A PRGPL 730L ‘ 93 ‘i ‘36“? fi' 1-: I m...» {rm k t F nu. Ln." D: if! \. ITY PHVERS PLOGLE l l a. .l EL T: EGAN SEA 5 I PMCH P. r Ea E R.R' CLYDD u.’ nhu «bu 10$ :ufibt‘ 0-169 This is to certify that the thesis entitled A Proposed Model for Renal Blood Flow Control presented by Clyde R. Replogle has been accepted towards fulfillment of the requirements for __I:1’L_D_°_degree in Ph' SiOIOEY Major profe r Zfiz/é gaff Date July 28, 1967 0 Michigan Staff Univctsity Li. W. LIBRARY ‘ ‘1 s! I H A PROPOSED MODEL FOR RENAL BLOOD FLOW CONTROL By Clyde R. Replogle AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physiology 1967 AFLC-WPAFB—JUL 67 n ABSTRACT A PROPOSED MODEL FOR RENAL BLOOD FLOW CONTROL by Clyde R. Replogle Autoregulation of blood flow in the kidney was recognized by Rein in 1931, and its characteristics have been described by manv authors (Hartmann, et a1. 1936; Forster and.Maes, 19h7; Selkurt, 19h6; Hinshaw, et al. 1959; Scott, et a1. 1965). Although there are few arguments over the existence of and the description of renal autoregulation, there have been many hypotheses attempting to explain the mechanism of its operation (Harvey, l96h; Scott, et a1. 1965; Raddy and Scott, 1965; Hinshaw, 196h; Replogle 1960a; Schmid and Spencer, 1962; waugh and Shanks, 1960; Wells, 1960). At the present time, three theories are generally accepted: one suggests a myogenic principle, another indicates some role of meta- bolic end-products, and a third invokes collapse of interlobular and arcuate veins. Proponents of each of these theories cite various lines of experimental evidence to support their theories which, at first glance, are mutually contradictory. Evidence supporting a conclusion that the blood flow resistance change responsible for autoregulation is located in the interlobular and arcuate veins (Hinshaw, et al. 1959, 1961, 1963, 196h; Replogle, 1960a, 1960b, Wells, 1960) stimulated the study presented in this thesis. If resistance change concomitant with autoregulation is located in the renal veins, and is caused by a purely passive phenomenon of a trans- mural pressure difference causing collapse, the blood flow control mechanism must be non-linear. That is, fluid resistance at a given pressure in a section of collapsible vein is dependent upon blood 1 pressure upstream, and upon flow. Flow, in turn, depends on the re- sistance in the vein and resistance of the vasculature upstream. In such a control system it is difficult, if not impossible, to assess the effects of changes in the prevenous circulation on autoregulation in the venous circulation. Equations are derived to describe the control of fluid flow through a simplified model of a section of the renal circulation. This mathematical model, when solved by a RungeéKutta numerical technique, predicts (a) that autoregulation can occur by passive collapse of small renal veins, (b) that flow instability can arise as the result of a limit cycle, ie. oscillation between two stable states. (The end of the collapsible tube where the transmural pressure is highest, the tube collapses to com- pletely stop flow. Pressure immediately increases and tube reopens). In order to observe the control mechanism postulated by the mathematical model in a real system, a hydraulic model was constructed using Tygon tubing to represent the non-collapsible arterial circulation and penrose surgical drainage tubing to represent the collapsible venous circulation. Autoregulation occurred in this model and its form was similar to that predicted by the mathematical model. Measurements were made of input pressure, output pressure, flow, and pressure within a water-filled chamber surrounding the penrose tubing through which water was flowing. The penrose tubing was analogous to a vein immersed in interstitial fluid. Several conclusions can be drawn from analyses of these data: 1. Autoregulation of fluid flow, similar to that seen in the kidney, can occur by the passive response of collapsible tubing to transmural pre ssure . 2. Because of the non-linear relationship between the area of the collapsing tube and the transmural pressure, it is not necessary for chamber (interstitial) pressure to increase suddenly in order to pro— duce autoregulation. 3. Oscillation in outlet pressure can be caused by a limit cycle when the resistance just upstream of the collapsible tube is low and the flow rate is high. Measurement of blood flow rate, arterial pressure, and interlobu- lar venous pressure (by retrograde insertion of a small cannula) were made on eight dog kidneys. Results indicate that the resistance change responsible for autoregulation takes place in the interlobular veins and the form of the pressure-flow curves approximates the form predic- ted by the mathematical model. The hypothesis of non-linear venous control of renal blood flow is compatible with other findings such as the concomitant release of metabolic end-products (Scott, 1965) because venous control is very responsive to changes elsewhere in therenal circulation. References l. Forster, R.P. and Mass, J.P. "Effect of Experimental Neurogenic Hypertension on Renal Blood Flow and Glomerular Filtra- tion Rates in Intact Devervated Kidneys of Unanestheti— zed Rabbits with Adrenal Glands Demedullated," Amer. J. Physiol. 1503534, 1947. 2. Haddy, F.J. and Scott, J.B. "Role of Transmural Pressure in Local Regulation of Blood Flow through the Kidney," Am. J. Physiol. 208:825-831, 1965. 3. Hartmann, H. Orskov, S.L. and Rein, H. "Die Gefassreaktionen der Niere in Verlaufe Allgemeiner Kreislauf Regulatimsvor- gange," Pflueger Arch. Ges. Physiol. 238:239, 1936. 4. Harvey, R.B. "Effects of Adenosine Triphosphate on Autoregulation of Renal Blood Flow and Glomerular Filtration Rate," Circu- 10. ll. 12. 13. 1h. 15. lation Research, Suppl. I to Vols 1b and 15:1-178 - 1—182, 196A. Hinshaw, L.B. Day, S.B. and Carlson, C.H. "Tissue Pressure as a Causal Factor in the Autoregulation of Blood Flow in the Isolated Perfused Kidney," Amer. J. Physiol. 197:309, 1959. Hinshaw, L.B. and worthen, D.M. "Role of Intrarenal Venous Pressure in the Regulation of Renal Vascular Resistance," Circ. Res. 9:1156-1163, 1961. Hinshaw, L.B. Brake, C.M. Iampietro, P.F. and Emerson, T.E. "Effect of Increased Venous Pressure on Renal Hemodynamics," Am. J. Physiol. 20h(1):119—123, 1963. Hinshaw, L.B. "Mechanism of Renal Autoregulation: Role of Tissue Pressure and Description of a Multifactor Hypothesis," Circ. Res. 1h,15: Suppl. I, 120-131, l96h. Replogle, C.R. Wells, C.H. and Collings, W.D. "Pressure-flow-dis- tension Relationships in the Dog Kidney, Fed. Proc. 19: 360, 1960a. Replogle, C.R. "Pressure-flow-distension Relationships in the Dog Kidney, Master's Thesis, M.S.U., 1960b. Schmid, H.E. and Spencer, M.P. "Characteristics of Pressure-Flow Regulation by the Kidney," J. Appl. Physiology 17:201-20h, 1962. Scott, J.B. Daughtery, R.M. Dabney, J.M. and Haddy, F.J. "Role of Chemical Factors in Regulation of Flow through Kidney, Hind Limb, and Heart," Am. J. Physiol. 208:813-82h, 1965. Selkurt, E.E. "Relationship of Renal Blood Flow to Effective Arterial Pressure in the Intact Kidney of the Dog," Amer. J. Physiol. lh7z537, l9h6. waugh, W.H. and Shanks, R.G. "Cause of Genuine Autoregulation of the Renal Circulation," Circ. Res. 8:871-888, 1960. ‘Wells, C.H. "Estimation of Venous Resistance and their Signi- ficance to Autoregulation in Dog Kidneys," Master's Thesis, M.S.U., 1960. A PROPOSED MODEL FOR RENAL BLOOD FLOW CONTROL By , s \t Clyde R? R‘Replogle A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of IIDCTOR OF PHILOSOPHY Department of Physiology 1967 .I‘ l'.ll {I l Illl'l‘IlllIl-ll [ll-[III (l I I'll" I ll I l I! ll MM; 3 M33” 67 LIST OF ILLUSTRATIONS . . . . . LISTOFTABLES .. .. . LIST OF APPENDICES . . . . . . Chapter I. INTRODUCTION . . . . . . II. MATHEMATICAL MODEL . . . Concept . . . . . . . Derivation of Dynamic Model........ Steady Flow Equations Summary....... Computer Solutions . III. HYDRAULIC TEST SECTION . Pressure Supply . . . Test Section . . . . Methods of Measurement Results . . . . . . . IV. EXPERIMENTS 0N DOG KIDNEYS Methods.... ... Equations 2i: Contents of Motion and Boundary Conditions Results and Discussion . . . . V. SUMMARY AND CONCLUSIONS . BIBLIWMPIIY O O O O O O O O O APPENDI X 0 O O O O O O I O O 0 ii for Page iii 12 12 13 25 3o 30 37 37 38 1.1 M: 51 51 SS 70 72 76 Figure 0\ U1 4;” W Q 10 ll 12 13 1h 15 16 17 18 19 20 21 22 23 List 2: Illustrations contml MOdel C O O O O O O O O O O O O O O O O O O O 0 Linear Resistance Tube . . . . . . . . . . . . . . . . C ont r01 V0 lume O O O O O O 0 O 0 O O O O O O O O O O 0 ‘Closed Loop Configuration . . . . . . . . . . . . . . . Ehd Boundary Conditions of the Collapsible Tube . . . Computer Results for Closed Loop Operation . . . . . . Prediction of Stability . . . . . . . . . . . . . . . . Computer Results for Open Loop Operation . . . . . . . Pressure Distribution Along Collapsible Tube . . . . . Pressure Distribution Along Collapsible Tube . . . . . Pressure Supply . . . . . . . . . . . . . . . . . . . . Test Section . . . . . . . . . . . . . . . . . . . . . Pressure Fixture . . . . . . . . . . . . . . . . . . . Chamber for Collapsible Tube . . . . . . . . . . . . . Configuration for Measurement of Area as a Function of Collapsing Pressure . . . . . . . . . . . . . . . . . . Open and Closed Loop Configuration . . . . . . . . . . Apparatus forVMeasurement of Open Loop Characteristics Cross-Section of Collapsible Tube as a Function of Collapsing Pressure . . . . . . . . . . . . . . . . . . Area of Collapsible Tube as a Function of Collapsing Pressure 0 O O O O O C O O O I O O O O O O O O O O O O 0 Open Loop Results . . . . . . . . . . . . . . . . . . . Closed L00p Results . . . . . . . . . . . . . . . . . . Recording of Simple Oscillations . . . . . . . . . . . Kidney l Segmental Autoregulation . . . . . . . . . . iii Page 16 17 19 26 29 31 33 3h 35 36 38 39 39 LO h2 h3 1.3 44 145 h7 AB 51 56 2h 25 26 27 28 29 3o 31 Kidney 2 Segmental Autoregulation Kidney 3 Segmental Autoregulation Kidney h Segmental Autoregulation Kidney 5 Segmental Autoregulation Kidney 6 Segmental Autoregulation Kidney 7 Segmental Autoregulation Kidney 8 Segmental Autoregulation Venous Pressure as a Function of Distance iv 57 . 58 S9 60 61 62 63 6h Table I II List 9; Tables symb01$ I O O O O O O 0 Functional Notation . . Appendix List 2f Appendices Derivation of Resistance of an Elliptical Tube Computer Programs and Results . . . . . . . . Experimental Data from Hydraulic Test Section Experimental Data from Deg Kidneys . . . . . . Page . 76 . 79 . 96 A PROPOSED MODEL FOR RENAL BLOOD FLOW CONTROL I. Introduction The phenomenon of blood flow control intrinsic to an organ or seg- ment of the circulation, without external control centers, is generally termed autoregulation. Autoregulation has been described in the liver (Torrance, 1958), skeletal muscle (Folkow, 19h9), myocardium (Berne, 1959), brain (Rapela and Green, 196A), intestine (Johnson, 1960), and kidney (Selkurt, 19h6). It has been found in humans, dogs, cats, rats, and calves. The form of pressure-flow curves can be quite different in each of these organs, and, because of the differences in blood flow rate per gram of tissue, interstital pressure levels, and reactivity to vasodilator metabolites, it is unreasonable to assume that auto- regulation in the kidney and in other organs might have a common mecha- nism. Therefore, the model suggested in this thesis is proposed only for the kidney and autoregulation in other organs will be considered nowhere else in the thesis. The first experiments on renal autoregulation were performed by Rein (1931) and later by Unna (1935), Hartmann (1936), and Forster and Maes (19h7). These investigators concluded that an increase in systemic blood pressure causes an increase in resistance to blood flow resulting in less- than-proportional increase in renal blood flow, and that this phenomenon does not depend on neural cennection to the rest of'the body. Selkurt (l9h6) further characterized renal autoregulation as a means of control which only exists after arterial pressure exceeds 80 mm Hg. Below this pressure, blood flow is directly proportional to pressure drop across the kidney. Since these first descriptions of renal autoregulation, many investigators have presented experimental evidence to support various opinions concerning the mechanism by which it occurs. The hypothesized mechanisms which are still widely accepted today may be divided into two types: active, (i.e., requiring energy expenditure by an effector), and passive (i.e., reacting to forces present with- in the circulation. Three types of active control have been suggested: reaction to vasodilator metabolites, myogenic reaction to changes in transmural pressure, and a local reflex causing arteriolar vasomotion in re- sponse to pressure or flow within the kidney. All three of these types are active feedback control mechanisms. While the three mecha- nisms imply different sensing mechanisms to regulate flow, (response of general or specialized cells to decreased flow in the case of the metabolic mechanism, a flow or pressure sensor in the case of a local reflex mechanism, and inherent response to muscular arterioles in the case of the myogenic theory), and different control pathways (a chemi- cal substance for metabolic control, neural pathways for a local re- flex, and internal smooth muscle reaction for myogenic control), they all require a common effector, namely, the smooth muscle of the arteriole wall. There is a good deal of evidence to suggest that a vasodilator meta- bolite is somehow connected to renal autoregulation. In early experi- ments, Winton (1934, 1951) and Bickford and Winton (1937) cooled dog kidneys in an isolated perfusion arrangement and found a decreased regu- lation with a concomitant increase in intrarenal pressure. This response to temperature could easily be construed as affecting a metabolically dependent mechanism. Haddy and CO-WOerrS (19583) hypothesized that renal autoregulation was dependent on a metabolite, and that autoregulation of renal blood flow is dependent on blood flow rather than blood pressure. They based this conclusion on their findings that the onset of autoregula- tion appears at the same flow rate but at widely different arterial pressures, that lymph flow increases very little with increased ar- terial pressure, and that decapsulation does not decrease autoregula- tion. Harvey (l96h) showed that renal blood flow and glomerular filtra- tion rate (GFR) (estimated by the product of directly measured renal plasma flow and creatinine extraction) were held relatively constant in spite of large changes in perfusion pressure in the isolated dog kidney. He further showed that adenosinetriphosphate (ATP) when infused into the renal artery could produce these results. He concluded that ATP produced dilation of the efferent arteriole. His work shows that a vaso- dilator metabolite could be released as a result of a decreased flow or pressure. Scott et a1. (1965) substantiated this hypothesis by using the forelimb as an assay organ.‘nry'found that a vasodilator substance is re- leased from the kidney when perfusion pressure was decreased. Scott and his co—workers found that adenosine, adenosine monophosphate (AMP), and ATP cause dilation in the forelimb but only ATP dilates the kidney. Also, they found that when ATP is infused into a kidney with the contra- 1atera1 kidney as the assay organ, the target kidney shows dilation while the assay kidney shows constriction. As AMP has been found in re— nal venous blood after reduction in perfusion pressure (Gordon, 1962), Scott and his co-workers interpreted their finding to mean that ATP is released when perfusion pressure is decreased. It then causes vasodilation and is then quickly converted to AMP or adenosine. Winton (196h) attempted to show that autoregulation is "activa- ted" by changes in velocity of blood flow rather than pressure dif- ferences across blood vessel walls. He obtained pressure—flow curves from control, epinephrine treated (0.h to 1h mm3 of 10 to 20LJg ml-1 added as an impulse to each 0.6 m1 of blood perfusing the kidney), and kidney cooled to 6-630. He then plotted resistance versus flow and resistance versus pressure. His curves show a better functional relationship between resistance and flow than for resistance and pressure. He cites this as evidence to support "a fairly clear indication of a flow - dependent mechanism". Even if his curves did support this conclusion, his method of calculating resistance produces a function which is unrelated to usual concepts of fluid resistance. He calculated resistance as the slope of the line connecting two adjacent points on a pressure-flow diagram, referred to the pressure mid-way between the two relevant pres- sures, divided by the slope of the line joining the origin to the point representing blood flow at 100 mm Hg. This calculation yields a nor- malized value of dP/dQ which has no physical significance. For example, if one has a system in which flow is constant for all values of pres- sure, Winton's method would calculate an infinite resistance for all values of pressure and flow. Bayliss (1902) and Folkow (19h9) have shown that arteries and ar- terioles respond to an increase in intraluminal pressure by increasing the tone of their walls thereby decreasing their cross-sectional area and increasing their resistance. waugh (1958, 1960) proposed that this myogenic, vasotonic reaction occurring in the afferent arteriole is the mechanism responsible for renal autoregulation. He based his con- clusions on his findings that autoregulation is abolished by cyanide (suggesting an active process), that autoregulation is present after de- nervation (discounting a neural mechanism), and that there is a short delay of 2-3 seconds between the arterial pressure change and the onset of regulation. Semple and De Wardener (1959) measured arterial and venous pressure and flow in kidneys. By varying venous pressure they found that autoregulation depends on arteriovenous pressure difference rather than on the absolute pressure. They took this to indicate that the mechanism must involve the distension of small renal arteries and arterioles which causes a muscular reflex. Haddy and Scott (1965) investigated the relationship between blood flow, arterial pressure, venous pressure, and pressure in an occluded hilar lymphatic vessel and found that hilar lymphatic pressure rises greatly on elevation of venous pressure but is little affected by a change in arterial pressure. They cite these data as evidence that the change in resistance caused by an increase in arterial pressure occurs in the arterioles. In the same study, Haddy and Scott were able to show that renal arterial pressure increases transiently for 2-h seconds in response to a transient flow pulse lasting 0.5-2 seconds which they state could represent a constriction (arteriolar) elicited.by the stretch. They conclude that a myogenic response to a change in trans- mural pressure might assist the metabolic mechanism they have postu- lated (Scott, et al. 1965). It is new interesting to note that while Haddy and co-workers (the strong proponents of a metabolic mechanism) have begun to sus- pect that there may be some myogenic responses creeping into their theory, waugh (l96h) (the strong proponent of a myogenic mechanism) has "tentatively suggested" that, while he is not going to throw out the transmural pressure idea altogether, autoregulation (still located in the afferent arterioles) may be dependent on variable metabolic feedback. This metabolic factor in turn is dependent on tubular re-absorption of glo- merular filtrate and all are related finally to changes in transmural glomerular capillary pressure. His conclusions are based on some rather extensive observations on the fully isolated kidney; Measurement of intrarenal venous pressure (arcuate, interlobular) allowing division of renal resistance into prevenous and postvenous resistance indicate that the autoregulating resistance change is prevenous. Perfusion with a cell- free solution (e.g., 20% plasma - 80% PVPALocke) resulted in good autoregu- lation thus eliminating a hematocrit change as important. A step increase in arterial pressure resulted in a transient, almost critically damped, pulse in flow with about a five second.time constant, and a new, less-than- proportional, increase in steady state flow. waugh reports the observation of a "hunting-type" reaction of flow in response to a pressure step. But his curves show nothing more than a decrease in damping (approximately 3 cycles to half-amplitude rather than approximately 0.8 cycles to half-amplitude, for his "non-hunting" kidney). The resistance of the higher damped kidney was 1.10 PRU be- fore pressure step and 1.85 after, while the resistance of the under- damped kidney (treated with yohimbine to accomplish intrarenal sym- patholysis) was 0.78 before pressure step and 1.28 after. The dif- ference in his two responses could more easily be accounted for by the lower inertance and resistance of his treated kidney, rather than by a complex servomechanism."hunting" of a predetermined operating level. waugh further observed that the flow response to a pressure step in a kidney in which the vascular reactivity was completely abolished by treat- ment with chloral hydrate, was also a step which changed proportionally more than pressure. This indicated to waugh a mechanism which relies on vasomotion. By comparing the autoregulation capability of the same kidney one hour after isolation and three hours later (after mannitol diuresis) he was able to show that even though resistance had increased h-5 times and flow reduced greatly, autoregulation.responses were still good. He concluded that autoregulation is pressure dependent not flow dependent. Since glomerular filtration rate (GFR) is also autoregulated (Harvey, l96h; Schmid et al.,196h), Waugh contends that his observations confirm the concept of a preglomerular vasomotion change during regulation. waugh's final paragraph (Naugh, 196h) is as follows: "The reported experimental findings appear to sup- port the hypothesis that myogenic vasomotor changes in the renal arterioles, in response to transmural arteriolar pressure, underlie active renal circula- tory autoregulation. However, the experimental findings also support, perhaps more strongly, a new hypothesis that active renal circulatory autoregula- tion is accomplished by afferent arteriolar changes in resistance caused by tubular re-absorptive meta- bolism in response to flow of glomerular filtrate or to the level of glomerular transcapillary pressure." Meanwhile, Schmid, who with Spencer in 1962 postulated an active feedback control system, has now (Schmid, et a1, 196h) supported waugh's old theory. Their first postulate was based on their observation that the renal pressure-flow curves are sharply inflected which,they main- tained (without presenting evidence or 10gical argumenO, would exclude a passive mechanism. In their later experiments, Schmid and his co- workers found that blood flow is controlled by either a reduction in arterial pressure or increase in venous pressure and that GFR is also regulated in a like manner. They concluded that the regulation re- sults from reduced pressure gradient and the resultant fall in trans- mural pressure of the preglomerular vessels. In contrast to the active type of control demanded by the previous mechanisms, a passive feed-forward control has been postulated by Hin- shaw (1959, 1960a, 1960b, 19600, 1961, 1963a, 1963b, 196h), Replogle (1960a, l960b),and wells (1960). This mechanism is based on partial collapse of small intrarenal veins (interlobular) caused by a positive interstitial - intraluminal pressure difference. It is a passive con- trol because no energy expending actuator is postulated and it is feed- forward control because the controlling variable (interstitial pressure) is derived upstream from the controlled variable (interlobular vein area). The first evidence supporting this "tissue pressure" hypothesis came from two experiments (Hinshaw, 1960a; Replogle, 1960a, 1960b; Wells, 1960) using different indirect techniques to separate prevenous from postvenous resistance. These experiments show the autoregulating resistance changes occurring downstream from the glomerular capillaries (Hinshaw 1960a) and downstream from the peritubular capillaries (Replogle 1960a, 1960b; Wells, 1960). More recently HinShaw (1960) and Section IV of this thesis show the resistance changes are located in the inter— lobular or arcuate veins. Hinshaw (1959) blocked ureteral flow and, when ureteral pressure reached equilibrium, took the value of ureteral pressure as the pres- sure within Bowman's capsules Having determined this pressure value, it is possible to calculate the pressure droprbetween the renal artery and Bowman's capsule (glomerular capillaries), and the pressure drop between the glomerular capillaries and the renal vein. By plotting each of these pressure differences against flow, the amount of autoregulation in each section can be compared. Although the regulation in each segment is calculated in the same way, the man- ner in which an intrarenal pressure is obtained is different for each of the methods. Direct pressure values are obtained from a cannula inserted into an interlobular vein just short of the wedging point. The indirect method used by Replogle and Wells is more complicated. With the isolated kidney weighed continuously and flow blocked at the renal artery, renal venous pressure is increased in stages and a rela- tionship between distending pressure and distensiOn volume noted. With flow at zero, pressure throughout the kidney is the same and the distending pressure is taken to be the average pressure at the site of interstitial fluid formation. Flow through the kidney is then resumed and by measuring arterial pressure, flow rate, venous pressure, and kidney weight, the resistance of the segments of the circulation up- stream and downstream from the transudation site can be calculated. The disagreement between the direct small vein measurements of lO Waugh (1961.) and those of Hinshaw and Replogle, and the criticism of both techniques by Haddy and Scott (1965) are virtually the only contro- versies over data (see page 53 for discussion of Haddy‘s priticism and page 68 for discussion of Waugh's measurement). However, the contro- versy over possible mechanism is widening. Perhaps the difficulties encountered in the attempt to merge the existing postulates and supporting data into a simple, and possibly correct, hypothesis are that the "criteria" (See Johnson, l96h) that have been suggested to separate the various mechanisms, and the "reasoning" used to reach the conclusions, have been largely intuitive. By intuitive it is not meant that the proposed mechanisms and "critical experiments" were created from whole cloth, but that they have not'been examined in the light of a rigorous description of renal fluid mechanics. Koch (196h) presented certain physical relationships which would exist in each of the proposed mechanisms by examining solutions obtained from idealized models. His approach, designed to provide a physical basis for separating one hypothesis from another, is certainly correct as far as he went. He did not, however, go beyond this to an illustrative solution of an entire control system. The model proposed in this thesis is based on passive collapse of the interlobular veins. The study was stimulated by earlier work suggesting that the autoregulatory resistance changes take place in these veins, and by the desire to show how arteriolar changes would af- fect venous regulation. Some of the confusion surrounding explana- ll tions for autoregulation can be explained by the non-linear relation- ships which must exist in such a passive control system. The collaps- ing pressure at any point in the veins is the difference between the extraluminal and intraluminal pressure at that point. The intralumi- nal pressure is a function of a) the resistance upstream from that point, b) the arterial pressure, and c) the flow. When the vessel collapses, its resistance increases, thereby changing the flow, and changing the variables on which collapsing pressure depends. It is very difficult to assess the influence of all of the fluid flow para- meters in such a system by intuition alone. It is the purpose of this study to describe mathematically a sys- tem which could operate to control renal blood flow in the venous cir— culation by passive means. A model is described which predicts the effects of the remainder of the renal circulation. Analysis of a hydraulic test model is presented to verify the mathematical model. Measurements of prevenous and postvenous regulation in the dog kidney are presented to verify the site of resistance change. Pressure-flow curves obtained from dog kidneys are compared with curves obtained from the hydraulic test model and predictions from mathematical theory. Results obtained by other investigators are examined in the light of the model. II. Mathematical.Mode1 Concept The purpose of a model is to reduce a complicated system by im- position of assumptions and constraints to a simple system for which descriptions can be feund and from which productive predictions of behavior of the complicated system can be made. If the modeling pro- cedure has been correct, the mathematical description of the model is also the description of those aspects of the system that have been modeled. The model presented here is restricted to operate on a principle compatible with operation in the renal venous circulation. That is, it must employ no mechanism that would depend upon active vasocon- striction. The mechanisnlis based on renal interstitial pressure aris- ing by fluid leakage from the peritubular capillaries and renal tu- bules. An increase in interstitial pressure compresses the thin- walled veins. Although the model is concerned with a principle mechanism of change in resistance responsible for autoregulation lo- cated in the collapsible vessels, the resistances of the other seg— ments of’the renal circulation are considered as parameters affecting the control. A simplifying assumption is made in considering the converging and diverging vessels of various diameters and wall thickness as single tubes. This assumption does not compromise the model, because the goal of the model is to show a principle on which renal autoregu- lation could operate, and not to predict the actual magnitude of the blood pressure at a specific location in a vein. The model is so con- 12 13 structed that it is consistent with renal anatomy in the sense that the same functional relationships between variables, such as pressure and flow, exist in the kidney as they do in the model. It is also assumed that distributed points in the renal circula- tion can.be modeled by discrete points. The error inherent in this assumption is dependent on the distance between the points. (See Appen- dix B for error analysis.) It is further assumed that the flow through- out the system is laminar. This has been shown to be the case by Mc- Donald (1960). Symbols used in the development of the model are shown in Table 1. In addition to definition of physical units, a functional nota- tion (see Fig. l) is shown in Table 2. The functional notation is used to describe locations within the kidney or model, and is used as subscripts to variables. In the description of a segment of circula- tion, the notation describing this segment is taken from the first point of reference to the next point of reference. That is, for example, the average flow between the inlet (O) and the leak point (S) is Q”; the drOp in pressure Pg — PS = APg; the resistance of this section Rg. The control circuit segment is denoted by subscript c. Derivation o_f. manic guations 91 Motion for Model Figure 2 shows a linear tube which is called "linear" because the area does not change as a function of any system variable, but the area can be made to change as a function of other inputs not considered. 1h Table I Symbols Name Units area cmz minor axis of ellipse on major axis of ellipse cm circumference cm base of natural log acceleration due to gravity fluid inertance length of tubing pressure volume flow rate resistance to fluid flow resistance per unit length shear stress viscosity fluid flow velocity variable distance fluid density time frequency 2.718 approximate 980.6 Dyne cm'g/cm H30 cmHgO seca cm-a defined dQ so that AP = L EE-across an element on cm H20 3 —1 cm sec cm H30 cm"3 sec; P = R9 cm H90 cm“L sec; R = r2 -3 dyne cm -2 dyne sec cm cm sec-1 cm gm cm_3 sec radians sec P 1 Point a. 15 Table II Functional Notation and Systems Analogs Kidney ' Model Point of hydraulic input Point O. hydraulic input to to kidney - a point in model the renal artery just outside the kidney Point 1. distributed point of Point S. leak point exchange between cir- culation system and interstitial space. (glomerular capil- laries to tubules to interstital space and peritubular capil- laries to interstitial space). Point 2. distributed point. Point (2). beginning of Where the collapsa- collapsible tube. bility of the veins becomes large enough to allow them to change area with a pressure difference across the wall. Point DV. a point of measurement Point CT. a point within the within the interlobular collapsible tube veins, upstream hydrau- which represents lically from a point of an average tube area. maximum constriction. Point V. a point of measurement in Point 3. a point at outlet of the renal vein just out- system. side the kidney. Point I. interstitial space Point 0. any location within the pressure chamber surround— ing the collapsible tube. 16 poavoo pm opemmonm u «m penance canvas onammohm u om one» oapamamaaoo «o mcflccdwon pm onfimmonm n om pfisohao Hohncoo mo oocmpmemoa hmocaq n om pcfloa xmoa one peace coospmn cowpoom amocfia mo mocmpmemon ceasunchm pflzohflo Homecoo ca seam. pcfloa xmoH pm ohsmmoum a .m co Q coapoom swoops» scam u Q0 em ea Houoe 0p poacfl pm onsmmmhm a tobacco .m pcoeoam .a actom em mm oocmpmflmom pacemum pseqw Nm mm 6959 oanflmcmwaoo \\ pmocflq “a oocmpmflmom pmoceH \\ . \ A( ll All 1 1 HM ) 11H N a u Blind pfisopwu Hoapcoo Aomv oocmpmwmom amocwq Hobo: Homecoo H on:Mfih l7 Figure 2 Linear Resistance Tube Perimeter C According to Newton's second law of motion, force equals mass times acceleration. Applying this principle to the fluid within a tube. mfl = g P1,, A—gPOut A-cTt (11'. Since the term "cw" represents the frictional force, the term may be replaced by an equivalent force (g R(w) Q A) which is the pressure drop (due to friction) times the area. This yields: dv maz= gP,u A-gPout A-gR(uJ) QA. Further, m = 0A2, and v = %. then, pawl. (63—) = g P," A-g P0,, A-g R(w) Q A. dt Canceling the area and expanding the derivative .9. dt A on dQ -— -— = P - o R(w Q. gA d‘L in ut ) Solving for Pout 18 pt dQ P = P - R(D - - ~-—. out in ( )Q Ag dt 2 If %-is defined as L (fluid inertance), and a correction factor 3 (Stedman '56) is used to include the effect of frequency on the effec- tive mass, the equation becomes: P... = P... - Pm) Q-L(w) 3% (1) Applying this equation to each linear element (see Fig. 1) provides the required set of dynamic equations. These are: P. = P, - 12,0») Q, - 1,,(w) 1:? <2) - 9.9.- e-P.-mm)m-medt e) m=P.-mmm.-mm)§& d) dt In addition to the above, an equation of continuity may be written for the junction point (see Fig. 1): %=Q.+m. G) This completes the derivation of the continuity and momentum equations for the linear elements. Although an energy equation could be derived, it is not independent and provides no new information. The methods of Shapiro (1953), for compressible fluid dynamics, are used to derive the collapsible tube equations because a simple rela- tionship exists between compressible flow in incompressible tubes and incompressible flow in compressible tubes. The equations are written for the fluid in a control volume (Fig. 3) defined by the walls of 19 the collapsible tube. It has been assumed that the tube collapses in an elliptical shape of constant circumferencewhich holds for the values of pressure and flow where regulation occurs. The principle of conservation of mass may be applied to the fluid in the control volume to derive the continuity equation. That is, the time rate of change of mass in the control volume is equal to the rate at which it enters one end minus the rate at which it leaves the other (see Fig. 3c). Or, a — Adx = oAv - Av at (p ) ‘x p ‘x+dx Figure 3 Control Volume -'-'--(as in Fig. l) Ellipse of Constant Perimeter c ‘1’; v+iv P+dP P T Q A A+dA x+dx b. X 00 (cross section) (control volume) fifim WV 2O EXpanding oAv in a Taylor series about x, B .. OAV‘X‘l‘ as OAV‘X + 8—}: (OAV)Cl.X, which will yield 3— (pAdX) = :3 (on) dx. 3t Ax Since the fluid is incompressible and x is not a function of time, A ~5— + L“) = o. (6) at ox Newton's second law may be applied to yield the momentum equation. That is, the time rate of change of momentum in the control volume is equal to the net rate of momentum entrance plus the sum of the forces. Or, a _ 2 2 530— (pAvdx) — pAv \x- oAv \x+ + gPA‘x- gPA‘x+ + dx dx dP dA P+— ‘ - g(: 2 . dx CT ' Again, expanding in a Taylor series, a B — (pAvdx) = oAvgl - [ pAv2\ 4' -- (psz) dx J + at x x Bx ‘ gPM -g[PA) +_a_.(PA) dxj+gde+ atpv axov 'g 8 dx Pg-EaAdx de g 28x '° ° If the higher order differentials are neglected, and the derivative of 21 PA expanded, dP d -d -- (oAvdx) =--(oAv2) dx -gA-dx- chx. Ax dx At This reduces to o iléXl.+ o jL-(Avg) + g A i§-+ cT = 0. At Ax Ax Expanding the differentials, a! + A + A(A ) + EX_+ £2-+ OAAt 0v t 0v OAvdx gAAx CT-O and subtracting ov times the continuity equation (6) eliminates the terms indicated. The equation then becomes, Av Av AP pASE+pAv§+gA§£+cT=O. Since the term "cT" represents the forces per unit length due to fric- tion, it may be replaced by an equivalent force of the pressure drop due to friction per unit length times the area. Therefore, since cT = grQA, where r is the resistance of the tube per unit length, Av Av AP A - + A _ + A _ + = O. ‘0 At 0 v Ax g Ax grQA It can be shown (see Appendix A) that for an elliptical tube of con- 13 stant perimeter, r = r' AA—l—, where r' is the resistance per unit A3 length of the tube when it is circular and has the area A’. The equa- tion then becomes: a Av AP , (.1’)‘3 o A o v Ax g Ax gr A2 Q 0 (7) The volume flow rate can be expressed in terms of area and velocity: 22 Q = Av. (8) The three differential equations (2,3,4) for the linear elements are coupled to the two differential equations (6,?) for the fluid in the collapsible tube. One coupling relation is the mechanical proper- ties of the collapsible tube which will be called the "equation of state" and which must be determined experimentally (see pagelJD. The relationship is: A(x) = f (Po - P(x)). For 7a " diameter pen- rose tubing the function is: A(x) = 0.362 e_’°'159(P° ' P(x))a + 0.105 e"°'°°61(Pc - P(x))2 1 0.0398 + 0.03. (see page hé) (9) (Po " P(X) + 1)o .2 A second coupling relation can be deduced by relating the area of the collapsible tube to flow through the control circuit (0). Since the chamber containing the collapsible tube is rigid, the net volume of the chamber is a constant. Therefore, the flow equals the net rate at which the tube is contracting: - A t C = — A c Q At So dx (10) For the real kidney and in a model used to test the effect of the stiffness of the renal capsule, the increase in volume of the system must be measured and equation (10) becomes: _ l QC=JS Adx+gz, At 0 dt where v is the volume of the chamber. The third coupling relation is the overall energy rate equation 23 for the chamber containing the collapsible tube. For the purpose of this derivation, energy dissipated due to friction may be considered lost because it can never be converted back to mechanical energy with- in the system. Within the chamber as a control volume, the time rate of change of energy stored is equal to the net rate at which kinetic energy enters plus the net rate at which work is done on the system minus the rate at which energy is dissipated due to friction. Or, A ‘£ oAve oAgv: oAgvf 53°C +P.E.>dx= + - 2g 2g 23 OAgvg I (A’)3 2 where P.E. is the potential energy per unit length stored in the walls of the collapsible tube. The potential energy stored in the tube is equal to the work done on the tube in compressing it from the initial area A’ to the final area. That is, Pc - P P.E. = 30 (pressure difference) (change in volume) Pc-P =3 a-dA(o)-i 0 or since dA(O.) = BMQ) do, Ax P, "P AA(a) P.E. = on ad dd=h (PC-P). 21: Substituting Q Q vc =-£’ v8 =-."Q3 " Qfl, and A, A, Q v3 = Ag, and setting P3 = 0 reduce Eh. 11 to 3 2 a 1. pAV’ P 0 3 all +hu,-p)]a= 93+ Qa- 28 ZgAca ZgAsa 0 "Oz AI 3 3 Q¢ + Pc Qc + P2 Q3 " 3 r' ( 3) Q3 anv 2a). ° A In summary, the complete set of equations consist of: AQQ; P1=P¢-%Q¢-L¢‘g; AQ. P2=P1 -R,Q,-L,g:c— A P.-P.-P.Q.-L.-9-‘i At Qg = Q3 + Qc for the linear elements, and 35 + A(Av) _ At Ax Aav+ A Av+ AAP+ , (1’)"Q o O -— 0 v— — r = Ax g Ax g 2 Q = Av for the collapsible tube, and :b ll f (PC-P) (2) (3) (h) (5) (6) (7) (8) (9) 25 -3 z Qc = --S Adx At 0 3 3 3 3 33.1, {a OAV 0 QchQ 2s 2g A, A, A, “I. r, (A/)3 + Pc Qc + P9. Qs'g (12) O A3 for the coupling relations. This set of ten equations in ten unknowns could (in principle) be solved to establish the dynamic behavior of the system. Steagy Flow Equations and Boundagy Conditions When the flow is not changing with time, the equations describing the system are reduced considerably. Under these conditions, the dif- ferential equations may be integrated numerically with a Runge-Kutta technique or by direct integration if the relationship between area and pressure in the collapsible tube is simple. The set of equations for steady flow reduces to: Pc = P1 (13) P2 = P1 - R, Q (1h) Q = Av (8) d(Av) _ dx - o (15) 0Av dv AdP I (A')3 -—g— a + a-x— + 1‘ A2 Q = O (16) A = r (Pc - P) (9) 26 This set of equations may be modeled in the following manner. There is a relationship between P9, P0, and Q within the collapsible tube; Q = g(Pe, PC), ‘which is represented below. ch Collapsible tube Pg—F "*Q Q = 8(P23Pc) The linear element also has a transfer function P9 = P1 - R. Q Linear Element Pl-l'r R 4P2 P,=P,-QR 1‘ Q In the closed loop configuration these two elements are connected so that Pc = P8 (see Fig. A). Figure 4. Closed loop Configuration Collapsible Tube linear Resistance --.F6--4 ‘ 27 The method used to solve the system shown in Fig. h is to deter- mine the transfer function Q = g(P2, PC), and then close the loop Pc = P8 to find the relation QC = Q(P1). ‘Fig. h shows that there are actually two control paths. The input pressure is "fed forward" (a) to apply pressure to the collapsible tube and the flow rate is "fed back" (b) to the linear resistance. To solve the set of equa- tions (8,9,13,1h,15,l6), the differential equation for the collapsible tube must be put in a form suitable for the use of Runge-Kutta tech- niques. Expanding 15, dv dA _ + "— = o A dx v dx 0 or, dv _ v dA dx ’ ’ A dx Putting this into (16) yields PvedA+AdP+r’(A’)3Q = _.___._. 0. g (1X (1X A2 But from (8), =9. V A Hence, 0 2 I I 3 -iE—A-+AEE+£—£§_.—)—_Q.=O. (17) gAe dx dx A2 If it is assumed that the area of the collapsible tube is a simple function of pressure (i.e., such as A = Pc - P) then, dA _ dA dP dx dP dx Therefore, 28 - --- — —- + A + o. gA‘a dP dx dx A2 Solving for 533, dx dP -r'(A')3 Q dx A3 0Q2 dA (18) ' ? KP This is in the required form for integration by a Runge-Kutta technique (i.e., 1’1 = v Pc-P=2 P -P=T’ P -P=3” - :1 (cm H 0:) ° ’ c 9 P. P do The relationship between collapsing pressure and area for'a fig" pen- rose tube is shown in Fig. 19. This relationship is difficult to fit accurately with a polynomial. Without success, polynomials to tenth order were tried by a curve-fitting technique on an IBM 1620. It was found, however, that two decreasing exponentials and a power term could be used to fit the curve. The resulting equation of area as a function.of applied.pressure (Pc - P) is: hS Figure 19 Area of Collapsible Tube as a Function of Collapsing Pressure Pressure Pc - P (cm H20) 116 2 2 A = 0.362 e“ °'159(P° ’ P) + 0.105 e“ °'°°62(P° ' P) + 0.398 (Pc ' P + 1)o.2 + 0.03 The non—linear characteristics of this relationship are important to the operation of the system. As noted in the introduction, other investigators have stated that a change in the relationship between interstitial pressure and arterial pressure would have to occur at the pressures where autoregulation begins. This need not be true if the relationship between collapsing pressure and resistance of the collapsible tube is non—linear. Fig. 20 shows the open loop charac- teristics of the system for 3/8 " penrose tube and Fig. 21 shows the characteristics of a closed loop system. 12 11 10 117 Q Figure 20 Open Loop Results (cm:3 sec-1) P d~ Predicted Maximum Stable Flow Rate - \ \ s ‘\ \ f P ‘~~--“-\ I * \ - 6 6 \ \ l \ \ _ \ ‘~_ -"‘-~ i o o o _ 2 a a s m e a I I II II II I I 0 0 0 0 O 0 O a. a. a. n. a. a. a. _ P,2 (cm H30) h8 AOmm gov cm I am mnemmonm eoflaoo< mpasmom mooA Ummoao Hm ohswflh 0a 9.. 8 om 3 on ON .3 o «I q q u q q a u I 4‘ I'll m3 u .m M i o O 01 _ 9.0 u .m . 1 + 6‘ [Gilli o 3.0 u .m o o .. o 0 was u .m gvlv a. 1.7» 10w 1 8.0 n .m L Aaloom me. Le Several observations carleasily be made. The system in closed loop configuration can regulate quite well (Fig. 21). In fact, using the proper materials and values or the series resistance, the system has application as a fluid flow controller. This control is accom- plished passively with a fixed relationship between input pressure and collapsing pressure determined by the input resistance Rfl, the series resistance (R.), between the leak point and the collapsible tube and the flow (Q). The difference in results between open loop and closed loop systems emphasizes the importance of the resistance to fluid flow between the cardiovascular system and.the interstitial space. The dependence of the shape of the closed loop control curve on R. is also important. When the difference in viscosity between blood and water is accounted for, it can be seen that for a certain value of R. regulation will not occur within pressure ranges studied in the cardiovascular system. This is significant when one considers that the series resistance in the kidney contains the efferent ar- terioles which can.be affected by a wide variety of drugs and the au- tonomic nervous system. Further analysis shows that the input resis- tance, Rfl’ can also affect regulation. If R , represented by afferent arterioles and small arteries in the kidney is high, much of the pressure will be lost before reaching the venous circulation. Since the relationship between pressure and resistance in the venous circula- tion is non-linear, different operating characteristics are found at a lower venous pressure. It can be seen from the theoretical development and from direct measurements in a hydraulic test section designed to simulate the renal 50 blood flow control mechanism, which parameters can be expected to affect regulation if, indeed, the postulated model does represent actual renal hemodynamics. Although it is postulated that the autoregulatory resis- tance change occurs in the venous circulation, the form of this regula- tion (shape of the pressure-flow curves) is affected by preglomerular resistance, efferent arteriolar resistance, capillary permeability and venous collapsibility. The oscillatory behavior of the system is shown in Fig. 22. This is the simplest mode of oscillation showing an equilibrium flow of 9.5 cm3 sec-1 with an amplitude of i3.§ cm;3 sec-1 and a frequency of 2.8 Hz. The flow is leading the pressure with a phase shift of 26° . IV. Experiments 23 Dog Kidneys Observations were made on the kidney for three purposes: (1) to find the section of the renal circulation in which blood flow control occurs, (2) to describe the pressure distribution in the renal vein, and (3) to compare regulation in the venous circulation with the mathematical model predictions. Methods Experiments were run on eight kidneys of one year old dogs rang- ing from 25-30 kg. The dogs were anesthetized with sodium pentobarbi- tal at a dose of 30 mg. per kg. of body weight and the kidney exposed through a retroperitoneal flank incision. The renal artery and vein were isolated with a minimum of manipulation. After the preparation was completed, the dog was given h mg. of heparin per kg. of body weight and 5 mg. per hour thereafter. Renal artery pressure was measured from a needle in the renal ar- tery and renal arterial blood flow was measured with an appropriately sized electromagnetic transducer of the same type used in the model measurements (page hl). Venous pressure at the renal vein and within the kidney was measured by retrograde insertion of a drawn polyethy- lene cannula. Polyethylene tubes of various sizes were drawn in air over a small electric heater to tip sizes ranging from 0.1 mm to 0.7 mm and inserted through a needle fixed in a rubber tube inserted into the renal vein, taking care not to wedge the cannula. Although this technique has been in use for some time by other investigators (Hinshaw, 1963, l96h) and S2 53 would seem to be almost routine, some difficulty was encountered in ob- taining a cannula that had a small enough tip to pass into the deep venous circulation and still have enough structural ridigity to pre- vent it from folding over in a small vein. A great many sizes of tube can be drawn to a still greater variety of tip sizes all of which can be passed into the renal venous circulation. The criterion used for satisfactory placement of the cannula was a recording of a high pressure just before wedge. As venous pressure increases in a rather smooth exponential for the first h0-50 mm up the venous circulation, the arbitrary decision of accepting a particular pressure, obtained with a given cannula, is difficult. If the cannula is not placed deeply enough, the resistance changes will be upstream from the measurement point. By using a slightly smaller cannula, the resistance changes will occur downstream of the measurement point. Using a smaller can- nula presents another difficulty. If the tip is too fine, it will in- variably bend over and the folded tube inserted to the wedging point. Also, with too small a cannula, it is almost impossible to detect wedging of the tip. Pressure and flow were measured with the same instruments used to mea- sure the same variables on the mechanical test section (see page hl). Hydrau- Ik:occluders (Jacobson and Swan, 1966) were used on the aorta upstream and downstream with respect to the kidney to vary renal arterial pressure. Certain objections to direct measurement of pressure within small renal veins by means of retrograde insertion of a cannula have been brought forward by Haddy (1965). Haddy has pointed out that it is im— possible to calculate the actual values of'resistance of segments of Sh small veins. This is because of the fact that flow within the vein at the site of pressure measurement is not known. This observation is certainly correct. In order to calculate the resistance of the seg- ment of small vein, one would have to assume that flow is uniform in all the parallel veins and know the total number of veins which shared the flow measured at the renal artery. This assumption is not neces- sary in the case where the only information sought is whether the resis- tance changed upstream or downstream of the catheter tip. The only way that such calculations could be improperly based would be if flow through the particular vein containing the cannula decreased when per- fusion pressure increased. Although this specific reaction is quite unlikely, another artifact can occur if the cannula is blocking flow as also mentioned by Haddy. If there are very few side branches interconnecting the interlobular veins and flow is blocked, it is possible to measure capillary, even, arteriolar pressure with a cannula in a vein. That this is very un- likely in the case of interlobular pressure measurements can be seen by inspecting Figure 31. It can be seen that the slope of the curve relating deep venous pressure to distance within the kidney is constant near the deepest pressure measurement. This indicates that resistance per unit length is constant, which is likely because of the almost con- stant cross section found in that segment of the circulation, or that cannula was interfering with flow in such a way that resistance per unit length appeared exactly constant. Also, if the cannula were wedged into the vein preventing flow, the measured pressure would not increase with distance but would re- 55 main constant. If the cannula were influencing flow, a bend toward the pressure axis would be evident at the deepest measurement. It is also unlikely that all measurements were taken just at the wedge point be- cause the cannula was withdrawn slightly from the point of maximum insertion. Results and Discussion Data from the kidney studies are tabulated in Appendix D and plotted in Figures 2h-31. In each figure, the curve marked "A" is the pressure-flow diagram of the entire renal circulation while "B" and "C" are pressure-flow diagrams of the circulation before and after the interlobular veins respectively. Autoregulation is shown in pressure- flow curves by'a change in slope. If the change is toward the pressure axis, it indicates an increase in resistance to maintain constant flow. Location of autoregulation in the arterial or venous circulation can be done by inspecting curves "B" and "C" to see which has the greater change in slope. Each of the kidneys exhibits autoregulation, and in each kidney the regulation occurs downstream of the deep-venous pressure measuring point. In Figures 23b-29b a slight increase in resistance is evident at the high end of the pressure range and in Figure 30b regulation is present upstream from the venous measurement point. There is no evidence to suggest that this change in resistance, upstream of the measurement point, occurs in the arterioles. In each measurement it was noted that if the venous cannula was not placed far enough into the venous circulation, the resistance change responsible for regulation was upstream from the point of measurement. When a Awm EEV m so u A V anoe om oa om om oH w a u i a i u 3 .a O o . aw 0 Adv amv ADV H mocowx a mobhoo soamuonsmmoam Hmpcmemom mm seamen rm Aaloom mEoV 57 8v Ea oe s 23 Ba - .a OS 3 .a HH honog n mobnso soamnoasmmoam agnosmom A5 am ensues g cm . OOH o: On Amm .55 m am \0'1' ON P . : AH lomm 080v o r 58 Amv Amm 2.5 m HHH access more 8 so awnopzmm ohm H Seesaw om mm 0.»:th 59 an aov a co om 04 on ow 0H 1- E Ba - .a News em: 8 oe 3 cm 3 .a . 3 E as E heaven n noise onMJogdmmoum Hmecoswom cm became 66 En Esv m >Q on a ow " _ Amy >Qm I am. ON—fl E .a A3 > ISSUE n merge Scanoaommgm Hopooemom em oedema 61 8V Ba oo ONH Amv >Qm I .m G: om OOH H> mono...» n 8350 Scanopsmmohm Hmpcoemom mm meow E A8 Awm EEV m om 2 cm HH> hmcvflx : mmbuno zofiuumndmmmpm prcmewmm 8 8% E AHnomm mEoV rm 63 nmm EEV m l N AHIOGm MEOV a o HHH> hwfinflm n maize zofllmhdmmmpm chwemwm om mafia soapmHSQMflo muocw>nfica nflm>_h0Humpxm song moampmao Aeev x 6 LL mflanOfipmHon Hdoasopmcw mumEflxouaad 0 OH om cm 0: _ . r. nfimb cfim> mqfimb mcamb noflnopxm pmnoahmvcH mpmzoud hmHsnoahoan cflm> Hmcmm mnp_cfi coapnndnpmfin whammohm Hm magmam om ..0m 1.0m w on am as >n 65 smaller cannula was substituted, the resistance change was downstream from the measurement point. This suggests that the slight changes in resistance upstream of the interlobular veins seen in the first seven kidneys are still in the venous circulation but above (upstream of) the position of the cannula, and, in kidney VIII, it probably was not possible to work the cannula in far enough. Therefore, all the resis- tance would occur downstream of the measurement point if a cannula could be made to pass far enough into the interlobular veins. Some comparisons can now be made between the regulation predicted by the mathematical model and the regulation measured in the mechanical model and kidney. From Figure 6 it can be seen that the initial slope of the pressure-flow diagram and the maximum flow rates are dependent on the series resistance (RS). RS is identified exactly in the mathe- matical and mechanical models as the resistance between the point where pressure is fed forward to collapse the thin-walled tubing, and the be- ginning of the collapsible tube (See Figs. 1 and 3). The analogous series resistance in the kidney (Fig. l and Table II) must be more vaguely described. The section of the renal circulation representing Rs is the section between the distributed site of tissue fluid formation and the beginning of the collapsible veins. Resistance of the efferent arterioles and part of the peritubular capillaries, and permea- bility of filtration membranes, proximal tubules, and peritubular capillaries can all be parts of the series resistance. Unfortunately it is difficult to assess the contribution of any of these possibilities except to note that the efferent arterioles have a high vasomotor reactivity. 66 Without being able to identify E structurally with precision within the kidney, it can still be identified functionally. In order to compare pressure-flow curves from the model to the curves from the kidneys, a correction must be made to account for the difference be- tween the viscosity of blood and of water. In hydrodynamics, flow characteristics are usually expressed in terms of the Reynolds number which is a function of viscosity (Prandl and Tietjens, 1931;; Schlichting, 1960). If this expression had been used, it would be apparent that the pressure-flow curves predicted by the model for R. > 5.0 (Fig. 6), measured in the hydraulic test section with R. > h.5 (Fig. 21), and measured in the kidney (Figs. 23a-29a) would have the same form. The use of a dimensionless quantity such as the Reynolds number would have made it difficult for physiologists to compare the kidney results with those of other investigators. To make a comparison between model curves and kidney curves, it is necessary, therefore, to multiply the pressure in the model curves by four (Ll blood '- h'u water). The conclusion that can be drawn from the pressure-flow curve comparison is that the series resistance in the kidney is probably higher than the uncollapsed venous resistance. ‘Ihis conclusion must be qualified because the collapsibility of the penrose tubing used is much higher than that of the small renal veins. Collapsibility is a function of both wall structure and radius of the vessel. The diffe- rence in wall elasticity between interlobular veins and penrose tubing is not easily measured, but the veins are certainly smaller than the smlleet penrose tubing available. Because collapsibility varies as the cube of the radius, greater regulation is expected using penrose 67 tubing in the test section. A similar comparison can be made between the computer predictions for pressure distribution along the collapsible tube (Fig. 10) and measurement of venous pressure as a function of distance along a venous vessel within the kidney (Fig. 31). The pressure distribution is also affected by R. and the model also predicts that the series resistance in the kidney is high compared to the uncollapsed venous resistance. Some investigators have made a few intuitive guesses of how the renal blood flow control system would work if it operated on a "tissue pressure" principle. The mere existence of the mathematical and mecha- nical models is enough to show that these attempts at theory were mis- guided. Swann (196h) and.Winton (196h) have stated that a passive mecha- nism could not explain autoregulation because interstitial pressure in- creases proportionately with arterial pressure. They have stated that a disproportionate increase in interstitial pressure relative to arterial pressure would be required to collapse the renal veins significantly in the pressure range in.which autoregulation occurs. Without a model or physical description they did not realize that their "criteria" would only hold if the area of the collapsible veins were a linear function of collapsing pressure. This linearity cannot exist in a geometry of a collapsing tube as shown in Appendix.A. Johnson (1961.) stated that autoregulation should not be abolished by agents which paralyze vascular smooth muscle. He was not aware of (or concerned with) the effects achange in the resistance of muscular arterioles could have on passive autoregulation. Schmid and Spencer (1962) stated, "The linearity of the (pressure-flow) relationship and 68 the sharp inflection of the curve...are indirect evidence against a passive mechanism." Figure 21 shows pressure-flow curves measured in a passive system made up of penrose tpbing. These curves show more linearity and a sharper inflection than measured in kidneys. Other investigators have hypothesized control functions to explain their observations that could be more easily explained by a passive control. waugh (196h) has shown that it is possible to elicit a damped second-order response in flow to a step input of pressure. He attributed this response to a "hunting type" reaction. The same response is seen in passive svstems and depends on control loop time constants. waugh noted that the oscillating response is abolished when renal vascular reactivity is abolished with chloral hydrate. He interpreted this as proving that the oscillations were caused by smooth muscle reactivity. Another explanation could be that by changing vascular diameter with the chloral hydrate, he changed the control loop time constants. Haddy (1965) found that renal arterial pressure increases transiently for 2-h seconds in response to a 0.5—2 sec flow pulse. He interpreted the reaction as vasotonic, but the same reaction could occur easily in a passive system. A flow pulse forced upon a system which is designed to regulate flow can cause in- ternal pressures to increase markedly. If the fluid transudation time constant is high or the system is surrounded by an elastic capsule, the control system can display a long time constant to equilibrium. Regulation of GFR (Harvey, 1961;; Schmid, 1961.) and regulation of lymphatic pressure (Haddy, 1958a) are more difficult to explain with a passive control system. It could be that the tubules and lymphatic 69 vessels also regulate by collapsing just as hypothesized for the small veins. waugh (1964) has based many of his conclusions on his observations of deep venous pressure during autoregulation. His results show the resistance change located upstream from the catheter tip instead of downstream. One explanation is that waugh failed to insert his can- nulae far enough into the venous circulation. This can easily be done if the proper sized tubing is not chosen and proper technique is not used in drawing the tubing to correct taper. V. Summary and Conclusions The results of the computer solution of a mathematical model and the direct measurements on a hydraulic test section to verify the model show the existence of a possible passive mechanism for the renal blood flow control which can operate in the renal venous circulation. The model also provides an insight into the role of arterial and arterio- lar resistance and capillary permeability might play in influencing this resistance. Autoregulation of blood flow in the kidney is postulated to be caused by an increase in resistance of the interlobular and arcuate veins, caused by their collapse with increasing arterial pressure. The cross~sectional shape of these veins is a function of their structure and collapsing pressure. Collapsing pressure, in turn, is the difference between pressure at the site of formation of intersti- tial fluid and pressure within the collapsing vein. At a given blood flow rate, the magnitude of the collapsing pressure depends on the resistance (R5) of the segment between the site of formation of inter- stitial fluid and the collapsing veins. Any factor which can effect Rs can change the regulation characteristics of the collapsible veins. Because the relationship between resistance of the collapsible veins and the absolute magnitude of the collapsing pressure is nonwlinear, and because the drop in pressure across the small arteries and pre- glomerular arterioles determines the pressure at the site of formation of interstitial fluid, the resistance (Ra) upstream of the leak point affects regulation. Other factors predicted to influence autoregulation include ca- 70 71 pillary permeability and collapsibility of the veins. Capillary per- meability influences regulation time constants and series resistance. Collapsibility of the veins is a function of both stiffness and ra- dius. As the vein size increases from the venules to renal vein, the collapsibility as a function of radius increases. But, as they be- come larger, the wall thickness increases and they become stiffer, de- creasing collapsibility. Collapsibility is a function of distance along the venous circulation and has a maximum somewhere between the ends. Measurement of renal arterial pressure, blood flow rate, and deep venous pressure within the dog kidney support earlier findings (Hinshaw, l96h) that the site of autoregulation resistance changes is in the in- terlobular or deep arcuate veins. The findings indicate that the principle mechanism of renal auto- regulation is a passive collapse of interlobular veins. This passive type of control is very sensitive to changes in resistance upstream from the veins, however, and could be augmented or controlled com- pletely by arteriolar changes, particularly efferent arteriolar changes. The model in no way proves how autoregulation works but its existence shows that a pasaive mechanism is possible. With a mathematical de— scription from which to work, it may be possible to design more mean- ingful experiments. 10. 11. 12. Bibliography Bayliss, W. M. "On the Local Reactions of the Arterial wall to Changes of Internal Pressure," J. Physiol. (London) 28:220. 1902. Berne, R. M. "Cardiodynamics and the Coronary Circulation in Hypothermia," Ann. N.Y. Acad. Sci. 803365, 1959. Bickford, R. G. and Winton, F. R. "Influencecd‘Temperature on the Isolated Kidney of the Dog," J. Physiol. (London) Folkow, B. "Intravascular Pressure as a Factor Regulating the Tone of the Small Vessels." Acta Physiol. Scand. 17: 289 i 19490 Forster, R. P. and Maes, J. P. "Effect of Experimental Neurogenic Hypertension on Renal Blood Flow and Glomerular Filtra- tion Rates in Intact Denervated Kidneys of Unanesthetized Rabbits with Adrenal Glands Demedullated," Amer. J. Physiol. 1502534, 1947. Gordon, B. "Vasodilator Products in Blood Resulting from Nucleotide Breakdown in Ischemic Tissue," Intern. Congr. Physiol. Sci. 22nd. Leiden. pg. 196, 1962. Haddy, F. J., Scott, J., Fleishman, M. and Emanuel D. "Effect of Change in Flow Rate Upon Renal Vascular Resistance," Amer. J. Physiol. 105-111, 1958a. Haddy, F. J., Scott, J., Fleishman, M. and Emanuel D. "Effect of Change in Renal Venous Pressure Upon Renal Vascular Resistance, Urine and Lymph Flow Rates," Amer. J. Phy- siol. 195297, 1958b. Haddy, F. J. and Scott, J. B. "Role of Transmural Pressure in Local Regulation of Blood Flow Through the Kidney," Am. J. Physiol. 208:825—831, 1965. Hartmann, H., Orskov, S. L. and Rein, H. "Die Gefassreaktionen der Niene in Verlaufe Allgemeiner Kreislauf Regula- timsvorgange," Pflueger Arch. Ges. Physiol. 238:239, 1936. Harvey, R. B. "Effects of Adenosine Triphosphate on Autoregula- tion of Renal Blood Flow and Glomerular Filtration Rate," Circulation Research, Suppl. I to Vols. 14 and 1521-178~ 1-182, 1964. Hinshaw, L. B.. Day, 8. B. and Carlson, C. H. "Tissue Pressure as a Casual Factor in the Autoregulation of Blood Flow in 72 I in ll 11! III! ,II 'II. III! III. I III all! Illl‘lalll II II aillllllv 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. Hinshaw, Hinshaw, Hinshaw, Hinshaw, Hinshaw, Hinshaw, Hinshaw, 73 the Isolated Perfused Kidney," Amer. J. Physiol. 197: 309! 19590 L. B., Flaig, R. D., Logemann, R. L. and Carlson, C. H. "Mechanisms of Autoregulation in Isolated Perfused Kidney," Proc. Soc. Exp. Biol. Med. 103:373, 1960a. L. B., Flaig, R. D., Logemann, R. L. and Carlson, C. H. "Intrarenal Venous and Tissue Pressure and Autoregulation of Blood Flow in the Perfused Kidney," Amer. J. Physiol. 198:891, 1960b. L. B., Flaig, R. D., Carlson, C. H. and Thuong, N. K. "Pre- and Postglomerular Resistance Changes in the Isolated Perfused Kidney," Amer. J. Physiol. 199:923, 1960c. L. B. and worthen, D. M. "Role of Intrarenal Venous Pres- sure in the Regulation of Renal Vascular Resistance," Circ. Res. 9:1156—1163, 1961. L. B., Brake, C. M., Iampietro, P. F. and Emerson, T. E. "Effect of Increased Venous Pressure on Renal Hemodya- mics," Am. J. Physiol. 204(1)xll9—123, 1963a. L. B., Page, B. B., Brake, C. M. and Emerson, T. B. "Mechna- nisms of Intrarenal Hemodynamic Changes Following Acute Arterial Occlusion," Amer. J. Physiol. 205L 1033. 1963b. L. B. "Mechanism of Renal Autoregulation: Role of Tissue Pressure and Description of a Multifactor Hy- pothesis," Circ. Res. 14, 15: Suppl. I, 120—131, 1964. Jacobson, E. D. and Swan, K. G. "Hydraulic Occulder for Chronic Johnson, Johnson, Koch, A. Electromagnetic Blood Flow Determination," J. Appl. Physiol. 21(4):l400-1402, 1966. P. C. "Autoregulation of Intestinal Blood Flow," Amer. J. Physiol. 199:311, 1960. P. C. "Review of Previous Studies and Current Theories of Autoregulation," Circ. Res. 14, 15: Suppl. I, 2-9, 1964 o R. "Some Mathematical Forms of Autoregulatory Models," Circ. Res. 14, 15: Suppl. I, 269-278, 1964. 4cDona1d, D. A. "Blood Flow in Arteries," Edward Arnold Pub. London, 1960, pg. 282. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 7L. Prandl, L., Tietjens, O. B. "Applied Hydro~ and Aeromechanics," ficGraw-Hill Book Co., Inc. 1934. Rapela, C. E. and Green, H. D. "Autoregulation of Canine Cere- bral Blood Flow," Circulation Research 15 (Suppl. I): 1—205, 1964. Rein, H. "Vasomotorische Regulation," Ergebn. Physiol. 32:28, 1931. Replogle, C. R., wells, C. H. and Collings, W. D. "Pressure-Flow- Distension Relationships in the Dog Kidney, Fed. Proc. 192360, 1960a. Replogle, C. R. "Pressure-Flow-Distension Relationships in the Dog Kidney, Master's Thesis, M.S.U., 1960b. Schlichting, H. "Boundary Layer Theory," McGraw Hill Book Co., Inc. 1960. Schmid, H. E. and Spencer, M. P. "Characteristics of Pressure-Flow Regulation by the Kidney," J. Appl. Physiology 172201- 2014 , 1962. Schmid, H. B., Garrett, R. C. and Spencer, M. P. "Intrinsic Hemo- dynamic Adjustments to Reduced Renal Pressure Gradients," Circ. Res. 14, 152 Suppl. I, 170-177, 1964. Scott, J. B., Daugherty, R. M., Dabney, J. M. and Haddy, F. J. "Role of Chemical Factors in Regulation of Flow Through Kidney, Hindlimb, and Heart," Am. J. Physiol. 208:813-824, 1965. Selkurt, E. E. "Relationship of Renal Blood Flow to Effective Arterial Pressure in the Intact Kidney of the Dog," Amer. J. Physiol. 1472537, 1946. Semple, S. J. G. and De wardener, H. B. "Effect of Increased Renal Venous Pressure on Circulatory "Autoregulation" of Iso- lated Dog Kidneys," Circ. Res. 7(4)2643~648. 1959. Shapiro, A. H. "Compressible Fluid Flow," Vol. I. New York: The Ronald Press Co., 1953. Stedman, C. K. "Alternating Flow of Fluids in Tubes," Statham Inst. Notes No. 30, 1956. Swann, H. G. "Discussion in Hinshaw, L. B.: Mechanics of Renal 39. 40. 41. 42. 43. 2m. 45. 47. 48. 49. 75 Autoregulation: Role of Tissue Pressure and Description of Multifactor Hypothesis. Circ. Res. 14, 15: Suppl. I, 120-131, 1964. Torrance, H. B. "Control of the Hepatic Arterial Circulation." J. Roy. Coll. Surg. Edinb. 42147, 1958. Unna, K. "Arterieller Druck und Nierendurchblutung," Pflueger Arch. Ges. Physiol. 2352515, 1935. Venard, John K. "Elementary Fluid Mechanics," (Third Ed.). New York: John Wiley & Sons, 1957. waugh, W. H. "Myogenic Nature of Autoregulation of Blood Flow in the Absence of Blood Corpusles," Circ. Res. 62363—372, 1958. Whugh,'W. H. and Shanks, R. G. "Cause of Genuine Autoregulation of the Renal Circulation," Ciro. Res. 8:871—888, 1960. 'Waugh, W. H. "Circulatory Autoregulation in the Fully Isolated Kidney and in the Humorally Supported, Isolated Kidney," Circ. Res. 14, 15: Suppl. I, 156-169, 1964. wells, C. H. "Estimation of Venous Resistance and their Signi— ficance to Autoregulation in Dog Kidneys," Master’s Thesis, M.S.U., 1960. Winton, F. R. "Intrarenal Pressure and its Variations Due to Cool- ing and "Reducing" the Isolated Kidney of the Dog," J. Physiol. (London) 82:27P, 1934. Winton, F. R. "Intrarenal Pressure and Blood Flow," In Trans- actions Third Conference, Rena1.Function New York, Josiah Macy, Jr. Foundation, p. 51, 1951. Winton, F. R. "Pressures and Flows in the Kidney," In Modern Views on the Secretion of Urine, Cushny Memorial Lectures, London, J.and H. Churchill, Ltd., p. 61, 1956. Winton, F. R. "Arterial, Venous, Intrarenal, and Extrarenal Pres- sure Effects on Renal Blood Flow," Circ. Res. 14, 15: Suppl. 1, 103-109, 1964. Appendix A In this section the pressure-flow relationship for fluid flowing in elliptical tubes is derived. It is shown that for an elliptical r! (AI)3 A3 tube r as "A" is the area. where "r" is the resistance per unit length and A thin shell of fluid in a circular tube is shown below. The velocity (v) is parallel to the axis (x) of the tube and is a function of the distance (h) from the axis. The shearing stress (T) on the inner surface is (assuming laminar flow), T1 = m viscosity. The stress on the outer surface is To Expanding ’1' O in a Taylor series about h, Av A 3v T =u-— +—— u— dh. 0 oh h Ah 5h) The net axial shearing force is dF = 2nh£('r, -To). 76 av . Sh‘z’ where L1 is the _ dv‘ Lldx h+dh° 77 Substituting for T and expanding the derivative, 2 one dF=-2nh£n &L This force must be balanced by the net axial pressure force. 2 (P1 —P2)2nhdh =dF=2 2nh£u 2 dh. Ah Or, PI'P2_ Rev A Lion? Pl'Pe d In the limit as 1 approaches zero, —-Z———— becomes E;-and dP 82v _=-H~ dx Aha More generally, <11) 2 a‘x'=-Wr v That is, 0A) where‘%? is the radial Laplacian operator in the coordinate system being used. An elliptical tube is shown below. ./ OB) This equation has the solution 2 2 v = c(:1 - 3.”.X__ 22 b2 ’ providing 1 a3 b2 dP C = O 2U. 32 + b2 dX Therefore, 1 2 2 P 3 3 V = a b d (1 _ 3:.— _y__) The volume flow rate (Q) is Q_&b§bdd_dp TTaabs '- "ov y 27d)! 4L1 a2+b2 Since the resistance per unit length is defined to be the pressure drop divided by the volume flow rate, _ dP _lul a2+b2 1'- dx /Q - Tr as b3 . For a tube of constant perimeter "a2 + b3" is nearly a constant, hence, combining the constant terms r = K’/aa b3 For an ellipse, a3tP =.A§/n‘3, and therefore, r = K/A3 where K = hu (.12 + b2)n2 This leads directly to the required result, 3 I I a r’(A')3 . . . r A = r (A ) or, r = ---—-—'whlch is used in the theo- A3 retical development. I'lllr‘ I ‘11:? Appendix‘B The computer program logic and programs for solving the open loop, closed loop, and pressure distributions are presented in this section. Each program is followed by a table of the results which are used in the body of the thesis. The logic proceeds as follows: 1. 10. 11. Set Pc and Rs. Guess a value of Q. Check to be sure Q is not too large. Use a third-order Runge-Kutta technique to step along the tube increments of Ax from x = 0 to x = 2. Check to see if the value of Q meets the end boundary condition. If it does, print out results, index the applied pressure and use the current value of Q for the next operation at the new value of PC. If the end boundary condition is not satisfied, check to see if cycle has been performed more than once. If this is the first guess at Q, check the sign of the error at the end boundary condition and guess at a new value of Q accord- ingly. See if the error has changed in sign if the cycle has been per- formed more than once. If the error has changed in sign, Q is somewhere between the 01d value and the present value so start searching between these two points. If the error has not changed in sign, guess at a new value of Q ac- 79 1 12 20 24 600 80 cordingly. The final search routine consists of a simple interval slicing technique. The parametric fluid flow control curves predicted by the computer program are shown in Fig. 6. The results plotted are for a x = 0.5 cm for a 30 cm tube length. ‘When x was reduced to 0.25 cm, the values of flow changed a maximum of 0.05% showing that the procedure of lumping the tube into 0.5 cm lengths is sufficient and this value was used thereafter. The same area subroutine listed below was used for each program. SUBROUTINE AREA (C’AgcA, IF‘C‘15.110v1C9125 0 CA=C*C*.159 CB=C*C*o0061 A=.362/EXP(CA) +.105/EXP(CB) +.0398/(C+1o)**o2 +o03 DA=.115*C/EXP(CA) +.00128*C/EXP(CB) +.00796/(C+1o)'*lo2 GO TO 600 5 IF‘C“50.)2019201'24 1 CB=C*C*.0061 A=.105/EXP(CB) +.0398/(C+1.)¢*.2 +.03 DA=o00128*C/EXP(CB) +.00796/(C*1o)**1o2 GO TO 600 A=.0398/(C+1.)"02 +.03 DA=000796/‘c+10)‘*102 RETURN END 81 Listed below is the computer program used to solve the closed loop curves. 25 READ(5.ICO) RoDX9P.N0 100 FORMAI(3E16.8.IIO) WRITE(6,103) R.Dx,N0 103 FCRMATClHla7X916HSULUTION FUR R HoFS-anHp OX 3 9F4.ZQ7H9 NO x, 115) 032.66 PC=1. DC 1 K=1920 IO=O 805 [=0 NCO [P(IO-SO)3¢3:11 3 PA=PC-02R E¢Q*DX!.000155 FBQ'Q'.OOIOZ [FCPA14.4,5 1' 030-01 60 ID 3 5 DC 6 L=11N0 C=PC-PA CALL AREA (CothAl 0ENO=Ai03—F*DA IE!DENO)804¢804,801 801 RKAa-E/DENO P‘63PA+05.RKA CsPC-RAG CALL AREA (CQAQD‘) DENOsbifia-FGDA IF!DENO)804.804¢802 804 IO=IO§1 030-.051 65 TO 805 802 RK83-E/DENU PAGSPA+ZoGRKB-RKA ChPC-PAG CALL AREA (C.A,DAI DENOtAOG3-FODA IF(DEN0)8049804:803 803 RKCz-E/DENO 6 RA=PA4lRKA+4.£RK8+RKC)I6. N=N+l IF‘N-7515051505011 505 CrABStPC-PA) 82 CALL AREA (C.A.DA) 502 PA'PA'4o*F*Zo*F/A 1F‘ABS‘PA)’H)11¢11:12 12 IF(N-1117917114 17 IF(PA)15115116 15 0330 PAB=PA 030‘005 GO TO 3 16 0830 PAB‘P‘ 030*005 GO TO 3 14 [P(PACPA8118213013 13 IF(I)17217220 18 I=I+1 QA=0 PAASPA GO TO 19 20 0880 PAB‘PA 19 Q808+.3*(QA-QB) IF(ABS(QA-QB)-.00111121123 11 WRITE(621021 PCsNto 102 FORMAT‘1HOa4XQ4HPC 39F501011X93HN 3.13113X93HQ 39F905) IF(PC-9o)70017002701 70° PC3PC*10 GO TO 1 701 PC=PC+6o 1 CONTINUE 1F1R‘100125025026 26 STOP END Solution for Closed Loop Operation 83 Tabulated Results of Computer Series Resistance 5.0 2.0 1.0 0.7 0.6 0.5 0.1 0.3 0.2 P1 = P .(cm-H, O) 2 0.36 0.96 1.82 2.52 2.91 3.11 1.11 5.26 7.21 1 0.71 1.58 2.78 3.66 1.10 1.69 5.18 6.63 8.16 6 0.91 1.91 3.16 1.01 1.18 5.05 5.83 6.95 8.72 8 1.16 2.22 3.16 1.32 1.75 5.32 6.07 7.17 8.90 10 1.35 2.18 3.71 1.51 1.96 5.50 6.21 7.30 8.99 16 1.65 2.87 1.07 1.81 5.23 5.73 6.13 7.11 9.01 22 1.76 3.00 1.19 1.91 5.31 5.81 6.18 7.17 Unstable 31 1.88 3.15 1.31 5.07 5.13 5.91 6.57 7.55 16 2.00 3.29 1.18 5.20 5.55 6.02 6.67 7.63 58 2.11 3.12 1.62 5.32 5.66 6.12 6.76 7.71 70 2.23 3.51 1.75 5.11 5.78 6 22 6.85 7.78 Resulting Flow Rate (cma/sec) 81 Listed below is the computer program used to solve for the open loop curves. The logic of the program is basically the same as that used for the closed 100p curves except that the boundary condition at x = 0 is independently controlled. 25 READ‘501001 RcDXQHoNU 100 FORMA1(3El6o8g1101 PC=10o DC 1 K3197 02.001 PAK=lo HRITE‘bolOB) PC 103 FORMAT(1H1:7X026HOPEN LOOP SOLUTION FOR PC'oF5o21 IC=O 805 130 N80 IF‘IO“501313011 3 PA=PAK 530*DXO.000155 F=Q.Q.000102 5 DO 6 L=1oNU CSABSCPC-PA) CALL AREA (C,A,DA1 DEN085593-Fsoa [P(DEN0180498049801 801 RKA=~EIDENO PAG=PA+o59RKA C?ABS(PC'PAG1 CALL AREA (CoAvDA) DENO=A§*3'F*DA 1F‘DEN0180498041802 804 ICSIO+1 QQQ’OOSI GO TO 805 802 RKBz‘E/DENU PAG=PA+2.*RKB-RKA C=ABS(PC‘PAG1 CALL AREA (C'A,DA1 DEN03A993-F*DA 1F‘DENO18049804g803 803 RKC3‘E/DENO 6 PA=PA+(RKA+4.‘RKB*RKC)’6. N=N+l IF!N-75)5050505911 505 C$ABSCPC-PA) CALL AREA (CvoDAI 502 PA=PA-4.-F+2.¢F/A‘ IFCABSIPA1-H111011012 12 1F1N-1117017014 17 1F(PA)15915916 15 0830 85 PAB=PA 030-005 60 IO 3 16 08=Q PABtPA 010+.05 GO TO 3 16 [FIPACPA8118s13o13 13 1F11111’17'20 18 13191 QA3Q PAABPA 0C 10 19 20 0030 PAB'PA 19 03080.3CIQA-081 IFIABSIQA-QB1-o00111191103 11 “81151691021 PAKoNOQ 102 FCRHAI(1H0,4X,4H P 39F5o1911Xo3HN 3013013X03HQ 8gF9o51 PAK=PAK+1. IF‘PAK‘PC180590059701 701 chpC*IOo 1 CONTINUE IFIR'100125025926 26 SIOP ,END 86 Tabulated Results of Computer Solution for Open Loop Operation Pc(cm-H20) P2(cm—HQO) Q(cm3/sec) Pc(cm-H20) P9(cm-H20) Q(cm3/sec) 10 1 0.33 10 36 1.71 10 2 0.73 10 37 5.91 10 3 1.21 10 38 8.51 10 1 1.77 10 5 2.11 50 20 0.51 10 6 3.31 50 30 0.85 10 7 1,91 50 35 1.20 10 8 8.08 50 10 2.09 10 9 11.83 50 12 2.75 50 11 3.66 20 5 0.36 50 15 1.22 20 8 0.80 50 16 1.93 20 10 1.28 50 17 6.09 20 12 1.98 50 18 8.59 20 11 2.93 20 15 3.53 60 25 0.61 20 16 1.30 60 35 0.89 20 17 5.59 60 10 1.08 20 18 8.30 60 15 1.13 60 50 2.31 60 52 2.97 30 10 0.31 60 52 2.97 30 15 0.70 60 51 3.86 30 20 1.60 60 55 1.11 30 22 2.29 60 56 5.11 30 21 3.22 60 57 6.21 30 25 3.80 60 58 8.72 30 26 1.51 30 27 5.77 70 35 0.81 30 28 8.38 70 15 1.12 70 50 1.30 10 15 0.11 70 55 1.65 10 20 0.60 70 60 2.52 10 25 0.96 70 62 3.17 10 30 1.85 70 61 1.05 10 32 2.53 70 65 1.60 10 31 3.15 70 66 5.28 10 35 1.02 70 67 6.39 70 68 8.80 tions is listed below. closed loop program are used in this program. I 0 U1 ‘3 *- m U ’1 (1 6 125 8C1 802 803 26 87 The computer program used to solve for the pressure distribu- REA01501001ROPC90 F0RNAT13E16081 HRITEI601031RaPCvQ FORMAT!1H197X929HPRESSURE DISTRIBUTION FOR R 8gF5o2g8Ho 1F602,7H9 Q 329F905, HRITE1691041 FORHATI1H 913X01HX015X01HP1 x300 £202.000077S F80909o00102 PAaPc-Q'R 00 6 L’1a63 WRITEI601051 XcPA FORMATIlH oTX9F50201OXOF12051 CSABSIpc-PA1 CALL AREA (C9A90A1 DENO=A'*3'F*0A RKA8-E/DEN0 PAG‘PA*05'RKA C=ABSIPC'PAG’ CALL AREA IC.A.0A) DEN0=AI03-F¢DA RKB3*E/DEN0 PAG’PA+2.*RKB-RKA C3ABSIPC-PAG1 CALL AREA (C9A90A1 0ENO=A!#3-FIDA RKct-E/DENO x3x*05 PASPA+IRKA+40*RKB‘RKC1/6o IFIR‘130125025026 STOP END The flow rate solutions obtained from the PC 88 Computer Results for Pressure Distribution for R8 = 5.0 and values of Pc = P1 (cm-HQCD Distance x em = 10 . = 16 . = 22 . = 31 Pc = 52 0.0 .27 7.77 13.22 21.59 11.72 1.0 .93 7.22 12.55 23.76 10.60 8.0 .56 6.60 11.78 22.77 39.15 12.0 .18 5.89 10.87 21.52 37.01 16.0 .76 5.07 9.71 19.81 33.11 20.0 .31 1.07 8.26 17.18 26.17 22.0 .07 3.18 7.31 15.20 21.68 21.0 .82 2.81 6.13 12.17 16.50 25.0 .69 2.13 5.12 10.77 13.83 26.0 .56 2.02 1.61 8.86 11.11 27.0 .12 1.58 3.68 6.77 8.31 28.0 .28 1.09 2.60 1.55 5.51 29.0 .13 0.55 1.31 2.21 2.70 29.5 .06 0.25 0.61 1.06 1.27 89 Computer Results for Pressure Distribution for PG = 16 (om—H.cn and Values of Rs Distance R, = 5.0 R, = 2.0 R, = 1.0 R, = 0.5 R, - 0.2 X cm 0.0 7.77 10.26 11.93 13.13 11.19 1.0 7.22 9.62 11.36 12.83 11.05 8.0 6.60 8.90 10.62 12.37 13.89 12.0 5.89 8.08 9.73 11.63 13.66 16.0 5.07 7.10 8.67 10.53 13.33 20.0 1.07 5.90 7.35 9.12 12.60 22.0 3.18 5.16 6.51 8.25 11.61 21.0 2.81 1.29 5.55 7.19 10.11 25.0 2.13 3.78 1.97 6.56 9.76 26.0 2.02 3.21 1.30 5.81 9.03 27.0 1.58 2.56 3.50 1.91 8.17 28.0 1.09 1.80 2.53 3.71 7.07 29.0 0.55 0.90 1.29 2.07 5.12 29.5 0.25 0.38 0.51 0.87 3.08 Appendix C Ehperimental Data from Hydraulic Test Section ——~‘ Applied Pressure Displaced Resulting Change (cmeHQCD Vol (cma) in.Area (cma) Area (cma) 0.0 0.0 0.0 0.537 1.0 0.8 0.027 0.510 1.5 1.6 0.053 0.181 2.0 1.2 0.110 0.397 2 2 6.0 0.200 0.337 2.5 8.0 0.267 0.270 2.8 9.0 0.300 0.237 3.2 9.9 0.330 0.207 1.0 10.8 0.360 0.177 1.9 11.2 0.371 0.163 10.0 12.8 0.127 0.110 18.0 11.0 0.167 0.070 27.0 11.5 0.181 0.053 10.0 11.8 0.191 0.013 Above 100 15.2 0.507 *0.030 *Assumed for the case when the tube is completely collapsed. 90 91 Ihta for Open Logp Curves P0 = 70 cm-HQO P2 cmeH20 P3 cm-HQO AP cmeHeO Q ems/sec 26.2 1.0 25.2 0.11 31.0 1.0 30.0 0.57 11.3 1.2 10.1 0.93 51.6 1.3 50.3 1.51 62.0 1.3 60.7 2.56 65.0 1.1 63.6 3.28 67.0 1.1 65.6 3.80 68.0 1.1 66.6 1.71 68.6 1.5 67.1 6.12 69.0 1.5 67.5 7.58 69.2 1.5 Unstable = 60 cm-H20 J cméHQO AP cm-HQO Q cmR/sec 55.8 1.1 51.1 3.61 56.6 1.1 55.2 3.92 57.7 1.1 56.3 1.12 58.5 1.1 57.1 5.52 58.8 1.1 57.1 7.25 12.6 1.2 11.1 1.67 18.5 1.2 17.3 2.22 31.0 1.1 32.9 1.11 21.5 1.1 23.1 0.67 15.0 1.1 . 13.9 0.33 59.0 1.5 Unstable Pc = 50 cm-HQO P2 cm-H20 P3 cmrHQO AP cmeHQO Q cmR/sec 15.0 1.1 13.9 0.11 22.0 1.1 20.9 0.76 30.3 1.2 29.1 1.33 36.5 1.2 35.3 1-75 13.0 1.3 11.7 2.77 17.0 1.1 15.6 1.23 18.7 1.1 17.3 7.13 19. 1.5 Unstable 92 Pa = ’40 cm-HQO AP cm-HQO Q ems/sec P3 cm-HQO P;2 cm-HQO Runvdxd/vtL47:o/RVQJ9_ A0717:C/Q/9.719.7111d2 QJQJQJQJQI312ndqlnl b 5555hh332 2 1 m 111111111111 350018020531 8897539hQ/269 3333332211 Pc = 30 cm-HQO Q ems/sec AP cm-HQO P 2 cm-H 2.0 P 3 cm-H 9’0 RufitvuAu11nud/ .wloQOAw77. 112222 620578902 1.9.9.9.9.9_9_ Pc = 20 cm—H20 AP cm-HQO Q ems/sec P"3 cm-HQO P 2 cm-H 20 5023716 51w1.2(.788 111111 93 Pc = 10 cm—HQO AP cm-H20 Q cmé/sec P9 cm-HPO Data for Closed Loqg Curves Q ems/sec 131.3.a AP== Rs = 0.73 AP = P1 - P3 cm-HQO Q cmé/sec PQ cm-HQO Pa cm-HQO 91 R5 = 0.10 AP = P1 - Pa cm-H,O Q ems/sec P2 cm-H20 Pa cm-HQO P1 = Pc cm-HQO 0568 28 3217 1.1...8886151 876h211 8888888888 &111111111 8285980000 1410147339152 887/Olu311 83h/OOQ/COQ/5 hon/66051363 88765321 R3 = 0.28 AP = P1 - P3 cm—HQO Q ems/sec P2 cm-HQO Pq cm-HQO 00802983887078 65952250.]“09601 21 122358 88888888888888 11111111111111 011252059800181» 6595236 9.1409601 21 122358 88680781685851“ 7616.1»13/1621723 211 1123358 95 R3 = 0.21 Q ems/sec AP = P1 - P3 cm-HDO cm-H 20 P3 P , cm-H 20 P1 = Pc cm-HQO 7830017 . . . . . . .ifl2233632 1358817 ........ 1 27688123 11233568 888888888888888 111111111111111 0162230 75556076 21.1.4588 1827688223 1 11233568 5618009520011th 3570/0/29hQ/8003h5 1 112luluS/08 Kidney I Kidney II Kidney III Kidney IV Appendix D Experimental Data from Dog Kidneys 30 58 72 93 120 132 31 62 83 108 126 139 1116 25 3h 52 61 78 90 103 111; 123 131 139 29 52 62 7'? 86 99 107 1211 138 \ON OxUIJI'w 16 2h P - P 7 2h 50 62 81 101 107 96 2h h? 63 79 83 92 91 DV 0 .17 .75 .oo .16 .70 .83 .95 MMNNHQO .25 1.55 2.59 11.05 5.00 5.27 5.h2 5.58 r‘r‘F’F’H’r’h’hJ \1 N .NNHrHrp (I) bu Kidney V Kidney VI Kidney VII Kidney VIII 15 36 6h 97 115 12 3 137 lhh 35 68 8L1 97 112 12 5 13h L12 51 68 81 93 100 111 12h 132 1&0 1118 22 32 52 60 71 85 9b 102 116 123 135 lhh 97 13 27 LL? 68 77 77 85 88 31 58 71 76 Bu 92 39 61 70 79 80 85 9h 100 101; 107 O MNNNNHOO wwwWWMI-‘H f’h‘k‘f‘f‘h’h’h‘kJ§DSD m)n3n1n3n1n>n3thJh4CDCDC>