Computational Framework for Generating Transport Models from Databases of Microvascular Anatomy

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1 Annals of Biomedical Engineering, Vol. 29, pp , 2001 Printed in the USA. All rights reserved /2001/29 10 /837/7/$15.00 Copyright 2001 Biomedical Engineering Society Computational Framework for Generating Transport Models from Databases of Microvascular Anatomy DANIEL A. BEARD Department of Bioengineering, Box , University of Washington, Seattle, WA (Received 30 November 2000; accepted 26 June 2001) Abstract Quantitative descriptions of transport and exchange in physiological systems should make use of the emerging wealth of data on vascular anatomic structure. These descriptions may take the form of computational models which then must be incorporated into the comprehensive database of knowledge of microcirculatory physiology being developed under the title, The Microcirculation Physiome Project. Toward this end we present a simple and efficient computational method for simulating transport advection, permeation, diffusion in tissues containing microvascular structures of arbitrary complexity. The method is convenient because transport is simulated on a regular Cartesian lattice, and efficient because features of the anatomy are resolved within individual volume elements of the lattice. As a result, relatively low-resolution lattices yield accurate results. Therefore the method provides a feasible approach for studying a general class of transport problems in the context of realistic representations of vascular anatomy Biomedical Engineering Society. DOI: / BACKGROUND AND STATEMENT OF PROBLEM The Microcirculation Physiome Project 9,10 is an international collaboration of scientists aimed at organizing structural and functional data and mathematical models relevant to microcirculatory physiology into a coherent and accessible database. Of critical importance to this effort is the design of mathematical modeling tools and anatomic databases which are functionally compatible. An additional goal is that the mathematical methods for simulation of physiological processes be accurate and efficient. Efficiency is paramount because if the models are to be used for data fitting and parameter estimation multiple solutions must be obtained in a reasonable time frame. Here we present a computational framework for generating blood tissue exchange transport models based on the three-dimensional microvascular anatomy as described by a structural database. The method is based on a discrete representation of the tissue on a cubic lattice. However, in contrast with standard finite-difference and finite-element schemes, each element in the discretized Address correspondence to Daniel A. Beard, PhD, Bioengineering Department, Box , University of Washington, Seattle, WA system can represent an inhomogeneous combination of extravascular and intravascular tissue in the method presented here. In this inhomogeneous volume IVol method the microvascular anatomy need not be deformed to match the computational grid. Thus the IVol method can be applied to microvascular anatomies of arbitrary complexity and is more efficient than standard finitedifference and finite-element methods that require the computational grid be fine enough to resolve the anatomy. Realistic three-dimensional representations of the microvascular structure may be necessary for accurate simulation, depending on the particular tissue and substances being considered. For example, microvascular networks from the cerebral cortex do not lend themselves well to the Krogh description. 15 Even in tissues where the network structure is much more regular, such as skeletal muscle, investigators point out the influence of network structure on transport phenomena. 2,5 In terms of its relevance to the Microcirculation Physiome Project, this work represents one feasible strategy for approaching the second of Bassingthwaighte s three essential, early goals of the Physiome Project: 1 to construct models which integrate the available data into a functional description of the physiology. Not surprisingly, in developing a framework for generating models based on databases of anatomy we gain insights into improvements that can be made in the databases themselves. Thus, as Bassingthwaighte 1 suggests, the exercise of working towards goal No. 2 constructing models based on the databases feeds back information necessary for goal No. 1 the construction of the databases. The databases themselves may take the form of a list of statistical properties of whole-organ vascular networks like those developed by Kassab and Fung, 6,7 they may be a library of structural descriptors of observed network anatomies like the data made available by Secomb and colleagues, 16 or they may be some combination of both approaches. Recent modeling work on transport in realistic vascular networks 2,3,5,13 15 takes advantage of both types of structural data. Since this paper deals not with

2 838 DANIEL A. BEARD FIGURE 1. Data structure for the microvascular transport problem. A Shown is a diagram of the microvascular network depicting vessel wall boundary, and the outward unit normal vector nˆ. Solute concentrations are C v x,t and C e x,t in the intravascular and extravascular spaces, respectively. The velocity vector v x is nonzero only in the intravascular space. B An inhomogeneous volume element in the cubic lattice is illustrated. Since a blood vessel partially obscures the element, volume V ijk e is less than x 3, where x is the lattice spacing. Similarly, area A ijk e is partially obscured by the blood vessel. C Shown is a blood vessel segment, indexed by l, and an element of surface area ds lm associated with the vessel segment. the construction of networks based on a morphometric database, but instead with the simulation of transport processes within a tissue containing such a complex three-dimensional microvascular network structure, the methodology may be applied to modeling efforts which utilize either type of data set. The general class of physiological problems which we are interested in characterizing here are those related to transport and exchange as described by the following equations: C v C t v D v 2 C v G v, C e D t e 2 C e G e. In Eq. 1, C v is the concentration of some solute within a microvessel, as depicted in Fig. 1 A. The extravascular concentration is denoted by C e. Solute advects within the microvessel with the velocity v(x), where x denotes the position vector. Molecular diffusion occurs within the vessel and in the extravascular space with diffusion coefficients D v and D e, respectively. The terms G v and G e represent chemical consumption or production of solute. Exchange between the intravascular and extravascular spaces determines the boundary condition on, the vessel wall see Fig. 1 A : D v nˆ C v D e nˆ C e Q C v,c e, 1 2 where the unit vector nˆ is normal to, and Q(C v,c e ), the mass flux density across, is assumed to be a function of the intravascular and extravascular concentrations. For example, passive permeation implies Q p(c e C v ), where p is the permeability of the vessel. For vessels in the microcirculation we make the assumption that advective transport is characterized by the average flow across a vessel cross section. Concentration gradients are considered only in the direction of flow radial concentration profiles are assumed to be negligible. The IVol method for simulating advection, diffusion, and permeation under this assumption is outlined below. The procedure is organized into two sections: 1 The section on setup and data structure outlines the construction of the grid for IVol calculations. 2 The section on numerical methods for simulation of transport processes details the IVol scheme for simulating advection in the vascular space, permeation between the vascular and extravascular spaces, and diffusion in the extravascular space. The methodology sections are followed by examples of applications of the methodology to simulating transport in the two networks shown in Figs. 2 and 3. The first example is the simplest possible network, a straight tube, used to study the convergence of the computational method. The second example is an anatomic model of a microvascular network of the rat cerebral cortex 8 obtained from a database of microcirculatory anatomy available on the Microcirculatory Physiome Project web page. 17 SETUP AND DATA STRUCTURE Typical finite-difference schemes are based on a spatial and temporal discretization of the governing differential equations which requires that each point in the computational grid be governed by either the equation for transport in the intravascular space or the equation for transport in the extravascular space. Thus such schemes require interpolation of the anatomy onto a regular discrete grid. In the method developed here, each volume element represents an inhomogeneous combination of the anatomic structures that lie within the element. Therefore the given anatomy is not altered by the adoption of a numerical grid for discretizing the governing equations. Such a volume element is illustrated in Fig. 1 B for a two-dimensional system. For three-dimensional systems we index the cubic lattice with three integer indexes, i, j, and k, representing position along the x, y, and z directions. The variable C ijk e represents the extravascular solute concentration in the ijk volume element. Brackets, C e, are used to represent all members of the set. Similarly, extravascular volumes of the elements are stored in V e. While the maximum value of

3 Transport Models from Microvascular Anatomy 839 FIGURE 2. Simulation of the impulse response washout function from a tissue region supplied by a single straight microvessel. A The network consists of a single 5- m-diam vessel in a 400 mã50 mã50 m region. B The tracer washout following impulse injection into the inflow was calculated with various levels of grid resolution xä5, 10, and 25 m as indicated in the figure. The inset shows the early part of the computed washout functions. These curves were obtained with the extravascular diffusion coefficient of D e Ä500 m 2 s À1, the vessel permeability of pä10 ms À1, and tissue perfusion of 0.5 ml min À1 per ml of tissue. V ijk e is x 3, where x is the spacing of the cubic lattice, lower values are obtained when blood vessels intersect with a volume element, as depicted in Fig. 1 B. Next we introduce the areas, A ijk x, A ijk y, and A ijk z, which are the extravascular areas of the ijk volume element in the positive x, y, and z faces, respectively. These areas attain the maximum value x 2 when no blood vessels intersect ijk with the ijk volume element. In Fig. 1 B, A y is a fraction of A ijk x because the blood vessel obscures some of the surface area on the positive y face. To summarize the data structure for the extravascular space, the variable C e stores the extravascular solute concentration on the cubic lattice, while the geometry of the extravascular space is described by the volumes V e and the areas A x,a y,a z. Because blood vessels occupy only a fraction of the tissue space, intravascular transport is not discretized onto the same volume filling grid as transport in the FIGURE 3. Simulation of the washout function from rat cerebral microcirculation. A The network consists of 50 vessel segments making up two topologically distinct networks perfusing a 150 mã160 mã140 m region. The smaller network indicated by the arrow has no bifurcations while the larger network is more complex. B The tracer washout following impulse injection into the inflow was calculated with various levels of permeability pä5, 10, and 20 ms À1 as indicated in the figure. The left panel shows the early part of the washout function on a linear scale, and the right panel shows the long-time behavior on a semilog scale. These curves were obtained with the extravascular diffusion coefficient of D e Ä500 m 2 s À1, and tissue perfusion of 0.5 ml min À1 per ml of tissue. extravascular space. Instead, the microvascular network is discretized into a finite number of interconnected segments Fig. 1 C. The blood volume and flow in the l segments are stored in the variables V s and f l, where the integer l 1,N s indexes the set of all microvascular network segments, and N s is the number of segments. Concentration in segment l is denoted C l v. Incorporation of radial intravascular concentration gradients would require a further discretization of the intrasegment concentration. To model permeation, the surface of each vessel segment is discretized into a finite number of surface area elements ds where ds lm is the mth element of surface area associated with the lth vessel segment. We define I(l,m), J(l,m), and K(l,m) as the functions which map the indexes of surface area elements to the indexes of the finite volume cubes. In other words, the

4 840 DANIEL A. BEARD indexes i I(l,m), j J(l,m), and k K(l,m) identify the volume element in which the surface area element ds lm is located. NUMERICAL METHODS FOR SIMULATION OF TRANSPORT PROCESSES Based on the data structure described above, we now introduce the following evolution equations for the concentration variables C v and C ijk e l : l C v A l P l t v, C e ijk t P e ijk D ijk, where the terms A l, P v l, P e ijk, and D ijk represent advection in the vessel segments, permeation into the vessel segments, permeation into the extravascular volume elements, and molecular diffusion within the extravascular volume elements lattice. The mathematical form of each of these terms is described in the following sections. Advection Advection in the vessel segments is discretized using the standard upwind differencing scheme: A l L Lu l f L V s l C v L C v l, where L u (l) represents the set of upstream vessel segments connected to segment l. Thus the summation in Eq. 4 is over all indexes L associated with segments which are connected to segment l and are upstream of segment l. Permeation For passive permeation the mass flux density mass per unit area per unit time out of a vessel is given by Q p p(c v C e )nˆ, where p is the permeability of the blood vessel wall. The flux out of the vessel lth vessel segment is discretized by summing permeation over all area elements ds lm : P l v 1 l V s p ds lm C l v C I l,m J l,m K l,m e. 5 m Similarly, the summation of all passive permeation into extravascular volume element ijk is given by 3 4 P e ijk 1 V e ijk p l,m ds lm C v l C e ijk ii l,m jj l,m kk l,m, where ab is the usual Kronecker delta function: ab 1, a b 0, a b. The summation in Eq. 6 is over all surface area elements in the network, but the Kronecker delta functions insure that only surface area elements ds lm located within the volume element ijk are counted in the sum. Diffusion The diffusive mass flux density in the extravascular space is given by Fick s Law: Q d D e C e. If we approximate the gradient vector with the finite-difference approximation C e 1 x C e i 1,j,k C ijk i, e,c j 1,k e C ijk i, e,c j,k 1 e C ijk e, 8 then continuity implies that the diffusive term D ijk is given by D D ijk e V ijk A e x ijk x C i 1,j,k e C ijk e A i 1,jk i 1,j,k x C e C e ijk A y ijk C e i, j 1,k C e ijk A y i, j 1,k C e i, j 1,k C e ijk A z ijk C e i, j,k 1 C e ijk A z i, j,k 1 C e i, j,k 1 C e ijk. APPLICATION TO TEST PROBLEMS Single-Vessel Network As a first test of the IVol method, we consider the simple network shown in Fig. 2 A a straight 5- mdiam vessel positioned in the center of a 400 m 50 m 50 m region. The length of the vessel is 400 m with end points located at the Cartesian coordinates 0, 25, and 25 m and 400, 25, and 25. We set the flow in the vessel to m 3 s 1, which corresponds to a perfusion density of 0.5 ml min 1 per ml of tissue. The extravascular diffusion coefficient D e was set to 500 m 2 s 1 and the permeability set to p 10 ms 1. Applying no-flux conditions to the boundaries of the region, we calculated the tracer washout impulse response function for this simple system at various levels of spatial resolution of the computational lat

5 Transport Models from Microvascular Anatomy 841 tice. The time step for integrating Eq. 3 is limited by the advection step and is set to the segment volume divided by the flow. Plotted in Fig. 2 B are computed impulse response functions for this system. An impulse of solute is injected into the inflow of the vessel at time t 0, Eq. 3 is numerically integrated, and the impulse response function h(t) is the calculated concentration at the outflow. For these curves the grid sizes x of 25, 10, and 5 m indicated in Fig. 2 B refer to the resolution used for the volume mesh as well as the size of the vessel segments. We note that even at the relatively coarse resolution of 25 m, the calculated impulse response function is surprisingly similar to the converged high-resolution solution. Differences in the computed results are apparent in the early part first 1 s of h(t) shown in the inset of Fig. 2 B. Other than in the shape of the early peak, the results for x 10 m and x 5 m are not distinguishable. This peak occurring near 0.9 s corresponds to a bolus of tracer that remains intravascular in its passage through the system. For grid resolutions of 5 and 10 m the intravascular bolus peak corresponds to a single time step and the area under the curve is a tiny fraction less than 1/1000 of the total outflow curve. specified at the boundary nodes. Thus closer attention needs to be paid to hemodynamics for practical applications. However, for generating a nontrivial flow distribution which is useful for a test problem, the linear treatment is sufficient. Impulse response outflow curves for this network are shown in Fig. 3 B for various levels of microvessel permeability. Transport curves were calculated using a grid resolution of 10 m, which was found to be sufficiently small for generating converged results. In the left panel, the early part of the curve is plotted on a linear scale. For these simulations D e was set to 500 m 2 s 1. As the permeability increases, the peak decreases in magnitude as a greater fraction of tracer permeates and diffuses into the extravascular space. The tails of the curves become exponential as demonstrated in the semilog plot in the right panel of Fig. 3 B. As is apparent in Fig. 3 B, the initial appearance of the high-permeability tracer precedes the lowerpermeability tracers. This early appearance is due to diffusional shunting of the tracer from inlet vessels to proximate outlet vessels. Such pathways for shunting are not available in one-dimensionally distributed systems. Washout from a Complex Network To test our methodology with a more realistic network, we accessed the virtual library of microcirculatory network structures made available by Secomb and colleagues 16 and downloaded the coordinates of a threedimensional microcirculatory network from rat brain. 8,15 The structure, shown in Fig. 3 A, consists of two topologically distinct networks in a 150 m 160 m 140 m region. The larger network has ten boundary input or output nodes, and the smaller network indicated by the arrow has no bifurcations and thus only one input and one output node. To simulate blood supply to the network, we designated the five boundary nodes associated with vessels of 6 m diam and larger in the larger network structure as input nodes. The remaining five boundary nodes 5 m diam and smaller were designated as output nodes. Input nodes were all set to the input pressure of 1 in dimensionless units and output nodes to 0. For the smaller network structure the pressure drop was set to 0.1. We fixed the total flow into the system to be m 3 s 1 corresponding to a perfusion density of 0.5 ml min 1 per ml of tissue, and calculated the flows throughout the network by assuming that vessel resistance is proportional to length divided by the fourth power of the diameter. We have ignored any potential nonlinearities associated constitutive properties of the perfusate such as the effective viscosity of blood. In addition we lack physiologically based justification for how the pressures are Steady-State Problem with Consumption To quantify the accuracy of prediction of the extravascular concentration field, we consider the test problem with zero-order consumption in the tissue: C e D t e 2 C e G e. 10 With a constant input concentration, C in 1 M, and constant consumption G e Ms 1, a spatially varying steady-state concentration field is established in the tissue, as illustrated in Fig. 4. For this calculation we set tissue perfusion to 0.5 ml min 1 per ml of tissue, D e 500 m 2 s 1, and p 10 ms 1. Shown in Fig. 4 A is the microvascular network studied in the previous section with a slice at the z 70 m plane indicated by a 10 m grid. Figure 4 B shows the concentration profile over the z 70 m slice predicted using the computational grid resolution of x 10 m; Fig. 4 C shows the profile obtained with x 5 m. The x 10 m solution differs from the x 5 m solution by a root-mean-square deviation of 3.6%, and by less than 7% everywhere in the computational domain. Further refinement reveals differences of less than 1% between solutions obtained on x 5 m and x 2.5 m grids.

6 842 DANIEL A. BEARD CONCLUSIONS AND RECOMMENDATIONS FIGURE 4. Predicted solute distribution for steady-state transport with metabolic consumption. A A twodimensional slice through the computational domain, located at zä70 m is illustrated using a grid in the x y plane with 10 m line spacing. B The extravascular solute concentration C e at zä70 m, obtained for the steady-state problem with consumption with xä10 m is plotted. C The steady-state concentration obtained for xä5 m is shown. The lower-resolution xä10 solution differs from the xä5 m solution by less than 4%. Parameters used for these simulation are D e Ä500 m 2 s À1, pä10 ms À1, C in Ä1 M, and G e ÄÀ7Ã10 À3 Ms À1, and the perfusion is 0.5 ml min À1 per ml of tissue. As the use of anatomically reasonable structural data as the basis for modeling and simulation is being recognized as fundamentally important for the accurate description of physiological transport processes, 2 5,13,14 and the undertaking of the Microcirculation Physiome Project is beginning to provide an infrastructure for organizing a functional database of microcirculatory physiology and anatomy, 9,10 the need for accurate and efficient computational tools for simulating transport which are compatible with the anatomic data is clear. We have presented one such tool, the inhomogeneous volume method, which proves efficient at describing the processes which we study here tracer washout of a permeating diffusible tracer from three-dimensional microvascular networks. Surprisingly, the required resolution of the computational lattice, around 5 10 m for the rat brain network Fig. 4 A, is larger than the microvessel radii. The method remains accurate at this relatively low resolution because features of the network anatomy are resolved within the elements of the computational lattice. Thus the IVol method is expected to consume substantially less CPU time and memory than a standard finitedifference scheme operating at a similar level of accuracy. Simulation of the washout for the rat brain network requires less than 30 s of CPU time per 60 s of simulated transport on a desktop PC 366 MHz Pentium II. The assumption that the cross-sectional concentration gradient in the microvessels is negligible is undoubtedly not valid for a number of solutes, including oxygen. This limitation is associated not with the IVol discretization of the extravascular space, but with the treatment of the intravascular concentrations. Radial intravascular gradients could be incorporated explicitly with a finer discretization of the intravascular space, or implicitly via an empirical parametrization of the intracapillary transport resistance, as implemented by Goldman and Popel. 5 Our simulations of transport in the rat brain microcirculation made use of a database of anatomic structures. 16 While the structure of the network was straightforward to deduce from the available data, hemodynamic properties e.g., flows were not. The assignment of a fixed pressure drop between input and output nodes, while providing us with a reasonable flow structure for testing our transport simulation methodology, certainly does not provide an accurate description of the hemodynamics in the microvessel network. The structural database should be modified to contain functional information, such as observed flows and hematocrits in the vessel segments, where those data are available. Barring the availability of such data, information on which boundary nodes are input upstream nodes and which are output downstream nodes would be helpful in calculating flow and hematocrit based on microscopic hemodynamic laws. 11,12

7 Transport Models from Microvascular Anatomy 843 In other words, adding more functional information to the anatomic information in the structural database would aid the development of simulation tools. We find that accurate prediction of the spatial concentration field in the extravascular space requires a finer grid resolution than is required for the calculation of the washout of an inert tracer. Although the predicted outflow concentration curve converges for grid resolution of 10 m and less, the 10 m grid produces an error of about 4% in the predicted steady-state concentration field. While the numerical convergence is satisfactory for the steady-state problem studied here when the grid size is reduced to 5 m, different network geometries and the introduction of nonlinear metabolic consumption and/or biochemical transformation may influence the convergence and accuracy of the IVol method in practice. REFERENCES 1 Bassingthwaighte, J. B. Strategies for the Physiome Project. Ann. Biomed. Eng. 28: , Beard, D. A., and J. B. Bassingthwaighte. Advection and diffusion of substances in tissues with complex vascular networks. Ann. Biomed. Eng. 28: , Beard, D. A., and J. B. Bassingthwaighte. Fractal nature of myocardial blood flow described by a whole-organ model of arterial network. J. Vasc. Res. 37: , Chinard, F. P. Water and solute exchanges. How far have we come in 100 years? What s next? Ann. Biomed. Eng. 28: , Goldman, D., and A. S. Popel. A computational study of the effect of capillary network anastomoses and tortuosity on oxygen transport. J. Theor. Biol. 206: , Kassab, G. S., and Y. C. Fung. Topology and dimensions of pig coronary capillary network. Am. J. Physiol. 267:H319 H325, Kassab, G. S. The coronary vasculature and its reconstruction. Ann. Biomed. Eng. 28: , Motti, E. D. F., H. G. Imhof, and M. G. Yasargil. The terminal vascular bed in the superficial cortex of the rat. A SEM study of corrosion casts. J. Neurosurg. 65: , Popel, A. S., A. R. Pries, and D. W. Slaaf. Microcirculation physiome project. J. Vasc. Res. 36: , Popel, A. S., A. S. Greene, C. G. Ellis, K. F. Ley, T. C. Skalak, and P. J. Tonellato. The Microcirculatory Physiome Project. Ann. Biomed. Eng. 26: , Pries, A. R., T. W. Secomb, P. Gaehtgens, and J. F. Gross. Blood flow in microvascular networks, experiments and simulation. Circ. Res. 67: , Pries, A. R., and T. W. Secomb. Microcirculatory network structures and models. Ann. Biomed. Eng. 28: , Secomb, T. W., R. Hsu, M. W. Dewhirst, B. Klitzman, and J. F. Gross. Analysis of oxygen transport to tumor tissue by microvascular networks. Int. J. Radiat. Oncol., Biol., Phys. 25: , Secomb, T. W., and R. Hsu. Simulation of O 2 transport in skeletal muscle: Diffusive exchange between arterioles and capillaries. Am. J. Physiol. 267:H1214 H1221, Secomb, T. W., R. Hsu, N. B. Beamer, and B. M. Coull. Theoretical simulation of oxygen transport to brain by networks of microvessels: Effects of oxygen supply and demand on tissue hypoxia. Microcirculation (Philadelphia) 7: , Secomb, T. W. et al. Microvascular networks: 3D structural information. network.html 17 The Microcirculation Physiome Project.