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The diffusive transport of a substance between a parallel capillary network and the surrounding tissue is investigated. The consumption/production rate of the substance in the tissue is assumed to be constant (zero-order chemical kinetics). The solution of the diffusion problem which describes the distribution of the substance in the tissue and along the capillary network is found in an analytical form. A rather general assumption regarding the symmetry of capillary network makes it possible to formulate a Neumann-type boundary-value problem in a rectangular domain. The solution of the diffusion problem in the rectangle allows the capillary-tissue fluxes to be expressed linearly in terms of the concentrations in the capillaries, and hence leads to ordinary differential equations for those concentrations. Several examples are considered with different network geometry and concurrent or countercurrent flow conditions. The solution makes it possible to investigate the effect of capillary interaction on mass transfer in various microcirculatory units.
The diffusive transport of various substances between capillary blood vessels and surrounding tissue structures has been studied extensively by numerous investigators both theoretically and experimentally. The theoretical work in this area has been recently reviewed by Middleman  and Leonard and Jørgensen .
The first simple mathematical model of the molecular transport between blood capillaries and surrounding tissue cylinders, based on the assumption that the diffusion flux vanishes at the external boundary of the tissue cylinder, was proposed in Krogh’s paper . This model, due to its simplicity, is widely employed in physiological studies concerning the estimates of the supply conditions in tissues.
Krogh’s model has been modified by several authors. Thews , for example, took into account the axial diffusion in the tissue cylinder, and Reneau et al.  solved the transport equations for the tissue cylinder numerically, taking into account the intracapillary diffusion and the axial diffusion in the tissue region.
In order to describe capillary interactions, Wilson  proposed a model of the tissue cylinder with a boundary condition on the external surface of the cylinder which differed from Krogh’s. Wilson set the concentration on this surface equal to a constant though unspecified value, and the integral of the diffusion flux through the surface equal to zero. This model leads to a concentration distribution in the tissue being significantly different from that predicted by Krogh’s model; however, it is difficult to interpret Wilson’s boundary condition in terms of the capillary network geometry.
Diemer , Bailey , and Reneau and Knisely  demonstrated different approaches to the problem of the mass transfer for countercurrent capillary-flow conditions. It was shown that countercurrent flow provides more homogeneous supply to the tissue than concurrent flow does.
Metzger [12, 13], Grunewald , and Grunewald and Sowa  considered the diffusion in tissues perfused by two- or three-dimensional capillary networks. The diffusion equation was solved numerically by the finite-difference technique. Results of the computer simulation of the oxygen transport in skeletal muscle appeared in a recent paper .
In the present work we will derive an analytical solution to the problem of the diffusive transport in tissues for a three-dimensional capillary bed. Certain simplifying assumptions make it possible to present the solution in a closed form and to investigate a number of examples that are of importance for physiological applications.
We consider the problem of molecular transport in a tissue supplied by an infinite array of parallel capillaries [Fig. 1(a)]. A rectangular coordinate system (x,y,s) is introduced such that the s-axis is parallel to the capillaries. The capillary distribution in the x–y plane is illustrated in Fig. 1(b).
The steady-state diffusion in the tissue is governed by the equation
where c is the local concentration of the substance, D is the diffusion coefficient in a plane perpendicular to the capillaries, Ds is the diffusion coefficient in the direction along the capillaries, and g is the local consumption rate (g > 0) or production rate (g < 0) of the substance.
Assuming that the axial diffusion of the substance in the blood can be neglected and that the concentration is uniform in the cross-section of the capillary (cf. Aroesty and Gross ), the convective transport in the capillaries can be described by the equations
where the index j identifies the capillary, Qj = πrjūj is the volumetric blood flow rate in the capillary j, rj is the capillary radius, ūj is the average velocity of blood in the capillary, j is the intracapillary concentration of the substance dissolved in the plasma, χ(j) is the concentration of the substance bound to hemoglobin, and Jj is the volume flux of the substance through the capillary wall per unit length of the capillary.
In order to find an analytical solution to the problem, the following assumptions are made:
The location of the capillary blood vessel can be given by the relationships
where xj,yj are the coordinates of the center of the capillary, sj is the coordinate of the “lower” end of the capillary, and Lj is the capillary length. If Qj is the capillary blood flow (Qj > 0 if the flow is in the positive s-direction, Qj <0 if it is in the negative s-direction), and cj0 is the concentration of the substance at the capillary inlet (i.e., at the arterial end of the capillary), the set of geometrical and physical parameters characterizing the capillary will be written as
With this notation in mind we can describe the symmetry of the capillary bed required for the solution in the following form: a rectangular coordinate system (x,y,s) with the s-direction parallel to the capillaries, and constants a and b can be chosen so that for any capillary in the rectangular cylinder Ω (0 ≤ x ≤ 2a, 0 ≤ y ≤ 2b, −∞ < s < ∞) described by the set (4) there is an infinite number of capillaries outside of Ω, characterized by the sets
where m and n are integers, and all possible combinations of plus and minus signs in (5) should be considered.
The symmetry condition (5) describes an infinite double periodic capillary array. It enables us to reduce the two-dimensional problem in the infinite plane (x, y), governed by Eq. (1) with the s-derivatives neglected, to the corresponding problem in the fundamental rectangle ω (cross-section of the rectangular cylinder Ω).
Taking into consideration the symmetry conditions (5), one can infer that the concentration possesses similar symmetric properties, namely,
The symmetry (6) permits us to formulate the no-flux boundary condition on the boundary Γ of the fundamental rectangle in the form
On the capillary-tissue interface Γj,
Here n is the unit vector normal to Γj pointing in the external direction, and v = S/Sb where S and Sb are the solubility coefficients of the substance in the tissue and blood, respectively.
The equations (2) will be solved using the boundary conditions
Since the capillary radii are small in comparison with the intercapillary distances, we can further simplify the problem by replacing the capillaries by line sources (or sinks) with the intensity Ji per unit length. Therefore, Eq. (1) can be rewritten in the form
and δ(x,y) is the two-dimensional Dirac delta function.
Generally speaking, replacing the capillaries by the line sources or sinks makes it impossible to satisfy the boundary condition (9), since the latter requires that c = const on Γj. We will approximately satisfy (9) by equating
where Γ̃j denotes the portion of the capillary-tissue interface confined within ω, and |Γ̃j| denotes the length of the corresponding arch.
Let us introduce a new coordinate system so that the fundamental rectangle ω is determined by the inequalities
into (12) leads to the Poisson equation
with the Neumann-type boundary conditions
where n is the internal unit normal to the boundary Γ.
where F(z) = Σqjδ(z−zj), c* is an arbitrary constant, and the second integral in (20) is taken over the area of the rectangle ω.
The necessary condition for the resolution of the Neumann problem
In order to find the Neumann function N(z′,z), we build up a transformation which conformally maps the rectangle (15) onto the upper half plane Im ζ ≥ 0, where
This transformation is effected by the Jacobian elliptic function 
where K(k) is the complete elliptic integral of the first kind,
and the parameter k is a root of the transcendental equation
In the following equations the standard notation used will be
When the functions K or K′ are written without specifying the argument, then the argument k is implied. The same agreement is also valid for the Jacobian elliptic functions, namely, sn u = sn(u, k).
The inverse transformation mapping the upper half plane Im ζ ≥ 0 onto the rectangle (15) in the z-plane is effected by the incomplete elliptic integral of the first kind,
The analytic function (24) transforms the boundary Γ of the rectangle ω into the straight line Im ζ = 0 so that the points
correspond to each other.
Introducing the notation
After transformation to the new variables (ξ, η), (Eq. (17) takes the form
where ζj = sn(zjK/a,k). The property of the delta function
has been used.
Denoting the image of the boundary Γ as Γ* and taking into account the relationships
we can rewrite the solution (20) in the form
where ω* denotes the domain Im ζ ≥ 0, and
The Neumann function for the upper half plane Im ζ ≥ 0 can be constructed using the method of images :
where is the complex conjugate of ζ.
Calculation of the distortion coefficient of the conformal mapping (28) yields
Therefore the concentration c is given by
It can be shown that
In order to verify (46) we notice that as z→2bi, ζ → ∞ and the functions u1 and u2 can be expanded as
The condition of conservation of mass (21) causes the coefficient at ln(ξ2 + η2) in (45) to vanish. Hence, the sum of the first three terms in (45) goes to zero as z→2bi, which proves the statement (46).
and using the tables of elliptic integrals , we obtain after some simple though lengthy calculations:
In order to satisfy the boundary condition (14) we have to integrate the concentration c expressed by (45) over the capillary-tissue interface Γj. However, this is very difficult to do because of the complexity of relationships (31).
In order to avoid this computation we will again use the fact that the capillaries are narrow in comparison with the intercapillary distances and, therefore, with the dimensions of the rectangle ω. Since a circle z = zj + rjeiθ in the z-plane is transformed by the conformal mapping (24) into a curve
we can use the condition rjK/a1, and expand the function (58) in a Taylor series as
where and are constants.
We consider three different cases:
The relationships (61), (63), (66), and (67) together with the boundary condition (9) and Eq. (21) permit us to express the capillary-tissue fluxes q1,q2,…, qM and the parameter c* in terms of the capillary concentrations 1,2,…, M, where M is the number of capillaries in the domain ω. Substitution of these expressions into the set of equations (2) yields a set of M ordinary differential equations with respect to 1,2,…, M. The boundary conditions for this system are given by (10) and (11).
It should be noted that in some cases integration of (2) can be performed analytically, in particular when the function χ(j.) is linear.
Let us consider a capillary distribution in the planes s=const with the symmetry shown in Fig. 3(a). The corresponding fundamental rectangle is shown in Fig. 3(b). This configuration can describe different types of structures in the s-direction, which will be specified later.
According to the formulas derived in the preceding section, the concentration distribution in the tissue can be written in the form
The conservation of mass of the substance (21) yields
where the area of the rectangle ω is denoted by
In order to simplify the following calculations we assume that the capillary radii are equal, i.e., r1 = r2 = r.
The following assumptions led Krogh to the solution of the problem :
In addition, Krogh used the boundary condition
where ρ is the distance from the capillary axis and rt is the radius of the tissue cylinder.
Assuming that the ratio r/rt is small, we can write Krogh’s result  in the form
where is the capillary concentration,
The case of concurrent flow in capillaries located in the corners of squares as shown in Fig. 4 is described by the relationships (68), (73)–(75) with q2 = 0 and a = b. The expression (74) with q2 = 0 yields the concentration at the point A in Fig. 4:
where the subscript 1 at the capillary concentration is omitted. The relationship (75) yields, for c*,
We introduce the Krogh cylinder radius rt so that the cross-sectional area equals the area supplied by each capillary in Fig. 4; this definition gives
whereas (78) to the same order can be rewritten as
It can be shown that for the case g > 0 (the substance is consumed in the tissue),
which could be expected from the geometrical picture shown in Fig. 4.
If the Krogh cylinder radius is chosen as , then , where is calculated using (78); if , then . Therefore we have the useful estimates
For simplicity in the present paper we restrict ourselves to the case of linear functional relationships between χ and j in (2). The particular case χ=0 corresponds to the condition in which the substance is not bound to hemoglobin. Using the notation
we rewrite the equations (2) in the form
If α1 + α2≠0, the general solution of (90) can be expressed in the form
and A, B are arbitrary constants. In the case α1 + α2=0, which according to (89) describes countercurrent flow in the capillaries with equal absolute values of the blood flow, the general solution of (90) is
where α = α1 = −α2, and A, B are arbitrary constants.
We consider a microcirculatory unit with concurrent capillary blood flow shown in Fig. 5(a) (the three-dimensional capillary structures similar to those shown in Fig. 5 were considered earlier by Grunewald; cf. [14).
for 0 ≤ s ≤ L. It is easy to show that in this case the concentration varies monotonically along the capillaries.
in accordance with the Krogh-cylinder-model solution .
The concentration at the venous end of the capillary equals
The relationship (98) can easily be obtained by consideration of the mass balance between a capillary and the corresponding volume of tissue supplied by this capillary. In contrast, the solution (96) (97) reflects more complicated phenomena, namely, the interaction between the adjacent capillaries. The formulas (96) and (97) conclude the solution of the three-dimensional mass-transport problem for the geometry specified in Fig. 5(a); substitution of (96) and (97) into (68) together with (73)–(75) makes it possible to calculate the spatial distribution c(x,y,s).
The derivation of the numerical values of the concentration at an arbitrary point in the fundamental rectangle involves computations of the Jacobian elliptic functions and complete elliptic integral. Rapidly converging series expansions can be used for this purpose ; detailed tables of these functions are also available .
The countercurrent flow conditions are realized in the microcirculatory unit shown in Fig. 5(b). We assume that the blood flow rates in both capillaries are identical; α1 + α2=0. Placing the origin of the coordinate: system at the midpoint between the arterial and venous ends of the capillaries, applying the boundary conditions
The concentration at the venous ends of the capillaries is
for the first and second capillary, respectively.
Therefore in the case of fully countercurrent flow the capillary concentration can achieve an extremal value at the point between the midpoint of the capillary and the venous end, but not between the arterial end and the midpoint.
In the microcirculatory unit shown in Fig. 5(c), the flow in the capillaries is concurrent or countercurrent depending on the spatial region.
The solutions (91), (92) and (94), (95) enable us to describe a situation with different capillary-inlet concentrations and capillary blood flow rates. For brevity, we consider only the case of equal capillary-inlet concentrations and equal blood flow rates. We write out the values of the capillary concentration at certain points of the unit, separated by the distance L/2 in the s-direction:
Here the notation (a +) or (a −) means
Since the capillary structure is periodic in the s-direction with period 2L, the functions 1(s) and 2(s) for 0 ≤ s ≤ 2L completely solve the problem.
The expressions (106.1), (107.2), (108), and (111.1) show that the blood going into venules is a mixture of blood from two kinds of capillaries with the concentrations
Since the blood flow rate in all capillaries is the same, the average concentration in the venular blood can be found as the arithmetic average of (112):
A model of steady-state mass transport in tissue has been developed, which can be considered as a generalization of the Krogh cylinder model.
The solution has been obtained with the assumption that the capillary blood vessels are parallel to each other and that their distribution in a plane s = const is double periodic, so that the diffusion problem can be stated in a rectangle instead of the infinite plane. It was also assumed that the capillary radii are small in comparison with the intercapillary distances, and the latter are in turn small in comparison with the capillary lengths; the latter assumption permits us to neglect the axial diffusion in the tissue.
Practical computations of the concentration distribution can be done according to the following format:
The solution presented is only valid if c > 0, since this condition enables us to assume a constant consumption/production rate g. If c = 0 in certain domains, the problem becomes non-linear, and the derived solution is no longer valid.
The present model makes it possible to study the mass transport of various substances (glucose, O2, CO2, etc.) in the tissue. Further, the model takes into account such important physiological effects as interaction between capillaries with concurrent and countercurrent flows, interaction between “fast” and “slow” capillaries, and mass transport between tissue regions supplied by different arteriolar vessels.
The author is indebted to Dr. J, F. Gross, Dr. H. D. Papenfuss, and Dr. H. S. Chen for helpful discussions and critical comments, and to Mrs. Rhoda G. Miller for typing the manuscript.
This study was supported by NIH Grants HL 17421 and NO I-CB-63981.