized 1n n(L,t) data. L’ Every two consecutively sampled PSDs were used to generate one growth rate datum; the differentiation of In n(L,t) with time was estimated as the difference of two ln n(L,t) points over time. The differentiation of In n(L,t) with size was calculated from the splined 1n n(L,t) data using IMSL routines. The computer program is shown in Appendix G. The average growth rates for the data between eight to 16 r are representative of the quasi-steady state and are shown in Table 4-5 for various experimental conditions. The growth rate extracted from the time-dependent growth rate results from Approach A for t - 12 r are also reported in the second column in Table 4-5. In comparing the results from these two approaches, growth rates of the same order of magnitude are obtained. The higher values obtained from Approach B might be due to the smaller slopes of the ln n(L,t) data due to the hand-fitting of the ln n(L,t) PSDs, especially where only few particles were observed in the quasi-steady and steady state periods. As seen from Equation (3-17), smaller slopes result in higher calculated crystal growth rates. This hand-fitting of the PSDs was less difficult for the samples containing more crystal particles. Because crystal numbers increase with increased calcium and oxalate inlet concentrations, the calculation error from Approach B due to underestimated slopes may be reduced. This may be the reason why identical growth rates are calculated from Approach B for the 120 experiments with 2 mM, 3 mM, and 4 mM calcium and oxalate concentrations. With increased calcium and oxalate concentrations, the calculation error decreases; therefore, the growth rate calculated from Approach B approaches the value obtained from Approach A. Table 4-5 Growth rates at quasi-steady state. GL (pm/min) Experiment with Approach A Approach B 2 2- [Ca +]-[0x ]: 1 mM 0.344 0.42 2 mM 0.466 0.67 3 mM 0.538 0.67 4 mM 0.653 0.67 Residence time: 6 min 0.571 0.80 16 min 0.308 0.54 PGA additive: 1 ppm 0.445 0.635 10 ppm 0.401 0.40 100 ppm 0.304 0.315 4-4-3. Com a so of Growt Rates The growth rate results for calcium oxalate crystallization under the effect of PGA additive may be compared with previous growth rate data. Drach, Sarig, et a1. (1982) measured calcium oxalate growth rate in urine-like solutions with added PGA. The steady-state linear population density model was adopted, and the growth rate data were reported as shown in the third column of Table 4-6. The second column of this table shows the growth rates extracted from time-dependent results of this work for time equal to 12 residence times. It is seen 121 that these two sets of growth rates are of the same order of magnitude. The higher growth rates of this work may be due to the higher inlet concentrations of calcium and oxalate used in this work. The urine- like solutions may exhibit an inhibitory effect on crystal growth. The molecular weight of PGA used by Drach and associates were 14,000; the PGA molecular weight is 26,500 in this work. DeLong (1988) found that PGAs of different molecule weights show identical inhibitory effect on COM crystal growth at the same molar concentration. The higher growth rates of this work may be also due to the lower PGA molar concentrations at the same weight concentration of PGA addition due to the difference in molecular weight. Table 4-6 Comparison of growth rates with previous work. GL (pm/min) PGA concentration This work Previous work (Drach et a1., 1982) 0 ppm 0.466 0.391 1 ppm 0.445 --- 2 ppm --- 0.125 10 ppm 0.401 0.136 100 ppm 0.304 --- For an MSMPR crystallizer operating at steady state, the linear growth rate may be obtained from the slope of the semi-log polulation density distribution as described by 1n n(L) - 1n n(0) - —ch_ (1-29) L Average growth rates obtained from the PSDs after 20 r for various experimental conditions are shown in Table 4-7. The steady state growth rate extracted from Approach A are also presented in the second 122 column in this table. When considering the uncertainty in the determination of a linear curve fit to the curved distribution, the steady state growth rates from Approach A and from the linear population density model are fairly consistent. Table 4-7 Growth rates at steady state. CL (pm/min) Linear population Experiment with Approach A density model 2 - [Ca +]-[Ox 1: 1 mM 0.308 0.321 2 mM 0.394 0.395 3 mM 0.472 0.408 4 mM 0.572 0.543 Residence time: 6 min 0.518 0.504 16 min 0.296 0.330 PGA additive: 1 ppm 0.379 0.377 10 ppm 0.365 0.315 100 ppm 0.298 0.326 The steady state growth rate for the experiment with 1 mM calcium and oxalate concentrations may be compared with the MSMPR crystallizer study of Garside et a1. (1982). Garside et a1. obtained a value of 0.18 um/min for the COM crystal growth rate at 8 equal to 10 at COM,c 37°C. In this work, the COM growth rate of 0.308 pm/min was obtained at 8 equal to 7.92. The higher value obtained in this work may COM,c be due to the higher degree of turbulence generated by the propeller type of stirrer compared to the stirrer with flat blades used by Garside et al. For the same total calcium concentration in the 123 filtrate, the supersaturation calculated in this work may be lower than the value calculated by Garside et a1., since Garside only considered the complexation of H2C20‘, HCQO‘-, and CaC20‘. Many more complex species were considered in this work, thus giving a more realistic measure of the driving force for crystal growth. Garside et al. (ibid) correlated growth rate to a second order dependence on calcium oxalate supersaturation-- a supersaturation based on the acitvity of the associated calcium oxalate species (as per Equation (2-4)). As seen in Table 4-4, the growth rates in this work are correlated to first power of supersaturation sCOM,c' Since the associated calcium oxalate species is large relative to free ions, it is reasonable that growth rate would show a first order dependence on S similar to the growth rate correlations observed for molecular COM,c crystal growth with higher supersaturations. The second order growth rate observed by Garside et al. may also have been due to crystal aggregation effects which are unaccounted for in the linear population density model. 4-5. Discussion The two approaches for the calculation of crystal growth rate described in this chapter show consistent results between each method and with previous work. Approach A described was to obtain growth rates by comparing PSD curves for time-dependent growth rates for the unsteady state, quasi-steady state, and steady state periods.' With this method, the effects of biopolymers on crystal growth may be isolated from their effects on crystal aggregation. 124 In the study of the effect of PGA additive, growth rate inhibition by PGA was observed. The inhibition is most likely due to the adsorption of PGA on the growing calcium oxalate crystals. The growth rate data for the entire duration of MSMPR crystallizer operation may be further used in the studies of crystal aggregation with or without biopolymer additives. The integro-differential population balance equations which describe the crystal aggregation phenomenon (Chapter 3) may be solved by numerical methods with the input of the known crystal growth data obtained by the methods described in this chapter. CHAPTER 5 THE EFFECT OF'HDCK CELLS ON CALCIUH.OXALAIE GROWTH IH'A.FLOH GHAHBER 5-1. Introduction The objectives of the work described in this chapter have been to investigate the effect of kidney tissue on calcium oxalate crystal growth and to relate the inhibition or promotion of crystal growth to tissue injury and to the presence of a biopolymer additive. The observation of the crystal growth in an isothermal growth chamber was first adopted by Valcic in the study of the growth rate of saccharose crystals (Valcic, 1975). This type of controlled constant- temperature growth chamber has been used in the growth rate study of crystals formed from secondary nucleation (Garside and Larson, 1978; Berglund, 1981; Shiau and Berglund, 1987). In nephrolithiasis research, DeLong and Briedis (1985) and DeLong (1988) used a chamber of similar design in a continuous-flow system to observe growth rate of single calcium oxalate crystals. This work has contributed significantly to the investigation of the effects of solution chemistry and biopolymeric additives on the formation of renal stones. Gill et al. (1980) have been pioneers in using catheterized rat bladders to study the effect of living tissue on the crystallization of calcium oxalate. Their results show that the nature of the container surface in which crystal growth experiments are preformed, whether it is glass, normal urothelium, or chemically injured urothelium, has marked effects on the adhesion of crystals and on the metastable concentration limit of calcium oxalate. The healthy urothelial surface exhibited a protective action against calcium oxalate nucleation and 125 126 adhesion. This protective property was destroyed by injurious agents such as dilute hydrochloric acid, Triton X100 (a nonionic detergent), or papain (a proteolytic enzyme). Gill and associates (1982) found that heparin was the only saccharide they tested that restored the protective effect of urolithial cells injured by HCl or by Triton X100. The saccharides they examined were two sulfated glycosaminoglycans (heparin and chondroitin sulfate C), a sulfated glucose polysaccharide (dextran sulfate), a nonsulfated polysaccharide with repeating subunits similar to heparin (hyaluronic acid), and the nonsulfated monosaccharide subunits of heparin (D-glucuronic acid and N-acetyl-D-glucosamine). It was also found that heparin bound to the HCl- and Triton X100- injured urothelium but not to the normal urothelium. The special effect of heparin in restoring the anticrystal adhesion may have been due to the negatively charged carboxylate and sulfate groups on heparin (Figure 5- 1). The negatively charged groups may preferentially adsorb on the 2 Ca + ions on the crystal surface and may chemically or sterically prevent crystal adhesion. COO CHZOSO3 l_4 l—__o /| \ /| \ H / H \ H H / H \ H \\on H// \\OH H// l I l muso" 030 3 3 Figure 5-1 Repeating unit of heparin. 127 Wiessner et a1. (1987) isolated the rat renal inner papillary collecting tubule (RPCT) cells and studied the interaction of the cells with preformed COM and COD crystals. Cultured RPCT cells were exposed to media containing COM or COD crystals, which were rinsed off after 15 minutes. The crystals, especially small ones, were found adhering on or around the rounded-up cells. This suggests a possible interaction between crystals and unhealthy cells. In further work, Mandel et a1. (1987) studied the crystal- membrane interactions by measuring the red blood cell hemolysis induced by crystals. The results showed that the several known inhibitors of calcium oxalate crystal growth and aggregation inhibited the potential for COM and COO crystals to interact with membranes. The methods used in the studies of Wiessner et al. and Mandel et a1. were done in a batch, stagnant condition and did not take into account the fluid dynamics of renal flows. The work described in this chapter considers both crystal-membrane interactions and flow dynamics. In this work, the integration of the photomicroscopic technique and experimentation on living cells will be addressed. A monolayer culture of Maden Darby Canine Kidney (MDCK) cells was incorporated in a modified growth chamber to study the effect of epithelium on the growth of single calcium oxalate crystals. The original cylindrical growth chamber described earlier was redesigned to a rectangular configuration so that the flow pattern in the chamber was well characterized. This permitted quantitative analyses of the mechanistic effect of the flow on epithelium. The rectangular growth chamber maintains the advantage of the original growth chamber for direct microscopic observation of crystal 128 nucleation and growth. Its design and assembly are discussed in the following section. 5-2. Precipitator Design and.Assemb1y The flow chamber precipitator is a multi-layer assembly held tight with screws (Figures 5-2 and 5-3). The chamber was designed by Alan L. Powell and was constructed in the Machine Shop of the College of Engineering. It contains two main chambers divided by transparent plastic; the upper chamber is for the flow of solutions, and the lower chamber is for the circulation of water for temperature regulation. The diagrams for upper and lower chamber pieces are shown in Figures 5- 4 and 5-5. When the growth cell is fully assembled, the experimental solution is introduced into the cell laterally as shown in Figure 5-2. It then flows upward to the upper chamber. The upper chamber is of a rectangular cross section with width, b, of 1.27 cm and height, a, of 0.397 cm (Figure 5-6). The total volume of the upper fluid pathway, including the horizontal flow chamber and the inlet and outlet portions of the chamber, is 5.5 m2. A thermistor is located at the downstream end of the cell in the solution side for monitoring of temperature. A cover glass is positioned on the plastic divider and provides a surface of glass or of cultured MDCK cells for growing crystals. The monolayer of cultured epithelial cells may be treated chemically or enzymatically to simulate tissue damage. This configuration allows the chemical effect of the presence of living cells (normal or injured) on crystal formation or the flow mechanistic effect on the cells to be observed by photomicroscopy. 129 Circulating Solution water Solution outlet outlet inlet mg: —» ME: 32:: -fiho . O ‘ v v "s a O“ h- 1 ”‘0. I‘Ha ‘ our. k (D H q M“. 3. NC‘ ‘..* b J-‘ola rh- - O :3 / / Temperature probe ---cflooourcub- ___.c.":‘.’::"::’fL.. - Circulating water inlet Figure 5-2 Top view of the growth chamber. 130 S A B C D E F T. J K L N r .m am“ p 1 J 4 J J 1 1 .m w 1001 v8! A fiUUIIL I run II'IuI'L t t A 9 o I I A 1 ..... III A IV 00“] 0 IHIIIIIII e r t S .-: U.-. . fl k e e J . s b k s .L . a m s e s ... a ..- ... 8h Mom he. “a! I ICIQCMII M u c m amom . nnnnn i nus... d r o a .c H n "-.:..- n . ...:u i e t e t . 4 1 nnnnnn V W t S S . fl \"II .1. O O a . rt» “UTLnun.9. IVs Tilk ..-; Thu.”- "UA 0 IUIIIHIHIIVI on e. o. to o. Amt-00L 10‘“! I IIIII a I to; IIIIICI I J K L M . . . . w o .1 c G h . t . . w. vol-L L YCPII l VAIHHI'Iu'- V UIHA '01" I I .... I e 3 E O '0'] -..-l 1.0-fill! I T ..... A A -----l d / .1 l 0 © H n . .A f v ..... 1 M .m c a .n . "an“; n ... "HG...” a i k or X s- '4 t p s . © 8 l 2 o a '01 I r- 'l r r r m g t-L r u r e e o r t-.. e C C r e v.'T lv..'ll f. Y- ..... I u 11.. LY """ l w M. ”no“ W I '0 I .E-E. l.‘ .l "I“. .l I. "l.' "l.l H 1Y0! c r801 I Ulnlll-nu¢ j v04 T 000000 U S S M C [L i D f L A B c D E (1 7/32" long) N: Lower retainer Figure 5-3 Exploded side view of the growth chamber. F: Upper chamber holder G: Cover glass H: Plastic divider 131 Top view <3 C) <3 <3 C) <3 Side view (2X) 'l I J 0' {0; “EL “““ r “““““ ili‘ ““““““ g 1'1 1 Ezzzzzp ”1 I I Figure 5-4 Upper chamber holder. 132 Top view {Is \a“! ‘0 . .....U .-...3.-. 1.3.3.3."- - - r u cu...» - . O "nun“...uuh v .>3& .x/H‘... . mwlnu I.‘.\I_urU—Uni SVWKA - . -- . . .13.. L O O ..---.L w ....... L o o WWW/xxx“ T©i r. ....I..; Side view Figure 5-5 Lower chamber holder. 133 -fl-’ -- ------n---------I I 'Q 7 "Q- ---r / “'"7’ I I‘---7 X + ...... J I“'::' 1 guy I, War I \vl l ‘1 Direction of flow Figure 5-6 Modeling coordinates of the flow chamber precipitator. 134 5-3. Fluid.Mechanics Properties in Upper Chamber In order to quantify the mechanistic effect of flow on cells, calculation of some of the flow characteristics in the upper chamber is necessary. These are average velocity of the flow, Reynolds number, transition length, velocity profile, and shear stress. Calculations were done based on the geometric dimensions of the solution chamber and the physical properties of the test fluid. The physical properties of the model renal fluid used in experiments are shown in Appendix D. According to the design, the flow chamber precipitator may be modeled as a rectangular slit with dimensions and defined coordinates shown in Figure 5-6. The horizontal x-axis is the direction of flow. Excluding two discontinuities in the chamber floor (Figure 5-4), the total length in the x-direction is 4.5 cm. The average velocity of the flow, V (cm/s), is related to the 3 volumetric flow rate, Q (cm /s), according to V - Q/A (5-1) 2 where A is the cross-sectional area of the flow (cm ). For the chamber of rectangular cross section, A - ab (5-2) where a and b are the thickness and width of the upper chamber. 5-3—1. Re 0 ds Numbe The characteristic dimensionless number for flow, the Reynolds number, Re, is the ratio of inertial forces to viscous forces and is expressed as (5-3) 135 s where p is the fluid density, g/cm , and p is the fluid viscosity, g/cm-s. The equivalent diameter of this noncircular conduit, de’ cm, is calculated as 4 ab de ' 2(a+b) (5-4) a definition related to cross-sectional area and wetted perimeter of the conduit (McCabe et a1., 1985). The magnitude of Re determines whether the flow pattern is laminar or turbulent. For Re below 2100, the flow is laminar; above about 4000, the flow is turbulent. The transition region lies between 2100 and 4000 where the flow depends on roughness of conduit surface and configuration of the conduit entrance. Reynolds number for the flow chamber designed in this work was calculated for various flow rates shown in Table 5-1. In all cases flow was laminar. The transition length and maximum shear stress shown in Table 5-1 are discussed in Section 5-3-2 and 5-3-4, respectively. Table 5-1 Reynolds number, transition length, and maximum shear stress for flow chamber precipitator. Flow Rate Re x r t max 2 (ml/min) (cm) (g/cm-s ) 10 28.4 0.859 0.0403 20 56.8 1.718 0.0807 25 71.0 2.147 0.1008 30 85.2 2.577 0.1210 40 113.6 3.436 0.1613 50 142.0 4.295 0.2017 99 281.2 8.503 0.3993 136 5-3-2. rans 10 e t Transition length is the length required in the entrance region for the boundary layer to reach the center of the flow and the velocity profile to become fully developed. In laminar flow, it is empirically formulated as (McCabe et a1., 1985): xt: D - 0.05 Re (5-5) where xt is the transition length and D is the diameter of the flow pipe. For the flow chamber precipitator of noncircular cross section, equivalent diameter was used for D to calculate transition lengths as shown in Table 5-1. To assure a well developed, well characterized flow field over the region where crystal growth was photomicroscopically observed, Table 5-1 shows that the flow rate for experiments was limited to 50 ml/min or below because of the length of the chamber. 5-3-3. Velogity Pzpfiilg In laminar flow, no lateral mixing is assumed. For a fully developed laminar flow in rectangular conduits, the velocity profile in x-direction has been given by Timoshenko and Goodier (see Knundsen and Katz, 1958): 2 - 4a g -1 “NW _ __3_2_ .3. z _1_3 (4)“ W x x p n-1,3,5 n _ gosh§pgy[a) _ [1 cosh(nxb/2a) cos(nnz/a) (5 6) where P is the hydraulic pressure of the steady flow fluid. The pressure gradient due to friction, dP/dx, is related to the friction factor, f, according to (ibid): 137 _2_o_—V f (5-7) where the Fanning friction factor, f, has been derived theoretically (ibid): 2 f [a ] [1 -"Lz'2—a—tanh(%') --13=-gcanh(§1'§3) + ...] - 6 de « b Re (5-8) With known a, b, and de’ the friction factor was calculated for the flow chamber to give 12,312 f - Re (5-9) The expression describing the velocity profile in the chamber may be derived from Equation (5-6) and is expressed as ux(y.2) - -’-—'-—°-3§ 5: 2 ‘13 (-l)(n'l)/2 x n (1a b n-1,3,5 n m [l - cosh(nnb/2a) ] cos(n«z/a) (5-10) Figure 5-7 illustrates the velocity profiles calculated from Equation (5-10) for 2 at H pm above the bottom wall of the chamber for a flow rate of 40 ml/min. Since hyperbolic cosine is an even function, ux(y,z) is an even function of y; i.e., velocity profile is symmetrical with respect to y. Figure 5-7 shows only the curves for y a 0. From Equation (5-10), it is seen that velocities at constant 2 or constant H are each the same function of y, i.e., a summation consisting of [l-cosh(n«y/a)/cosh(nnb/2b)]. Therefore, curves in Figure 5-7 have the same profile when they are presented in terms of percentages of the maximum value of each curve. The vertical lines in this figure mark the percentages of the values in terms of their 138 0.15. - -- 99 41 3‘ --- 99.11 98 21 T; \ E ’2" + N \ 'fl 0 I' t 'I- "i I 1 -~ 3 .1 33 '1 l I u 43". 5:5 a? ”x 005 ' ' ”a. fi-‘i' - I I H - 20 pm «$2113 A.“ Figure 5-7 Illustrative velocity profiles; flow rate at 40 mfl/min. 139 maximum values, which occur at y - 0. Nearly constant velocities are found within y - $0.2 cm. 5-3-4. Shea 8 es The shear stress at the interface between the flow and epithelial cells is a quantity of importance in observing the mechanistic effect of flow on living cells. It is defined as (Bird et a1., 1960) r ( z--a/2) - - aux(y'2) (5-11) zx y’ p 62 z--a/2 Substituting Equation (5-10) into Equation (5-11) for ux(y,z) and taking the derivative with respect to 2 gives: zx(Y. z--a/2) _ _lé_1§29_ 2 _1_ ( l)(n- l)/2 I 2de 2b n-l, 3, 5 n2 m . [1 - cosh(n«b/2a) ] Sln(-nn/2) (5‘12) Figure 5-8 shows the shear stress calculated from Equation (5-12). Since shear stress in laminar flow also represents a flux of momentum, negative values of shear stress in Figure 5-8 denote that the momentum is transferred towards the direction of negative 2. Since Equation (5- 12) contains the same function of y as that in Equation (5-10), shear stress at the surface may also be presented in terms of percentanges of the maximum shear stress, the maximum absolute value of shear stress, which also occurs at y - 0. Let rzx(y-0,z--a/2) - rmax(Q)’ then _ll_1229_ .1. 1 ax(Q) - « zde 2b i_1 3 5 n2 [1 - cosh(nnb/2a) ] (5.13) Table 5-1 shows the calculated maximum absolute shear stress at several flow rates. 140 flu... .. fir GIJFIDlnTmJudIFdd_ NN.mmiu lllllllllllllllllllllll 2.3% ...: - null}..- --.. u-.uu-.-..- Risa. ...... _ uuuuuuuuuuuuuuuuuuuu «e.g.,---uu..- -------u----n.uuuunn N~.m&..l .l I I .I illvlilllilll 5.:-..-..l-|....-....-..-u-..--..-u ILWlOI- all... < 4 4 d # q q 4 q A a A q a J 1 0 5 o 5 0 1 1 O .0 .0 A 2.23 a}... scan. .5 .3 -05 -07 y (cm) Figure 5-8 Shear stress at interface surface; flow rate at 40 mE/min. 141 5-4. Experimental Materials and.Methods The experimental technique and procedures were developed and implemented to test the hydrodynamic effects on cell viability and to observe the effects of cell presence on crystal growth. 5-4-1. a b a The Maden Darby Canine Kidney (MDCK) cell line was chosen for use as it is one of the best characterized epithelial cell lines available for study (Taub and Suier, 1979). The cells present the morphological and enzymatic properties of the epithelium from the distal tubule of the kidney. In this work, facilities for culturing MDCK cells were provided by Dr. William S. Spielman of the Department of Physiology. Monolayer cell cultures were successfully grown on an ordinary cover glass. The cells were cuboidal in shape. For consistency in experimental runs, the cell samples were used four days after culturing; at that time, they were confluent and multicellular blisters or domes were always observed. Formation of multicellular blisters in confluent MDCK cultures is an indication of healthy cells that are ready for water and solute transport (ibid). The culturing technique used is briefly described below. Dr. Spielman provided the starter culture of MDCK cells in a 100x15 mm culture dish. For these cells, 5 m2 of phosphate buffered saline (PBS) was added to remove calcium ion from the original medium residue. Three m1 of 0.1% trypsin in a 0.05% EDTA solution was used to disaggregate the cells. After 10 to 15 minutes, the cell dish was tapped gently to ensure the detachment of the cells. They were then washed twice each with 5 m2 DulBecco Modified Eagle's medium (DMEM) and collected in a centrifuge tube. After centrifuging at 1500 rev./min 142 for 5 minutes, the supernatant in the tube was removed, and 5 m2 of 10% fetal bovin serum (FBM) in DMEM was added. Three drops of the well- mixed cell medium mixture were then transferred onto a cover glass (briefly sterilized with a flame) and put in a 35x10 mm culture dish. With careful addition of 1.5 m2 10% PBS in DMEM to the culture dishes, the cut cells were placed in an incubator to achieve confluency. 5-4-2. We}: For the MSMPR crystallizer system described in Chapter 4, calcium oxalate was chosen as the model urinary precipitating agent; equal concentrations of calcium and of oxalate were used. Likewise, equal molal concentrations of calcium and of oxalate were also used in this study. In order to prepare solutions for the experiments with cultured MDCK cells, a modified Earle's medium containing compounds necessary both for cell maintenance and for calcium oxalate precipitation was required. Table 5-2 lists components of the unmodified Earle's medium (Gibco Laboratories), the calculated ionic strength, sodium to potassium ratio, and total species concentrations. These values were calculated by assuming that all the ingredients were completely dissociated in solution and no complex formation among ion species occurred. Using Earle's medium as a basis, the compositions of experimental solutions were determined to accommodate several criteria: 1. adjustment of Earle's medium to include equal calcium and oxalate concentration in each preparation of solutions; 2. adjustment of medium osmolarity to prevent water loss from the intracellular compartment via osmotic flux (Cooney, 1976); 3. control of medium Na/K ratio and Na concentration to prevent significant nonequilibrium distribution of ions via cells’ sodium- potassium pump (Berne and Levy, 1983); 4. addition of fewest ingredient species possible to avoid 143 Table 5-2 Original Earle's medium recipe (1X). Molecular Concentration Weight (g/l) (M) INORGANIC SALTS: CaC12 110.99 .2 .0018020 Nazczo, 134. KCl 74.56 .4 .0053648 MgSO‘-7H20 246.48 .2 .0008114 NaCl 58.44 6.8 .1163587 NaHCO3 84. 2.2 .0261905 NaH2P0‘-H20 137.99 14 .0010146 OTHER COMPONENTS: D-Glucose 180.16 1. .0055506 Phenol red 354.38 .01 .0000282 AMINO ACIDS: L-Arginine-HCl 210.67 .126 .0005981 L-Cystine 240.3 .024 .0000999 L-Glutamine 146.15 .292 .0019979 L-Histidine HC1~H20 209.63 .042 .0002004 L-Isoleucine 131.18 .052 .0003964 L-Leucine 131.18 .052 .0003964 L-Lysine HCl 182.65 .0725 .0003969 L-Methionine 149.21 .015 .0001005 L-Phenylalanine 165.19 .032 .0001937 L-Threonine 119.12 .048 .0004030 L-Tryptophane 204.23 .01 .0000490 L-Tyrosine 181.19 .036 .0001987 L-Valine 117.15 .046 .0003927 VITAMINS: D-Ca pantothenate 476.54 .001 .0000021 Choline chloride 139.63 .001 .0000072 Folic acid 441.41 .001 .0000023 i-Inositol 180.16 .002 .0000111 Nicotinamide 122.13 .001 .0000082 Pyridoxal HCl 203.63 .001 .0000049 Riboflavin 376.37 .0001 .0000003 Thiamine HCl 337.27 .001 .0000030 CALCULATIONS: Sum(Cj) .3441467 Ionic strength .2150339 Na/K 26.76027 144 complicating the experimental results due to the presence of multiple ionic species while still maintaining MDCK cell viability. Many species used in the original Earle's medium are present to provide for cell growth and multiplication; the objective herein was only to maintain the cells in a viable state for the duration of the experiments (one to two hours). Therefore, the inorganic salts chosen for the solution were CaC12-2H20, Na2C20‘, KCl, and NaCl, and the only organic species was D-glucose. Table 5-3 shows the composition of solutions prepared for experiments with [Ca2+] - [C20‘2-] - 0.001 M and its derivation from the original Earle's medium. In the table, recipe A excludes all organic components from the medium except D-glucose. Recipe B shows a recipe with dihydrous calcium chloride replacing anhydrous calcium chloride and with MgSO‘-7H20, NaHCOa, and NaHPO‘-7H20 omitted to avoid effects of these ingredients on experimental results. The amount of NaCl was increased to maintain identical total Na concentration. The column for recipe C shows that the concentrations of KCl and NaCl were increased proportionally by multiplying the ratio of Sum(C ) in the J column of recipe A to that of recipe B. Since single crystal experiments require mixing of the cationic (containing Ca2+) and anionic (containing C2042-) solutions, these two solutions were prepared separately but each contained equal concentrations of KCl, NaCl and D-glucose. Upon mixing of the two solutions, the total Na concentrations and Na/K ratio were the same as those in recipe C. Weights of chemicals for solution preparation are also shown in Table 5-3. For experiments with calcium and oxalate concentrations other than 0.001 M, weights of CaC12-2H20 and Na2C204 were changed according to 145 Table 5-3 A typical solution composition for 2+ 2- [Ca ] - [C 0 ] - 0.001 M. 2 O Inorganic Salts Recipes Experimental A B Weight in C Conc.,after Solutions M.W. (M) (M) (M) Mixing (M) g/2 1 g/4 2 g/6 1 CaC12-2H20 147. --- .00180 --- 0.001 .5881 1.1762 1.7643 CaCl2 111. .00180 --- --- --- --- --- --- Na2C204 134. --- --- --- .001 .5360 1.0720 1.608 KCl 74.55 .00536 .00536 .00588 .00588 .8773 1.7546 2.6319 MgSO‘.7H20 246.5 .00081 --- --- --- --- --- --- NaCl 58.45 .11634 .14354 .15742 .15542 18.1683 36.3365 54.5048 NaHCO3 84. .02619 --- --- --- --- --- --- NaH2P04-H20 138. .00101 --- --- --- --- --- --- D-glucose 180.2 .00555 .00555 .00555 2. 4. 6. Sum(Cj)* .3386 .3089 --- .3342 Ionic Strength* .2150 .1543 --‘ .1673 Na/K 26.756 26.756 26.756 26.7565 pH 7.2 7.2 7.2 ** Ionic Strength .1489 .1613 *: Assuming all ions were dissociated and no reaction took place. 2+ **: Assuming all ions were dissociated and reaction consumed all Ca 2- and C20‘ . 146 the required concentrations, but the amount of NaCl was reduced to maintain the same total Na concentrations and Na/K ratio as those in recipe C. All the chemicals used in this study were of reagent grade. CaC12-2H20 was obtained from J. T. Baker Chemical, Na2C20‘ from Fisher Scientific, and ROI, NaCl, and D-glucose were from Mallinckrodt Chemical. Solutions were prepared with doubly distilled deionized water. The solution pH was adjusted to 7.2, an optimum pH for living cells, immediately before use. The Metrohm E632 pH-meter and a Metrohm combined pH glass electrode were used in monitoring pH values in solution. 5-4-3. me t et-U Figure 5-9 shows the experimental set-up for the crystal growth flow system. Calcium and oxalate solutions were pumped separately through filters, flowmeters, heating coils, and then into the flow cell. There growth of single crystals and mechanistic effects of the flow on living cells or on crystals were observed microscopically. The (R) microscope used was the 110 Microstar microscope manufactured by A0 Scientific Instruments. A 10X objective with a 9.1 mm working distance was used to accommodate the design of the flow chamber precipitator. When the microscope was in use with the flow chamber precipitator, the objective extended into the rectangular opening of the upper retainer of the precipitator. This configuration and the dimensions of the opening and the objective allowed the microscope to focus on objects only within a range of approximately y - $1.25 mm in the flow chamber. From Figures 5-7 and 5-8, it can be seen that both the fluid velocity and shear 147 .Emumxm zopc suzocm pmamxgo mum «camp; imam: new; Loam: ocean—aucpo .1 mumpmxo umpazo =o_u=_om u --.l- 4. mcmuppa mass; .8955 538. . _._...__ b a _ mcowum>gmmno : <5 <5] ovaoomocowsoaoga F_ mcmumezopm _r _ Esau—mu 148 stress are within 99% of their maximum values within this range of y - £1.25 mm. It is therefore reasonable to assume that the experimental observations were implemented under a fairly constant flow field. An Olympus OM-G camera and an automatic winder, Olympus Motor-Winder 2, were mounted on the microscope for photomicroscopic observation. Photography was automated by the coupling of a timer with the automatic winder. The pumps used in this work were Cole-Farmer Masterflex variable speed peristaltic pumps. The flowmeters were Cole-Farmer variable-area flowmeters. For the filters, 0.2 pm cellulose acetate membrane of 25 mm in diameter, manufactured by Micro Filtration Systems, were used with in-line holders. The glass heating coils were made in glass shop of the Department of Chemistry. An E-3 Haake circulator was used to control the temperature in the water bath. A stainless steel thermistor probe (YSI series 400) of 1/8 inch in diameter was inserted into the flow chamber and connected to a multimeter (Keithley Instruments, Model 179 A TRMS). The temperature of the solution in the flow chamber was indicated by converting the resistance reading of the multimeter to temperature. The desired solution temperature was 37°C to simulate human body temperature. Because of the heat loss from the liquid flowing out of the water bath to the flow chamber, the temperature in constant temperature water bath was usually several degrees higher than 37°C. A T-connector, placed immediately upstream of the growth chamber, allowed the mixing of the calcium and oxalate streams and introduced the mixed solution into the cell. The flow configuration of such a T- junction was believed to provide thorough mixing of two solutions based on the kinetics of ion complexation (DeLong and Briedis, 1985). 149 (R) Tygon tubing was used for the connecting lines throughout this experimental set-up. 5-4-4. Expegimental Scheme Experiments in this work included testing the viability of MDCK cells under flow shear stress and observation of single crystal growth on MDCK cells. Monolayer MDCK cells grown on a piece of cover glass, 18 mm square, were installed in the flow chamber to simulate a physiologically realistic interface between the model renal fluid and renal epithelial tissue. For observations of crystal growth on glass, a plain cover glass was used. The cover glass, with or without MDCK cells, was placed in the chamber so that the observed single crystals were in a fully developed flow field according to the transition lengths calculated. 5-4-4-1. Viability Tests on MDCK Cells Testing of MDCK cell viability was studied using three flow rates, 25, 49.5, and 99 ml/min. At the flow rate of 99 mfl/min, the transition length (Table 5-1) was longer than the length of the chamber, i.e., the velocity profile within the chamber was not fully developed. However, the boundary layer effects could be considered to be similar to fully developed flow, especially at a region close to the wall where boundary layers develop. The flow shear stress on MDCK cells is an effect of the boundary layer at the wall; its value may be calculated from the equation for flow of a developed velocity distribution, i.e., from Equation (5-13). Solutions for the shear tests were prepared with low calcium and oxalate concentrations, 0.0002 M, to avoid crystallization of calcium oxalate from the solution. Therefore, only one solution stream was 150 necessary. After testing the cells under shear stress for several hours, the solution pump was turned off, and 1 to 1.5 m1 of 0.4% trypan blue stain (Sigma Chemical Company, Catalog number T 9520) was injected from upstream of the solution chamber. The blue dye was allowed to flow into the chamber and was left in the chamber for about 10 min. Dead cells with ruptured cell membranes would take up the dye. After replacing the blue dye with clear solution, dead MDCK cells were easily distinguished from the living ones. Specific results of the viability tests are discussed in Section 5-5. Satisfactory viability tests of the MDCK cells made possible the growth rate study of calcium oxalate crystals on MDCK epithelium. 5-4-4-2. Growth Rate Experiments: Effects of Additive and Tissue Surfaces The objective of studying the growth rate of single calcium oxalate crystals was to observe the interaction of kidney epithelial cells and COM crystals as a possible presursor to the formation of urinary stones. These studies were performed under controllable conditions through a simulated renal flow system while previous work had been done only in a static environment (Mandel et a1., 1987; Wiessner et a1., 1987). For the requirement of fully developed flow, flow rate for experiments was also limited by the length of the flow chamber according to the transition length calculated. The flow rate for this study was set constant at 40 mI/min, almost the maximum flow rate attainable for a well characterized flow field. Before beginning an experiment, solutions were prepared and connected to solution pumps. The circulator for the water bath was turned on. Air bubbles in solution lines, mostly trapped in the filters and flowmeters, were 151 purged out of the system while the growth chamber was not in line. After the chamber precipitator was assembled, it was placed on the microsc0pe stage and connected to the solutions and to the circulation water. The pumps for solutions and for heating water were turned on, and the flow rates were increased gradually using the pump controllers. Both the upper and the lower chambers of the precipitator were filled so that there were no gas bubbles trapped in the chambers. Tilting and tapping the chamber accomplished this. The flow chamber was then observed under the microscope with the objective focused on the solution-surface (glass or cells) interface. While waiting for the formation of crystal nuclei to occur, the solution flow rates and the precipitator temperature were monitored. After the induction time, nuclei of single crystals were formed and observed. Timed-sequenced photomicrographs of crystals were taken to follow the course of crystal growth. The photographic slides were developed and images of the crystals were transferred to paper. Equivalent circular diameters, Leq (pm), of the crystal images were then calculated from the area of crystals measured with an image- analyzer (the HipadTM Digitizer, Houston Instrument), i.e., Leq - /4A/1r (5 - 11.) Growth rate G, pm/min, of each crystal was then calculated: ALe G - ‘3329— (5-15) where At is the time interval between photographs (in min). To test the effects of solution chemistry on the growth rate of crystals growing on MDCK cells, experiments with various concentrations of equimolar calcium and oxalate were conducted to observe the effects of supersaturation. Values of supersaturation were calculated 152 according to the equations in Chapter 2. For comparison, experiments with crystals growing on plain glass were performed with the same solution conditions as for the crystal growth experiments on MDCK epithelium. One of the other significant effects of solution composition is the inhibition or promotion of crystal growth on MDCK cells in the presence of biopolymers. This could provide evidence for kidney stone etiology in terms of the concentration or species of the constituents in renal fluid. Each individual candidate biopolymer may be added into experimental solutions to examine its effect on the rate of crystal formation. In this work, heparin was chosen to model the effects of biopolymer additives in experimentation. The sodium salt of heparin from Sigma Chemical Company (H 3125) was added to the anionic solution to prevent interactions of calcium cations and the anionic heparin. Since heparin would have been filtered out by the in-line filters, preparation of the anionic testing solution was altered. The solution was filtered in batch before addition of heparin, and the solution pH was adjusted afterwards. The in-line filter for the anionic stream was then removed from the system. Concentrations of heparin from 2 mg/2 up to 100.mg/£, in terms of the total volume of mixed solution, were tested. In order to understand the effect of damaged tissue surface on the possible mechanism of renal calculi formation, MDCK epithelium injured with HCl and with papain were included in the photomicroscopic crystal growth study. Papain is a proteolytic enzyme, which hydrolyses a variety of peptides, amides, and esters (Barman, 1969) and is used in tissue disaggregation (Schwartz and Azar, 1981). Although the specific 153 action of HCl is not known, it was suspected of denaturing cellular surface proteins. The chemical concentration and length of time of MDCK cell exposure for chemical injury were tested to assure cell viability during the time period of an experimental run. Appropriate procedures were identified for HCl and for papain injured cells; healthy MDCK cells on a cover glass were soaked in 0.1 N HCl solution for 4 minutes and in a prepared papain solution for 35 seconds. The papain soaking solution was prepared as follows: 0.0333 g crude papain powder (Sigma Chemical Company, type II, Catalog number P 3375) was added to 100 m£ H20, then shaken occasionally for 30 minutes. One m2 of the papain decantant, 1 m! 83.3 ppm EDTA solution (as an activator), and 0.666 m2 of 0.5 M acetate buffer solution, pH 6.5, were mixed with H20 to a total volume 50 m2. To test the system's performance and to understand the mass transfer phenomena in the system, growth rates of crystal grown on glass surface were also investigated in comparison with DeLong's work which was conducted with a cylindrical flow chamber (DeLong and Briedis, 1985; DeLong, 1988). The ionic strength of solutions were kept at 0.15 M by addition of KCl instead of the compositions used to prepare the modified Earle’s media. Potassium oxalate was used in place of sodium oxalate, and solution pH was adjusted to 6.0. For some of the runs, effluent samples from the flow chamber were collected and diluted 1:10. Total calcium concentrations were then analyzed with an atomic absorption spectrophotometer (Varian AA-375). The concentrations of free calcium ions were measured with a calcium (R) electrode (Orion, Model 93-20) on the Fisher Accumet pH meter (Model 154 825 MP). The reference electrode used was a single-junction electrode (Orion, Model 90-01). 5-4-4-3. Induction Period in Closed Vessels For calcium and oxalate solutions mixing in a closed vessel, an induction or incubation period is usually observed before occurrence of nucleation of crystals (Walton, 1967; Nielsen, 1955). In order to understand the nucleation phenomena in the flow chamber system, auxiliary experiments on induction period were performed by direct mixing of calcium and oxalate (potassium salt) solutions in beakers. The ionic strength of solutions were kept at 0.15 M by adding KCl. D- glucose and NaCl were not added. The induction period was defined as the time period between mixing of solutions and occurrence of visible turbidity. 5-5. Results and Discussion 5-5-1. Cell Viability After the live epithelia were exposed to the flow shear, some of the cells became rounded and eventually detached from the supporting surface. Stathopoulos and Hellums (1985) studied the effects of shear stress on human embryonic kidney cells and suggested an argument that the cells remaining attached are all viable. This was also assumed for the discussion that follows. At the three flow rates used to test for cell viability, the results show significant effects of the flow shear stress on MDCK cells (Table 5-4). After the application of trypan blue stain after each experimental run, no intake of the blue dye was observed for the cells remaining attached, though they were rounded to some extent. It was 155 also discovered that the length of time that cells were outside of the incubator also played an important role in cell viability. As a result of these observations, the conservative conclusion made was that, with careful handling, MDCK cells remained viable through the time period for experimental runs in the study of single crystal growth, which took 1 to 2 hrs at a solution flow rate of 40 mfi/min. Table 5-4 MDCK cell viability with shear stress. Flow Rate 1 Cell Viability max (mi/min) (g/cm.sz) 25.0 .1008 intact for 7 hrs. 49.5 .1997 some cells detached after 2 hrs, the rest are fine for 6 hrs. 99.0 .3993 some cells detached at the onset of the run, the rest are fine for 6 hrs. 5-5-2. b e t' n t ta 5 During the experiments of single crystal growth, observations on crystal nucleation and crystal habit were made and are discussed in this section. 5-5-2-1. Nucleation It was found that single crystals grew all over the surface of the upper flow chamber, from the T-junction, where solutions were mixed, to the tubing that leads solution out of the flow chamber. It is interesting to compare the nucleation observed in the flow chamber with - ’n‘J-.__r 156 the homogeneous nucleation in a closed vessel which occurs after an induction period after mixing of calcium and oxalate solutions. Figure 5-10 shows induction period data for the formation of calcium oxalate from the literature (Nielsen, 1955; Skrtic et a1., 1984) and from experiments conducted in this work, in which E is the intial reactant concentration or the geometric mean concentration of the mixed solution. Although the data were obtained by mixing calcium and oxalate from different salt sources with diverse experimental conditions (temperature, ionic strength, stirring condition), the data exhibit a consistent monotonic decrease of induction time with increasing concentration and a fairly consistent increase of induction time with increasing ionic strength. The experimental data for ionic strength of 0.15 M were fit to yield a linear regression line shown in the figure. From this regression line, the induction periods calculated for concentrations used in the single crystal experiments are tabulated in Table 5-5. These induction period data could be used to help identify the nucleation mechanism occurring in the flow chamber precipitator. Table 5-5 Calculated induction periods at the reactant concentrations used in single crystal experiments. Reactant Concentration, 3 Induction Period, t (mM) (8) 0.5 3630 0.75 965 1.0 377 1.25 182 1.4 126 1.5 100 157 4 3 - 2—4 l-. r, E :1 0 U) ,9. -1 J U +- + DD -24 U-u- -3‘llllllijlljflTT—f -3.3 -2.9 -2.5 -2.1 -1.7 109 [5 (M)J : Ca(CHSCOO)2 + (NH,)20204, 23°C, no salt added (Nielsen, 1955) : Ca(CH3COO)2 + (NH‘)2C204, 23°C, ionic strength 0.5 M by adding NaCl (ibid) : CaCl2 + Na2C20‘, 6°C, ionic strength 0.5 M by adding NaCl (ibid) : CaCl2 + Na2C20‘, 25°C, ionic strength 0.3 M by adding NaCl, constantly stirring (Skrtic et a1., 1984) : CaCl2 + ch,o,, room temperature, ionic strength 0.15 M by adding KCl, without stirring (experimental data of this work) Figure 5-10 Induction period for the formation of CaC20‘. 158 5-5-2-2. Heterogeneous Nucleation If homogeneous nucleation was the only mechanism of crystal nucleation in the growth cell no nucleation should have been observed in the chamber, because the induction period for homogeneous nucleation (Table 5-5) was significantly longer than the solution residence time in the chamber (9 seconds at a flow of 40 ml/min). The nearly instantaneous nucleation observed at the inner surface of T-junction may have been due to a nucleation mechanism entirely different from the homogeneous, or near-homogeneous, nucleation in a closed vessel. Two effects probably played important roles in affecting crystal nucleation in the growth cell. The first is the turbulent agitation resulting from the mixing of two solution streams at 180 degrees in a small contact space in the T-junction (inside diameter of approximately 3/32 to 1/8 inches). The second effect is heterogeneous nucleation that occurred in the flow chamber due to the presence of good nucleation surfaces. Experimental evidence and theory show that heterogeneous nucleation occurs at a lower supersaturation level than that for homogeneous nucleation (Walton, 1967). In the flow chamber system, the relatively high surface area to solution volume ratio compared with other precipitator systems (e.g., a stirred tank reactor) may provide abundant sites for surface nucleation. Surface nucleation on the wall of the chamber, one route of heterogeneous nucleation, probably prevailed in this system. 5-5-2-3. Nucleation on Glass and on MDCK Cells Nucleation of calcium oxalate on MDCK cells was somehow inhibited. The inhibition of nucleation on MDCK cells, healthy or injured, was observed as 1) a prolonged "induction period" for the first appearance 159 of calcium oxalate crystals on the MDCK cell surface and 2) from the low number density of single crystals formed on the MDCK cell surface. The "induction period" lasted over an hour in some cases; therefore, in experiments with live cells, nucleation had to be initiated by initial use of solutions of higher supersaturation (usually a concentration of 1.5 mM) or had to be enhanced by using elevated solution flow rates. This was never necessary in experiments in which a glass surface was used as the nucleating surface nor where crystals were grown directly on the plastic divider. Single crystals were found to preferentially nucleate on glass and on the plastic chamber divider than on MDCK cells. Figures 5-11 and 5- 12 are photographs taken at the end of an experimental run with healthy MDCK cells in which several spots of bare glass were observed by photomicroscopy in the same run. These figures show the comparison of nucleation on the spots with and without MDCK cells. It is seen that more crystals are evenly dispersed on the glass surface without MDCK cells than with MDCK cells (Figure 5-12). The enhancement of nucleation by flow might suggest a secondary nucleation mechanism on the epithlial cells. That is, new nuclei may have been formed by sweeping away the nuclei or embryos on the crystals that grew on an upstream portion of the chamber surface (McCabe et a1., 1985). The delayed nucleation, enhanced nucleation in the presence of elevated flow rates, and the likelihood of nuclei being generated from the turbulent flow at the T-junction all support the possibility of a secondary nucleation mechanism. After the first stage of nucleation, new crystals were observed occasionally throughout the experiments, except in the experiments with HCl injured cells. For the untreated (Figure 5—11) and papain injured 160 D " .. 'u so‘ . Figure 5-11 COM crystals on MDCK cells; Run 1102087, T a - T0x - 1.5 mM, taken after experiment. z/ y’Q “ ":1 Figure 5-12 COM crystals on glass in an experiment with MDCK cells, a spot with cells detached; Run 1102087, TCa - TOx - 1.5 mM, taken after experiment. .. IIUIII ..t ,V- ,‘ i". .. c 3" 0,214 f. 5: " e- . ..: ~ .I.:-. - I} . iASEW3¥F \ 1' 'qun .. ~. ~- . ' 0 ' I" . e s" 30'5“".9' 1 ‘ 33° 11.5% ~ - '0 v , I ‘-." :?‘¢2;\fi «AL ' Figure 5-13 COM crystals on MDCK cells treated with papain; Run 2110487, Tca - Tox - 1.25 mM, t - 15 min. \ . NX ' . ‘ ‘ 0‘ 7.3‘ ...")..‘O‘r b‘ I ' . \ ~ ,.~, ‘ a 1 . v.” : ..o . _ ‘ . \ ‘ ‘u C ‘ o .. r .3 . ‘. V ... . 3 . ." '\ O I .:\:\ .‘~.,. e . |~ ‘ . , '°;.“\T .- ‘. ‘ c' . s .a, . ' 'l - ‘ J 1'2: .~ ,H. .., _ \ ‘ a;~air. :A ‘- ' \t -‘ .. ‘ ‘0llllllllllllhlll|||l|||l| lluulm Figure 5-14 COM crystals on MDCK cells treated with HCl; Run 2102087, TCa - T0x - 1.5 mM, t - 27 min. 162 \ ' .‘°'~ 0H5”: 34‘}'.§H ' 1m nu nu ”Wm: [”11”an mum“! ' Juum .‘ '-::x a 4& '7 ’ 30 1:: - co m m 0 ‘ y. % a. ‘3‘ Q Figure 5-15 COM crystals on glass; Run 2092387, T - T - 1.25 mM, t - 35 min. Ca OX 00- ‘~- ~‘ \- \-‘- . ~ . J..- ‘~ \\\\ X“; ‘_ . \ c '. - L3: ‘fia’ ..W. :3“? - .2..-"‘\ a. ‘ - .. ; ‘*" ’ , ‘ \' .'~.-- \ ‘ c: ~- 2‘ m. . vwe xk ‘- ', - “23i‘ ’: . . ; ‘ \‘ '?‘I . G x K ‘ .“. . c . Figure 5-16 COM crystals on MDCK cells with 100 mg/l heparin; Run 1082587, T - T - 1.0 mM, t - 40 min. Ca 0x 163 (Figure 5-13) epithelia, it appeared that nucleation of new crystals occurred around some of the "not-so-healthy" (rounded) cells. The healthy cells seemed to be immune from calcium oxalate nucleation. For the experiments with HCl injured epithelium cells, there was no further nucleation seen after the nucleation was initiated by enhanced supersaturation and by high flow rates. Only growth of a small number of crystals was observed through a run (Figure 5-14). Because rounded cells were also observed for HCl injured epithelia, the inhibition of post-nucleation (nucleation after initial nucleation activated by elevated supersaturation and by high flowrate) on HCl injured cells may be attributed to the removal of "active sites" for nucleation with HCl treatment. 5-5-2-4. Crystal Habit Besides the nucleation characteristics, the appearance of growing calcium oxalate crystals was also influenced by the nature of supporting surfaces. Figure 5-15 shows crystals formed on plain glass surface. The single crystals are transparent hexagons with distinct angles and have been identified as calcium oxalate monohydrates (DeLong, 1988). Crystals of both convex (Q) and concave (E) hexagons were usually seen. When comparing Figure 5-15 (growth on glass without previous MDCK attachment) with Figure 5-12 (growth on glass with previous MDCK attachment), it is surprising to see the convex type of crystals predominating on the glass surface where epithelial cells were detached. Whether it is a coincidence or whether this is an indication of the changes in the properties of the glass surface due to previous epithelial attachment remains unknown at this point. 164 The crystals growing on untreated and papain injured cells (Figures 5-11 and 5-13) show more convex type of crystals and fewer concave hexagons with two unequal parallel edges. Fewer convex type single crystals were seen on epithelial cells treated with HCl (Figure 5-14). In general, crystals growing on live cells, untreated or injured, exhibited less distinct angles than those on glass surfaces. For all the various conditions, imperfect or distorted crystals were also observed on the epithelial cells. They were darker in comparison to the transparent single crystals. Addition of heparin to the experimental solution also changed the appearence of crystals growing on MDCK cells. As seen in Figure 5-16 for the crystals formed with the addition of 100 mg/l heparin, the convex hexagons show significant increase in the width to length ratio of the crystals. This aspect ratio increased with increase of heparin concentrations, while the frequency of occurrence of the concave hexagons decreased. These results are consistent with the results of DeLong's work (1988), in which the aspect ratios of crystals grown on glass surface were investigated under the influence of heparin additive. 5-5-3. u e satu at o ev w hambe Since supersaturation is required in correlating crystal growth quantitatively with its driving force, an effort was made to estimate solution supersaturation around the observed crystals grown in the flow chamber precipitator. For some of the experiments performed with KCl as an ionic strength buffer, calcium concentrations of the effluent samples were analyzed with AA and with a calcium selective eletrode. TCa e denotes 165 the total calcium concentration measured with AA, and [Ca2+]e is the concentration of free calcium ions measured with calcium eletrode. Table 5-6 shows the results for chamber outlet concentrations in comparison with the total inlet concentration of calcium, TCa,i’ prepared for each run. A consistent trend was found among the data, 2+ 2+ that is, TCa,i> TCa,e> [Ca le' The [Ca ]e values were lower than TCa e due to formation of calcium-complex ions in the 1:10 diluted effluent samples; a decrease in concentrations from the inlet T to Ca,i outlet TCa e streams indicates consumption of solute within the flow chamber. Calculation of (T T )/TCa i showed a 4-15% Ca,i- Ca,e concentration drop. As a result, supersaturation level in solution around the observed single crystals must be lower than that calculated from TCa,i‘ * Table 5-6 Calcium concentrations of some effuent samples . R N '1' ‘1' 02+] TC'-TC, F1 Rt un o. Ca,i Ca,e [ a e géil a e ow a e T Ca,i (mM) (mM) (mM) (%) (ml/min) R1020787 0.5 0.4666 --- 6.7 40 R1020587 0.75 0.7221 0.583‘ 3.7 40 R1013187 1.0 0.8737 0.809 12.6 40 R1020387 1.5 1.2768 1.121 14.9 40 R1012987 1.0 0.9056 0.823 9.4 99 R1012487 1.0 0.917 0.865 8.3 98 R1012587 1.5 1.335 1.25 11.0 99 * Effluent samples were diluted 1:10 after collection. 2 The measured calcium concentrations TCa e and [Ca +]e are shown in Figure 5-17 for the 40 mi/min runs. Linear regression of T data Ca,e Calcium'concentrations (mM) 1.5 1. 2 166 fl: TCa,e measured with AA 2+ x: [Ca ]e measured with Ca-electrode I I I I I I I I I I I I I 0 .2 .4 .6 .8 l. 1.2 TCa,i (“1“) Figure 5-17 Effluent calcium concentrations. I 1.4 I 1.6 167 versus T is shown as a dashed line. The solid line is T .. The Ca,i Ca,i total calcium concentration in the solution around the crystals, TCa b’ must be somewhere between T and T Since the crystals were Ca,i Ca,e located at the downstream end of the flow chamber, it may be logically assumed that T was close to but lower than the average value of Ca,b TCa,i and TCa,e° It was assumed that TCa,b - 0'6TCa,e + 0.4TCa’i. With the substitution of the linear regression result for T , T Ca,e Ca,b may be expressed as ,5 TCa,b (M) - 0.8754 TCa,i (M) + 5.523x10 (5-16) Calcium oxalate supersaturations, either (SCOM,c)b or (SCOM,i)b’ calculated from the modified EQUIL programs with T from Equation Ca,b (5-16) were used in characterizing the driving force for crystal growth rate discussed in the following section. 5-5-4. owth a e u de d't'v a d Tissue Su ac Following the experimental scheme described in Section 5-4-4, experiments were conducted to observe growth rates of single calcium oxalate crystals under the effects of biopolymer additives and epithelial cells. The original crystal growth data obtained from image analysis were equivalent circular diameter of crystals versus time, of which the slopes were used to calculate growth rate as per Equation (5- 15). For the discussion that follows, size-independent growth rate was assumed. 5-5-4-1. Effects of Tissue Surfaces For crystals growing on glass and tissue surfaces, the results show that growth rates, G (pm/min), may be correlated to the or (S supersaturation driving force, (8 such that COM,c)b COM,i)b’ 168 G - kc(S (5-17) COM,c)b or G - ki(S (5-18) 2 COM,i)b where the subcript b denotes supersaturations in the bulk solution, and the rate constants, kc and k were obtained from regression with the i’ appropriate supersaturation. Figures 5-18 and 5-19 show the growth rates and such correlations for four different growth conditions, COM crystal growth on glass, on healthy MDCK cells, on HCl-treated MDCK cells, and on papain-treated MDCK cells. The rate constants of both correlations are also listed in Table 5-7. A consistent decrease of the rate constants for both correlations in the order of crystal growth on glass > MDCK cells treated with HCl > MDCK treated with papain > healthy MDCK cells was observed. Each of these two correlations (Equations (5-17) and (5-18)) provides a direct comparison of growth rate from the rate constants; i.e., the higher the rate constant, the higher the growth rate for a given supersaturation. Therefore, the rate constants in Table 5-7 indicate fastest crystal growth on glass, followed by HCl-treated cells, and then papain-treated cells. Crystal growth on untreated MDCK cells was slowest. Table 5-7 Rate constants of COM growth. Correlation rate Growth conditions constant Glass MDCK/HCl MDCK/papain MDCK « (SCOM,c)b kc 0.0137 0.0090 0.0081 0.0071 2 « (SCOM,i)b ki 0.0201 0.0134 0.0119 0.0105 169 ,‘ (a) 5 . E ; glass 5 I Ei '5 4 f : MDCK cells .3 I i E ,3 . S a I I o S— (D ,1 . O 10 20 30 40 .6 (b) MDCK cells treated with 361 Growth rate (um/min) ;. o 10 I 20 3o 40 ,‘ .6 C 'g .(c) MDCK cells treated with papain \ g 04 " m H .. RU L. S 02 ‘ H 3 2 + cs 0 r 0 I r I If I 0 10 20 30 40 (SCOM,c)b Figure 5-18 Growth rate versus (SCOM,c)b; (a) crystal growth on glass and on healthy MDCK cells, (b) crystal growth on HCl treated MDCK cells, (c) crystal growth on papain treated MDCK cells. 170 (a) f . glass I Growth rate (pm/min) E; (b) MDCK cells treated with HCl 5 5 .4 . 0) .. H 2 g .2 d 4.) 3 o + 35' 0 - . 1 0 2 4 A .6 .5 (c) MDCK cells treated with papain E i 04 d m .1 4.) M S. J: .2 . 4.» :3 o u 3.. cs 0- . . . 0 2 4 Figure 5-19 Growth rate versus (SCOM,i)b; (a) crystal growth on glass and on healthy MDCK cells, (b) crystal growth on HCl treated MDCK cells, (c) crystal growth on papain treated MDCK cells. 171 Confidence in the above results was tested according to the Student's-t test described by Steel and Torrie (1960). The rate constants of crystal growth correlated with (SCOM,c)b along with the upper and lower bounds of a 90% confidence interval are shown in Figure 5-20. The growth rate inhibition from healthy MDCK cells over that from HCl-treated or papain-treated cells was not dramatic. However, inhibition of growth by healthy MDCK cells over growth on glass was significant. The growth rate result for crystals growing on glass may be compared with previous work. DeLong (1988) reported growth rate constants for three faces of COM growth using the second order growth rate correlation as per Equation (5-18). The rate constants were 0.0238, 0.00777, and 0.0109 for 110, 010, and 101 crystal faces. The rate constant (0.0201) obtained in this work for crystals growing on a glass surface (Table 5-7) is consistent with DeLong's result. 5-5-4-2. Effects of Heparin The addition of heparin, a model urinary biopolymer, had a distinct inhibitory effect on COM growth. Figure 5-21 shows the inhibition of COM crystal growth on untreated MDCK cells by heparin. This result is the same as that found by DeLong (ibid) for COM growth on glass surfaces. Since the concentration of heparin added in the experimental solution was not in a great enough amount to cause significant complexation with calcium or oxalate ions, the marked inhibitory effect of heparin on COM growth rate may not be explained by the decrease of calcium oxalate supersaturation due to the addition of heparin. The inhibition of crystal growth rate is likely due to the adsorption of heparin on crystal surface which retards calcium oxalate ion Rate constant .015 172 ... 0.... "1 "‘ r... "1 a. o__ II-i ". ... .1 “... .. I a- In" a... . "a ‘5‘. ”I - “I. °'. F. '- I d ...I '5'. ’0' .' .- '\_. '.. h. I " 5‘3 \ ‘5." s.- .‘.\_ . «'4.- - :: o g. \\ 'w ‘5. T 0.. fi" .' I pf I 1" ‘5‘ \J ..' I "3 I' 7'. ’1 . ,- ,/ x” ,7_ f. f _ ,J "o I F I I... A... C C A \ 0’ J i I -. I. g, ’4' (I\ 54 . \ v I. .1 '0 '. . h‘. f. .- . ‘ I. ,o’ ._ I K’s". ‘ V f "a. \ 1'23?! If 3'”)- \'-. Vii’xfl V!" J" .. '7' ‘5"?! If If; \\ d I f ... 5..JD..'..(.!._.- ... in \' 0‘1 .. .,' i".- '_l ._ o“ ‘.I f N .1? V .J' ."".\ 'I. Pita; ' .I' ”5“ ..K, 50;);f u" I I, a” '1‘ waif/".6 .' {1"\\ r; {(1 , a. . . . . y f"; I . If .1 ' . ..0 - “'/ via 5.44:?" "3' 2i“- \ r’ " "a "R 'i~.“ ’ 21": ’ Ix -. :zt'crI r‘ " , ., "{‘Ifllffr'fri r' .1 I» ‘~ " {4,923,111 r’ ‘ “WE: flNfi‘ifii’Ei .1 f.’)_.o.\".‘ 'II..'{$:_,{ r" .«";¢._."1_:.{55,51 .4r _..’ ..:). ~4.’,{f{§g / +5. \-.,;::,.:;.§¢1 .1 . I".I a. ..I I. "313:." VI. .1. ‘ ..‘0 .P ilk}: a", .... ' ,fifij .. I. .1 g. ’.:P.'-‘..:" hflwfififl 'QN %% Iflf:“&& cvhwma l I 15:71 1 IA, Rate constant MDCK/HCl SN as.“ Upper limit of 90% conficence interval MDCK/papain interval Figure 5-20 Comparison of growth rates at any constant (S Lower limit of 90% confidence COM,c)b' 173 .16 .14 ~ E E \ E 3 Q3 {-3 ‘U S- 4: U H 3 2 w .06 ‘ ab .04 " .02 I I I I I I I I I 441 I I I I I 0 20 40 60 80 100 120 140 160 Heparin concentration (mg/l) Figure 5-21 Inhibition of COM crystal growth on MDCK cells by heparin; TCa - Tox - 1.0 mM. 174 incorporation into the crystal lattice. If the growth rate of COM crystals, G, is proportional to the total crystal surface uncovered by heparin and the total crystal surface on which heparin is adsorbed is proportional to the concentration of heparin, [heparin], then G - G. (1 - kads[heparin]) (5-19) where G. is the crystal growth rate in the absence of heparin additive, G is the growth rate in the presence of heparin, and kads is the proportionality constant relating the fractional coverage of crystal surface to the concentration of heparin. Equation (5—19) may be rearranged to give Go, 1 '--—-— - (5-20) G. - G kads[heparin] The growth rate data read from the solid line in Figure 5-21 were replotted in Figure 5-22 in the form of G./(G.-G) versus 1/[heparin]. The linear relationship between G./(G.-G) and 1/[heparin] in Figure 5- 22 indicates that the inhibitory effect of heparin on calcium oxalate crystals is due to a saturation or adsorption-like phenomenon on the crystal surface; the adsorption of heparin follows a Langmuir adsorption isotherm mode. 5-5-5. x e menta es t t ne Formation H othes 5 Observations of calcium oxalate crystal formation and growth rate results from above suggest a biopolymer-inhibition hypothesis for urinary stone formation in 3129. For the growth inhibition in the presence of MDCK cells, a biopolymeric inhibitor of the type similar to sulfated glycosaminoglycans such as heparin that consists of negatively charged carboxylate and/or sulfate groups may be present on the surface GO/(Go - G) 175 11 3i 1 [Heparin (mg/2)] Figure 5-22 Inhibitory effect of heparin on COM crystal growth follows Langmuir adsorption isotherm. 176 of MDCK epithelial cells; this may interfere with the growth process. Biopolymer molecules may incorporate in the crystal lattice and change the crystal habit so that crystals become less transparent (see Figure 5-11). This biopolymer may be part of a structural protein on the MDCK cell membrane or a substance secreted from the cells. Chemical treatment of the MDCK cells may either destroy the active inhibitor structure on cell surface or close or disturb secretion of the active inhibition substance from the cell so that crystal growth on HCl- or papain-treated cells is less inhibited than growth on healthy cells. The inhibitor present in the solution, e.g., heparin, affects crystal growth more dramatically but in a similar fashion probably because the effective biopolymer concentrations in solution are greater than those on the tissue surfaces. Nucleation on epithelial surfaces may also be affected by a biopolymeric substance. The absence or presence of the biopolymer around the rounded, unhealthy cells may be the cause of more nucleation around these cells. At the present time, the exact effect of HCl on epithelial cells is not understood; however, the distinct inhibition of nucleation on HCl-treated cells is not likely to be due to the presence of an excess of biopolymer. Nucleation inhibition would probably be caused by the removal of nucleation sites on epithelial surfaces by HCl. Dr. Robert Roth, a pharmacologist in the Department of Veterinary Medicine, has suggested that HCl might completely strip the cell surface of all proteins, glycoproteins, or other biopolymers. The cell appearance in the photomicrographs and the lower degree of inhibition of crystal growth rate on HCl-treated cells than by papain-treated or untreated cells are strong evidence for this hypothesis. 177 From the above results and discussions, it appears that both tissue surface and solution composition play important roles in nucleation and growth of COM crystals. Since crystal nucleation depends greatly upon availability of nucleation sites and biopolymers may actively interfere with both nucleation and crystal growth, kidney tissue injury or the lack of nucleation inhibitors may be the first step in early stages of stone formation. Once the crystal nidus is formed, it may provide sites for crystal growth where crystal growth takes place in a relatively lower supersaturation than crystal nucleation. The nidus on a tissue surface may also be available for attachment of other small crystals formed from upstream in the urinary tubule due to crystal affinity for aggregation or due to clogging of the urinary tract from the existing nidus. The absence of protection from biopolymer inhibitors in urinary fluid or on tissue surfaces may result in the growth of kidney stones to sizes that become physiological problem. CHAPTER 6 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 6-1. Summary and.Conclusions This study of calcium oxalate crystallization was undertaken to develop a fundmental understanding of the physical and chemical mechanisms of kidney stone formation. The important aspects of this work are summarized and conclusions from the work are presented in this section. The driving force for the crystallization process was quantified by the calculation of supersaturation of calcium oxalate in solution. The supersaturation was defined in two forms: in terms of the concentrations of calcium and oxalate ions and in terms of the concentration of calcium oxalate complex species for each hydrate phase. The concentrations of calcium and oxalate ion species and of the associated calcium oxalate complex were calculated with computer programs by assuming an equilibrium state among all the free ions and the associated species in the solution. Calcium oxalate crystallization was studied with a MSMPR crystallizer and with a flow chamber crystallizer. The MSMPR crystallizer allowed determination of fee-solution phenomenon of calcium oxalate crystallization and the flow chamber crystallizer allowed study of crystal-tissue interactions. For the work conducted with the MSMPR crystallizer, particle size distribution (PSD) of the suspension sample was one of the major properties characterizing calcium oxalate crystallization. A particle sizing instrument, the ELZONE(R) particle data analyzer (Particle Data, Inc.), was used to size the particles for the determination of PSD. The instrument counted data were correlated to PSD; limitations of the ELZONE(R) 178 179 system were found. Because of these limitations, the implementation of the ELZONE's data analysis was modified for obtaining reliable PSD data. A modified calibration procedure was also developed for an improved understanding of the basis for the calibration procedure. This modified calibration procedure offers a more straightforward determination of the calibration constants than the original technique provided by the ELZONE manufacturer. Calibrations of the ELZONE(R) system were conducted to show the validity of the modified calibration procedure and the "Normalize" function of the instrument's operation. Calcium oxalate crystallization in the MSMPR crystallizer was characterized by the determination of particle size distributions and the total calcium concentration in the suspension and in the filtrate. Crystal hydrates and crystal phases were identified with x-ray diffractometry and optical microscopy. The results show that calcium oxalate trihydrate was the major hydrate formed in the reactor for the calcium and oxalate feed concentrations above 2 mM. Calcium oxalate monohydrate formed at 1 mM calcium and oxalate concentrations. Addition of PGA (with calcium and oxalate concentrations 2 mM) favored the formation of calcium oxalate dihydrate and monohydrate. From the measurement of PSD and calcium concentrations with time, the crystallization process in the MSMPR was characterized by unsteady state, quasi-steady state, and steady state periods. Crystal aggregation, nucleation, and crystal growth occurred in the unsteady state region. Crystal aggregation and nucleation decreased as crystallization progressed with time. At steady state, crystal growth was the only significant mechanism in the crystallizer. The unsteady state MSMPR crystallizer operation was of interest because of the 180 ultimate goal of understanding crystal aggregation mechanism as well as the crystal growth mechanism. For the unsteady state period in which crystal aggregation and crystal growth were superimposed, an isolation approach for determining crystal growth rate was developed based on shifts of PSDs along the x- axis (particle size) with time. The growth rate for the entire MSMPR crystallizer operation, including unsteady state and steady state, were obtained. The growth rate results were compared with other growth rate data and with other MSMPR crystallizer work modeled with linear population density analysis. A good correlation was observed. The growth rate for the process was correlated linearly with supersaturation calculated from the calcium concentration in the filtrate. The correlation is written in the form: GL - k x supersaturation (4-3) where GL is the linear growth rate and k is the growth rate constant. The addition of PGA, a model urinary biopolymer, in the MSMPR crystallizer experiments increased the total crystal retention and supersaturation in the reactor, and reduced crystal growth rate. The biopolymer PGA was thought to affect the calcium oxalate crystallization by adsorbing on the crystals and inhibit crystal growth units from integrating into the crystal lattice. Low crystal growth rate in the crystallizer in the case of PGA addition results in 4 higher supersaturation, which may generate bursts of nucleation and be the reason for the wave-like appearance observed in the PSD data. The inhibitory PGA adsorption on crystal surface may also prevent crystal deposition on the wall of crystallizer; as a result, the total crystal retention in the reactor is higher. 181 The flow chamber crystallizer experiments were conducted to investigate the effects of both healthy and injured cultured epithelium (Maden Darby Canine Kidney (MDCK) cells) on the growth of calcium oxalate single crystals in the presence of flow shear. Chemical injury by HCl and by papain was used to simulate histological damage. The healthy MDCK cells was found to inhibit crystal nucleation. The calcium oxalate monohydrate (COM) crystal growth rates decreased in the following order: crystal growth on glass > MDCK cells treated with HCl > MDCK cells treated with papain > healthy MDCK cells. The biopolymer additive, heparin, exhibited inhibition on COM crystal growth in the presence of healthy MDCK cells. Inhibition may also be due to the adsorption of the negatively charged biopolymer on the crystal surface. The growth rate obtained from the MSMPR crystallizer experiment is approximately twice of the growth rate resulted from the flow chamber crystallizer experiment. The growth rate constants resulting from these two experimental systems for the unhibited COM crystal growth are 0.037 (Table 4-4, [Ca2+] - [C20,2‘] - 1 mM) and 0.0137 (Table 5-8, crystal growth on glass surface) for the correlations with SCOM,c' The lower growth rate constant for the flow chamber crystallizer may be due to a certain degree of mass transfer resistance in the laminar flow, while the MSMPR crystallizer was substantially turbulent. The supersaturation in the MSMPR crystallizer was calculated from the calcium concentration in the filtrate. Because of the unavoidable time delay in filtration, the calcium concentration and supersaturation may be underestimated, which also results in a higher rate constant observed for the crystal growth in the MSMPR crystallizer. From this study, the paucity of kidney stone formation in healthy individuals may be understood as follows: inhibitory effect of healthy 182 MDCK cells on crystal nucleation and crystal growth and the inhibitory effect of PGA and heparin on crystal growth suggest a membrane-crystal interaction via some biopolymers on the epthelial surface or in solution that prevent stone formation. Since stone-formers seem to be lacking these biopolymers, they also lack the stone formation inhibitors. A crystal-membrane interaction with damaged kidney epithelium may also be the precursor to stone formation. 6-2. Recommendations for Future Research Suggestions for the future research are: 1. Using the growth rates obtained in the MSMPR crystallizer experiments in the study of crystal aggregation with numerical analysis. 2. Testing of other biopolymers on crystal growth and crystal aggregation model to understand kidney stone formation. 3. Finding the mass transfer coefficients for calcium oxalate, so that the growth rate in the flow chamber crystallizer may be characterized with known mass transfer resistance. In this case, the growth rate correlations with the two supersaturation definitions may be discerned one from the other. APPENDICES APPENDIX.A EQUIL COMPUTER PROGRAMS Two EQUIL computer programs were written in BASICA to calculate calcium oxalate supersaturations for the experimental conditions in this work as described in Chapter 2. Program EQUIL-NA.BAS was written for the solutions used in the MSMPR crystallizer experiments (Chapter 4), which contain Ca(N03)2, K2C204, and the ionic strength buffer KNO3. The other program EQUIL-M7.BAS was written for the modified medium used in the study of crystal growth on MDCK cells (Chapter 5). The modified medium contains CaClz, Na2C20‘, KCl, NaCl, and D-glucose. D-glucose was assumed not to affect the solution supersaturation through complexation. The flow diagram of EQUIL-NA.BAS and program listings of EQUIL-NA.BAS and EQUIL-M7.BAS are shown in the following. The flow diagram of EQUIL-M7.BAS is similar to the flow diagram of EQUIL-NA.BAS. The latter is shown as an example. 183 184 Flow Diagram of EQUIL-NA.BAS egig 2 Define dimensions of variables;I [set a tolerance EPS i Input stability constants; / input thermodynamic solubility / constants. KSOM (COM) nd KSOT COT KCOM-KCan0KSOM , KCOI-KCaOg-KSOI, Input total calcium and oxalate conc'ns,I TOx and TCa; input KNO3 conc'n, KNO3 (use the experimental conditions; this is for calculating TR and TNO3) I. Calculate total K+ conc'n, TK, TK-TOx-ZfKNOB calculate total N0; conc'n, TN03 TNO3-TC§°2+KNO3 I, Input filename, FILNlS, for the / filtrate concentrations; / input filename, FILN2$, for the // stora e of calculated supersaturations I, Read in from FILNl$ the total no. of data,/ NT, and ((T(in), FRCA(in)), in-l,NT) / where T(in) is the time the filtrate was ' I sampled and FRCA(in) is the total calcium / conc’n in the filtrate in the unit of mM; ./ calculate the CA in / L_‘ In the unit of M , set H-6.0I I. L Loop 510, IN-l,NT; alculate supersaturations for each FRCA(IN) (cont’d) 185 (cont’d) Let TCa-FRCA(in), TOx-FRCA(in), activity coefficients, Y1: yz, and y3-0.5; let guess values, GCa, GOx, GK, and GNO3, be one tenth of their total concentrations; 2 2 .... - calculate y1 , y2 , [H ], and [OH ] v + Calculate free ion conc'ns, [K ], [NOa-I. 2+ 2- . [Ca ], and [C204 ], from equations similar to Eguationslin44llthrough (2-48 If any of the relative difference of the guess conc’n and the calculated value > EPS No Yes Let the guess conc'ns be the new calculated conc'ns; calculate conc'ns of associated species; Calculate supersaturations according to Equations (2-6) ans (2-7). calculate ionic strength, I; using this I value to calculate y1 and y2; .. Store the calculate //supersaturations in the file FILN2$ S? (_sc§) 7 186 Listing of EQUIL-NA,BAS 10 CLS:PRINT "PROGRAM: EQUIL-NA, (EQUIL-KNO3 SYSTEM)":PRINT SPC(10) "Calculate SCOM and SCOT for the filtrates from MSMPR":PRINT:PRINT STRING$(50,61):PRINT 20 DIM T(20),FRCA(20),SCOMC(20),SCOTC(20),SCOMI(20),SCOTI(20) 30 RPS-.0001 40 KHOX-21008.74 50 KKOX-lO! 60 KCAOX—l869 70 KCA2OX—71.6 80 KCAOX2-17.3 90 KH20-1/3.8905E+13 100 KHZOX-l9.12 110 KCAOH-29.5 120 KCAHOX-64.592 130 KCAHOX2-312.8 140 KHNO3-4.938556E-02 150 KCANO3-4.7863 160 KKN03-.47863 170 KSOM-2.514E-09 180 KSOT—7.88E-09 190 KCOM-KCAOX*KSOM 200 KCOT-KCAOX*KSOT 210 INPUT "Ca(N03)2 (M) - ";TCA 220 INPUT "K20x (M) - ";TOX 230 INPUT "KNO3 (M) - ";KNO3 240 TK-TOX*2+KNO3 250 TNO3-TCA*2+KNO3 260 INPUT "Filename for filtrate concentrations";FILN1$ 270 INPUT "Filename for output results";FILN2$ 280 OPEN FILN1$ FOR INPUT AS #1 290 INPUT #1,NT:FOR IN-l TO NTlePUT #1,T(IN),FRCA(IN):FRCA(IN)-FRCA(IN)*.001:NEXT:CLOSE 300 PH -6! 320 FOR IN-l TO NT:TCA-FRCA(IN):TOX-FRCA(IN) 330 340 350 360 370 375 380 390 400 187 F1-.5:F2-.5:F3-.5 GCA-TCA*.1:GOX~TOX*.1:GK—TK*.1:GNO3-TN03*.l IT-l ' Fll-F1*Fl F22-F2*F2 FH-10“(-PH)/Fl:FOH-lO“(LOG(KH20)/LOG(lO)+PH)/Fl FK-TK/(l+KKNO3*GNO3*F11+KKOX*GOX*F2) FRNO3-TNO3/(1+KHNO3*FH*F1l+KKNO3*GK*Fl1+KCANO3*GCA*F2) FCA-TCA/(1+KCAOX*GOX*F22+KCAOH*FOH*F2+KCAHOX*KHOX*FH*GOX*F22+KCAHOX 2*(KHOX*FH*GOX*F2*F1)A2*F2+2*KCAOX*KCA20X*GCA*GOX*F22+KCAOX2*KCAOX*GOXA 2*F22+KCANO3*GNO3*F2) 410 FOX-TOX/(1+KHOX*FH*F2+KCAOX*GCA*F22+KCAHOX*KHOX*GCA*FH*F22+2*KCAHOX 2*(KHOX*FH*F1*F2)A2*GCA*GOX*F2+KCAOX*KCA20X*GCAA2*F22+2*KCAOX2*KCAOX*GC A*GOX*F22+KHOX*KH20X*FH 420 430 440 450 460 470 480 490 500 510 520 530 IF ABS((GCA-FCA)/FCA)>EPS GOTO 550 IF ABS((GOX-FOX)/FOX)>EPS GOTO 550 IF ABS((GK-FK)/FK)>EPS GOTO 550 IF ABS((GNOB-FRNO3)/FRNO3)>EPS GOTO 550 CAOX-KCAOX*FCA*FOX*F22 SCOMC(IN)-CAOX/KCOM-1 SCOTC(IN)-CAOX/KCOT-1 SCOMI(IN)-(SQR(FCA*FOX*F22)~SQR(KSOM))/SQR(KSOM) SCOTI(IN)-(SQR(FCA*FOX*F22)-SQR(KSOT))/SQR(KSOT) NEXT IN OPEN FILN2$ FOR OUTPUT AS #2 FOR I-l TO NTzPRINT #2,T(I),FRCA(I),SCOMC(I),SCOTC(I),SCOMI(I), SCOTI(I):NEXT:CLOSE 540 550 560 570 580 590 600 610 620 630 END HOX—KHOX*FH*FOX*F2 CAOH-KCAOH*FCA*FOH*F2 CAOX—KCAOX*FCA*FOX*F22 CAHOX—KCAHOX*FCA*HOX*F2 CAOX2-KCAOX2*CAOX*FOX CA20X-KCA20X*CAOX*FCA KOX-KKOX*FK*FOX*F2 CANO3-KCANO3*FCA*FRN03*F2 IS-(FH+FOH+FK+FRNO3+HOX+CAOH+CAHOX+KOX+CANO3+(FCA+FOX+CAOX2+CA20X) 188 *4)/2 640 FOR I-O TO 1:FF(I)-10A(-.523*(I+1)“2*(SQR(IS)/(l+SQR(IS))-.3*IS)): NEXT 650 F1-FF(O):F2-FF(1):GCA-FCA:COX-FOX:GK-FK:GNO3-FRNO3:IT-IT+1:GOTO 360 Listing of EQUIL-M7,BAS 10 CLS:PRINT "PROGRAM: EQUIL-M7, (EQUIL-MEDIUM7, MODIFIED EARLE'S MEDIUM, Version 7)":PRINT ”(TCA-TOX, with varied TCA)":PRINT:PRINT STRING$(70,61):PRINT 20 DEFDBL A-G,K,P,S,T:DEFINT I,N 30 DIM ATCA(400),ACA(400),AOX(400),SCOMC(400),SCOTC(400),SCOMI(400), SCOTI(400),AF2(400),IIT(400) 40 BPS-.0001 50 KHOX-21008.74 60 KKOX-10# 70 KCAOX—l869 80 KCA20X-71.6 90 KCAOX2-l7.3 100 KH20-l/3.8905E+l3 110 KHZOX-l9.12 120 KCAOH-29.5 130 KCAHOX—64.592 140 KCAHOX2-312.8 150 KNAOX-l3.3 160 KNACLP.3981072 170 KKCL—.1737801 180 KSOM-2.514E-09 190 KSOT-7.88E-09 200 KCOM-KCAOX*KSOM 210 KCOT-KCAOX*KSOT 220 INPUT "Concentrations of CaC12 and Na20x (M) vary from";TCAI 230 INPUT "How many values of concentrations to be calculated (<-401)";NC 240 INPUT "Constant concentration difference-";DC 250 INPUT "Filename for data storage (Maxmium 7 1etters)";FILEN$ 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 189 PRINTzPRINT "Program is running. Please wait!" TK-5.8833lE-03:PH-7.2 FOR N—O TO NC-l:TCA-TCAI+DC*N TOX-TCAzTNACLP.15741706#-2*TCA Fl-.S:F2-.S:F3-.5 TNA-TOX*2+TNACL TCL-TCA*2+TK+TNACL GCA-TCA*.1:GOX-TOX*.l:GK-TK*.l:GNA-TNA*.1:GCL-TCL*.1 FH-10“(-PH)/F1 FOR-10“(LOG(KH20)/LOG(lO)+PH)/F1 IT-l Fll-F1*F1 F22-F2*F2 FK-TK/(l+KKCL*GCL*Fll+KKOX*GOX*F2) FRNA-TNA/(1+KNACL*GCL*F11+KNAOX*GOX*F2) FCLPTCL/(1+KNACL*GNA*F1l+KKCL*GK*F1l) FCA-TCA/(l+KCAOX*GOX*F22+KCAOH*FOH*F2+KCAHOX*KHOX*FH*GOX*F22+KCAHOX 2*(KHOX*FH*GOX*F2*F1)A2*F2+2*KCAOX*KCA20X*GCA*GOX*F22+KCAOX2*KCAOX*GOXA 2*F22) 430 FOX-TOX/(1+KHOX*FH*F2+KCAOX*GCA*F22+KCAHOX*KHOX*CCA*FH*F22+2*KCAHOX 2*(KHOX*FH*F1*F2)A2*GCA*GOX*F2+KCAOX*KCA20X*GCA“2*F22+2*KCAOX2*KCAOX*GC A*GOX*F22+KHOX*KH20X*FH 440 450 460 470 480 490 500 510 520 530 540 542 550 560 IF ABS((GCA-FCA)/FCA)>EPS GOTO 620 IF ABS((GOX-FOX)/FOX)>EPS GOTO 620 IF ABS((GK-FK)/FK)>EPS GOTO 620 IF ABS((GNA-FRNA)/FRNA)>EPS GOTO 620 IF ABS((GCL-FCL)/FCL)>EPS GOTO 620 ATCA(N)-TCA CAOX-KCAOX*FCA*FOX*F22 SCOMC(N)-CAOX/KCOM-1 SCOTC(N)-CAOX/KCOT-1 SCOMI(N)-(SQR(FCA*FOX*F22)-SQR(KSOM))/SQR(KSOM) SCOTI(N)-(SQR(FCA*FOX*F22)-SQR(KSOT))/SQR(KSOT) ACA(N)-FCA:AOX(N)-FOX AF2(N)-F2:IIT(N)-IT:NEXT N OPEN FILEN$+"a.csv" FOR OUTPUT AS #2:FOR I-O TO NC-lzPRINT #2,ATCA(I) "," SCOMC(I):NEXT:PRINT #2,"END,DATA":CLOSE 190 S70 OPEN FILEN$+"b.csv" FOR OUTPUT AS #2:FOR I-O TO NC-lzPRINT #2,ATCA(I) "," SCOTC(I):NEXT:PRINT #2,"END,DATA":CLOSE 580 OPEN FILEN$+"c.csv" FOR OUTPUT AS #2:FOR I-O TO NC-lzPRINT #2,ATCA(I) "," SCOMI(I):NEXT:PRINT #2,"END,DATA":CLOSE 590 OPEN FILEN$+“d.csv" FOR OUTPUT AS #ZzFOR I-O TO NC-l:PRINT #2,ATCA(I) "," SCOTI(I):NEXT:PRINT #2,"END,DATA":CLOSE 600 OPEN FILEN$+"e.csv" FOR OUTPUT AS #2:FOR I-O TO NC-lzPRINT #2,ATCA(I) "," AF2(I):NEXT:PRINT #2,"END,DATA":CLOSE 610 OPEN FILEN$+"f.csv" FOR OUTPUT AS #22FOR I-O TO NC-1:PRINT #2,ATCA(I) "," IIT(I):NEXT:PRINT #2,"END,DATA":CLOSE 612 OPEN FILEN$+"g.csv" FOR OUTPUT AS #2:FOR I-O TO NC-l PRINT #2,ATCA(I) "," ACA(I):NEXT:PRINT #2,"END,DATA":CLOSE 614 OPEN FILEN$+"h.csv" FOR OUTPUT AS #2:FOR I-O TO NC-1:PRINT #2,ATCA(I) "," AOX(I):NEXT:PRINT #2,"END,DATA":CLOSE:END 620 CHOX-KHOX*FH*FOX*F2 630 CAOH-KCAOH*FCA*FOH*F2 640 CAOX—KCAOX*FCA*FOX*F22 650 CAHOX-KCAHOX*FCA*CHOX*F2 660 CAOX2-KCAOX2*CAOX*FOX 670 CA20X-KCA2OX*CAOX*FCA 680 KOX-KKOX*FK*FOX*F2 690 CNAOX-KNAOX*FRNA*FOX*F2 700 SI-(FH+FOH+FK+FRNA+FCL+CHOX+CAOH+CAHOX+KOX+CNAOX+(FCA+FOX+CAOX2 +CA20X)*4)/2 710 FOR I-O TO l:FF(I)-lO‘(-.523*(I+1)“2*(SQR(SI)/(I+SQR(SI))-.3*SI)): NEXT I 720 F1-FF(0):F2-FF(1):GCA-FCA:GOX-FOX:GK—FK:GNA-FRNA:GCL-FCL:IT-IT+1: GOTO 370 APPENDIX.B CALIBRATION DATA OF THE ELZONE(R) SYSTEM AND THEIR.APPLICATIONS This appendix consists of the calibration data of the ELZONE particle data analyzer and will briefly review the applications of these calibration data to the practical operation of the instrument. The calibration and particle size measurement of the ELZONE(R) system are strongly influenced by the current-trim control setting. Since changing the current-trim control setting requires a new calibration for the instrument (Particle Data, (a)), the current trim setting was set at 52 for all the calibrations and sample measurements in this work. Calibrations of the ELZONE system for several aperture tubes and some log span settings were conducted following the modified calibration procedure described in Chapter 3. The data obtained directly from calibration are the slope and intercept of the regression line of the (I+G) values versus the recorded median channels (Equation (3-32)) for the calibration particles of size Dst From the d' slope and intercept, the (I+G) value at channel 128, (I+G)12,, and the true log span, Ls’ may be calculated from Equations (3-32) and (3-35), respectively. The L8 and (I+G)128 values thus obtained are listed in Table B-1. To obtain the sizes of each counting channel for a specific aperture tube and log span setting used in particle size measurement, the appropriate LS and (I+G)123 values may be chosen from Table B-1. With known (I+G) value, the particle size of each counting channel may be calculated from Equation (3-36). In the operation of the instrument, the choice of proper log span, current, and gain settings was found to be difficult. Tables B-2 191 192 through B-6 were prepared to gain a broad view of the variation of size ranges with respect to the I and G settings for a particular aperture tube and log span setting. The lower and upper size (equivalent spherical diameter) limits, D2 and Df, of the counted particles were calculated according to Eq. (3-36) with the channel number, ch, equal to 1 and 128. Small debris in the electrolyte solution or the electronic interference may also be counted as particles, called noise, in particle size measurement. The noise usually overwhelms the first few channels in the small size range. Therefore, the actual lower size limit is larger than the particle size of the first channel, Dz' The D2 and Df values shown in Tables B-2 through B-6 may serve as ideal maximum sizing windows for particle counting. The measuring range for a selected aperture tube as provided by the intrument's manufacturer is 1.5-3% to 40-50% of the diameter of the orifice on the aperture tube, Dap' On the other hand, 5% to 20% of the aperture diameter is the suggested range for standard particles used in calibration. Therefore, 2%, 5%, 20%, 40%, and 50% of the diameter of the aperture tubes are also included in Tables B-2 through B-6. 193 Table B-1 Calibration constants with current-trim control setting - 52. Aperture: 76 pm (Dap) D : 10 pm (13.6% Dap) Log span setting: 10 12 14 Ls: 10.018558 11.94107 13.645753 (ItG)123: 11.835012 11.888911 11.837729 Aperture: 95 pm (Dap) 10 pm (10.53% Dap) Log span setting: 8 10 12 14 Ls: 7.9995131 9.9590889 11.972443 13.643723 (ItG)128: 9.7976095 12.856577 12.894593 12.870295 Aperture: 150 pm (Dap) D : 20.27 pm (13.51% Dap) Log span setting: 10 12 14 Ls: 10.070323 11.973205 13.741679 (ItG)128: 11.993836 12.029075 11.956434 Aperture: 300 pm (Dap) 44.66pm (14.89% Dap) Log span setting: 8 10 12 Ls: 8.0204586 10.105556 12.092228 (ItG)128: 8.9026537 11.991407 12.029926 Aperture: 480 pm (Dap) D : 91.2 pm (19.0% Dap) Log span setting: 10 12 Ls: 10.204376 12.174651 (ItG)123: 12.137953 12.163773 194 Table B-2 Maximum sizing windows for 76 pm aperture tube; current-trim control setting - 52. Aperture tube: 76 pm Dstd: 10 pm (- 13.16% Dap) 2% D : 1.52 pm aP 5% Dap' 8 pm 20% D : 15.2 pm 3? 40% D : 30.4 pm aP 50% D : 38.0 pm aP Log span setting: 10 12 14 L5: 10.018558 11.94107 13.645753 (I+G)128: 11.835012 11.888911 11.837729 I+G Dz Df Dz Df Dz Df 3 7.61 77.01 4.94 77 97 3.29 77.06 3.5 6.78 68.61 4.40 69.47 2.93 68.65 4 6.04 61.12 3.92 61.89 2.61 61.16 4.5 5.38 54.45 3.49 55.14 2.33 54.49 5 4.79 48 51 3.11 49 12 2.07 48 54 5.5 4.27 43.22 2.77 43 76 1.85 43.25 6 3.80 38.50 2.47 38 99 1.65 38.53 6.5 3.39 34.30 2.20 34.73 1.47 34.32 7 3.02 30.56 1.96 30.94 1.31 30.58 7.5 2.69 27 23 1.75 27.57 1.16 27.24 8 2.40 24.26 1.56 24.56 1.04 24 27 8.5 2.13 21.61 1.39 21 88 .92 21 62 9 1.90 19 25 1.24 19 49 82 19 26 9.5 1.69 17 15 1.10 17 37 73 17 16 10 1.51 15 28 .98 15 47 65 15 29 10.5 1.34 13 61 .87 13 78 --- --- 11 1.20 12 13 .78 12 28 --- --- 11.5 1.07 10 80 .69 10 94 --- --- 195 Table B-3 Maximum sizing windows for 95 pm aperture tube; current-trim control setting - 52. Aperture tube: 95 pm Dstd: 10 pm (- 10.53% Dap) 2% Dap' 1.9 pm 5% D : 4.75 pm ap 20% D : 19.0 pm aP 40% D : 38.0 pm aP 50% D : 47.5 pm 3? Log span setting: 8 10 12 14 L8: 7.9995131 9.9590889 11.972443 13.643723 (I+G)128: 9.7976095 12.856577 12.894593 12.870295 I+G Dz Df Dz Df Dz Df Dz Df 2 9.54 60.60 --- --- --- --- --- --- 2.5 8.50 53.98 --- --- --- --- --- --- 3 7.58 48.09 --- --- --- --- --- --- 3.5 6.75 42.85 8 70 86.87 5 51 87.64 3.73 87.15 4 6.01 38.17 7 75 77.39 4 91 78.08 3.32 77.64 4.5 5.36 34.01 6 91 68.95 4 38 69.56 2.96 69.17 5 4.77 30.30 6 15 61.43 3 90 61.97 2.63 61.62 5.5 4.25 26.99 5 48 54.72 3 47 55 21 2.35 54.90 6 3.79 24.05 4 88 48.75 3 09 49 18 2.09 48 91 6.5 3.37 21 42 4 35 43.44 2 76 43 82 1.86 43.57 7 3.01 19.09 3 88 38.70 2 46 39.04 1.66 38.82 7.5 2.68 17.00 3 45 34.47 2 19 34.78 1.48 34.58 8 2.39 15.15 3 08. 30.71 1 95 30.98 1.32 30.81 8.5 2.13 13.50 2 74 27 36 1 74 27.60 1.17 27.45 9 1.89 12.02 2 44 24 38 1 55 24.59 1.05 24.45 9.5 1.69 10.71 2 18 21 72 1 38 21.91 93 21 79 10 1.50 9.54 1 94 19.35 1 23 19.52 --- --- 10.5 1.34 8.50 l 73 17.24 1.09 17.39 --- --- 11 1.19 7.57 1 54 15 36 .97 15 49 --- --- 196 Table B-4 Maximum sizing windows for 150 pm aperture tube; current-trim control setting - 52. Aperture tube: 150 pm Dstd 20.27 pm (- 13.51% Dap) 2% Dap' 3 0 pm 5% Dap' 7 5 pm 20% D : 30.0 pm 8P 40% D : 60.0 pm 8P 50% D : 75.0 pm aP Log span setting: 10 12 14 s: 10.070323 11.973205 13.741679 (I+G)123' 11.993836 12.029075 11.956434 I+G Dz Df Dz Df Dz Df 3 15.81 161 93 10.27 163.25 6.71 160.54 3.5 14.08 144 26 9.15 145 44 5.98 143.02 4 12.55 128 52 8.15 129.57 5.33 127.42 4.5 11.18 114.50 7.26 115 44 4.74 113 52 5 9.96 102.01 6.47 102.84 4.23 101.13 5.5 8.87 90.88 5.76 91.62 3.77 90.10 6 7.90 80.96 5.13 81.63 3.35 80.27 6.5 7.04 72.13 4.57 72.72 2.99 71.51 7 6.27 64.26 4.07 64.79 2.66 63.71 7.5 5.59 57.25 3.63 57.72 2.37 56.76 8 4.98 51.00 3.23 51.42 2.11 50.57 8.5 4.44 45.44 2.88 45 81 1.88 45.05 9 3.95 40.48 2.57 40.81 1.68 40.13 9.5 3.52 36.07 2.29 36.36 1.49 35.76 10 3.14 32.13 2.04 32.39 1.33 31 85 10.5 2.79 28.63 1.81 28 86 --- ~-- 11 2.49 25 50 1.62 25.71 --- --- 11.5 2.22 22 72 1.44 22.91 --- --- 197 Table B-5 Maximum sizing windows for 300 pm aperture tube; current-trim control setting - 52. Aperture tube: 300 pm Dstd: 44.66 pm (- 14.89% Dap) 2% Dap' 6.0 pm 5% D : 15.0 pm 8P 20% D : 60.0 pm 3? 40% D : 120.0 pm aP 50% D : 150.0 pm aP Log span setting: 8 10 12 Ls: 8.0204586 10.105556 12.092228 (I+G)128: 8.9026537 11.991407 12.029926 I+G Dz Df Dz Df Dz Df 3 27 38 174.67 34 52 356 57 22 01 359 76 3 5 24 39 155.61 30 76 317 67 19 61 320 51 4 21 73 138.63 27 40 283 01 17 47 285 54 4 5 19 36 123.51 24 41 252 13 15 56 254 39 5 17 25 110.03 21 75 224 63 13 87 226 63 5 5 15 37 98.03 19 38 200 12 12 35 201 91 6 13 69 87.33 17 26 178 29 11.01 179 88 6 5 12 20 77.81 15 38 158 83 9.80 160 25 7 10.87 69.32 13.70 141 51 8.74 142 77 7.5 9.68 61.75 12.21 126 07 7.78 127 19 8 8.62 55.02 10.87 112 31 6.93 113 32 8.5 7.68 49.01 9.69 100 06 6.18 100 95 9 6.84 43.67 8.63 89.14 5.50 89.94 9.5 6.10 38.90 7.69 79.42 4.90 80.13 10 5.43 34 66 6.85 70 75 4.37 71.39 10.5 4.84 30.88 6.10 63 03 3.89 63.60 11 4.31 27.51 5.44 56.16 3.47 56.66 11.5 3.84 24 51 4.84 50.03 3.09 50.48 198 Table B-6 Maximum sizing windows for 480 pm aperture tube; current-trim control setting - 52. Aperture tube: 480 pm Dstd: 91.2 pm (- 19.0% Dap) 2% Dap' 9.6 pm 5% D : 24.0 pm aP 20% D : 96.0 pm 3? 40% D : 192.0 pm aP 50% D I 240.0 pm 3? Log span setting: 10 12 LS: 10.204376 12.174651 (I+G)128: 12.137953 12.163773 I+G Dz Df Dz DE 5 44 91 474 51 28 65 477 34 5 5 40 01 422 74 25 53 425 27 6 35 64 376 61 22 74 378 87 6 5 31 75 335 53 20 26 337 53 7 28 29 298 92 18 05 300 71 7 5 25 20 266 31 16 08 267 90 8 22 45 237 25 14 33 238 67 8 5 20 00 211 37 12 76 212 63 9 17 82 188 31 11 37 189 43 9 5 15 88 167 76 10.13 168 77 10 14.14 149 46 9.03 150 35 10 5 12.60 133 15 8.04 133 95 11 11 23 118 63 7.16 119 34 11 5 10 00 105 68 6.38 106.32 12 8 91 94 15 5.69 94.72 12 5 7 94 83 88 5.07 84.38 13 7 07 74 73 4.51 75.18 APPENDIX C COMPUTER PROGRAM BOXYABCV.BAS The computer program 80XYABCV.BAS was written in BASICA to transmit the particle data counted by the ELZONE particle data analyzer to IBM PC as described in Chapter 3. The data transmitted are the n 's of 128 channels. In program execution, 80XYABCV.BAS requires ch inputs of the calibrated volume of the mercury volumetric siphon, Vv’ the dilution factor in sample preparation, DF, and the sample I.D. for the filename for data storage. The program stores the nch c data calculated from Equation (3-19) to the diskette, for which the drive is specified with the input filename, e.g., szilename or Azfilename. 199 200 Flow Diagram of 80XYABCV,BAS Input Vv, DF, and sample I.D., ID$; / give an instruction to restart the program if 1/ Vv or OP is different from the previous values; a check guestion for the sample I.D. is asked, it can be changed Yes If ID$ - "end" 4 to- \\\\or "END" No Open a communication file; give an instruction to start transmitting the data; i transmit 154 data according to the protocal 1 described on page 47, User's Manual; ‘gedefine the filename with an extention ".CSV" i .-—.——— 7 Open the the file for data storage;' w ite the first 24 transmitted data The 19th datum is the preset time, Tps; the 20th datum is the elapsed time, Te; the 24th to 151th data are nch’s for channel 1 through channel 128; the 152th datum is the total number of n ch; 'calculate nch c for 128 channels and the écorrected total number of nCh c in the unit of '#/channe1-m1 according to Equations (3-37) and .(3-19) Write the calculated data to the file for data storage; «close the file 201 Listing of 80XYABCV,BAS 10 DIM V$(153) 20 CLS:PRINT SPC(20)"80XYABCV: 80XYAB ----> CSV DATA FILES.":PRINT: PRINT "---- Transmit the data from area A or B in 80XY to IBM and store them as Comma Separated Value files."‘ 25 PRINTlePUT "Volume of mercury volumetric siphon in m1; VV";VV 27 PRINTlePUT "Dilution factor (>-l); DF";DF 3O PRINTzPRINT "Restart the program whenever VV or OP is changed.": PRINTzPRINT "VV - ";VV,;";",;"DF - ";DF 31 PRINTlePUT "Sample I.D.";ID$:PRINT:INPUT "Is above I.D. correct ";A$:IF A$<>"Y" AND A$<>"y" GOTO 31 40 IF IDS-"end" OR ID$-"END" THEN PRINTzPRINT "End of program." :END 50 OPEN "COM1:2400,E,7,1,CS,DS,CD" AS #1 60 PRINTzPRINT "Press ( then) on TP50 when ready to transmit the data." 70 FOR I-O TO 153:1NPUT #1,V$(I):NEXT:CLOSE 80 IE$-ID$+".CSV":DM$-" " 9O OPEN IE$ FOR OUTPUT AS #2 100 FOR I-O TO 23 STEP 6 110 FOR J-I TO I+4zPRINT #2,V$(J)",";:NEXT:PRINT #2,V$(J):NEXT 120 TPS-VAL(V$(18)):TE-VAL(V$(19)) 130 PRINT #2,DM$ "," DM$ "," DM$ "," DM$ "," "Differential Population [#/channel.m1]," 140 FOR I-24 TO 152:DP-VAL(V$(I))*TPS*DF/(TE*VV):PRINT #2,DM$ "," DM$ "," DM$ "," DM$ "," STR$(DP) ",":NEXT 150 PRINT #2,V$(153) 160 CLOSEIPRINTzPRINT "Finish with " ID$;".":PRINT:PRINT STRING$(79,45) :GOTO 30 APPENDIX D PHYSICAL PROPERTIES OF SOLUTIONS The solutions used in the MSMPR experiments contain the ionic buffer KN03 to maintain solution ionic strength at 0.15 M. The physical properties of 0.15 M KNO3 (z 1.517% KNOs) were used to estimate the physical properties of the experimental solutions. For the study of the effects of MDCK cells, the modified media used contain approximately 0.0059 M KCl and 0.1574 M NaCl in addition to equal molar of CaC12 and NaC204 (varied from 0.00075 to 0.0015 M) and D-glucose (0.00555 M). The physical properties of 0.165 M NaCl (z 0,9644% NaCl) were used. The property data for both systems are shown in Table D-l. Table D-l Physical properties of solutions at 37°C. Physical Solutions containing Reference properties 1.517% KN03 0.9644% NaCl Density, P 1.00223 0.99984 a 3 (g/cm ) Viscosity, p 6.897 7.039 b (10-3 g/cm-s) a: Perry and Chilton, 1973. b: International Critical Tables, Washburn et a1., 1930. 202 APPENDIX.E COMPUTER PROGRAH.NTH3.FOR The FORTRAN program NTM3.FOR was written to calculate the total particle number, N and the third moment, M3, for calcium oxalate T’ crystallization in the MSMPR system as discussed in Chapter 4. The input data are v(L,t) PSDs, Le, x, and number of sets of PSDs, lower particle size limit, and maximum upper particle size of each experiment. CSAKM, CSITG, and CSVAL are the IMSL subroutines used to manipulate data with a spline-fit approximation and to calculate integrals and the approximated values. 203 204 Flow Diagram of NTM3.FOR ' v(O, t) - 0 / Input Lc , n, / / //number of PSDs per run, NRUN(I-l, 9) d(lower limit of L), and ,max. Lb (upper limit of L); BR(IR) Loop 100 ‘\ - 9 f0 9 ns Set 601 grid points ranging from Le to BR(IR); XGCIel 601) 1? / Open a data file for storing NT and Ma/ (NTM3* DAT: * code for the run) Loop 90, calculating NT and M3 from each PSD, v(L,t) A V / Input v(L,t) (NINTV data points) / x (including v(0,t)-0, there are NDATA data' (I-1.NDATA). Y(I-1.NDATA)) a J. 'I Spline on X. 1 using QSAKM h 0 Calculate Ma using CSITG, 6 M3 - T JLb V(L,t) dL L . c .L 1 (cont’d) (cont’d) 205 (cont'd) (cont’d) Find the XG(J) immediately below Lb: if XG(J) - Lb, then NSPL - J, Nswitch - if XG(J) < Lb, then NSPL - J+l, N XG(J+1) - Lb, i (store the original XG(J+1) as XGJPl) jwhere NSPL is the total number of n(L,t) data ,calcuiated from v(L,t) and to be used to calculate ET. 0 switch - l v Calculate NSPL values of ! ; n(L,t) - 6 v(L,t) using CSVAL : 5 «L v 1 S ine on n C) using CSAKM H Calculate NT using CSITG, - - .. --.”---_ NT - Jib n(L,t) dL c Write NT, M3 in NTM3*.DAT, where * is the code identifying the run \\“ g 1 ////, If Nswitchy' / Yes No gRestore XG(NSPL) - XGJPl' gand set N switch - A ‘7 i 606) ) HO O ( Stop 5 206 Listin of NTM FOR C********************************************************************** C PROGRAM NTM3.FOR C C *** Calculate total number of particles (NT) and the third moment C *** (M3) from v(L,t) data. C C SYMBOLS: C X(i); L (micron) C Y(i); v(L,t) C XG(i); grid points on L C YN(i); n(L,t) C C; Lc- .0013 microns C BR(i); max. Lb of each experimental run C (Lb- upper limit of L for a PSD) C D(i); Ld- lower limit of L for an experimental run C TM3; M3- third moment C PNT; NT C N1; grids between Lc and Ld C N2; grids between Ld and BR(i) C WD; w at Ld C DW; delta w C NRUN(i); number of PSD data per experimental run DIMENSION X(lOl),Y(lOl),XG(601),YN(601),BREAK(601),CSCOEF(4,601), + NRUN(9),D(9),BR(9) CHARACTER NMCODE*19,FILDAT*9,FILENM*7 DATA PI,X(1),Y(l),C/3.l41593,0.,0.,.0013/ DATA NMCODE/'ABCDEFGHIJKLMNOPQRS'/ DATA (NRUN(I),I-1,9)/17,15,15,16,14,13,16,19,15/ DATA (D(I),I-1,9)/3.,lO.,6.,6{,6.,12.,6.S,6.5,6.2/ DATA (BR(I),I-1,9)/38.5,78.8,74.3,71.2,69.6,89.5,74.8,35.3,68.3/ DO 100 IR-l,9 C ~repeating for 9 experimental runs C C ..................................................................... C *SET 600 GRID POINTS BETWEEN Lc AND BR(IR)* C ..................................................................... N2-INT(.5+600./(l.+(D(IR)-C)/2./(BR(IR)-D(IR)))) Nl-600-N2 BDC—BR(IR)/C WD-LOG(D(IR)/C)/LOG(BDC) DW—WD/Nl DO 12 J-O,Nl XG(J+1)-C*BDC**(DW*J) 12 CONTINUE DW-(l.-WD)/N2 207 D0 14 J-1,N2 XG(J+N1+1)-C*BDC**(WD+DW*J) 14 CONTINUE c ..................................................................... C *OPEN A FILE FOR DATA STORAGE* c ..................................................................... FILDAT-’NTM3'//NMCODE(IR:IR)//'.DAT' 0PEN(UNIT-5,FILE-FILDAT,STATUS-'NEW’) WRITE(5,20) 20 FORMAT(' ',6X,'NT',13X,’M3') DO 90 I-1,NRUN(IR) FILENM-'A’//NMCODE(IR:IR)//NMCODE(I:I)//'.DAT' c ..................................................................... C *INPUT PSD; v(L,t)* c ..................................................................... OPEN(UNIT-6,FILE—FILENM,STATUS-’OLD’) READ(6,30)NINTV 30 FORMAT(I4) NDATA-NINTV+1 DO 50 J-2,NDATA READ(6,40)X(J),Y(J) 40 FORMAT(2E16.0) 50 CONTINUE CLOSE(6) C ..................................................................... B-X(NDATA) CALL CSAKM(NDATA,X,Y,BREAK,CSCOEF) C -spline v(L,t) TM3-CSITG(C,B,NINTV,BREAK,CSCOEF)*6./PI C ..................................................................... C *CALCULATE NT AND n(L,t) FROM SPLINED v(L,t) DATA* C ..................................................................... D0 52 J-601,N1,-1 IF (XG(J) .LT. B) THEN NSPL-J+1 XCJPl-XC(J+1) XG(J+l)-B NSWCH-l GO TO 60 ELSE IF (XG(J) .EQ. B) THEN NSPL-J NSWCH—O GO TO 60 END IF END IF 52 CONTINUE 60 DO 70 J-1,NSPL YN(J)-CSVAL(XG(J),NINTV,BREAK,CSCOEF)*6./PI/XG(J)/XG(J)/XG(J) 70 CONTINUE 208 CALL CSAKM(NSPL,XG,YN,BREAK,CSCOEF) NINTV-NSPL-l , PNT-CSITG(C,XG(NSPL),NINTV,BREAK,CSCOEF) c ..................................................................... WRITE(UNIT-5,FMT-80)PNT,TM3 80 FORMAT(2E16.7) IF (NSWCH .EQ. 1) THEN XG(NSPL)-XGJP1 NSWCH—O END IF 90 CONTINUE CLOSE(S) 100 CONTINUE STOP END APPENDIX.F SHIFTS OF HIE v(L,t) CURVES Examples of the v(L,t) shifts due to crystal growth as described in Approach A for finding crystal growth rate (Chapter 4) are shown in Figures F-l through F-6. In these figures, the origin (zero size) of each PSD is marked. The scales show the relative magnitudes of particle size and particle volume, v(L,t). For each curve, the PSD below the lower particle counting limit are neglected and shown as zero values (a straight line on the x-axis). The matches of the characteristic shapes of the curves are shown with arrows on these figures. 209 210 Sampling time: A: 3 min f“”NAJ\A/V\P\v*\/\_Al__ B: 11 min 0' 10 C: 19 min D: 27.6.m1'n 5 ”W 3x50 E: 35 m1n F: 43 min 6: 51.6 min C v(L,t) (pm’lpm-m1) N 3 F .J G o 10,pm Figure F-l Shifts of v(L,t) for the experiment with PGA - 100 ppm; sampling time: 3 to 51.6 min. Arrows and "*" identify points compared among the PSD curves. 211 Samp1ing time: G: 51.6 min H: 59.5 min I: 68.5 min J: 83.03 min K: 99 min L: 115.25 min M: 131 min '3. I E fi E £3 7E _.l 3: N :2 A I J I ' L 0 v M O I III 10,um Figure F-2 Shifts of v(L,t) for the experiment with PGA - 100 ppm; sampling time: 51.6 to 131 min. v(L,t) (um’lum-ml) 212 Samp1ing time: : 131 min : 147 min : 164.38 min A : 179 min : 195.5 min : 201 min : 206 min . M MJUO'UOZZ 1e7 O ‘ - 10.um Figure F-3 Shifts of v(L,t) for the experiment with PGA - 100 ppm; sampling time: 131 to 206 min. v(L,t) (um’lum-ml) 213 Samp1ing time: A: 2.5 min B: 10.5 min C: 18.5 min D: 26.5 min E: 34.5 min F: 42.5 min Ax10 20 um Figure F-4 Shifts of v(L,t) for the base run; sampling time: 2.5 to 42.5 min. Sam KL—IID‘H v(L,t) (urrF/um m1) . 42.5 min : 50.5 min : 58.5 min : 70.5 min : 82.5 min : 98.5 min 214 p1ing time: Hx1.5 ' Ix2.5 0 Kx7 Figure F-S Shifts of v(L,t) for the base run; sampling time: 42.5 to 98.5 min. v(L,t) (um3/um-m1) 215 Samp1ing time: : 98.5 min 114.5 min 130.5 min 146.5 min : 162.5 min : 178.5 min : 194.5 min 1 01302er 20.um Figure F-6 Shifts of v(L,t) for the base run; sampling time: 98.5 to 194.5 min. APPENDIX.G COMPUTER PROGRAN.GHSHPR The computer program CMSMPR was written in FORTRAN to calculate the linear growth rate, G for the MSMPR crystallizer operating at L’ quasi-steady state as discussed in Chapter 4. Equation (3-17) was used in the calculation. The required input data are the digitized PSD (1n n(L,t)), sampling time, filenames of these PSD data, filenames for storing the calculated growth rates, residence time, total number of PSD data being used in the calculation (NA), and the lower and upper size limits of the PSD data (XI and KP). The list of principal variables, flow diagram, and the program listing for the experiment with calcium and oxalate concentrations equal to 1 mM are shown in the following. 216 P bo X(I) F(I) XA(I) YB(I) YA(I) C(4,30) and BREAK(30) G(167) T(20) TAU DN(20) GN(20) DX XX XI XF NA NX NP DM IM 217 V b es Defiriitian x-value of the input 1n n(L,t) distribution. y-value of the input In n(L,t) distribution. x-value of the grid points on L. y-value evaluated by an IMSL subroutine, DCSVAL, for x - XA(I). YB(I-l), I is the index counting the sequence of sampling time of the PSD data. Spline coefficients of DCSAKM, the IMSL spline subroutine. Calculated growth rate. Sampling time, t (min). Residence time, r (min). Filenames for the digitized 1n n(L,t). Filenames for the calculated growth rate. Increment of L of the points growth rates are to be calculated. Difference of length L in calculating 6(ln n(L,t))/8L. Lower limit of L. Upper limit of L. Number of sets of PSDs in each experimental run planned for growth rate calculation. Number of digitized points of each PSD. Total number of points at which the growth rates are calculated. NP-3. 6(ln n(L,t))/8L. A dummy variable. 218 fitJfiuflJL,) Define dimensions of variables;§ defipg types pf variables 7 11' Input filenames GN(I) and DN(I); / input T(I), NA, NX, XI, XF, DX, txxi_and TAU = Calculate NP, M; set the abscissa XA(I) for evaluating 1n n(L,t) by spline approximation; set J-l (counting the no. of PSDs have beep psgg in palguiation) A :1. 1" /Input ln n(L,t) data LIIXIIIAIKIIAI_I:11NX)./ Spline approximation on PSD data, (X(I),F(I)), using DCSAKM routine; e <(ZIS J-1“> YesI/ No YA(I-l,M)-YB(I-1,M)‘= J-2 J . FCalculate 8L , DM; ‘calculate GL by Equation (3-17) / Store the calculated GL data _V//\\\ v~éYA(I-1,M)-YB(I-1,M)i: ' If J-NA“ I J-J+1 ‘ /“ Yes C §top 1 10 30 40 45 48 50 219 tin o GMSM R DIMENSION X(30),F(30),XA(501),YA(501),YB(501),C(4,30),G(l67), & T(20),BREAK(30) DOUBLE PRECISION X,F,C,XA,YA,YB,TAU,DM,XX,T,G,XI,XF,DX,BREAK INTEGER NA,NX,IC,M,I,J,IM,NP CHARACTER DN(20)*11,GN(20)*12 DATA NA,NX,IC,XI,XF,DX,TAU,XX/10,30,29,2.5,24.5,.5,8.,1.E-4/ DATA GN(I),GN(2),GN(3),GN(4),GN(5),GN(6),GN(7),GN(8),GN(9)/ +'G408T409.DAT','G409T410.DAT','G410T411.DAT','G411T412.DAT', +'G412T413.DAT','G413T414.DAT','G414T415.DAT','G415T4l6.DAT', +'G416T417.DAT'/ DATA DN(l),DN(2),DN(3),DN(4),DN(5),DN(6),DN(7),DN(8),DN(9), +DN(10),'DGT4080.DAT','DGT4090.DAT','DGT4100.DAT','DGT4110.DAT', +'DGT4120.DAT','DGT4130.DAT','DGT4140.DAT','DGT4150.DAT', +'DGT4160.DAT','DGT4170.DAT'/ DATA T(l).T(2).T(3).T(4).T(5).T(5).T(7).T(8).T(9).T(10)/ +56.,64.,80.,96.,112.,128.,l44.,160.,176.,192./ NP—INT((XF-XI)/DX+1.1) M—NP*3 D0 10 I-0,NP-1 IMs3*I+2 XA(IM)-XI+DX*I XA(IM-1)-XA(IM)-XX XA(IM+l)-XA(IM)+XX CONTINUE J-l 0PEN(UNIT-11,FILE-DN(J),STATUS-'0LD') DO 45 I-1,NX READ(UNIT-1l,FMT-40)X(I),F(I),IM FORMAT(F13.0,2X,F13.0,2X,12) CONTINUE CLOSE(ll) CALL DCSAKM (NX,X,F,BREAK,C) D0 48 I-1,M YB(I)-DCSVAL (XA(I),IC,BREAK,C) CONTINUE IF(J .EQ. 1) THEN DO 50 I-1,M YA(I)-YB(I) J-2 GO TO 30 END IF DO 60 I-0,NP-l IM-3*I+2 220 DM-(YA(IM+1)~YA(IM-l)+YB(IM+1)-YB(IM-l))/XX/4 IF (ABS(DM) .LT. 1.E-25) THEN G(I+l)-0 GO TO 60 END IF G(I+1)-~((YB(IM)~YA(IM))/(T(J)-T(J-l))+1/TAU)/DM 60 CONTINUE OPEN(UNIT-ll,FILE-GN(J-l),STATUS-'NEW', &ACCESS-'SEQUENTIAL',FORM-‘FORMATTED') WRITE(UNIT-ll,FMT-65)XX 65 FORMAT(' XA(I)',10X,'G(I) (XX- ',E8.l,')') DO 80 I-1,NP WRITE(UNIT-11,FMT-70)XA(3*I-1),G(I) 70 FORMAT(F8.3,5X,E13.6) 80 CONTINUE ENDFILE(UNIT-11) CLOSE(UNIT-ll,STATUS-'KEEP') IF(J .EQ. 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