vl 0'12. MA {17 rtw 34H _ ..l N... .r... , 1:1, .‘.‘..40 F .u 53;” a (\t .. . . v.3 :5: 2a. .. xv quads: :52.‘ 1.. . .2 O. . .Lmqufivmfiiai , as ‘ .i. .%§van. R a l k- km» s . .1... . 2.x... , .96... L: .73,“ g EA. .. ix , n .. L: 31.3.94; a: 2.. avianmumwfih r‘kksfiwgfl hunk» ‘0 ,. 3.3.... x0.‘.... 5‘...- .z.. vi. 5&6th ') h 15‘1” 5... . i ) a . ‘ ssvnv “5.3;: “.032... 63...}: .. 3b»??? .151. .1 . 1.4.1533. W’s ' LIBRARY "» :33 Michigan State University This is to certify that the dissertation entitled MATHEMATICAL ANALYSIS OF MULTICAPILLARY SUPPLY REGION presented by Liang Sun has been accepted towards fulfillment of the requirements for the Doctoral degree in Applied Mathematics 070—, Major Proféssor’s Signature I 1/4/03 Date MS U is an Affirmative Action/Equal Opportunity Employer PLACE IN RETURN BOX to remove this TO AVOID FINES return on 0 MAY BE RECALLED with earlier checkout from your record. r before date due. due date if requested. DATE DUE DATE DUE DATE DUE Due. indd 5108 K:IProilec&PrelelRC/Date j— MATHEMATICAL ANALYSIS OF MULTICAPILLARY SUPPLY REGION By Liang Sun A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Applied Mathematics ‘2008 ABSTRACT MATHEMATICAL ANALYSIS OF MULTICAPILLARY SUPPLY REGION By Liang Sun This dissertation presents a mathematical analysis of substrate diffusion and con- sumption in a capillary-tissue bed. The model contains multiple non-homogeneous capillaries convecting different amount of solute surrounded by a tissue region. In the first section of this dissertation, solutions are obtained for a rectangular supply region where there are multiple capillaries. Both of its steady and transient states are corn- pletely analyzed. The time-wise evolution shows that hypoxia occurs during an early state and is restored as time increases. Furthermore, effect of axial diffusion in a 3-D model is analyzed in an extension where multicapillaries are embedded in a cylindri- cal tissue region. Perturbation method is employed to solve the associated governing equations. Finally, the dissertation introduces a numerical compartmental model for analyzing multicapillary free boundary supply regions. Hexagonal compartments are used to solve the diffusion-consumption governing equation. Capillary-tissue domains are reported of three types of boundary: rectangular, cir— cular and with free boundary. A matching technique is used to analytically solve the problem with regular domain boundary (i.e. rectangular or circular), in which the so- lution is expanded into an infinite eigenfunction form with unknown coefficients. The resulting equations inverted using Fourier series methods together with the Neumann boundary conditions are solved for the remaining unknown coefficients. Moreover, the time-dependent diffusion is investigated numerically and the solution to its equation is approximated by the Ficks law of diffusion. Hexagonal compartments are utilized for simple implementation. ACKNOWLEDGMENT I would like to express my gratitude to my thesis advisor, Professor C.Y. Wang, for bringing me into the world of Mathematical Biology, and for his guidance along this long journey. I would like to thank my parents, my brother and my sister in law, for their support and love. Without them I could not have reached this point. and gone further. I am also thankful to the kind people and all my friends in East Lansing who have helped me and provided me the experiences that will benefit my life. I enjoyed every moment I spent there in the past few years. iii TABLE OF CONTENTS Chapter 1 Introduction 1.1 Krogh Model ............................... 1.2 Capillary Supply Region and Capillary Domain ............ 1.3 Literature Review ............................. Chapter 2 Multi-Capillary Supply within Rectangular Domains 2.] Steady State of Multi-Capillary Supply in a Rectangular Region . 2. 1.1 Formulation ............................ 2.1.2 Matching Technique and Scheme ................ 2.1.3 Solution .............................. 2.1.4 Examples and Discussion ..................... 2.2 Transient State of Multi—Capillary Supply in a Rectangular Region . . 2.2.1 Formulation and Solutions .................... 2.2.2 Analysis and Example ...................... Chapter 3 Substrate Ti‘ansport in Multi-capillary Beds with Axial Diffusion 3.1 Problem Review and Analysis ...................... 3.1.1 Formulation ............................ 3.1.2 Perturbation of Substrate Concentration ............ 3.1.3 Small 2 Boundary Layer ..................... 3.2 Further Discussion on Matching Solutions ............... Chapter 4 A Compartmental Model of Multi-Capillary Supply Re- gion 4.1 The Basic Problem ............................ 4.2 Unsteady Diffimien-Consumption .................... 4.3 Essence of the Compartmental Model .................. 4.4 2D Compartmental Model ........................ 4.5 Examples and Discussion ......................... Chapter 5 Discussion Appendix A Analytical Solution for Oxygen Distribution in Circular Domains Appendix B Substrate Distribution within Capillaries Bibliography iv 38 39 39 42 46 51 63 63 67 68 73 77 88 91 94 97 2.1 2.2 2.3 2.4 2.5 2.6 2.7 4.1 4.2 4.3 LIST OF TABLES Coefficients of Example 1 for a rectangular domain up to n = 30 with azlandfizl .............................. 21 Locations and flux strengths of three capillaries in Example 2 . . . . 22 Coefficients of Example 2 for a rectangular domain up to n = 30 with a=1andfi=1 .............................. 22 Locations and flux strengths of nine capillaries in Example 3 ..... 23 Coefficients of Example 3 for a rectangular domain up to n = 30 with a=1andfl=1. ............................. 23 Coefiicients of Example 3 for a rectangular domain up from n = 40 to n. = 60 with 0- =1 and [3 =1. ...................... 23 Locations and flux strengths of three capillaries ............ 34 Radius of the circular supply region ................... 77 Parameters for compartmental model with different choices of (1:1? . . . 82 Locations and flux strengths of three capillaries reprised ....... 87 1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 LIST OF FIGURES IV’lulticapillaries in a tissue region where Voronoi boundaries are drawn 2 Geometry of the Krogh Cylinder: a single capillary surrounded by a coaxial cylinder of tissue ......................... 3 Some special cases of capillary supply boundaries which are also Voronoi boundaries ................................. 5 (a)V0ronoi boundary, (b)capillary supply boundary. (Size of dot rep- resents source strength) .......................... 5 (a) A general distribution of capillaries in a rectangular region, (b) the coordinate system. O represents its origin of the Cartesian coordinates. 11 Oxygen concentration in its steady state with one single capillary in- side a sufficiently oxygenated rectangular domain, a = 1 and t3 = 1 Example 1. ................................ 24 Oxygen concentration in its steady state with three capillaries inside a sufficiently oxygenated rectangular domain, a = 1 and B = 1 Example 2. 25 Oxygen concentration level curve in a sufficiently oxygenated rectan— gular domain with nine capillaries, steady state, a = 1 and [3 = 1 Example 3. ................................ 26 Oxygen concentration level curve at t = 0, transient state, three cap- illaries with random characteristics ................... 35 Oxygen concentration level curve at t = 1, transient state, three cap— illaries with random characteristics ................... 36 vi 2.7 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 Oxygen concentration level curve at t = 10, transient state, three cap- illaries with random characteristics ................... Uneven locations and diffusion strength N capilary bed surrounded by a cylinder of tissue ............................ (a) General distribution of capillaries in the circular perpendicular cut, (b)The coordinate system. O reprensents the origin of the polar coor- dinates ................................... Arbitrary multi-capillaries in a cross—sectional tissue region ...... Oxygen supplied and ischemic regions as result from Eq. (4.4) Blockage of one single capillary ..................... Result compared with solutions from Eq. (4.17) at. t=0.5, t=1 and t=2 Oxygen diffusion into a consuming region at t=0.1, t=0.2, t=0.5, t=1 Compartments of a) square b) hexagonal c) triangular ......... Oxygen concentration profile with one single centered capillary at t = 0.01 .................................... Oxygen concentration profile with one single centered capillary at t = 0.1 Oxygen concentration profile with one single centered capillary at t = 1 Radial oxygen concentration distribution error as results from example 1 compared to Eq. (4.45) ........................ Schematic regular supply region with I = 4 .............. Oxygen concentration profile at t = 0.1, numerical results shown in an insulated capillary-tissue region ..................... Oxygen concentration profile at. t = 1, numerical results shown in an insulated capillary-tissue region ..................... oxygen concentration ratio comparison with Example II in Chapter two a) Normalized concentration comparison at source one where y = 0.821 b) Normalized concentration comparison at source two where .r = 0.243 Errors are relative low at each source point .............. vii 37 39 41 64 78 79 81 83 84 86 Chapter 1 Introduction Capillaries are tiny vessels connecting arterioles with venules and forming networks through the body. Oxygen diffusion from the microcirculation through capillaries and its consumption by tissue cells is basic in all higher organisms. A basic mathematical model to estimate the oxygen concentration profiles for highly regular capillary bed of skeletal muscle was originally introduced by Krogh [1]. Oxygen orientation is modeled by a central capillary surrounded by circular tissue cylinder. Extensions of the Krogh cylinder model have been made, including polygonal regions of supply, capillary effects, axial dependence, oxygen pressure etc. Krogh cylinder among with many other models approximated oxygen distribution generated from homogeneous capillaries within regular capillary boundaries. Imper- meable barriers and zero interactions among capillaries were also assumed. But in real cases, the capillaries may not be parallel, may be arranged more or less randomly in normal tissues. The oxygen concentration, along with its time-wise distribution, may be uneven. Therefore heterogeneity is considered much more common. An important case of heterogeneity is abnormal perfusion. This may be caused by local ischemia due to blockage such as in heart attack, ischemic stroke, local collapse due to pressure and trauma. Where does such hypoxia occur? Would neighboring capillaries provide enough oxygen to the anoxic regions? How soon and how much? These are the questions to be answered. Note again we are concerned about the scale of distance between capillaries and not global scale problems such as increase in demand due to exercise or reactive hyperemia after a tourquinet is released. 0 '— o . ’ b ' to. ‘ O ' O ' hi I» 3 a :4 C 29 ‘1’ O o 1 ' o o o L Figure 1.1: h’Iulticapillaries in a tissue region where Voronoi boundaries are drawn The last two cases mentioned above are primarily transient problems rather than heterogeneity problem. In transient problem time dependence is an essential factor and the effect is in global scale. For example, the observed periodicity in capillary flow rate is a transient problem if all capillaries are in phase and it is an heterogeneity problem if the capillaries are not in phase. As we shall see later the solution to a. steady state heterogeneity problem may be sometimes obtained more easily through a transient process. Consider a Long and parallel capillaries such as skeletal muscle. The time depen- dent development of anoxia can be simulated following occlusion in a multi—capillary domain with same (or different) diffusion solute. Some closer area to the capillary is more severely affected by such effect than the boundary area of the domain over a time interval. Remarkable advantage of application on diffusion equation is seen in such unsteady state, possessing its mathematical complexity and biological importance. 2 1 . 1 Krogh Model Krogh[1] originally introduced the model for the study of oxygen distribution in the regular capillary beds of skeletal muscle. He used a perpendicular bisector to describe the oxygen transportation and concentration in tissues around blood capillaries. A hexagonal region was first considered. A circular region was addressed then for the boundary region assumedly only small error would occur if the hexagon is replaced by a disk. The steady state equation for the oxygen concentration in the tissue, C(T‘, z, t) is then derived as following: ‘ -” ' Figure 1.2: Geometry of the Krogh Cylinder: a single capillary surrounded by a coaxial cylinder of tissue ac 1 0 ac 0% Where Dr and D; are the. diffusion coefficients along the direction of r and 2. AI is the constant consumption rate per volume of tissue. The longitudinal diffusion of oxygen can be neglected because the length of the cylinder is comparably 100 times longer than the radius of the cylinder. By consid- ering only the steady state diffusion and neglecting the axial transportation among capillaries and in the tissue, the governing equation (1.1) then becomes a one dimen- 3 sional diffusion-consumption equation: 1 d dc _._._. ‘_ = ‘A/ . Drrdr (7 (17‘) I (12) with boundary conditions C(T‘C) = CO (1.3) dc _ .2 z 1.4 (17‘ '7 7'1: 0 f ) where TC is the capillary radius and rt gives the radius of the tissue cylinder. Krogh gave a analytical solution under the simplified conditions: 1. Substrate concentrations were assumed to be in steady state 2. Axial movement was neglected: 3. The concentration is CO in the capillary. 4. No flux comes in or out of tissue cylinder The solution to Eqs. (1.2) to (1.4) is the Krogh-erlang equation [1]: Ill/I 2 "r r2 — r3 r, As we see in the next few chapters, i) unsteady state is far more desirable es— pecially in cases of hypoxia and anoxia. ii) a single capillary supply region could not be approximated in the multi-capillary domain, and multi-capillaries with arbi— trary characteristics yield more complex results and the 1-D equation becomes a. 2-D equation. 1.2 Capillary Supply Region and Capillary Domain For multiple capillaries, the geometric area or volume allotted to each capillary is usually not the same as the capillary supply region. The former is called the capillary domain which is a descriptive measure of capillary spacing ([49], [60]). The latter is a. 4 functional measure of the capillary diffusion region and would change with changes in perfusion, such as partial occlusion of some capillaries. Voronoi polygons are usually used to best describe capillary domains, which coincide with capillary supply regions if the capillaries are of the same strength and even distribution. Some special uneven periodic distributions are also possible(Fig 1.3). ’As I o I ': . I . I .: ' .‘r. . I*\ /J*‘ l——--.'.---1'--—-4 A : ’k‘ I’ Y I o ' o I o I 1’ ‘s ,’ x i o I O I I : J J' 5‘ . 1” ‘ . ’1 -------- an--- | \ ~ ’ (I ‘Y’k‘Y’ E“ : I . | .Y. : .Yo ' I ° I 0 I s 0 I 0 : O: K\ : ’A‘ i ’) K‘ ’J\‘ I) h""' """""" \+’ ‘\.’I v v Figure 1.3: Some special cases of capillary supply boundaries which are also Voronoi boundaries However, if the capillaries are of different characteristics, i.e. different location and strength, then Voronoi boundary could not be the capillary supply boundary. Consider two adjacent nodes of different source strength. The voronoi boundary is the perpendicular bisector(Fig 1.4(a)). But the concentration of the solute is not the same at the midpoint. Consequently the capillary supply boundary must shift toward the weaker node. It may also be curved as in Fig. 1.4(b). Figure 1.4: (a)Voronoi boundary, (b)capillary supply boundary. (Size of dot. repre- sents source strength) Due to arbitrary characteristics of multicapillaries inside a micro piece of tissue, where the shape and the boundary of capillary supply region are highly related to the location and strength of its capillary, it is important to study such a feature which best describes the substrate diffusion. 1 .3 Literature Review Krogh first proposed the conceptual model to describe the oxygen distribution and exchange by substrate molecules from capillaries to tissue. This model was later known as Krogh cylinder which is presented in section 1.2. In Krogh’s work, diffusion equation was used and simplified to obtain a analytical solution of concentration for oxygen inside a cylindrical skeleton tissue region. Since then considerable effort has been made to develop more a elaborate analysis of the model. Significant simplifica- tions has been made to keep the mathematics tractable, resulted from the complex nature of the physiological phenomena. Based on the Krogh cylinder, Jacob Blum [40], in 1960, presented a paper on the mathematical analysis of a model for substrate concentration in tissue in the American Journal of Physiology. His model analyzes the oxygen perfusion from a single capillary into its surrounding cylinder of tissue. Other works, for example, [20], [22], [38], [39] and [58] involved the effects of myoglobin, the effects of axial diffusion within tissue,on the single capillary model. Apelblat, Katzir-Katchalsky and Silberberg [2] gave a mathematical analysis of capillary-tissue fluid exchange in 1974. Most of the previous works have simplified the diffusion equation by neglecting the axial diffusion, which has relatively small effect compared to the radial diffusion. Mathematical analysis of oxygen transport to tissue which included axial diffusion within the tissue was studied by Salathe, Wang and Cross [24]. Their model used perturbation techniques to determine the axial diffusion effect for a single capillary in a Krogh cylinder. Fletcher [33], [34], [35] and Lih [52], reviewed several accounts 6 of mathematical studies of substrate and oxygen transport inside the Krogh cylinder. Significant simplifications has been made due to the complex nature of the governing equations. Numerical analysis in solving the full equations can also be found in works of Fletcher et. al. [33], [34], [35], Hyman [64] and Reneau et al [14], [15], [16]. In 1972, Gonzalez—Fernandez and Atta [42] numerically computed the non-circular equilateral triangular, square and hexagonal cylinders. It was found that the corners of the polygons are most susceptible to hypoxia (lethal corner). The stacking of reg- ular polygonal cylinders still require both (1) all capillaries are same (flow, substrate diameter) and (2) capillaries are evenly distributed in a regular array. Schmidt-Nielsen and Pennycuik [46] discovered a triangular regional feature for white skeletal muscles with capillary-to—fiber ratio approximately equal to one. Opitz and Schneider [18] used a square region of supply for brain capillaries. These authors used polygonal region for capillary supply rather than a disk and estimated minimum tissue pressure within the supply region. But their target tissue contains only one single capillary and they assumed a uniform distribution of capillary locations across the whole tis- sue region. Such a case might exist in the real situation due to arbitrary capillary characteristics. A single Krogh cylinder would represent all by symmetry. While single capillary models give useful information about oxygen exchange between tissue and capillary, they overlook the intensive large scale interactions among the many capillaries com- prising a vascular bed. Capillaries, the microscopic size blood vessels, link the muscle with the cardiovascular system by a common arteriole and drained by a common venule. Each muscle cell may have from three to eight capillaries directly in contact with it. Blood flow along with substrate and oxygen distribution from capillaries to tissue through such a network of multi-vessels is critical. Therefore, the delivery of oxygen from the richly perfused to a poorly perfused region may definitely involve in- teractions among the neighboring capillaries. Thus studies the capillaries of different 7 strength and unevenly distributed throughout the tissue region are urgently needed to reflect the actual situation. Numerical methods were used extensively to solve for multi-capillary models. A.S. Popel [8] studied the effect of multi-capillary network anastomoses an tortuosity on oxygen transport using computational methods to treat a reasonable sized array of capillaries, as well as some later work [6] he has done on heterogeneous oxygen delivery on oxygen distribution in skeletal muscle. In 1978, AS. Popel published a paper in analysis of capillary tissue diffusion in multi-capillary systems. MSharan et a1. [53] provided a finite-element analysis of oxygen transport in the systemic capillaries in 1991. All previous work faced the problem of requiring large amount of computation time and imposing restrictions on the number of capillaries that can be included. Salathe [18] used a homogenization approach to analyze a large number of capillaries in skeletal muscles, where capillaries are lined relatively long and parallel to each other. However, homogeneities are usually not the real case and heterogeneity is much more common. Previous work which studied uneven capillaries include Clark et al. [55] who used a method of imaging. However, in his study capillary supply regions were not con- sidered. Hoofd et a1. [50] and Hoofd [48] required only the net flux on the boundary be zero. Diffusion within the planes perpendicular to the capillaries was determined using iterative method in Hoofd’s study. However, without any control over the re- sulting boundary conditions the solution may be non—unique, therefore difficult to implement. R. Hsu and T.W. Secomb [57] used a green’s function method for analy- sis of oxygen delivery to tissue by microvascular networks. It allows multi-capillaries with arbitrary characteristics within the tissue region, but becomes intractable when the integral equation is approximated by a system of algebraic equations. In 1999, Wang [12] studied capillary supply region for multiple capillaries (evenly or unevenly distributed) in a disk region and gave fully analytical solutions to the oxygen concen- 8 tration and discovered that the capillary supply region shapes substantially different from Krogh cylinder. More recently, Salathe [19] presented a multi-capillary approach which includes exchange of substrates in a two dimensional array of capillaries. In his work, substrate concentration is determined in an rectangular or square tissue area consisting of a. single capillary, where flux satisfies the outer boundary of the region as a result of interaction among all adjacent capillaries. By matching the substrate concentration along the boundaries, it provides a system of ordinary differential equations and can be solved analytically. Homogeneity is again assumed here. The method described in chapter two in this thesis exploits a matching tech- nique which solves the substrate diffusion within a rectangular tissue domain where multi-capillaries are unevenly or randomly distributed. Both steady state and un- steady state of the capillary supply are presented. In chapter three, axial diffusion is considered, treated as some perturbation to substrate concentration along the axial direction of multi capillaries. And in chapter four, a compartmental numerical model is introduced where hexagonal compartments are employed as functional units. Sub- strate diffuses along the borders of each compartment from the sources of substrate governed by F ick’s diffusion law. Substrate concentration can then be approximated for unevenly distributed multi capillaries with free boundary domains. Chapter 2 Multi—Capillary Supply Within Rectangular Domains Single capillary supply region has been modeled historically as in circular, hexagonal or polygonal regions. A uniform distribution of capillary locations across the whole tissue region was assumed and minimum tissue pressure within the supply region was estimated. However such case might not be suitable and is far from being complete when interactions among capillaries exist and multiple capillaries are considered. In this chapter, a mathematical analysis of the capillary-tissue exchange of substrate in microcirculation in rectangular regions when multiple capillaries are embedded with arbitrary characteristics will be presented. A matching technique is used to help solve the associated governing equations. Furthermore, the time-dependent unsteady state in a rectangular region will be modeled and examples will be illustrated. As we will see, sudden abolishment of steady state creates hypoxia area which is recovered and restored slowly during capillary-tissue substrate diffusion. Such a process is reversible if sufficient amount of substrate flux is convected through the capillaries. 10 2.1 Steady State of Multi-Capillary Supply in a Rectangular Region This sections focus on oxygen concentration for multiple capillaries in a rectangular tissue region. We start with discussion of steady state. The ideal case is to have diffusion equation V20 = 1 with no flux in or out of the boundary. The pinpoint. results benefit from 1) Filing of circular regions leave voids in the tissue area and thus is not suitable to be extended to a greater tissue area. 2) in certain human functional capillaries like brain tissues, a region of square or rectangular is more suitable the case (Opitz and Schneider [18]). 3) it is easier to take account of unevenness, heterogeneity and flow variations of multi-capillaries when we study this supply region. 2.1 . 1 Formulation (a) (b) Figure 2.1: (a) A general distribution of capillaries in a rectangular region, (b) the coordinate system. 0 represents its origin of the Cartesian coordinates. Consider a large rectangular region Q of length a and width ,3, containing N capillaries of uneven locations and flux strength. We assume a uniform consumption rate [‘6 per volume across (2. Let qj be the rate of oxygen pcrfusing into tissue area from the jth capillary source, with radius pg centered at (I7 = :1: y = yj. Considering 7x. 11 the steady state of the oxygen distribution, we have the mass flux balance which give s the condition on q]- N qu=fl°0w73 (2.1) 1:1 The governing equation in two dimensional is given by diffusion equation: DrV - VC(.1:,y) = 15:, C(.'r.,y) E Q (2.2) Where Dr is the diffusion coefficient of oxygen. On the boundary of the region there is no flux 80 _ = 2" 017 [:r—>O,oz O ( 3) BC — = 2.4 8y [3140,73 ( ) Let (pj, cpj) be local cylindrical coordinates centered at each capillary. Due to con- sistent flux into tissue area from capillary wall, the value of r; is zero. The equation for each capillary source is a? 1 a Dp (—§+——-C—) =0 6p]. Pj apj The boundary condition for each capillary is 27r 8C Qj=—Dp',0r/0 8—,0; (1993', (2-6) Pj—Wr where 01) is the diffusion coefficient in capillary and pr is the radius of capillary. Normalize the flux by (1,3 and the difiusion by DI), the equations for diffusion in capillaries are 12 , (92c 1 8c _0 0,03. 103-37)] 271' 5 ac pr —-—— —— , 52— 1 2 l q] 7r/0 OpJ-IE y] R << ( 7) N qu=1 i=1 L Normalize the concentration in (2.2) by rt/Dr, the equation for oxygen diffusion in region Q is AC(.r, y) = , C(.-r,y) E 9 (2.8) with boundary condition : 5 277 80 q, =__/ _| 7r 0 apj 5 a _ 80 a .— :0 3'1? 1?—>0,(1 ()y y——>0,L’3 2.1.2 Matching Technique and Scheme Diffusion equation in a rectangular region Q with Dirichlet or Neumann boundary condition has been studied and can be solved by using free Green functions and method of image. Notice boundary condition (2.9) is case specific due to a constant flux out of muti-capillaries at different locations, i.e singular points as e ——> 0 within the region Q, it’s no longer a simple Neumann problem. Here we use a different strategy for solving Eqs (2.8) to assume general solution in the form N C(JIT, y) = (1‘2 + y2)/4 — Z qj/2 -lnpj + T(.r,y) (2.10) i=1 The first term is a particular solution due to uniform consumption. The second term is the combination of fluxes from Eqs (2.7). The last term T is the homoge- 13 neous solution which is constructed in order to satisfy the boundary condition on the rectangular region 0. To satisfy the boundary conditions with no substrate flowing across,we use the process of matching of boundary conditions and apply the series method to construct a suitable function T (r y). Quote from Philip M Morse, Herman Feshbach (1953) ”The process of fitting of boundary conditions is somewhat analogous to the process of solving an ordinary dif- ferential equation. In neither case is the procedure straightforward; We must assume some general forms, which seems likely to satisfy our requirements, and then adjust things to satisfy them in detail (if, indeed, this turns out to be possible!) Many of our guesses for solutions of difierential equations are very general in form such as power series or integral representations, but we find the specific solution by inserting the assumed form into the diflerential equation and then trying to make it fit.” The boundary conditions are the particular Neumann conditions on a closed rect- angular boundary. We separate our solution to satisfy the required boundary condi- tions on four walls along :17 = 0, (c = a, y = 0 and y = ,5. Consider, to begin with, that a series solution has flux zero along a: = a, y = 0 and y = B and has an arbitrary sequence of values 103(5):) (inhomogeneous conditions) along £1: = 0. After separation of variables from homogeneous equation associated with Eq. (2.8), the solution of the .1: equation which has derivative zero at :1: = o: is coshwflhr — a), where n is integer and tag is the frequency representing n7r / .13, and the solution of the y equation which has derivative zero at y = 0 and y = [3 is cos way. Consequently the most general solution of V2,], = 0, in two dimensions, which satisfies the homogeneous Neumann conditions that it! has derivative zero along 1* = a, 14 y = 0 and y = ,8 can be represented by the series 00 ii) = Z An coshwfi(:r — a) coswfiy (2.11) n=0 The crucial step is to justify the functions in its above series form can satisfy all pos- sible Neumann conditions along a: = 0 and thus it can represent all possible solutions satisfying the zero-flux conditions along the three sides. Because we know Neumann conditions along a closed boundary specify a unique solution up to an additional con- stant. Then if we find another form of conditions, we can be sure that this new form corresonds to the series and, vice versa, that the series can represent the new form. (Theorem 2.1) A least squares representation of a piecewise continuous function F (z) in Q can be achieved if the sequence of eigenfunctions is a complete set. i.e. mutually orthogonal. And therefore the series form can represent general solution to the differential equation. We) — F12dw = 0 The eigenfunctions cos(n7r y/fl) as a factor in the partial solution (2.11) giving its dependence along the boundary surface depends on a separation constant as well as on position y along the boundary a? = 0. This factor also satisfy the boundary conditions at the ends y = 0 and y = [3. Here the constant are specially determined to fit the conditions at the chosen boundary sides. The other factor cosh [Bf—(r —— (1’)] is then adjusted to fit the conditions at the other end of the enclosed region Q (.r = 0 in our case), and the complete solution is then the sum of these products for all permissible values of the separation constant. We have so far only considered the portion :1: : 0 of the boundary to have boundary values if) to have boundary flux different from zero. To fit the conditions where w is different from zero along other parts of the rectangular boundary we can use obvious 15 modifications of the functions used in series (2.11). For instance, for fitting conditions along y = 0 we use the series 00 2 Dn coswaz cosh wa(y — {3) (2.12) n=0 where tag 2 nit/a. For :1; = a and y = ,3, adding the individual series to obtain the final solution. Therefore corresponding unit function can be derived here to fit the integrand for any boundary values at any point along the rectangular boundary. To solve the Eq.(2.8) and find the unit function in the series, we start with the choice of T for general solution to Eq. (2.8) T = tb‘L +1/2R +wT +'¢”)B (2.13) From previous discussion, the four terms can be seen as representation of general solutions against four boundary walls(left, right, top and bottom). Each term involves an unknown coefficient which is to be determined by combination of nonhomogeneous part of equation. And conveniently the four solution terms has property QZ _ a"(’L (LT _ Lilli (214) 8:1: :r—>0 — 0:1? :r—>0’ 8:2: ;r—>0z _ (9x r—mz ' ‘ 80 at: 6_T = IT Q = _.7*3 (2.15) 8y y—afi 0y y—ni By 31—0 0y y—*0 2.1 .3 Solution The general form of T is oo 00 T = Z An coshwfim: — Oz) coswfiy + Z Bn cosh wflr cos wfiy n=0 oo n.=0 00 + 2 Ca COS wail? C0511 way + 2 Dn cos war cosh wa(y —— 3) n=0 n=0 16 (2.16) The relation between pj, .Tj, yj is (Fig (2.1)(b)) p3 = (as—mp2 + (y—y,->2 (2.17) In order to determine the unknown coefficients An, Bn, Cn and Dn , we look at the boundary conditions given by (2.3) and (2.4). The oxygen concentration along each capillary walls depend on characteristics of capillary itself, therefore the radial outward normal at boundary Q of (2.10) gives N 0C ;. fl" 21' _2‘T' 8T 5,. :£_Z.J_ 2 J 2+_=0 (2.18) and 0C . N (1' y-Ir 0T —"[ .=g— J— '] +,——=0 (2.19) 01/ y—fld 2 i=1 2 (.r—JJ)2+(y-y])2 01/ (2.16) gives or 00 00 E = Z Anujfl sinhwfi(.r — a) coswfly + Z Bum/3 sinhwfir coswfly .-_-O - e=0 J I if 00 oo _ Z ana sin was: cosh way — Z ana sin was: coshwa(y —— )3) e=0 - e=0 , Iii 17/ (2.20) As a: —+ 0 the last three terms tend to approach zero, I I ——> 0, I I I ——> 0,1V —> 0 . therefore 17 8T — 209: A to" sinh’ ( 01) coswvv' — 8W“ (2 21) 0:1“. ar—>0 _ 0 n it} “Off L fay _ 0.17 :1:—>0 I n: AS$—>a,I—>0,III—+0,IV-—>0 8T 00 . 074" 0—1 ar—m = 2: anfl smh wfla coswfly = BIB .r—m' (2.22) n=0 (2.21) and (2.22) gives the expression for zero flux flowing towards boundary :1? ——> 0, a respectively, similarly for the vertical boundary lines y —> 0, (3 8T oo- oo 2.)— = Z —Anwfi(.r — 01) coshwfihr - a) sinwfiy — 2 anfi coshwfia‘ sinwfly f f1 00 00 + Z ana cos was: sinh way + Z ana coswax sinhwa(y —— ,3) e=0 - e=0 m 171 If, (2.23) As y —> 0 I —> 0,11 —> 0,111 —> 0. therefore 00 , ' T , a :1 2— : Z ana sinh wav(—3) coswaa: = —LT— ,. (2.24) Asy—*;3,I—>0,II—->0,IV—>0 8T 00 6 1?— = Z ana sinhwafl coswar 2 (SB . (2.25) y 3)....)3 ":0 y y-+0 Eqs. (2.21), (2.22),(2.‘24),(2.25) show that the form of solution (2.16) satisfies (2.14) and (2.15). It’s then used to obtain the values for unknown coefficients An, Bn, Cu. and Dn . (Proposition 2.1) Under {2.3) and (2.4), the general solutions to Eqs (2.8) with 18 zero flux on boundary of the rectangular region bounded by 0 S :1: S a and 0 S y g 3 r _ 7271' nm An COSh —(T — (I) COS ___'l #3 3 RF. _ nay C( )_ .732 +112 _ iv: 31 l , + i ‘f‘ Bn COSh 73—1 (OS-73— 1‘ y — 4 2 ftp] mm: mm 3:1 n=0 + CncosT cosh _a_ nun n7 + Dn, cos — cosh [—I(y — 3)] a a (2.26) Use the boundary conditions (2.18) and (2.21), as :r —> 0 N 00 0C qj 21]” n7r , [7m ] nay — . = 0+ -— , + An.,—- Slnh ,—,(—o') cos—,—‘— 2 0 ()1 '1—>0 ng 4 13 + (y _ yj)2 712:0 3 3 3 (2.27) :> 171' ”r N (I T Z Am—j smh [7(—o)] cos—:B—y Z —J 2 J 2 (2 28) 910/) . , i3 ., 111mm sinh [—(—a)] '71“ = /3 g1(,i/)-cos:,[—31ydy (2.29) ’ (I' —I' . , , , . VVhere 91(y) : Z3~=1 727— p2+(q3yJ-)2‘ Let Y = (y __ (”)2‘ hq..(224) ylelds J ' 19 mrr mrr Am? smh [73—(—a)] 7r N . :3 . _ q] t ‘17] mrr may . 771.71 - _ 32-1—2— _3W(COS—l—3_YCOS 7y] — sinm— 3 —7rYsin—— 13 yj) dY N . ,3 . q ,1 —:r. 2 E E]— / 7% COS %Y C08 2% .dY N 3 (11' m7r 1 1 mu 2 — —-:r.- 2cos— —y- / —— cos—YdY j=12( J) t3 J 0 mJZHY)? 1’3 (2.30) Thus N 1 17m 71 1 Am = , "m E q r-)cos—- —-y / ~cosmydy . . , _ . ]( —‘T'J J 7r2 mrr smh 73—( a) .___1 ,3 0 $715] 52+(y)2 (2.31) Here y— - $1”, and again (:1: :1:j, yj) gives the location of the jth source Similarly, we achieve the values for the unknown coefficients Bm, Cm and Dm N 1 mn 7r cos my Bm = , 71171 E q (a — :r ) cos —. y,/ dy . . , . J J ,1 J 2 ' mrr smh 73—(0) j=1 l3 O 1; (a _ 3.1.)2 + ((1)2 1 (2.32) 1 N mir 7f cos ma: (7,, = . _ E 3_ ' i ' l, 2.33 m mrr sinh ’an) ._1(1j( 'Jj) (os— 01 .rj/O (.l.)2+7r 7T2 312”“ ( l J— " 5231,) N D 1 (_ ) m7r [7f cos 711.1: d m = 7777]- _ q ' y COS —'I ' I ma sinh —. —-13 , J J (1 J - 2 1 (2.34) 20 2.1.4 Examples and Discussion Oxygen delivery to tissue has been studied in various ways. Many researches were limited to a single capillary. Popel [7] used evenly distributed domains for multi- capillaries with heterogeneous flow. However, with different oxygen demands multi- capillary supply region can not be approximated by single capillary capillary supply regions. Circular region has been studied by Wang [12] and analytical solutions was given. However, tiled circular regions will leave voids in tissue area and therefore can not represent the entire tissue region. However, rectangular regions can be tiled without voids into sub rectangular tissue regions. Thus we have used a matching technique to achieve complete solutions for oxygen concentration through a rectan- gular region. Three examples will be illustrated in the following I. A single capillary in a well oxygenated diffusion-consumption rectangular region with (1 =1 and [3 =1. Consider a single capillary in the center of the region with flux strength q = 1. The results for coefficients are displayed in Table (2.1) to show the convergence of the series to the entire rectangular region. We also show the plot of oxygen concentration C(r, y) in Fig. (2.2). We have truncated to only the first 40 terms in each set of coefficients. It reaches our accuracy demand. 11:1 n=2 11:10 11:20 n=30 An —0.1019 —0.1006 1.066x10—5 —1.796><10-6 —3.001x10-g Bn —0.1024 —0.1023 1.080x10‘5 —-1.659x10_6 —-3.025><10"8 on —0.1024 —0.1023 1.080><10_5 —1.659><10_6 —3.025><10_8 Dn —0.1019 —0.1006 1.066x10—5 —1.796><10_6 —3.001><10—9 Table 2.1: Coefficients of Example 1 for a rectangular domain up to n = 30 with 01=1and13=1 II. Three random single capillaries in a well supplied diffilsion-cons11mption rect- 21 angular region with 01 = 1 and 1’3 2: 1. Consider three capillary of uneven strength in a rectangular region. Locations and flux strengths of the capillaries shown in table (2.2) are randomly generated with the sum of flux equal to 1. The results for coefficients are shown in Table (2.3) to show the convergence of the solution to the entire rectangular region. We also show the plot of oxygen concentration C(:r, y) in Fig. (2.3). We have truncated to only the first 40 terms in each set of coefficients since it reaches our accuracy goal. k 9311 111k (11.- 1 0.721 0.637 0.176 2 0.812 0.243 0.412 3 0.175 0.320 0.412 Table 2.2: Locations and flux strengths of three capillaries in Example 2 71:1 n=2 71:10 71:20 71:30 An 0.0035 ”-0.0003 —1.160 x 10—6 ——-1.310 x 10"8 1.227 x 10‘H) Bn 0.0037 —0.0006 —5.216 x 10"7 -—1.804 x 10-8 -3.078 x 10—10 On —0.6731 —0.0012 8.478 x 10—5 —5.249 x 10—8 4.606 x 10—9 Dn —0.0011 0.0003 1.153 x 10—7 —7.940 x 10“9 1.049 x 10—10 Table 2.3: Coefficients of Example 2 for a rectangular domain 11p to n. = 30 with a = 1 and 13 = 1 III. Nine random single capillaries in a well sufficed diffusion-consumption rect- angular region with a = 1 and fl = 1 with plot of contour. Consider nine capillary of uneven strength in a rectangular region. Locations and flux strengths of the capillaries are randomly generated shown in table (2.4) with the sum of flux equal to 1. The results for coefficients are given in Table (2.5) and (2.6) to show the convergence of the solution to the entire rectangular region. We also show the plot of oxygen concentration level curve in Fig. (2.4). We have truncated to only the first 40 terms in each set of coefficients. 22 yk. (11.. CO ooqozcnuxwwp—aw 0.501 0.721 0.242 0.091 0.633 0.867 0.425 0.664 0.723 0.498 0.235 0.754 0.046 0.923 0.354 0.301 0.316 0.833 0.044 0.089 0.133 0.133 0.089 0.023 0.133 0.178 0.178 Table 2.4: Locations and flux strengths of nine capillaries in Example 3 71:1 71:2 71:10 71:20 71:30 An -—0.2884 0.0065 1.534 x 10“4 2.615 x 10“6 8.110 x 10-8 Bn —0.4366 0.0060 —5.671 x 10"4 —6.174 x 10"6 1.099 x 10"7 0,; —0.2591 —0.0047 1.790 x 10—4 —2.513 x 10—6 —6.557 x 10'8 Dn 0.2254 0.0049 1.906 x 10-4 2.441 x 10—6 3.121 x 10—8 Table 2.5: Coefficients of Example 3 for a rectangular domain up to n = 30 with 01 = 1 and 13: 1. 71:40 71:50 71:60 An 1.690 X 10——12 3871 X 10—14 8.775 X 10—16 371 8.553 x 10'14 3.693 x 10-15 1.263 x 10"16 C71 2.184 x 10"12 —8.853 x 10“16 —1.608 x 10-15 071 —8.142 x 10—13 —8.648 x 10—14 4239 x 10_15 Table 2.6: Coefficients of Example 3 for a rectangular domain up from n. = 40 to 71:60 with a=1an(1,1’3=1. 23 4 4 4 4 4 4 4 4 4 4 4 4 4 \ \ \ x \ \ \ \ 4 11111 4 11111 4 IIIII 4 11111 x \ x x 4 4 4 4 4 4 4 4 4 4 4 4 4 4 x \ \ \ \ 4 IIIII 4 11111 4 11111 41111141111151.114 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 \ x \ 4 11111 4 11111 41 1111411111W1111$11114 \ \ x 4 \ \ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 \ \ \ \ x x \ 4 x \ x \ \Illllx IIIII x lllll \ lllll 4 IIIII \l IIIIII 4 4 4 4 4 4 4 4 4 4 4 4 4 \ \ \ 4 \ x 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 ..... 2 ..... 31-1131111211112111121 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . 2 2 2 2.1,llllll1 2 2 2 2 2 2 2 2 2 2 2 2 21 11211 12 11111 2 11111 2 11111 21 11111 2 11 2 1 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2||I|l2 lllll 2 ||||| 2 lllll 2lllll2l lllll 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 11111 2 11111 2 11111 2 11111 2 11111 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 lllll 2 lllll 2 lllll 2 lllll 2 lllllllllll 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4. 4. 4. 4. 4. 4. 5. 3 5. 2 5. 5. 3 2 1 0 de 1118! its steady state with one single capillary a sufficiently oxygenated rectangular domain, 01 = 1 and ,6 = 1 Example 1. ion 1n Oxygen concentrat Figure 2.2 24 X 1es inside a 0 th three capillar 1 and fl = 1 Example 2. ts steady state wi ion in i Figure 2.3: Oxygen concentrat sufficiently oxygenated rectangular domain, a 25 \\ 0.3 0.8 2 0.4 — 0.7 5 0.9 0.8 0.7 0.6 0.5 0.4 0.2 0.1 Figure 2.4: Oxygen concentration level curve in a sufficiently oxygenated rectangular domain with nine capillaries, steady state, 0: = 1 and B = 1 Example 3. 26 2.2 Transient State of Multi-Capillary Supply in a Rectangular Region Study of transient state of the oxygen distribution in tissue area is crucial and even more important than learning the steady state of such concentration. Since in the beginning the tissue cells might consume more oxygen than necessary, it helps to see the change of hypoxic area as time increases. And sometimes a lethal corner may exist and time dependent diffusion has this outstanding advantage to discover this instinct phenomenon. 2.2.1 Formulation and Solutions Now let us consider again the rectangular tissue region Q of length a and width ,3, containing N capillaries of uneven locations and flux strength. We may neglect the longitudinal diffusion of solute and derive the governing equation conveying oxygen from capillaries to tissue, after normalization of concentration by 1.3/Dr 6C 2 — = I _ 1 2.3- at V C ( o) and boundary condition is q- _ 5 27‘ BCI (109' J _ —; 5—- 5 J 0 ”3 (2.36) 30 _ 0 El _ 8:1". :r—>0,a _ ’ 8y y—10fl T To set up a strategy for solving this equation over a rectangular closed boundary with multiple source points, we first consider the homogeneous solution and treat the time dependence using the method of separation variables 27 C(I. y, t) = T(l)Q(:r-. 31) (‘2-37) then for some unknown constant A, we have .}.T’ = —A (2.38) and AQ + AQ = 0 (2.39) The solutions of the Eqs.(2.38) is T = Ae_)‘t (2.40) The solution of T is exponentially decreasing along time. Without adding or subtracting a constant distribution CO(:r,y), our initial condition here (when t = O) is always the steady state of oxygen concentration. This complies the real case because 1. the consumption rate Ii across the tissue area is never zero, and 2. the. process of regaining steady state is important in biological metabolism. If necessary, we can subtract the steady state from the solution and add a constant 00 as required from the initial substrate distribution in the rectangular tissue area. The Eq (2.39) is a Helmholtz equation in two variables, with rectangular supports. We can set. the solution by ‘ n2w2 nay An cos [\//\ — , (.1? — (1)] C05 ,' 52 d 2 2 - + Bn cos /\ —— n (n 1" COS m 00 #2 ,5 Q = Z 2 2 (2.41) ":0 + Cn cosm cos A —— n g y a a 2 2 + Dn, Cos Eric cos I: /\ — n g (y — 3)] a a where .4”,Bn.Cn and D” are unknown constants bounded by the boundary 28 2 2 condition (2.36). Let 6(h) = /\ — BES— Then the homogeneous solution is 00 00 . Ch(t,.r,y) 2 f0 €_)\t Z (An, cos [6(d)(.r. — (1)] cos? + Bn cos (5(,;3);1r cosyg‘I—j n=0 ' ’l + Cn cos ”it: cos 6(0)}; + Dn cos [HT—L cos [6(a)(y — 8)]) d/\ a (I 943 To find the particular solution to Eq. (2.35), we can use a transform into Cp 2: v + Cps, which provides (91) at- = V2’U + V2Cp3 — 1 : V21) (2.43) The Ups is the particular solution in steady state and CI) is the particular solution in our modeling. Then the particular solution is thus 00 00 . . Cp(t,a:,y) = / 6—)‘t 2 (En cos [6(fi)(.r — 0)] cosn'f—y + Fn cosé(,d).tr cos-71% 0 n=0 ’ A + Gn cos? cos 6(a)y + HrLcosz?(:()s [6(o)(y—fi)]) d). 2,2' +$ :y QAM The source solution of j th capillary (2.7) and the Eq. (2.17) together with the homogeneous solution (2.42) and particular solution (2.44) yield to the general solution of oxygen diffusion where Smry G: fooo _e/\t ”2::0( An cos [6,( d)(.r—a)] cos-71%:q + Bncos 6(3):“ co I (A. + an cos-g£0 cos 6(a)y + Dn cosn—Zillcos [6(a)(y—L3)])dA (2.46) In the Eq. (2.45), the first term is particular solution due to uniform consumption. The second term is homogeneous solution and third term is the combination of sources from each capillary. Being still more specific, we can compute the required values of the coefficients 371,371,671, and En by using the boundary conditions at the four rectangular walls 7V 8C * q.- 2.1? — 2.17- 00 f. :32. _J_ . J 2++=0 (2.47) 01‘- ;I.—+0,(.1’ 2 j=14 (ff—TJ) +(y—yj) ()1 and ’V 8C ‘ 4' y — y- '9 __ /:£/__ _J_ J 24.20.20 (2.48) (3y y—->0d 2 j=1 2 (g; — L‘J) + ((1 —- yj) d1] (2.46) gives 00 00 _ 0° _ .5; 2/0 (2 )‘t(— 21471503) 8111 [o )3)(;1:—o)]cos-n—;y— e=0 . I 00 mt- — Z [3715(3) sin 6(fi)r cos—,.—y =0 *3 i? 00 mr mm 00 Mt —— Cn— sin—cosé(a )y —2 Dnn— sin—cos[6( a) (y— )3)])dA a a a a c=0 , e=0 , 1Y1 1y (2.49) 30 As .17 —+ 0 the last. three terms tend to approach zero, I I ——> 0,111 —’ 0, IV ——> 0 . therefore 80 00 _,\t 00 - . my 2 , A. 6 L3 6 t3 _ . —-.—dA 8—1 ’71—»0 /0 e 2:: n, ( )sm[ ( )0] cos )3 (2.50) ASI-—>C¥,I—*0,III—+0,IV—+0 BC e—At nay ——.— = Bn 6(3 6(3 os-—dA d H. [0°C Z (w) .3 (2.51) Similarly for the vertical boundary lines y —> 0, g3 0C} _ 00 —At 00 — E .. _ , mry 537 — f0 6 <— :Anfi cos [6(d)(.r 0)] , e=0 5, Y — Z Bn:33 cos 6(3);r 5mm _ X3 l3 9—0 - II 00 00 mm: CR6 cos—51n6(o)y Z Bn6(a)cos¥—4siri[6(a)(y —t3)])(IA a a .=0 - e=0 , [TI [17/ (2.52) As y —> 01—» 0,11 —> 0,111 —> 0. therefore T _ ' .1 ‘ ZL—U- y—+0 =/Oooe “\t : B7)6( )cos r121" sin [6(o )3] (1A (2.53) 31 Asy—+)’3,I—>0,II—>O,IV—>O a: — _/000 e—At ”E: Cn 6(a c)os——¥ sin 6(a )3dA (2.54) 05’ 'y—*fl Eqs(2.47) and (2.50) yield As :1: ——+ 0 2:13]- V ac 1 (Ij "?— =0 + — ()1? .r——>0 J; 4 13+(y—3/jl2 '30 -—At 00 - "7“! + [0 e Z A-n6(L3) sin [6(,.3)a']cos——"dA = O 71:0 5 (2.55) 00 . oo _ N q- —;r- / (7M 2: An 6(g3) sin 6()3) ngy dA- —— Z 722— 2 J .2 (2.56) 0 ”:0 1 3'21 1]. + (y - 11)) l\’Iultiply Eq.( 2.56) by cos 7—7513; and integrate from —,13 to )3 with respect to y _ ’3 Am ”[000 e—At Z( 6(,L3 )sin 6(;3 3)adA = //3 gl(y) cosgydy _/I Let. Y = (y —— yj)2 32 II M2 10ng I\ III. E; L; [\D + N) A O 0 (13‘2 7C7? 0 O U‘ Til 7T 7717f 77177" — sin —,—Ysin —,—' dY 7W 6 7W) K). II p—A ll M2 [0ng J —.r - . J m7r 7717f —— , .‘ dY W63+:; 7 7W k; H H (Ij m7r [(3 1 ma — —r- 2cos— .- —— cos,—Y(1Y 2( J) W 0 1124—0”)? (3 142 K.) II |—‘ Thus _ 29:1qu ‘ j)i:)s%7_ry] /7T2cos7111/2dy (2.59) 0 72 Am -" —At /)\_ 112 ”s? /_ 71:62 I2.+( ’3 f0 8 —'2—.(Id)\ 31:] (1,02 Here y = 1511/, and (:rj.yj) is the location of the jth source. Similarly, the values for the unknown coefficients Bm. Cm and Dm are achieved N - q ((1 WCOSm y 1 , Bm— _ j: __1 J T J cos my ‘ (1y —6f00 e-At,/A— ”2725 inn—M nifliadAW §g(a—rj)2+(y)2 ' (2.60) _ q (2— yj 2)CCS— "—mfl 0771:2321 J ”/2 ”Hm”; 31211.1- (261) EN q- (13— y-) cos m—‘zr 7r D _ 3:1 J . J '~ (1 _] / cos 771.1: (11‘ 777. - ' _ 22 22,, .0 ,.2 n2,_,.2 _(, fooe At A_na7r )_flgg_,6d/\ (.1) +303 yj) (2.62) 33 2.2.2 Analysis and Example Oxygen diffusion from capillary supply in its transient state has rarely been studied. Many authors have treated the problem as in its steady state with highly simplified profiles. A steady state is always achieved through solving its time-independent equations. However in practice, during oxygen transport, substrates are perfused constantly along with the time. Some of the tissue may never achieve its steady state due to constant change of flux strength or sudden occlusion of capillaries. For example, some tissues suffer from sudden lack of oxygen during physical exertion. The hypoxia appears and the consumption rate rises even though the amount of oxygen out of each capillary preserves the initial quantity. It is common to all creature metabolism and therefore important to study the time-dependent oxygen transport. To illustrate the results we get from above sections, we use the following example with again a = 1 and {3 = 1. C1: yr. qr. 0.721 0.637 0.176 0.812 0.243 0.412 0.175 0.320 0.412 “Mk-*FT' Table 2.7: Locations and flux strengths of three capillaries Consider a rectangular region with three capillaries. The location and solute amount of each capillary are shown in table (2.7). We also plot the oxygen concen- tration level curves at t = 0, t = 1 and t = 10 as shown in Fig (2.5), (2.6) and (2.7). 34 0.9 l flr—H—mfiw 0. 8 * f_P_—\fl\\ —1 / / \ \ )H’TTJIT / /”T‘\ \\\ 0. 7 C J, / ”T “\ \j fife—HIT,— // // r:\ \ \\\ /r’“”’ ‘ / ( <59?) ) .\ x 0'6 Ix__r/ / \g/ \\ \ /. > 0_5 / ,, M. \ 1 /‘/ /7 // \\ i) 0.4 — ,x’::,.T\\ f/fiw \ - // /// (,1’//,/::§\ \ . // / \\ \\g l O 3 J l iii-$31)}! ) l . t) // //—7—\\ \ \ ‘1 . & l) W. Xv ) ll 1 / K,” {/R\ \ \‘x \' . \ / / 7 l r’ {7 i \ i ‘w \\ T/ / l )5 ) Kip/WI J) l ' J o 2 ._ f / \ ‘\ \/ / 2 . l \ / x \\ \\ \‘wf/ / (I k. 77 7TH,“ \\ \ \ \m _ / /’fr/ ’ 0 1 L \\ \\‘ \\\ "K HIx__r_,/'//f /1 \J.m\\ \ /r ) \, \\ //’/ l 0 L l l 1 l 1 \ l l I ’4 l J 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X Figure 2.5: Oxygen concentration level curve at t = 0, transient state, three capillaries with random characteristics Fig. (2.5) displays the oxygen concentration level at t = 0 for its steady state. At time t = 1, we can see from the Fig. (2.6) that hypoxia area occurs at the upper left region where consumption rate increases and less oxygen is supplied to this area. Fig. (2.7) shows that the hypoxia area is restored and moves slowly to the top y boundary. The hypoxia area is recovered as time increases. This might be due to the reason that the cells are very susceptible to the change of circumstance before they get used to new environment. And they may consume more oxygen than necessary in the beginning state. (Go [27]) 35 1 7" T 1 an Y *r\ 1 fi‘ _. //T// //_’_/_,__¥\\\ \ \ \ \ \ \ 0 9 l“ / / / // \ \\\ \\\ a / / /// \ 1“ )1. /{ j// l \j \\ {I f” (,1 1,] 0.8 ” I; I," / /f ’1] ‘1 [ / » i l. f, / i 0.7 ’— W 1“ X, // _l .4 \ ‘ / r’“\ f \ \\ / f' /',..\ \‘ 1' g \ \ ’1’ r I} (13‘ x, \\ K/ ‘ X \ ’1 K \v "I 1/ 1 0 6 1 \. \, ,I/ I \\ l ”f 1 i\\ \ \\\ / I \1 K‘” r l a» \ . 7‘w {I /" >‘ O 5 L \\ // / ,f’ 4 \\ ’1'. / i 7T7 \\ / / // /// MIMI/”T“ \ \ ‘V/ /’ / ,1- J“ / / 12$: ‘_ “ I l y ,1 (33,41 3 . ‘1, \\ / / 1 /’”““\__.1 O 3 [F j \ \\:—_-"//// l H \\ ,/ / fl / (:R\\ --1 l / ,,/ 1 / / / I , {.I’I‘;?i\\ \ \ 0 2 P\ \‘ 7/ r/ (W \l \\\\::l/’ ,/ \ “i , \ \ \ [I/ a // K\ \\ 0 1 \ 1 l\ / l ‘1 \\ 1 \ 1 .. / .r ‘ 1 | \\ \\ \h/// l I! / l\ 1\ 1 \\ 4 1 1 1 / 1 1; _1 _.__.:f_m_.___z: 0 O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 2.6: Oxygen concentration level curve at t = 1, transient state, three capillaries with random characteristics 36 I \ 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 Figure 2.7: Oxygen concentration level curve at t = 10, transient state, three capil- laries with random characteristics 37 Chapter 3 Substrate Transport in Multi-capillary Beds with Axial . Diffusion Radial diffusion may have considerably more effect than axial diffusion for oxygen transport for long and parallel capillary bed. In Krogh’s initial work the tissue area was taken as a coaxial cylinder around a single capillary, and axial diffusion was neglected in the elementary mathematical solution. Salathe and Wang [24] studied and presented solutions including axial diffusion for a single capillary and its coaxial Krogh cylinder. Previous work from them suggests oxygen concentrations surround— ing single capillaries as well as linear oxyhemoglobin dissociation relationship clearly indicates the axial diffusion may have significant effect on oxygen transport profiles. It. is therefore tempting and beneficial to have a full analysis of substrate diffusion under effect of axial diffusion. In this chapter we will make attempts to extend the solutions with axial diffusion to multicapillary beds inside a tissue cylinder, where arbitrary characteristics includ- ing random locations and uneven oxygen strength are taken into account. Methods are used following the scheme presented by Salate and Wang in their paper. Review 38 of the model and further analysis for multicapillary region will be given first. Further discussion of the solutions for oxygen supply on multicapillary beds near the arterial end, in the central region of capillary and near the venous end will be introduced in the remainder of the chapter. The prime model we use here is similar to what has been introduced in the previous chapter except that a Z axis is added and circular cylindrical tissue around multiple capillaries is employed. In order for us to apply relatively small longitudinal diffusivities, perturbation method is utilized to solve the associated governing equations. 3.1 Problem Review and Analysis Figure 3.1: Uneven locations and diffusion strength N capilary bed surrounded by a cylinder of tissue 3.1 . 1 Formulation Consider n number of capillaries surrounded by a cylinder of tissue having radius Ru. The capillaries are parallel to each other and have length L and radius RE, where 1 S j g n. The oxygen is diffused from the capillaries to the tissue which consumes oxygen at a constant rate FE per volume of tissue. Let 2 and 7” denote the centered axes 39 along the capillary direction and radial direction both normalized with respect. to L, 0 S 2: S 1, and Ru,0 _<_ r S 1. The governing equation for the oxygen concentration in the tissue cub", 9, 2) follows 1 (9 06 1 age 202c —— — —— —=':, <1, 0 0. For 5 = 0, it reduces to a 2- dimensional problem neglecting axial diffusion which gives the dominant effect on oxygen concentration in tissue and capillaries. Perturbation technique will be used for higher order of 5 (5 << 1) and corresponding solutions will give the effect of axial diffusion.Let C(::) = C j(z). The solution for C(21) and C(r, 0, z) will be obtained in the form of an asymptotic series, for small 5. 0(2) ~ 00(2) + 5201(2) + c('r,9, 2) ~ 00039, 2) + 8261(7‘,9, z) + (3.7) The oxyhemoglobin dissociation relationship can be expanded for small 5 by using Taylor series expansion to obtain S(C0 +5201) = 5(00) + 52015761,) + (3.8) By letting a = 0, the equations for the leading terms C0(z) and c0(7', 0, 2) along with the boundary conditions at the arterial end, capillary wall and outer boundary of cylindrical cut are 1 a aco 1 82m __ _ _ — < z< 1 0?“ 0r ' r2 862 ’ r 1’ _1’ (3 9) (9c a—BIT=1 = 0, z 31 (3.10) 8c 8c 83'220 = 0’ a |~-1 = 7" £1 (3.11) 277 d(0 ( — C +VS C = f —— d" 3.12 [0 (0)] 0 0p PRU o ) 00(0) = 1 (3.13) We then need to give solutions to for (70(2) and c0(7‘, 6, z) to the above equations. For CO, the oxygen concentration in tissue, here we employ matching technique similar to the results from the previous chapter. It follows from Eqs (3.9) to (3.11) that N c0(1‘,6,z) 7- 72/4 + Z {004' — FE/4 ~ ln[r2 + a? — 2raj cos(6 — aj)]} j=1 ' 00 + 111- Z rn(An cos n0 + Bn sin 726) (3.14) n=0 where An and Bn are constant coefficients that (see Appendix A) N 1 n E, An — 2—7; . aj cos(naj), n >1 (310) .7=1 1 N _ _ U ' . Bn — 271. 2a] Sln(naJ), n 21 (3.16) J: The solution states that the oxygen concentration in tissue is a combination of oxygen diffusion from each capillary within the circular region 0 S r g 1. pj is the distance from the j th capillary. The effect of oxygen diffusion from j th capillary diminishes as p j increases. A0 is a constant due to Nuemann boundary condition and can be set to make CO > 0. This analytical solution gives sufficiently good description of oxygen concentration in the cylindrical cross section. Substitute (3.14) into Eq. (3.12). It follows from boundary condition (3.13) that the solution for C0, the leading term of C(:) by letting E = 0 C0 + VS(CO) = [m (R — g — N) + mm] .2; + VS(1) +1 (3.17) 43 where N is the number of capillaries. and ERR) follows (see Appendix B) = Z (1321; . Na + 0(a) for R << 1 (3.18) The oxygen concentration in capillary (70(25 ) can be obtained from Eq. (3.17) due to monotonity of the oxyhemoglobin dissociation function S (C ) The equations for the second term c1(r, 6, z) and CO(2) can be obtained by sub- stituting Eq. (3.7) into Eqs. (3.1), (3.2), (3.3), (3.5) and (3.6) and retaining terms of order 52. (3.8) gives 16 an; 1702c1_ 62c0 r 07‘T ——a—:-7 2062 — — (922’ 7‘ $1, 2 S 1, (3.19) %|,.=1 = 0, z s 1 (3.20) %Iz=o = , 382% = r .<.1 (321) —Cl[l+l/S C0)] —-7/027r08—:} _Rd¢+u::;0, 0< 2 <1 (3.22) 01(0) = 0 (3.23) which includes the effect of axial diffusion. The solution for c1(r, 6, z) to Eq. (3.19) satisfying boundary conditions (3.20) and (3.21)is found to be c1(r,6,z)— — 7‘2/4+ Z{CLJ-+C’ 0’2( )/4 ln[r 2+c12-—27'(z JCOS(9—0'j)l} j=1 oo .6’(z) - Z rn(An cos 716 + Bn sin 716) (3.24) n=0 where An and En are constant coefficients that 44 Kl.[\9H|’_, N "2:1" 0. cos( 7m]- n 2 1 (3.25) 1 N Bn, 2 2—— jZ—Ean -sin( naj n 2 1 (3.26) In the above solution, the zeroth order term C0(z) has already been found in (3.17). At this point, the oxygen concentration c = (:0 + 82 c1 at any location :1 in the tissue cylinder around N number of capillaries is completely known. For the oxygen concentration in capillary C = CO + 8201 including axial diffusion, similar to what we achieved above, substitute (3.24) into Eq.(3.22) with boundary condition (3.23). It follows 0m + VS,(C0(2))] = [m (R— m] .2 + (v—fi1R)+-%) ~106121—061 (1)] (3.27) \ + 01(0)[I + VS where N is the number of capillaries. and fi(R) follows N 51(12): 2 a]2 «12+ 0(a) for R <<1 (3.2s) j=1 Cl can be obtained from above because S (C) is monotone and concave down. The solutions are valid throughout the cylindrical region except when it is near 2: = 0. At the arterial end of the cylinder radial, diffusion Dr and axial diffusion Dp are of 2 can not be employed for 2 equal importance. Therefore the small perturbation 5 near zero, and the solutions from (3.27) and boundary condition (3.23) are no longer valid. The expansions needs to be adjusted for a small layer near 2 = 0 and to satisfy the boundary condition with no flux through the end of the cylinder. 45 3.1.3 Small z Boundary Layer For small :5, there exists a boundary layer where solutions shall be constructed dif- ferently. The length of the layer diminishes in the sense of E ——» 0. Instead of 2, use Z = 2/6 (2: —> 0 as a -——> 0 for fixed Z ) as the variable for this boundary layer. The new functions of oxygen concentration in tissue and capillary become now 5(7', 6, Z) = C(T', 6, eZ) and C(Z) = C(EZ) The governing equations are 16 as 1625 825 __ _ __ —_ = < > 2 Tarrar+r2862+822 , r 1 Z_0, (3 9) d ~ ~ 271 65 (126 — .1 VSC = ' — d —. Z > 0 3.30 dzl + f ) 57/6 8’0!sz (15 + sudZQ‘ _ ( ) (92" 0r|T_1 0, z _ 1 (3 31) 85 ,. 5(0) = 1 (3.33) Eq. (3.30) can be written as d[5’+VS(C)] [27f 85p+53+5h (165+ (125' Z > 0 (334) —— . = 8 5V—, _ . ' dz ’7 0 8p p=R dZ2 where 5p, 5p, 5;; give respectively the particular solution, source solution and ho- mogenous solution of oxygen concentration in tissue. Further discussion of the explicit expressions will be given in the next section. The solutions of 5(1", 6, Z) and C (Z) near 2: = 0 should be uniformly consistent to what has been achieved from solutions throughout cylinder outside the boundary layer. To satisfy 46 lim 5(7‘, 6, Z) = lim C(T‘, 6, z) (335) Z—>oo z—> lim 5‘ Z = lim C 2 3.30 Z—>oo ( ) 2—10 () ( ) it is necessary to look at the solutions of C(T‘, 6, Z) and C (Z ) near the boundary layer at z = 0 and match the inside solutions C ,E to the outside solution C ,c at the first few orders of 5. write c(r,6, Z) + 520(7‘,6,Z) and C(Z) + 520(Z) in terms of 1',6 and Z and expand in terms of 5 C(52) = 0(0) + 5,112 + 521122 + 0(53) (3.37) where 11.1 = C6(()) and mug = C6, (0) / 2. Expand C0(EZ) + VS (C0(€Z )) using Taylor expansion C0(sZ) + VS(C0(5Z)) =1+ v3(1) +5111(1+VS’(1))Z vs” 1 +52 [(1 + VS’(1))112 + 2( )fi] 22 + 0(53) (3.38) From (3.17), C0(€Z) + VS(C0(5Z)) can be explicitly written as (700:2) + VS(CO(5Z)) -_-1+ VS(1) + 8[7I' 7 (R — g — N) -+- ,36(R)] Z (33.39) Compare (3.38) with (3.39). #1 and #2 are found to be respectively 7W (R- 3"; -N) +1300?) 1+VS’(1) [1.1 = (3.40) 47 2 VS”17r'R—"3—N 77R #2 = _ (1] 11 1: 13+m1 1] (3.41) 2(1+VS'(1)) where «796(12) is given in (3.18). The oxygen concentration in capillaries are approxi- mated to the order of 52. Add the C1 term to the above C(z) ~ 00(8) +5201 2) ( ~ 1+ 5/112 + 52 (11222 + (31(0)) + 0(53) (3.42) where 111 and p2 are given in (3.40) and (3.41). The oxygen concentration in tissue c0(r, 6, z) + 52c1(r, 6, 2) can be expanded as c(r,6,z) ~ c0('r,6,z) + 52c1(r,6, z) N N N+ 72/4 — 113/4 2 ln[1“2 +11% — 27‘aj (20ng — 0' )l .7 i=1 00 N -+- r: 2 rn(An cos 716 + Bn sinn6) + 5 Z [11,]- Z n=0 j=1 2 2 2 N +5 {1122 +r /4+ 2 014(0) i=1 N + 112/2 211162 + a; —— 2raj cos(6 — aj)] 1:1 oo — 2,112 X rn(An cos 116 + Bn sin 716)} + C(53) (3.43) 7220 where (to j th capillary) 711112]- — Rh? _ N) + fimj) (3 41 “‘10 ‘ 1+vs'(1) ' ) To match the boundary layer solutions to outside solutions C and c, expansions 48 of C and E in terms of 8 must be in the form 5(8) ~ 1+ 5 11.12 + 52 H(Z) (3.45) N ’5(;) N N + 72/4 — 5/4- E: ln[7‘2 + (13— 2raj cos(6 — a‘j)] F1 00 + N. Z 7""(An cos 716 -+- Bn sin 716) 71:0 +5711(7‘,6,Z) + 52 ,0 (7‘, 0, Z) (3.40) H (Z), 712' (r, 6, Z) and (,9 (7‘, 6, Z) are to be determined. Use the expansions given above (3.45) and (3.46) substitute into the Eqs. (3.29) and (3.30) with boundary conditions (3.31) to (3.34). Match first and second order 5 terms corresponding to C and Z. The set of differential equations for H (Z), 1,”) (7', 6, Z) and up (7', 6, Z) are I. For H (8), after matching the 52 terms from Eq. (3.30) dH '7' 811') — = 2 Z —— —+ d 3.47 dZ “2 + 1+vs’(1) cpl/HR ‘9 ( ) H(O) = 0 (3.48) and in order to match the outside solution C (2) (3.42) for 0(5) terms lim H(Z) = 11222 + 01(0) (3.49) Z—>oo II. For 0‘) (7‘, 6, Z), after matching corresponding terms with respect to 5 1 a aw 1620'; 620'; _ _____. __ __,_. <1, z>0, 3'0 TUT‘TCT‘+7'2062+BZZ 0, 7“_ _ (0) 8w - 5;"721 = 0, Z 2 0 (3.31) 49 81> . ,. gilzzo = 0, 7‘ S 1 (3.02) A Caz/(736, Z) = #12, at p = R, Z Z 0 (3.53) where Cu“) is the capillary source concentration associated with 71') and in order to match the outside solution C(T‘, 6, z) (3.43) at a term, where 111, j is defined in (3.44) N lim 0') = - Z 3.54 23... £31713 < > 111. For 1p (7‘. 6, Z), after matching corresponding terms with respect to 5 1 a a; 1 82,9 82,9 __ _ _. —————— —— = < 1 > . . i r Orr (97‘ 7‘2 06‘ + BZ 0, 7' _1, 7 _ 0, (3 00) 017 - 019 _ < ‘1' CZIZZO 0, 7“ 1 (3 07) 130,112) = H(Z), at p = R, z _>_ 0 (3.58) where C99 is the capillary source concentration associated with c" and in order to match the outside solution C (2.) (3.43) for 0(82) terms N - . . 2 2 23110019 — #2Z + 7‘ /4+ 2 01,](0) 3:1 N + #1'2/2' 2111]?“2 + a? — 2raj cos(6 — aj)] j=1 oo _ 2,112 Z Tn(An cos 716 + Bn sin n6) (3.59) 71:0 50 From Eq.(3.47) and boundary condition (3.48) _ 2Z H(Z) _ 11.22 +1—1_+vs’(1/OZ 71%;] qudadZ (3.60) Substitute (3.60) into matching condition (3.49), then 01(0) = 1—_—_+vs'(1)/000 jig—$41,313.72 (3.61) 3.2 Further Discussion on Matching Solutions In order to successfully match the inner solution with the outer solution inside the cylindrical tissue region, we need to obtain the solutions for perturbation functions 112' and 1,9. To obtain the solutions for 1/1, let N =JZCJ- — z)1/2(lnpj — 111R) + T(r, 6, Z) (3.62) where C j(z) = 111, j Z . The first two terms in (3.62) gives the combination of oxygen diffusion from each capillary source. The function, T, defined in the semi infinite cylinder 7* g 1, Z 2 0 satisfies the Laplace’s Equation 62T 18T 182T 62T m+;a+;§5—0—2—+a—Z-§=07 7'31: 220: (3-63) T: 0 3711/2 2 In pj), at r = 1 (3.64) g: ‘ Notice that on the outer boundary 1" = 1, T has no flux in or out of the cylindrical region. T satisfies a well posed problem that can be solved first by separation of 51 variables. T(r,6, Z) = R(7‘)O(6)F(Z) (3.66) then for some unknown constant A, we have 1 7/ _ 2 1329 _ _ 2 . F F — A , 9662 — n (3.67) 282R 3R 2 2 T—a—TQ—+TE+('TA—H)R=O (3.68) The Eq (3.63) is reduced to a Helmholtz equation in two variables, with circular supports. T is then solved to be in a general form T(r,6, Z) = [000 6—92 Z OAn(/\){[ )sin (716) + Bn()\)cos(n6)] [1937307) + E31300] } .11 13-69) where J... and Y... are Bessel functions of the first and second kind. and An (A),Bn (A),Dn (A) and En(/\) are constants which are limited to the boundary conditions. The radial derivative of Eq. (3.69) is (29; Z [000 (AZ 2 0A{l”()’\ >81“ ("9) + Brz(/\)COS(n6)] .1[a,(,\) (13-1w) — Jn+1()\7‘)) + E710) (73-107) — Yn+1 07-1)] ] .11 (3.70) 52 The boundary condition (3.64) gives at 7' = 1 T r—a.-cos(6—a-) (:97: Z 2 2 32 6] .. (3.71) j=177 +aj -— raj cos( —a]) => g: l—aJ-cos(6—aj) j=11+ (13— 2aj cos(6—01]) 00 oo = / e—AZ 2 {[AMA )sin( (716) +BMA)cos(n6)] 0 _ 71—0 )Jn—1(’\)_J71+1()‘)) . A } (1A (3.72) +E:(Y_ < >— 13.10)) l\1ultipl.y the above equation by cos(m6) and integrate from 0 to 277 with respect to 6 yield 7" — aj cos(6 — a ) :2 7+2 3 (3.73) 27 a] cos(6 — (1]) :> f A ] A 00 AZ D771.(/\)(Jm—1(/\l - Jm+1()\)) Bn17f/ €- ' /\ A (1A +En0>(Ym_10> — 13310)) N 27f 1-—a-cos6—a~) - = Z / 2 J ( j cos(m6)d6 (3.74) _1 0 1+ (1]. — 2aj cos(6 — aj) Letting 19 = 6 — OJ and using integration formulas in Gradshteyn and Ryzhik [27], it yields F - 13mm (1.3-10) — Jm+10)) _ +En —Ym+10>) , A 00 an / e—AZ - A (1A 0 1—acos(6—07j) 277 = 3.2/01+ (12 cos m6d6 —2aj cos(6— aj) 2 iv: /27r 1 gaj cos(6 — aj) cos(m19) cos(maj) d6 J_1 0 1+ a]. ‘ 2aj 005(9 _ aj) —sin(m19) sin(maj) = Z nag-”cos(maj), m 2 1 (3.75) and thus, we can achieve A =ZN_1a mcos(maj) B 3.76 m Gm(Z> ( ‘ l where f A . 00 AZ Dm(’\)(Jm 10‘) Jm+1(/\)) Gm(Z) = /0 e- ' - A dA (3.77) In the same manner, multiply Eq. (3.74) by sin(m0) and integrate from 0 to 27r with respect to 6. It yields * 1 Dm(/\)(Jm—1()‘) _ Jm+1()‘)) A 00 Am7r / e—AZ . A A d/\ 0 +En (Ym_1(/\> — 1mm) N 27f 1—a-cos(6—a-) = / 2 ‘7 J sin(m6)d6 j=1 O 1+aj ~2aj cos(flwaj) N 2]?” 1 — aj cos(6 — ozj) sin(7m9) cos(maj) (119 j— 1 0 1+0 ‘13- 203‘ 003(9— aj) +cos(m19) sin(maJ-) N :Zlmj msin( maj m 2 1 (3.78) and thus, we can achieve A 237$ sin (maj) 3.79) m Gm(z ) ( The axial derivative of Eq. (3.69) is 0T N 5242—0 Z/‘1,j 3:1 A N . . 2 /00(_)\) f: Dn (A)Jn()\7‘)+En(A )Yn( ”>211” (Sln(naj)sm(n.6) (1A 0 7120 071(2) 3:1] +cos(nalj)(.-03(n6)) (3.80) Dn, and En can be solved in terms of an eigenfunction expansion involving Bessel 55 functions, where the set of eigenfunctions is {Jn(/\r) sin('n.6), Yn(/\7‘) sin(n6), Jn()\7‘) cos(n6), Yn()\7') cos(716)} Then if), which describes the oxygen concentration near the arterial boundary layer to the first order of E can be expressed as N a = Z 6j(z)—1/2(1npj —lnR]-) i=1 0° 00 AZ +/ :ZA'L'M )fl‘,(n7‘9)€ - d/\ (3.81) 0 2' 7120 where E71571 Jn(/\r) sin(n6) 21‘ E Y Ar) sin n6 BnDn Jn(/\r) cos(n6) " finfin Yn(/\7‘)cos(n€) The function H (Z) for the oxygen concentration in capillaries can then be deter— mined by substituting the solution for 211(7‘, 0, Z) into Eq. (3.60) H(Z)— QZ+C(0)+—7—/Zf@ — ”2 1 1+VS’(1) 0 8p The third term in 112 is well posed and has radial derivative in terms of Bessel (1 d2 3.82 [HR <2) ( ) functions. Denote f 505(21- fi,j)lp—>Rd¢ by 5, then H(Z) = “.32 + 01(0)— fiéigi“) /000 (:0 Z A’—“—-"—) e—AZ d/\ (3.83) To complete the boundary layer solution (3.46), Now consider the oxygen con- 56 centration (p in boundary tissue layer to the second order of E. This solution can be written in terms of a new variable W as N 99039, Z) = W(r, Z) + #222 + 72/4 + Z Old-(0) i=1 A 00 _ g /0 A(/\)e—)‘Z dA N 2 2 + #2/2. len[7~ + a]. — 2raj cos(6 — 013)] J: 00 — 2112 Z rn(An cos 716 + Bn sin 716) (3.84) 71:0 where An, Bn are defined in (3.15) and (3.16), and A 5 - '7 00 Al T2. = —— d A A = é 1+VS’(1) an M Z . A 71:0 2 From Eqs. (3.55) to (3.58), W(r, Z) satisfies 1 ('3 BW 82W A 00 2 -—)\Z 7 ; 377.77" + _BZQ ._ {/0 /\ A(/\)e (1)1 (3-80) 8W — _ = > . ' 87' lr—l 0, Z _ 0 (3 86) 8W) — —E/OO AA(A)d/\ . <1 (3 87) 82 2:0 _ 0 ’ r “ ' Notice that W does not depend on 6. As Salathe and Wang [24] has shown in their paper, the capillary source concentration is already included in the rest of the expression for (p in (3.84) to the second order of E as shown in (3.43). The solution to the poisson equation (3.85) with boundary condition (3.86) and (3.87) can be constructed by using eigenfunction expansions gn(r), in the form of 971(7) 2 Y0(/\n,R)J0(A-nr) — J0(/\nR)Y0(/\nr) (3.88) 57 where J0 and Y0 are the zero order bessel functions of the first and second kind. the eigenvalues An are obtained from boundary condition (3.86) the roots of Y0(AR)J1(/\) — J0(AR)Y1(A) 2: 0 (3.89) where J1 and Y1 are the first order bessel functions of the first and second kind. A solution to Eq. (3.85) with boundary conditions (3.86) and (3.87) can be found (Salate and Wang [24])by constructing W(r, Z) in the form 00 we. 2) = Z En(Z)9n(7') (3.90) 71, _AZ and gn,(r) are eigen— where En(Z) is a more general function of Z instead of 6 functions associated with the equation for 1' after separation of variables, defined in (3.89). The orthogonality property of the eigenfunctions gives frgm(r)gn(7')dr = 0, for m 76 n (3.91) 2 2 _ (mu) 2 _ , frgn(7')d7‘ — ——2——— — 332?, TI. —1,2.3.4... (3.92) After applying Eqs (3.91) and (3.92) to relation (3.90) using the orthogonality property and multiplying g-m(r) on both sides of (3.90). one can conclude that En(Z) = Pn - frW’(r,Z)gn(r)dr (3.93) with 2 (971(1)) 2 _1 58 Differentiating En twice with respect to Z gives 0214/0, 2) II _ -rnr Bum—P... f g<> 022 (17° (3.95) together with the governing Eq. (3.85), it yields a 3 135(2) = 42,, . %gn(T)ET-a—TVV(T,Z)(1T + 1372(5/000 A2A(A)e_’\Z dA) - if Tyne-W (3.96) By using the boundary conditions for W , applying integration by parts twice and using the Eq. (3.93), the first integral on the right side of Eq. (3.96) is reduced to (9 8 . fryfiflg'rEWU‘, Z)dr = —A722]{1‘W (r,Z)gn(7‘)d7‘ = —A,2,P,,‘1En (3.97) The second integral on the right side of Eq. (3.96) using properties for Bessel functions (Watson [26]) can be reduced to 2 frgn(r)dr = —— (3.98) Therefore, combining (3.96), (3.97) and (3.98) it gives 1 00 . 1345(2) = A213,,(2) — 32$ /0 flame—AZ d/\ (3.99) 7171' From property (3.93) and the boundary condition (3.87), another boundary con- dition for En(Z) is obtained 2313 00 5;,(0) = 5—7—1 f0 AA(,\)dA (3.100) Agflr 59 General solutions to Eq. (3.99) can be achieved given the boundary condition (3.100) The solution in cylindrical tissue c and the solution in capillaries C are computed outside the boundary layer where z is comparably small, as well as the other end boundary layer, which form the two end regions of our cylindrical model. The so- lutions obtained in the middle intermediate region are valid through the cylindrical tube except for the two end regions. For the arterial boundary layer, one can use perturbation method by letting Z = z/s to find the approximate solutions 5' and 5 suggested as shown above to the first and second order of 5. Similarly for the venous end region by letting Y = (1 -— z)/5 one can approach the solutions with respect to the first and second order of 5 as well. A single solution shall be constructed through the whole cylindrical region from 2 = 0 to z = 1 uniformly composed from the outer solutions C, c from section 1 and the inner solutions 5', E from section 2. This can be done by adding the outer solutions and inner solutions for both the arterial and venous boundary layers, and subtracting the common terms from the expansions. C = C + 5' — Ccommon Uniform €1,717;me = C + C — Ccommon The common solutions are needed in order to find the uniform solutions for capil- lary and tissue oxygen concentration. Observe the oxygen concentration in capillaries given in central region as approximated by (3.42) and in small 2 boundary layer region as given by (3.45). It gives Observe the oxygen concentration in tissue given in central region as approximated 60 by (3.43) and in arterial boundary layer region as given by (3.46). It gives N ccomm = N + 72/4 — rt/4- Z ln[7‘2 + (13— 2raj cos(6 — aj)] j=1 00 N + r; E rn(An cosnd + Bn sin n6) + 5 Z 1110- Z j=1 N + 52 {/1222 + 72/4 + Z Old-(0) j=1 N + 112/2 - Z ln[r2 -+— (13— 2raj cos(6 — (13)] j=1 — 2112 Zorn (An cos 719 + Bn sin n9)} (3.102) Therefore the uniformly composite solution for the capillary oxygen concentration through the central region and the smallz boundary layer is given from (3. 7) and above 0.... = Cot-z) + 5201(2) + (5 — Ccomm) (3.103) where 5—Crronllrl=52'{— :gV—S’Z—r—AOO (22:4: 71203 ")5” (1).} (3.104) So then from Eqs. (3.103) and (3.104), it gives Cucs = 00(3) + 52{C1(3) _____1 5);, [000 (Z 2 A2 —"') (AZ (1).} (3.105) n=03 This solution is valid throughout both of the central region and the arterial bound— ary layer region, for the last term of Eq. (1.105) vanishes as Z goes arbitrarily large, 61 meaning that the dominant effect is given by C0 + €2C1 which gives exactly the solutions for oxygen in capillary through the central region. For the uniform composite solution of the tissue oxygen concentration through the central region and the arterial boundary layer. Similarly from (3.7) and (3.102) it is given as CUC8(T7 9, Z) = C0(7‘, 6, Z) + E2C1 + (E— (Ycomrn) (3.106) This solution is valid throughout both of the central region and the arterial bound— ary layer region. The dominant effect is given by CO + 52 c1 and solutions of oxygen concentration can be unified through the mixed layer near the end of tissue cylinder. In conclusion, axial diffusion can be neglected for long and parallel capillaries. However, it may play an important role in oxygen transportation in tissue specially when a relatively short pathway is counted into one’s consideration. Analytical solu- tion for rectangular and circular regions are given and one can extend the discussion of effect of axial diffusion onto a cuboid capillary domain. For irregular shaped do- mains, numerical implement is necessary for analyzing oxygen distribution across the region. A compartmental numerical model is introduced in the next chapter. 62 Chapter 4 A Compartmental Model of Multi-Capillary Supply Region 4.1 The Basic Problem Consider a cross-sectional region perfused with N parallel capillaries at arbitrary locations and arbitrary diffusion fluxes q,- (1 S i g N). The axial variation is small and can be neglected. The diffusion-consumption equation suffices v20 2 1 (4.1) where C = Cu / C A is the normalized with respect to oxygen concentration of the arterial blood, CA. The oxygen consumption “0 in the surrounding tissue to the zeroth order, can be normalized by a nonhomogeneous term 1, and is equal to the sum of the fluxes out of the capillaries. It was found by Wang([12]) that due to uneven capillary characteristics, the capillary domains are highly contorted, differ from either the Krogh circular cylinder or Voronoi polygonal cylinders. 63 Figure 4.1: Arbitrary multi-capillaries in a cross-sectional tissue region For uneven consumption rate, we can write v20 = n(z,y) (4.2) where H(QS, y) is the consumption rate at (2:, y) and satisfies (4.3) ffrt(m,y)d:cdy_ _ N _ ”MW—“$3 The solution to such nonhomogeneous problem is still possible, using Green’s func— tions. But in practice [€(I, y) is unknown, usually related to the capillary strengths ‘11- For example, when one capillary of a well perfused region is experiencing obstruction, and the rest of the capillaries could not react in time, the delivery (and consumption) of the region in the neighborhood of the obstructed capillary would decrease, but we do not know where and how much. One reasonable model of the oxygen consumption of tissue is the all-or-none model. Each location of the tissue is ether fully supplied or completely ischemic. To illustrate 64 this model, noticing that in previous two chapters we discussed the case for insulated rectangular or circular regions with no flux in or out of the domain, we start first here with the case of a single capillary in an infinite domain. For unbounded domain with one single capillary, it does not supply the needs of the entire tissue area. The region is ischemic outside the circular supply boundary where 7‘ g R. Let C denote the oxygen concentration by perfusion from a single capillary with oxygen strength q at the center. The governing equation is v20 = 1 r g R (4.4) and the oxygen flux is zero at 1‘ = R 3C_ C = — — O, at r=R (4.5) 07' 0C 1' —2' — = 4.6 T121110 ( 7‘ (97 ) q ( ) The solution to Eqs (3.4) and (3.6) is 2 _ T_ _ 3 C — 4 21nr + p (4.7) where the constants R and p are found by Eq.(3.5) R = f. p = ganq— 1) (4.8) The oxygen concentration decreases from the center of the source through the tissue area to the boundary at ya. This region inclosed by the circular boundary is also referred as the capillary supply region. Fig. (4.2) shows the result. The all-or-none model is reflected by myocardial infarction where necrotic regions 65 C (r) 1111111311, ’r fully supplied ischemic Figure 4.2: Oxygen supplied and ischemic regions as result from Eq. (4.4) are clearly demarcated. In contrary the first order consumption model v20 = aC (4.9) or the hv'licliaelis-Menten model 2 (YO V C = . 4.10 0 + C ( ) would give solutions which yield non-zero concentration effects at r— > 00, which does not happen in reality. However if there are more than one capillary, the problem becomes very difficult since the boundary of the ischemic area is no longer circular and is not known a priori. The question is similar to the blockage of one capillary in Fig (4.3), where we have an ischemic area of unknown boundary. This area may be decreased by an increase in flux from the neighboring perfused capillaries. But such adjustments would unlikely be immediate. The mathematical problem now become difficult due to the unknown boundary 66 lschemia Figure 4.3: Blockage of one single capillary location. Our method is to approach the steady-state by a transient process, which paradoxically may be more efficient than the steady state problem. 4.2 Unsteady Diffusion-Consumption The unsteady equation for concentration is g = v20 — n(x,y, t) (4.11) where MI. 11. t) (4.12) The boundary of the perfused region p(.r._ y, t) = 0 is unknown. On this boundary OC ~ 1101.11.13) = 5099.2) = 0 (413) where n is the normal to the boundary. The problem is a. difficult moving boundary 67 problem or known as Stefan problem (Crank[28]), studied in the context of freezing and melting of materials and seepage flow in porous media. There exist some analytical solutions to Eqs. (4.11) in one dimension including axisymmetric. For two dimensional problems, such as the case for multiple capillaries, even direct numerical interpolation fails due to the unknown moving boundary. To locate the boundary for numerical computation, several methods are suggested. These include fixing a partial grid on the moving boundary, a flexible grid that include the boundary, grid lines that coincide with time step, and exchange of dependent and independent variables. All are impossible to implement in two dimensions. The 2D ischemic quasi-steady problem was attempted by Salathe and Wang [23] using a Krogh cylinder and a perturbation from steady. The best method to treat truly 2D unsteady problem with moving boundary is the truncation scheme prOposed by Berger, Ciment and Rogers [3]. They used a large, fixed domain, and set or truncate any negative values of concentration to zero. Since the domain is fixed, the usual finite difference numerical schemes can be used. The moving front is then the zero concentration curve. The method is applied to a 2D oxygen diffusion problem by Evans and Gourlay [54] who used a hopscotch finite difference method to increase stability. In what follows we shall propose a compartmental diffusion consumption model which is much simpler than the truncation-finite difference method of Berger et al([3]). 4.3 Essence of the Compartmental Model Consider the whole plane partitioned into contiguous finite tissue bins. Each bin would consume one unit of oxygen per unit of time, if sufficient oxygen supply is available. If oxygen is less than the amount to fully suffice the plane, the bin is hypoxic and therefore consumes all available oxygen. Some of the bins are designated as capillaries which produce relatively large amount of oxygen per unit of time. By F ick 68 diffusion law through membrane separations, those bins with high substrate/ oxygen concentration would difluse into adjacent bins where substrate/ oxygen concentration is low. Within each bin the oxygen content is even or well mixed. Therefore the whole system is a network of physical compartments. The resulting equations are similar to the discretized diffusion - consumption equations but are formulated from completely different principles. First let us look at some examples in 1D. 1. 1D Unsteady Diffusion Consider the one dmensional problem C(as,t), a: 2 0, where the concentration at 3: = 0 when t = 0 is suddenly raised and kept at a normalized value of 1. The governing equation is ac 320 _ = _ 4.14 with the boundary and initial conditions C(0,t) = 1, C(oo,t) = 0 (4.15) C(z,0) = 0, :r > O (4.16) we shall compare it with the exact solution .1: C(cc,t) = e'rfc(—) (4.17) 2x5 where er f c is the complementary error function. (Crank 1975) Instead of the partial differential equations approach, consider a domain .1: Z 0 which is composed of ordered compartments of width 62 in 1D, The ifh compartment is between (i — 1)A:I: and ‘iAz'. Each compartment ex- changes oxygen with adjacent compartments through Fick diffusion law. Let C,- be 69 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 O O 2 4 6 8 10 Figure 4.4: Result compared with solutions from Eq. (4.17) at t=0.5, t=1 and t=2 tration distributions from Eq. (4.21) at t = n - At. Fig (4.4) shows the result for A1: = 0.1, a = 0.1. We see that the dots computed from Eq. (4.21) coincides with the curve generated by the exact solution Eq. (4.17). 11. 1D Unsteady Diffsion—Consumption Let Q be the perfused region of tissue. The governing equation is 30 82C with the boundary and initial conditions 37C: = C = 0 on the boundary atr = b (4.24) :r C(0,t) = 1, (4.25) 71 h the concentration of the (it compartment. Thus dC' AAx—t—z- = kA(Cz° — Ci) + kA(Cl'+1 — Ci) (4.18) where A is the area of diffusion and k is the transfer coefficient. Eq (4.18) can be rewritten as C'i—l — QCZ‘ + Ci-i-I (Ax)? (4.19) In the limit of As —> 0 we see the last bracket is (92C / 8:132, thus (kAar) must be the diffusivity which we have normalized to unity in Eq. (4.14) Now consider very small time step At t = At - n (4.20) and replace dCi/dt by (Ci,n+l — Ci,n)/At where n denotes the time. Eq. (4.19) now becomes the algebraic equation Ci,n-+—1 Z Ci,n + a ' (Ci—1,71 — 203,71 + Ci+1,n) (4'21) where a = At / (Arr)2 which is in the order of 0(1). This is exactly the finite difference explicit discretization of Eq. (4.14) (Mitchell, 1964) In the beginning, all compartments are empty except the concentration of the zeroth compartment is maintained at 1. 0 z' # 0 Ci. = (4.22) I i = 1 Also €0.72 2 1, all n For given 0 << 1 and Ar, get At = a(A;r)2, we generate subsequent concen- 70 11 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 O O 0.2 0.4 0.6 0.8 1 1.2 1.4 2 Figure 4.5: Oxygen diffusion into a consuming region at t=0.1, t=0.2, t=0.5, t=1 The solution for the final steady state is $2 (4.20) (4.27) As shown in section (4.1), the region 0 S a: g 1/2 is fully perfused, elsewhere is- chemic. We shall use the compartmental method on the transient problem. Balancing mass on the ith compartment, we obtain .10., AAr—t— = kA(C,- — 0,) + 1~A(C,-+1 — 0,) — Am 72 (4.23) where A is the area of diffusion and k is the transfer coefficient. Thus Ci,n+l 2 Ci,” + 01 - (Ci—1,71 - 201'?” + C'i+1,n)_ Al; (4.29) If Ci,n+1 is found to be negative, it is set to be zero and the compartment is . labeled ischemic. Fig. (4.5) shows the spread of the oxygen diffusion from a plane into a consuming region. Note when t = 1 the computed results have reached the steady state exact solution in Eq. (4.27). For stability, similar to finite difference explicit methods, a shall be less than 0.5. Otherwise implicit or semi-implicit schemes would improve stability but introduce systems of equations to be solved. 4.4 2D Compartmental Model For one dimensional unsteady diffusion, the compartmental model leads to similar equations as the finite difference method. For 2D, the compartmental model has decisive advantages. Hexagonal compartments are utilized to cover the whole tissue region. Firstly, the compartmental model is quite simple to implement. In comparison, the 2D unsteady finite difference schemes are quite complicated. Secondly, finite differences are nor- mally used in a square grid, while the compartments used here can be of any shape. i.e. one can use tissue cell boundaries or Voronoi domains like shown in Fig 1.1. Here we shall use hexagonal compartments which are more natural than squares. Notice in Fig (4.6 a)) the spread of a square grid, having only four neighbors, is more distorted or uneven than the spread of a hexagonal grid which has six neighbors. The neighbors of the (i, j )th cell are (i- 1.j+1). (i.j+1). (73 - 1.3). (141.1). (213'— 1). (i+1.j- 1) (41-30) 73 i ' ,j+1 ] j j.. ...... .3 ...... .. j—l j... 1 : j—l' 1—1 2' 7+1 (a) (b) (c) Figure 4.6: Compartments of a) square b) hexagonal c) triangular Now consider the (1', j) cell of area A, height h with side length l. 2A3: l = 4.31 3 ( ) 2 A = 2V”) (4.32) \/§ Mass balance gives dCz'j Ah. dt = slh-(Ci_1,j+1+ Ci.j+1 + C-_1,j + 0241.7 + C1',j—1'l' C-i+1,j+1_ 6017) _ nAh (4.33) Here Cij is the concentration of the (i, j )th cell, H. is the transfer coeflicient, or permeability through the cell boundary, and K. is the consumption as in mass per volume per time. Comparing the above with the diffusion equation as _ D 320 + 620 K at 7 am? (By? = D (01,341 + 0241,)“ + 01—14 + Ci,j—1 ~ 4013') _ 1.; (Am)2 (4.34) 74 we find diffusivity 74. (A517) 273 Normalize all lengths by R which is a measure of the size of the region. the concen- rzl 6 2 (4.35) tration by sR2 / D and the time by R2 / D, Eq. (4.33) becomes dCij 2 (Z neighbors - (SCH) 1 (4 36) dt 2: 3 (A12)2 To discretize the time using explicit method, If time is discretized by t = n. - At Ci, 33,, H = 0%,, + a (Z neighbors — 60mm) — At (4.37) here a = 2At/3(A:r)2 and Zneighbors = Ci-l,j+l+C'i,j+l+Ci—l,j+Ci+I,j+Ci,j-I+C’i+l,j+l (4.38) The coordinates (.r, y) are related to (1,3) by , i y = (j + §)—A;r (4.39) For this free boundary problem, one can choose a large region which contains the oxygen well sufficed and consumed area. For example, if the outer boundary is not oxygenated, we can use a rhombic region where z' = —N, N, j = —N, N large enough to contain any perfused regions. On the rhombic boundary, one can prescribe C = 0 which is equivalent to the zero flux condition BC/Bn = 0. In the case the zero flux condition is truly needed, the concentrations of adjacent cells in the normal direction 75 can be prescribed equal. For example alone j = —N Cz',—N+1 = Ci,—N (4-40) or even better, remove the two outside neighboring bins Ci _N, 0.1-+1 _ N' Similarly for z' = —N one can use 1 C—N+1,j = 5(C—N,j + C—N,j+1) (4.41) or remove the two neighboring bins C__ N j’ C_ N 7+1. Similar relations can be obtained for Dirichlet boundary conditions y = constant or any curved boundary. Now consider one of the cell is a capillary or oxygen source. Suppose the flux is Q, nourishing a region of 7W2 in the steady state. This region contains 7rr2/A — 1 consuming cells. If each cell consumes one unit of substrate per time, the total flux needed from the capillary is 2 M“ Let r = J Am, then from Eq. (4.34) 7n/S Q = (_2 .12 _ 1) (4.43) For a radius of r = 1,A:r = .1, Q = 271, a radius of .5, Q = 67. Now by Fick diffusion law, the oxygen flux is evenly distributed through the membranae wall between each two adjacent cells, therefore Q / 6 amount of oxygen is supplied to each of the capillary neighboring cells in time of At. The governing equation of oxygen 76 concentration for capillary neighboring compartments (i, j) is 5cells Q Ci,j,n+1 = 0233;” + a( Z neighbors — 501,330+ 6At — At (4.44) 4.5 Examples and Discussion First, we look at the case with one single capillary in the center with flux strength sufficiently enough for a circular supply region with radius r = 1. From the results shown in previous section, as t —> 00 the concentration profile should approach [\3 r _4_ (4.45) which is from the steady state solution for Eq. (4.7). Here we use a- : 0.1,A1? = 0.01 and tt 2 0.5. We display the results as in Fig. (4.7) to (4.9) which shows the oxygen concentration profiles at t = 0.01, t = 0.1 and t = 1. When time is small, the spread is approximately hexagonal, showing the effect of the grid. For larger time the spread becomes circular. Table (4.1) shows the radius of the supply region against time. At t = 0.5 the radius reaches 0.972 which is close the steady state estimate of 1. Fig (4.10) shows the radial oxygen distribution error as results compared with Eq. (4.45). As the compartment partition becomes smaller, we can adjust the size of our capillary source by adding a fixed number of compartments so that the capillary wall is approximated by the outer region of the source compartment boundary. t=0.01 t=0.05 t=0.1 t=0.5 t=1 r 0.124 0.699 0.817 0.972 0.996 Table 4.1: Radius of the circular supply region 77 . \ ~ ~ \ . ~ ~ \ . Pl hl ~I .I N. .II N . . 4 .-x 5.55.... 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"0.... .....?........... ........:.... ...u.._“........# ..... .. .. 531.5,; ’2" . ...: .. ... ...: ...n I 0.2 ' t t = 1 ' ed capillary a t'on profile with one Single center entra 1 Oxygen conc 80 Figure 4.9 0.1 . . . . . r . 0.09- .. 008+ . 0.07- .. 0.06- . 0.05 ~ . . 0.04» ., 0.03 - ' .. 0.02- « 0'01 h “"0...“ l , °:~«m...,......m __________ , . 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 4.10: Radial oxygen concentration distribution error as results from example 1 compared to Eq. (4.45) 81 Using the compartmental model, we can construct boundaries with different shapes as well as for studying insulated closed regions. Due to the natural properties of hexagonal boundaries, the insulated region boundary can be of hexagonal or rhom- bic. We can also illustrate the oxygen concentration in its transient state using an approximated rectangular region. Fig. (4.11) shows a schematic rectangular region bounded by i = I , j = 21 — 1, j = 2(1— 1) — 1 and i = z' — I. There are 212 — 21 +1 cells in Fig. (4.11), I = 4 with 25 cells. The interior cells all have 6 neighbors. The boundary cells have less neighbors due to insulation from outside. There are four type of boundary cells. I. The four corner cells each has 2 neighbors, II. a: boundary cells each has 4 neighbors, III. y outer boundary cells each has 3 neighbors, and IV. y inner boundary cells each has 5 neighbors. We can treat these cells differently applying the governing equation from Eq. (4.37) and make flux zero on each boundary cell walls. Note that for Neumann boundary conditions such as the fully perfused insulated region, the steady concentrations are unique up to an additive constant, while for Dirichlet or mixed boundary conditions, such as partially ischemic cases, the steady concentrations are unique. This means the concentrations are independent of history, or how steady state is achieved. dcr. = 0.2 (1.7: = 0.1 drr = 0.01 (11 = 0.004 (LT = 0.001 Number of cells 41 181 19801 124501 1998001 Cells of type II 6 16 196 496 1996 Cells of type III 6 16 196 496 1996 Cells of type IV 8 18 198 498 1998 Strength Q 67 217 2.72 x 104 1.7 x 105 2.72 x 106 Table 4.2: Parameters for compartmental model with different choices of (1.1? 82 -- Cell type IV <:> -- Interior cell Figure 4.11: Schematic regular supply region with I = 4 83 Figure 4.12: Oxygen concentration profile at t : 0.1, numerical results shown in an insulated capillary-tissue region 84 Figure 4.13: Oxygen concentration profile at t = 1, numerical results shown in an insulated capillary-tissue region 85 0:03 I I I I I I I I 0.025 I I 0.02 0.015 I 0.01 - 0.005 - 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.04 I T T I v 0.035 r 0.03 ~ 0.025 r 0.02 - 0.015 - 0.01 - 0.005 \. . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (b) Figure 4.14: oxygen concentration ratio comparison with Example II in Chapter two a) Normalized concentration comparison at source one where y = 0.821 b) Normalized concentration comparison at source two where :r = 0.243 Errors are relative low at each source point 86 For multiple capillaries with insulated region. Using the source locations from Example II in Chapter two, we approximate the rectangular region by the hexagonal compartmental model. The following sources are given in table (2.6). Use d3: = 0.004, a = 0.1. The total flux strength from three capillaries are calculated to satisfy the metabolic needs of the whole rectangular region. Fig. (4.12) and (4.13) shows the oxygen concentration at t = 0.5 and t = 3.5 respectively. The difference between the profiles at t = 3.5 and t = 4 is very small and we assume a steady state of is well achieved. Fig. (4.14) shows the oxygen concentration comparison between the compartmental model and the analytical results presented in example 11 of chapter two. Minimum oxygen concentration can be obtained on the boundary using the exact solution provided in chapter two. The two results differ by a constant, and this constant can be eliminated from consideration due to Neumann type of the boundary conditions. One can, from the concentration distribution, determine the capillary domains of each source. The method is to find local minimums along the three grid lines 2' =const, j =const and z' + j =const. The boundary of the capillary domains must contain three local minimums. However, due to the approximate nature of the course grid, it may be difficult to discern a true minimum especially at the outlying areas where the concentrations are almost uniform. xk y]: We 0.721 0.637 0.176 0.812 0.243 0.412 0.175 0.320 0.412 Cowl—*PT‘ Table 4.3: Locations and flux strengths of three capillaries reprised 87 Chapter 5 Discussion The purpose of this dissertation is to provide a mathematical analysis of substrate diffusion for a multicapillary model which is different from the Krogh cylinder. The long and parallel multicapillaries lie inside a tissue region. Exchange of substrate between capillaries and tissue is fundamental in the function of most living creatures. Clustered multiple capillaries with arbitrary characteristics in a unit region i.e. rect- angular, circular or with free boundary region were modeled in this paper. This is more natural than multicapillary arrays with homogeneous characteristics (e.g. loca- tion of single capillary, flux strength and unit region), due to the following reasons: 1. Heterogeneity is more common than homogeneity. For example, the Voronoi bound- ary usually does not form a repeated pattern. 2. The tissue region boundary is not an impermeable barrier and there are interactions among capillaries. Using uniform boundary conditions for each array cell may leave anoxic lethal corner. 3. Establish- ment of a perfused steady state for multicapillary supply region is naturally achieved from interaction between capillaries and readjustment of supply regions rather than substrate diffusion through single capillary-tissue unit boundaries. The matching technique was first used in this paper to solve the governing equa- tions for rectangular multicapillary regions. N eumann boundary was assumed and the a linear combination of eigenfunctions with unknown coefficients were used to express 88 the general solution. Then these coefficients were determined by using the matching conditions along the four boundary edges. Both of the steady state and transient state for such a region were solved. In the case of time—wise evolution of anoxia which occurs during the beginning state of diffusion, the capillary supply area forms hypoxic region which slowly vanishes at the tissue to regain its steady state. This process is much slower than the abolishment of such a state from sudden blockage or clamping one of the capillaries. Except in special cases (e.g. Fig 1.3) the areas enclosed by each Voronoi polygon are of varied sizes. If Voronoi polygons also represent capillary supply areas, the capillary strengths, which are proportional to each area, would not be the same. Hoofd et al. [50] found the capillary supply areas are similar to Voronoi polygons. This is probably because in their simulation somewhat even strength and somewhat even distribution of capillaries are used, unlike the cases illustrated here. Clark et al. [55] used an image method to compute the concentration inside a circular region. Their results when computed are identical to Eq. (3.14). Hoofd [51] used a different boundary condition but his formulation is non-unique. Since capillary length is about 102 times capillary diameter, in most cases the longitudinal diffusion of solute may be neglected in comparison to radial diffusion and therefore can be treated as a small perturbation to the solution. Eqs. (3.17) and (3.27) are solved implicitly for substrate solution inside the capillaries. The order of small perturbation is determined by the longitudinal location. At the both ends of the capillary cylinder, axial diffusion constant and radial diffusion constant are in the same order. Axial effect should to be treated differently and a full three dimensional analysis is required. Then the two set of solutions, describing oxygen diffusion in the cylinder and near the two ends of the cylinder, need to be matched completely to obtain the effect of axial diffusion. Using insulated regions with regular boundaries such as rectangular and circular, 89 we found the general solutions for both steady and time-dependent unsteady states. The matching technique provides solutions in a fast convergent series form. Another research direction presented is the study of a diffusion-consumption model when the boundary is constantly moving(i.e. a Stefan problem). Using Fick’s law of diffusion, the hexagonal compartmental model was used to describe two dimensional unsteady diffusion-consumption. The method models transport among cellular units and is much simpler than traditional finite differences. 90 Appendix A Analytical Solution for Oxygen Distribution in Circular Domains The solution to Eqs. (3.9) to (3.13) is in a general form N 00(7',6,z) = 7‘2/4+ :{COJ —n/4-lnp§} - j=1 00 + It- 2 rn(An cosnO + Bn sin716) (A1) 1120 The first term is the particular solution due to uniform consumption. The second term is a combination of sources from each capillary. The last sum is the homogeneous solution which shall be treated using the matching technique similar to chapter 1. The relation between pj, cpj and r 9 is (Fig 3.2(b)) p2 = r2 + (13— 2raj cos(l9 — 07]) (A2) The coefficients An and Bn are determined by the boundary condition at 7“ = 1. 91 The radial derivative of Eq.(A.1) is N a 2, oo _87“ = i _ K/4. 2 —p2 —r + E 717" (An cosnt9+Bn s1nn6) 3:1 J n=0 27' — 2a-cos(6 — Oz r N =2_”/4'Z2 J j) j=1T + a; — 2raj cos(6 — aj) 00 + Z nrn—1(An cos 716' + Bn sin 710) (A3) n=0 At boundary there is no flux through 00 l—aj cos(6— 0]) Z n(An cosn6 + Bn sin n6): — — 7.3/2 2 2 2 (A.4) n=0 1 +aj— 2aj cos(O—aj) Multiply Eq. (A.4) by cos(nfl) and integrate from 0 to 277 gives K. nAnW— — Is/Z- :1 j n1 n 21 (A5) j: —0 where 27f 1—a-cosl9—a') Ijfll = f 2 2 J ( *7 cos(n9)d6 0 1 +aj — 2ajcos(0—aj) 27f 1 — ajcos(19) = f 2 [cos(m9)cos(naj) — sin(m9)sin(na J')]dz9 0 12 + a]. — 2aj 008(0) 77 1 — aj cos(6) : 2cos(naj)/ 2 cos(m9)d79 0 12+a- —2a-cos(19) J J 27r, n = 0 = (A6) 7m? cos(naj), n 2 1 Here 6 = 6 — ozj and we have. used integration formulas from Gradshteyn and 92 Ryzhik[27]. Thus from Eq. (A5) 1 N An — 271 E 00] cos(naj), n 21 J: Similarly multiply Eq. (A.4) by sin(n6) and integrate, N rt-an‘Irzn/2- ZJjfll’ n21 i=0 where Jan 2 2_ . _ . 1+aj 2aJcos(0 07]) 7r 1 — aj cos(19) = 2sin(na )/0 cos(m9)d79 '7 12 + a; — 2aj cos(19) 27r, n=0 n - , 77a]. sm(noz]), n 21 Thus 1 N _ _ , 7} ° . Bn — 272 E 00.3 sm(naJ), n 21 .7— Details and examples can also be found in [12]. 93 27f 1—a-cosB—a- J-. = f 3 ( J) sin(n6)d0 0 (A.8) (A.10) Appendix B Substrate Distribution within Capillaries To find the solution of 00(2) to Eq. (3.12), substitute c0(r, l9, 2) (3.14) into (3.12). d 27? a (13.1) where 00,}, is the particular solution, c0, 5 is the combination of capillary source solutions and CD It is the homogeneous solution. 00,1) = (p? + (1%- 2ajpj cos(¢))/4 N 00,5 = 210042) -K/2-ln(pj)l i=1 00 C0, h = Z rn(An cos 716 + By; sin n6) n=0 where An and Bn are constant coefficients N N _ _"3_ n .. . _ I: 7? - . . An — 2n 2:107 cos(naJ), Bn — 2n 2:127 sm(na]), for n 21 .7: .7: 94 (13.2) (13.3) (13.5) In order to find the right hand side of Eq. (B.1), we proceed /27l' 800,1) 0 apj 277 a 1' _ _ 2 2,_ . . ' do — 1/4/0 3 '(p] +0.3 209/1] COS(¢))lp--Rd0 pj:R 9] 'J 277 : U24 R—aj cos(gb)) dab =R 77 (8.6) 277 800 S, N 277 a ’ do = — C :: K: 2-ln do /0 0p] pJ—R =1 0 6p ( OJ() / (p1))lpj=R 277 2 = — — 4 d H/R 77 K/ Z/O 8p] (Pall/J :12 d? 7%] =—I{/R'7T—K./4Z/ _—_l' (19’) 2 - -= 277 R—Ei'cosfii A #J' j J ' ‘ ' = -7~t/R-77 — 7.722277 (13.7) 275.7 Here we have used integration formula from Gradshteyn and Ryzhik [27]. Thus /277 acm‘ dab: _n/R-n—N77=—7T'(£+N) (B8) 0 3P7 pjzR R For the last term on the right hand side of Eq. (B.1) 27f 8c N 00 2,”. / (907'hl _Rdgb = Z Z/ a—afrn(An(:osn0+aninn6)| --Rd¢ O p] pr j=1n 1 0 p] 7)]- . N 00 277 72/ 8 n n , 2 Z Z — —7‘ a cosn(6— an), (10‘) ° J J .2 j=1n=12n 0 6p] P] R (B9) Forn=1 277 . 277 n 8 n 0 2 2 2 - — —raj cos9— aj dgb=— —7° +a.-—2,0,-| (16’) 2/0 Bpj ( J)pj=R 4 0 6pj( J J) ijR 277 a : —— d' 2/0 6p_j(%p1a~7€0w)l =R 0 277 = —E/ cosqfidgb 2 0 = 0 (BIG) Forn=2 277 a 7,2 2 _ 26— - d :/0 b—pj aj cos ( a])lpj=R g0 277 a 2 2 2 7,2612 , = _ 2 6— d 12/0 a—pjfma‘m“ 77> jpj—ll 277 a __ 2_...-2_2 2_.. '2.) ' _ 4/0 apj (2(a) pJaJ cosgb) (p3 +77] 2pja] cosqp)a]) pjszCD 2 = 3/0 77 (2013— Ra JCOS¢)(_aJ-cosgb)— (R—aj cosgb)a 2) dd) . 277 _ E 2. ' — 2RaJ(/O (cos2q§+1)d¢) : mm? (8.11) for 77 2 3, the nth order in the CO h series representation has partial derivative of order 0(R). 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