A Mathematical Model for Understanding Fluid Flow Through Engineered Tissues Containing Microvessels

Transcription

1 Kristen T. Morin Department of Biomedical Engineering, University of Minnesota, Minneapolis 55455, MN Michelle S. Lenz Department of Biomedical Engineering, University of Minnesota, Minneapolis 55455, MN Caroline A. Labat Department of Biomedical Engineering, University of Minnesota, Minneapolis 55455, MN Robert T. Tranquillo 1 Department of Biomedical Engineering, University of Minnesota, Minneapolis 55455, MN; Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis 55455, MN tranquillo@umn.edu A Mathematical Model for Understanding Fluid Flow Through Engineered Tissues Containing Microvessels Knowledge is limited about fluid flow in tissues containing engineered microvessels, which can be substantially different in topology than native capillary networks. A need exists for a computational model that allows for flow through tissues dense in nonpercolating and possibly nonperfusable microvessels to be efficiently evaluated. A finite difference (FD) model based on Poiseuille flow through a distribution of straight tubes acting as point sources and sinks, and Darcy flow through the interstitium, was developed to describe fluid flow through a tissue containing engineered microvessels. Accuracy of the FD model was assessed by comparison to a finite element (FE) model for the case of a single tube. Because the case of interest is a tissue with microvessels aligned with the flow, accuracy was also assessed in depth for a corresponding 2D FD model. The potential utility of the 2D FD model was then explored by correlating metrics of flow through the model tissue to microvessel morphometric properties. The results indicate that the model can predict the density of perfused microvessels based on parameters that can be easily measured. [DOI: / ] Introduction The engineering of microvascular networks in vitro is of interest to basic scientists and tissue engineers alike. Much can be learned about endothelial cell biology from in vitro microvascular networks, which are more accessible and controllable than in vivo networks. In addition, engineered microvasculature is required to deliver nutrients to engineered tissue with high nutrient demand, such as myocardium. Flow of culture medium in vitro (and ultimately blood in vivo) through such an engineered microvascular network is an essential component of efficient nutrient delivery. Although a wide variety of mathematical models have been developed to understand fluid flow and nutrient exchange through native capillary networks [1 5], no models have been developed for engineered microvascular networks. Engineered microvasculature obtained via vasculogenesis is often quite dissimilar to native microvasculature. Capillarylike structures (microvessels) are often isolated from other cellular structures, and lumens often do not extend the entire length of the structure [6 9]. Branches may occur, but not necessarily at physiologically relevant angles. In addition, it may be that the ends of some microvessels are open, enabling fluid from the interstitium to flow through, while the ends of others are closed, creating lumenal space isolated from the interstitial fluid. Theoretically, if the ends of microvessels were open, interstitial flow would drive fluid through their lumens because the lumens would provide less resistance to flow than the surrounding extracellular matrix (ECM). A model describing fluid flow through such an engineered microvasculature could be useful for determining parameters such as the density of microvessels that actually carry flow, which may be difficult to directly measure. Therefore, we developed an idealized FD model to describe pressure and fluid flow through a tissue comprising engineered microvessels within ECM. The FD model is based on Poiseuille flow through a distribution of straight tubes (microvessels) and 1 Corresponding author. Manuscript received July 24, 2014; final manuscript received November 8, 2014; published online February 24, Assoc. Editor: Ram Devireddy. Darcy flow through the surrounding interstitium (ECM). To simplify the mass and linear momentum balances, the tubes have infinitesimal radius and therefore no interaction with the interstitial flow except to act as point sources and sinks of mass and momentum. A finite radius is admitted solely for calculating the pressure-driven Poiseuille flow. These assumptions result in a substantial reduction in computational time that allows for consideration of distributions of large numbers of tubes. The tubes can possess a single bifurcation (as a first step toward complex topologies), and they require fluid to enter or leave from the interstitium only at their ends (i.e., tubes have zero permeability). Accuracy of the FD model was assessed by comparison to a 3D FE model in two ways. First, the error associated with the FD model assumptions stated above was assessed by comparing predictions for a single tube using a 3D implementation of the FD model and the FE model. Then, the additional error associated with representing flow through a tissue with microvessels aligned with the flow but confined to the same plane using a 2D implementation of the FD model was assessed. The rationale for this 2D FD model is to examine flow through domains dense in tubes that are uniaxially aligned, as we study experimentally [10]. By restricting the aligned microvessels to lie in parallel planes (Fig. 1), which are consistent with flow occurring predominantly in the plane, we further reduced the computational time needed to understand the basic fluid mechanics of an idealized system using the 2D FD model. Our experimental experience with engineered microvessels was used to demonstrate the potential uses of the 2D FD model in understanding the relationship between metrics of flow through the model tissue, such as pressure drop, to microvessel morphometric properties that are difficult if not impossible to measure, such as the density of microvessels that possess open ends and are thus perfusable. Methods FE Model of Fluid Flow in Tissues Containing Engineered Microvessels. An FE model was formulated based on the Brinkman extension of Darcy flow through the interstitium (ECM) and laminar viscous flow through the straight tubes (microvessels) for a Journal of Biomechanical Engineering Copyright VC 2015 by ASME MAY 2015, Vol. 137 /

2 Table 1 Parameter Parameter values for the single vessel FE model Value Permeability a cm 2 Tissue length (x-direction) 250 lm Tissue width (y-direction) 200 lm Matrix anisotropy 1 Tissue porosity 0.95 Viscosity a 0.89 cp Fluid density 1000 kg/m 3 X-velocity at tissue inlet 1.33 lm/s Pressure at tissue outlet 0 Microvessel length 100 lm Microvessel radius 2.5 lm Microvessel angle 10 deg Microvessel wall thickness 1 lm a Parameter values obtained from Ref. [10]. Fig. 1 Diagram showing case of interest: Uniaxially aligned microvessels confined to the plane. Only two planes are drawn for clarity. Newtonian fluid, which is valid because cell culture medium would be the perfusate for in vitro studies. The FE model was created in COMSOL Multiphysics (v. 4.2a) using the Free and Porous Medium module. The Brinkman equations are defined by COMSOL as q ðv rþ v ¼r PI þ l rv þðrvþ T 2l ðrvþi e p e p e p 3e p l k þ b fjjþ v Q v þ F (1) qrv ¼ Q (2) where q is fluid density, e p is ECM porosity, v is velocity, P is pressure, l is dynamic viscosity, k is interstitium permeability, b f is the Dupuit Forcheimer drag coefficient, Q is volumetric flow rate associated with distributed mass sources, and F is the net external force. b f, Q, and F were set to zero given the problem of interest (distributed mass sources are modeled explicitly as described below). The Navier Stokes equations are defined by COMSOL as h i qððv rþvþ ¼ r PI þ l rv þðrvþ T þ F (3) qrv ¼ 0 (4) in which the symbols represent the same variables as in the Brinkman equations above but refer to the fluid in the tubes. Again, F was set to zero. The convective acceleration term in Eq. (3) was not set to zero a priori but the Reynolds number in the tubes was small for the parameter space examined (Re ¼ for the standard parameter values of Table 1). The analysis assumes the domain is normal to gravity, so the body force term is not applicable. The surface tension term was also neglected because it is not relevant to pressure-driven flow of filled channels at steady state, as assumed. At the interface between the end of a tube and the interstitium, the normal stresses were equated h i l rv þðrvþ T n PI ¼ l rv þðrvþ T PI tube e p ECM (5) Boundary conditions applied were no flux through the sides of the tissue, no slip at the sides of the tissue and on the tube surface, fixed and uniform pressure at the tissue outlet, and fixed, uniform, unidirectional flow at the tissue inlet. Only simple, prescribed geometries were characterized with this model to enable direct comparison of the FE and FD models. It was not possible to model more complex geometries in COMSOL due to the software s inability to mesh sharp angles between neighboring elements. In addition, more complicated geometries would take an unreasonably long time to converge to a solution, and our intent here is simply to assess validity of the FD model assumptions. Once the geometry was defined, a tetrahedral mesh was applied, and the equations were solved for the pressure and velocity fields. Standard parameter values for the FE model are given in Table 1. FD Model of Fluid Flow in Tissues Containing Engineered Microvessels. The FD model was developed by further assuming the tubes possess infinitesimal radius, that is, interaction of interstitial flow with the tubes was ignored. Flow of the fluid (culture medium) through the interstitium was governed by Darcy s Law v ¼ KrP (6) where v is fluid velocity, Kð¼ k=lþ is the hydraulic conductivity of the interstitium. Fluid flow within tubes followed the Hagen Poiseuille Equation for steady laminar flow of a Newtonian fluid in a smooth cylindrical tube Q ¼ pr4 DP (7) 8lL where Q is the volumetric flow rate, r is the radius of the tube, P is pressure, and L is the length of the tube. Fluid entering or exiting a tube was considered to be a point sink or source, respectively, of mass and linear momentum, which was handled by adding an additional term to the balance equations based on the length, radius (a finite radius for each tube was used to calculate resistance to tube flow), and orientation of the tube. The following are balance equations for the case of interest, a tube lying in the x y plane oriented with angle h relative to flow in the x-direction (macroscopic flow is always assumed to be 1D, in the x-direction): Mass Q t rv ¼d i D 3 (8) Momentum e x ðvþk rpþd i v t cos h ¼ 0 (9) e y ðvþk rpþd i v t sin h ¼ 0 (10) e z ðv þ K rpþ ¼ 0 (11) in which Q t is the volumetric flow rate through the microvessel, D is the distance between equispaced nodes, v is fluid velocity in the / Vol. 137, MAY 2015 Transactions of the ASME

3 interstitium, K is the hydraulic conductivity tensor, v t is fluid velocity through the tube, and d i signifies a tube end located at position corresponding to node i (minus sign applies to an upstream end and is associated with a point sink, and plus sign applies to a down stream end and is associated with a point source). As the equations indicate, at node i, the velocity through the tube and the interstitial velocity are assumed to be independent in terms of magnitude and direction; this additional assumption needed to yield a well-posed linear algebraic system is expected to be more valid in the limit of decreasing density and radius of the tubes, when the percentage of flow through a tube becomes smaller. For tubes containing one bifurcation, each portion of the tube was assigned a radius, and flow through each portion was determined by the pressure difference between the bifurcation point and the end open to the interstitium. The pressure at the bifurcation point was determined by enforcing mass conservation through the tube. Mass and linear momentum balances were evaluated at each node using first-order FD approximations (see the Appendix). The same boundary conditions as used in the FE model were used in the FD model, except no slip at the tissue sides was not enforced since Darcy s law was used, and no slip on the tube surface was not applicable (consistent with tubes of infinitesimal radii). The system of linear equations for the velocity and pressure fields was solved in MATLAB (Mathworks v. R2012b). Microvessel density was specified as a total number of tubes, and tube location and dimensions could be specified directly or generated randomly based on an average and standard deviation of each tube parameter (length, radius, and angle). Specified tubes were used in error assessment, and randomly generated tubes, based on parameters measured from engineered microvessels, were subsequently analyzed. The results from six randomly generated tube configurations were averaged to produce each data point presented. The standard parameter values used for the FD model are listed in Table 2. Error Assessment for the 3D FD Model. The error associated with the key FD model assumptions (i.e., tube ends as point sources/sinks, infinitesimal tube radius, Poiseuille flow throughout the tube, and no correlation between interstitial flow direction and tube orientation) was first assessed by comparing predictions for a single tube using the 3D FD model and the FE model. In both cases, the tube was located at the same location and orientation (10 deg with respect to the flow (x) direction in the x y plane). In Table 2 Parameter Tissue, fluid, and boundary parameters for FD models Value Permeability a cm 2 Tissue length (x-direction) 250 lm Tissue width (y-direction) 200 lm Matrix anisotropy 1 Node spacing 5 lm Viscosity a 0.89 cp X-velocity at tissue inlet 1.33 lm/s Pressure at tissue outlet 0 Defined microvessels (validation geometries with one or two microvessels) Parameter Value Microvessel length 100 lm Microvessel radius 2.5 lm Microvessel angle 10 deg Randomly generated microvessels Parameter Value Microvessel radius (standard deviation) 3 lm (0.125 lm) Microvessel length (standard deviation) 100 lm (10 lm) Microvessel angle (standard deviation) 0 (20 deg) Lumen density 275 (þ/ 2.5%) Percentage bifurcating microvessels 50% a Parameter values obtained from Ref. [10]. order to have a basis for comparison in terms of similar spatial resolution, the tissue thickness was set to 70 lm, which was the maximum problem size that could be solved in MATLAB in 3D for the chosen node spacing, and a coarse meshing was used in COM- SOL so that the node spacing was approximately the element size. Error Assessment for the 2D FD Model. The additional error associated with representing flow through a tissue with microvessels aligned with the flow but confined to the same plane (Fig. 1)usinga 2D FD model was assessed. Again, a single tube was first used in both the 2D FD and FE models for comparison. The only difference from the 3D FD model was that the z-momentum component of the FD model balance equations (Eq. (12)) was omitted. While there is z-momentum intrinsically associated with fluid flow into and out of a tube, the assumption is that for the case of interest (Fig. 1), there is no macroscopic flow in the z-direction. The rationale was this omission might incur acceptable error in order to substantially increase computational efficiency. For the purpose of comparing volumetric flow rates, a tissue thickness of one tube diameter was used. Parametric sensitivity analysis for both models (2D FD and FE) was conducted. Each parameter was examined at 5 7 values in a range that included its standard value. Parameters that were varied in both models included inlet velocity, interstitium permeability, fluid viscosity, tube radius, and tube angle relative to the x-direction. In the 2D FD model, the interstitium permeability anisotropy was also varied by decreasing the interstitium permeability in the y-direction while holding the permeability in the x-direction constant. This was not varied in the FE model because a permeability tensor (which describes interstitium anisotropy) was not supported in COMSOL s Free and Porous Medium module. The effects of separation distance between tubes for a dual tube geometry were also investigated for both models. The first tube was held at a constant location whereas the second tube was translated in the x(flow) direction to achieve varied distances between the outlet of the first tube and the inlet of the second tube, which were held at the same y value. Standard parameter values (Tables 1 and 2) were used for both tubes. Evaluation of the 2D FD Model Predictive Capability. To evaluate the predictive capability of the FD model for the case of interest (Fig. 1), two flow metrics were assessed over a range of parameter values: Pressure at the tissue inlet and an effective hydraulic permeability for the tissue, k eff (i.e., accounting for flow through both the ECM and tubes). k eff was obtained via Darcy s Law (Eq. (6)), with the velocity set to the x-velocity at the tissue inlet. Defined this way, the effective permeability is proportional to the inverse of the tissue inlet pressure. These were chosen because they could easily be measured experimentally. The input parameters that were varied included tube density, percentage of bifurcations, interstitium anisotropy, tube anisotropy index, and tube length. Tube density was calculated assuming the 2D FD model represented the projection of a 3D tissue with a square cross section. The number of tubes crossing each column (y-direction) of nodes was counted and averaged for the middle third of the tissue domain in the x-direction andnormalizedtotheassumedcross-sectionalarea.thevariationin microvessel densities calculated when using the same number of tubes is noted in Table 2. Each bifurcating tube was made from three straight tubes (one inlet and two outlets). The angle of each straight tube in the FD model was determined based on a random distribution of physiologically relevant angles around 0 deg (in the x-direction). So, the angle of the bifurcation was determined from the randomly generated angles of the two outlet tubes. The interstitium anisotropy was varied by decreasing the permeability in the y-direction. The tube anisotropy index, a measure of tube alignment in the direction of flow, was defined as P Lx AI ¼ P ; (12) Ly in which L x is the tube length in the x-direction, and L y is the tube length in the y-direction. A higher anisotropy index reflects a Journal of Biomechanical Engineering MAY 2015, Vol. 137 /

4 higher degree of tube alignment. For each analysis, only one parameter was varied; standard parameter values listed in Table 2 were used otherwise. Each data point represents the average of six runs in which the tubes were randomly generated based on an average radius, length, and angle relative to the x-direction. Results 3D FD Model Error Assessment. Example velocity vector plots and pressure maps using the standard parameter values (Tables 1 and 2) are shown for the (3D) FE model in Figs. 2(a) and 2(b) and the 3D FD model in Figs. 2(c) and 2(d). It can be seen that the 3D FD model qualitatively captures the exact solution. The exaggerated perturbation of the velocity and pressure seen near the microvessel inlet and outlet for the 3D FD model are reflected in quantitative differences in various flow field metrics listed in Table 3, comparing the 3D FD results and the COMSOL results for the thin tissue with coarser mesh (a coarser mesh was used so that the computations yielded similar spatial resolution for both models; COMSOL results for a finer mesh are included for comparison). The 3D FD model predicted a four times greater flow through the microvessel and a two times greater pressure drop across the microvessel. 2D FD Model Error Assessment. Accuracy of the 2D FD model was first evaluated using a single microvessel. Example velocity vector plots and pressure maps using the standard parameter values are shown in Figs. 3(a) and 3(b) for the 2D FD model and Figs. 4(a) and 4(b) for the (3D) FE model. As seen for the case of the 3D FD model (Fig. 2), there is a qualitative resemblance of the velocity and pressure fields. Interestingly, quantitative differences are less for various flow field metrics listed in Table 3 when comparing the 2D FD results and the FE results (thick tissue, finer mesh) versus comparing the 3D FD results and the FE results (thin tissue, coarser mesh) as discussed above. Parametric sensitivity results for the 2D FD model when varying the velocity at the tissue inlet are shown in Fig. 3(c). The linear relationships indicate that both the velocity through the microvessel and the pressure at the tissue inlet increased by the same factor as the velocity at the tissue inlet was varied. As the ECM permeability was decreased, the inlet pressure remained constant near zero until a permeability of the order of cm 2, at which point it rapidly increased (Fig. 3(d)). Fluid velocity through the microvessel increased as ECM permeability decreased until approximately the same permeability value, at which point further decreases in permeability did not increase the velocity. Variations in fluid viscosity had minimal effects on microvessel flow velocity, but high viscosities led to increased pressures at the tissue inlet (Fig. 3(e)). Microvessel radius was varied in a small, physiologically relevant range (Fig. 3(f)). Tissue inlet pressure remained constant across radii, but the fluid velocity in the microvessel decreased slightly with increasing radius. The angle of the microvessel relative to the x-direction had little effect on either the inlet pressure or the microvessel velocity (Fig. 3(g)). Little effect was observed on tissue inlet pressure when the ECM anisotropy (k x =k y ) was varied by decreasing k y, but with increasing anisotropy the microvessel flow velocity decreased (Fig. 3(h)). Parametric sensitivity for the FE model was similarly evaluated using a single microvessel (Figs. 4(c) 4(g)). The dependences for the FE model are qualitatively similar to the 2D FD model (cf. Figs. 3(c) 3(g) and Figs. 4(c) 4(g)), except the dependence on microvessel radius is much less pronounced (Fig. 4(f)), and there was a modest decrease in microvessel velocity with increasing angle (Fig. 4(g)). Fig. 2 Comparison of 3D FD and FE models for a single microvessel. Velocity vector plot (a) and pressure map (b) for the 3D FD model, and velocity vector plot (c) and pressure map (d) for the FE model. In (b) and (d), the pressure units are mm Hg. Values plotted are for the plane containing the microvessel. Parameter values used are from Tables 1 and 2. Tissue thickness lm / Vol. 137, MAY 2015 Transactions of the ASME

5 Table 3 Comparison of flow metrics between the FD and FE models V tube (cm/s) P tube, inlet (mm Hg) P tube, outlet (mm Hg) DP tube (mm Hg) P tissue, inlet (mm Hg) Q tube (10 9 cm 3 /s) % Q total through tube 3D FD thin tissue D FE thin tissue, coarser mesh D FE thin tissue, finer mesh D FD D FE thick tissue The error in the 2D FD model as a function of microvessel radius was further characterized. Several single microvessel geometries spanning several microvessel lengths (100 and 150 lm) and orientations (0 deg, 10 deg, and 35 deg) were also considered. The results for these different microvessels were similar, so only the results using standard parameter values are presented. An error minimum was observed for all flow metrics at a radius of approximately 2.5 lm (Fig. 5), which corresponds to half the set node spacing. Percent errors of the 2D FD model for various flow metrics were as follows, indicating reasonable agreement: Velocity through microvessel ¼ 1.6% error, percentage of total flow through microvessel ¼ 29.7% error, and pressure difference across tissue ¼ 17.5% error. Using higher order FD approximations had little effect on the errors (not shown). The error minimum similarly occurred for a node spacing of approximately half the radius when the radius was set and the node spacing varied (not shown). Assessment of a two-microvessel geometry was also performed to further define the error in the 2D FD model in advance of solving for higher microvessel densities (see below). Both microvessels were defined by the standard parameter values, and the distance between the outlet of the first microvessel and the inlet of the second microvessel was varied. The velocity vector plots from each model (Figs. 6(a) and 6(b)) for the smallest separation distance possible (one node spacing) show resemblance. The error was quantified for the velocity through the downstream microvessel and the pressure at the tissue inlet. Both errors decreased with increased distance between the microvessels and were relatively low for all separation distances (Fig. 6(c)). Evaluation of 2D FD Model Predictive Capability. Sample pressure maps for microvessel densities of approximately 25 and 550 microvessels/mm 2 are shown in Figs. 7(a) and 7(b). The pressure at the tissue inlet and the effective permeability of the tissue (i.e., including flow through both the ECM and microvessels) were computed flow metrics from the 2D FD model for six randomly generated microvessel geometries, which were then averaged, while individual parameter values were varied. The percentage of bifurcations, microvessel anisotropy index, and the microvessel length had little effect on either metric (Figs. 7(c) 7(e)). If the microvessels nearly spanned the tissue region (e.g., 230 lm in length), the outputs were affected, as expected (not shown). As anisotropy of ECM permeability increased, the tissue inlet pressure decreased, and the effective hydraulic conductivity increased (Fig. 7(f)). A similar trend was observed with increasing microvessel density (Fig. 7(g)), where the percentage of microvessels with end spacings less than 10 lm were 8%, 29%, and 70% for lumen densities of 25, 550, and 2250 lumen/mm 2, respectively (cf. Fig. 6). Discussion Overall, the results indicate that the FD model is reasonably accurate for experimentally relevant parameter values despite all of its assumptions, including microvessels modeled as tubes that have infinitesimal radii (in terms of effects on interstitial flow) acting as points sources and sinks. This was based on comparison of the 3D FD model with the FE model, which did not make these assumptions, among several others, for the case of a single microvessel (tube) embedded in an ECM (interstitium). It was also based on comparison of a 2D FD model motivated by the case of interest aligned microvessels as shown in Fig. 1 with the FE model. A comparison of the three models (Table 3) indicates the majority of the error is inherent to the FD model, not in using its 2D approximation relevant to Fig. 1. This is not surprising given the tradeoff for such a simple FD model is the paradox of defining two fluid velocities (ECM and microvessel) at the inlet and outlet nodes of each microvessel. Fortuitously, the error associated with the FD model assumptions was reduced for 2D FD versus 3D FD, at least for the single tube case. Parametric sensitivity analyses of the 2D FD and FE models were similar for the same single microvessel case. They indicated that the microvessel flow velocity was largely insensitive to changes in fluid viscosity and microvessel angle, as expected. Microvessel angle also had no effect on the pressure at the tissue inlet, but at high viscosities (beyond physiological), the pressure increased dramatically from near zero. This result is intuitive, as more viscous fluids would require larger pressures to drive the same flow rate. Microvessel flow velocity decreased with increasing ECM anisotropy, which may at first appear counterintuitive; however, examination of the velocity vector plots shows that fluid from a variety of y positions enters the tube. If flow is retarded in the y-direction, less fluid would enter the microvessel. Increases in the inlet velocity caused proportional increases in inlet pressure and microvessel flow velocity, as expected. One of the more interesting parametric sensitivities was ECM permeability. As the permeability was decreased, the microvessel flow velocity increased, which is logical because the ratio of resistance to flow between the microvessel and the ECM was decreasing; essentially the effective permeability of the tissue remained constant. However, near a permeability of cm 2, the microvessel velocity reached a plateau, and the tissue inlet pressure, which had previously been low, increased substantially. This result is also logical because a high pressure will be required to drive flow through a tissue with a very low permeability. The microvessel flow velocity plateau is somewhat less intuitive because the ratio of resistance to flow between the microvessel and the ECM was still decreasing. However, examination of the velocity vector plots at relatively high permeabilities indicates that the higher velocity through the microvessel required fluid to be diverted from a farther distance from the microvessel inlet. In the plateau region, a balance was apparently established between the decreasing ratio of resistance to flow and the increasing pressure required to divert fluid through the microvessel from further distances. The largest difference in parametric sensitivities occurred in the radius variation. In both models, the inlet pressure remained constant, but in the 2D FD model the microvessel velocity decreased with increasing radius, whereas in the FE model the microvessel velocity was nearly constant. Although the lack of agreement is somewhat disconcerting, the maximum difference is less than two-fold, a difference that may be acceptable for the computational speed enhancement and other benefits of the 2D FD model. Comparison of the 2D FD and FE model results demonstrated that the 2D FD model was most accurate for a small range of microvessel radii. This was expected since the assumption that v t was independent of v i at microvessel inlets and outlets is intuitively more reasonable as the microvessel radius decreases. For Journal of Biomechanical Engineering MAY 2015, Vol. 137 /

6 Fig. 3 2D FD model results for a single microvessel. (a) Velocity vector plot of the outlet of the microvessel for standard parameter values (Table 2). The thick line represents the microvessel. The arrows in the bulk represent interstitial velocities, whereas the arrow at the end of the microvessel represents the velocity through the microvessel. (b) Pressure map of the whole tissue. (c) (h) Velocity through the microvessel and the pressure at the tissue inlet plotted against the following parameters: (c) x-velocity at the tissue inlet, (d) ECM permeability, (e) fluid viscosity, (f) microvessel radius, (g) microvessel angle relative to the x- direction, and (h) ECM permeability anisotropy. For each plot, all other input parameters were kept at their standard values / Vol. 137, MAY 2015 Transactions of the ASME

7 Fig. 4 FE model results for a single microvessel. (a) Velocity vector plot of the region near the microvessel outlet for standard parameter values (Table 1, tissue thickness lm). The arrows indicating the velocity through the microvessel were removed to more clearly show the interstitial velocities. (b) Pressure map of the whole tissue. (c) (i) Velocity through the microvessel and the pressure at the tissue inlet plotted against the following parameters: (c) x-velocity at the tissue inlet, (d) ECM permeability, (e) fluid viscosity, (f) microvessel radius, and (g) microvessel angle relative to the x-direction. For each plot, all other input parameters were kept at their standard values. Journal of Biomechanical Engineering MAY 2015, Vol. 137 /

8 Fig. 5 Dependence of 2D FD model accuracy on microvessel radius for a single microvessel. The percent error in the microvessel fluid velocity relative to the FE model for varying microvessel radius is plotted. example, a smaller percentage of the flow in the ECM near a microvessel inlet enters it for a smaller radius, and so the directions of the local interstitial flow and tube flow become less coupled near the inlet. For a fixed node spacing of 5 lm, the minimum error occurred at a radius of 2.5 lm, which is within the physiologic range for capillaries (2.5 4 lm radius for human coronary capillaries [10]; the radii of engineered microvessels are generally higher by several microns but the physiological range is the target). Therefore, a radius of 2.5 lm was used for all subsequent computations. At a radius of 2.5 lm, the error for a variety of flow metrics was less than 30%, indicating good agreement. An error minimum at a node spacing equal to approximately twice the radius is most consistent with the pressures at the inlet and outlet nodes driving Poiseuille flow through a tube with its inlet and outlet centered on two nodes. Smaller nodal spacings should lead to larger contributions of pressures at adjacent nodes in the tube flow, but this effect is not captured in the FD model equations. Obviously, at larger nodal spacing the error increases because of a loss of spatial resolution in the interstitial flow and pressure fields. Finally, the error in the 2D FD model is expected to decrease for smaller (nonphysiological) radius as the infinitesimal radius assumption is better realized. At microvessel radii near 1.5 lm, the FD model predicted reverse flow through the single microvessel geometry, which was not corroborated by the FE model. These results were taken to be inaccurate, so radii near 1.5 lm were removed from the randomly generated microvessel geometries by using a small range of radii centered on 3 lm. Reverse flow was also occasionally observed in one outlet portion of a bifurcating microvessel regardless of radius. Attempts were made to validate these results with the FE model, but geometries resulting in reverse flow could not be recreated in COMSOL due to its limitations in prescribing microvessel bifurcation angle. Randomly generated microvessel geometries generated for the results of Fig. 7 yielding instances of reversing flow instances were not excluded because these instances were rare and not necessarily counterintuitive (the path of least resistance for flow from a tube outlet might be to the inlet of a neighboring tube located upstream for high ECM flow resistance and close proximity of the tubes). The error in the 2D FD model was also determined for a twomicrovessel geometry. As the separation distance between the microvessels was decreased, the error increased. However, even at the minimum separation distance of 5 lm (equivalent to one node apart in the 2D FD model), the error in several flow metrics was well under two-fold. In addition, the velocity vector plots showed reasonable qualitative agreement. These results suggest that the FD model has acceptable accuracy even when the microvessels were in close proximity. Note that projecting the microvessels into one plane, as was effectively done in the 2D FD model when generating the microvessel densities examined, exaggerated the frequency of microvessels with small separation distances. Fig. 6 Comparison between the 2D FD and FE models for two microvessels. (a) Velocity vector plot showing the FD model results for a separation distance of 5 lm. The thick lines represent the microvessels. Arrows in the bulk represent interstitial velocities and the arrow at the tip of the microvessel represents velocities through the microvessels. (b) Velocity vector plot results from the FE model for a separation distance of 5 lm. No velocity vectors for flow within the microvessels are shown so that the interstitial velocities can be seen. The white rectangles represent the microvessels. (c) The percent error in the downstream microvessel velocity and pressure at the tissue inlet between relative to the FE model for a variety of separation distances between the two microvessels. The rationale for developing the 2D FD model was for use in predicting the morphometric properties of microvessels within engineered tissues exposed to flow, which may be difficult to determine experimentally. With this goal in mind, the tissue inlet pressure and effective tissue permeability, which are both outputs of the model and easily measurable experimentally, were computed as microvessel morphometric parameters were varied. Several of these parameters had little effect on the inlet pressure or the effective permeability, at least at the lower lumen density used for this assessment (275 lumens/mm 2 ). These parameters include the percentage of microvessels containing a bifurcation, the / Vol. 137, MAY 2015 Transactions of the ASME

9 Fig. 7 Evaluation of the FD model predictive capability. Example pressure plots for tissues with microvessel densities near 25 (a) or 550 (b) microvessels/mm 2. The black lines represent the location of microvessels. The color scale is the same for both plots. Pressure at the tissue inlet and the effective hydraulic permeability are plotted for varying levels of the following parameters: (c) percentage of bifurcations, (d) microvessel anisotropy index, (e) microvessel length, (f) ECM anisotropy, and (g) (perfused) lumen density. (f) and (g) The data were fit with power equations, which are shown on the plots. microvessel anisotropy index, and the average microvessel length. These results suggest that neither the measured pressure at the tissue inlet nor the effective permeability can be used to predict any of these microvessel properties. It also suggests, however, that these parameter values need not be known in order to accurately predict other morphometric properties. Journal of Biomechanical Engineering MAY 2015, Vol. 137 /

10 With increasing ECM anisotropy (k x =k y, varied by changing k y ), the tissue inlet pressure increased and the effective permeability decreased. This suggests that the inlet pressure or the effective permeability could be used to predict the ECM anisotropy. However, ECM anisotropy is relatively simple to measure by taking permeability measurements in orthogonal directions. Thus, the main conclusion from this result is that the ECM anisotropy must be known to predict other morphometric properties. (Previous modeling has indicated that even in tissues with high levels of anisotropy, the permeability only varies approximately three-fold [11]; therefore, permeability differences greater than five-fold were not assessed.) As microvessel density increased, the inlet pressure decreased and the effective permeability increased. This result suggests that either the pressure or the permeability could be used to predict the density of perfusable microvessels. The FD model assumes all microvessels are perfusable, and since they have infinitesimal radius, the model predictions are independent of nonperfusable microvessels. The density of perfusable microvessels, although extremely important in understanding the flow through the tissue, is extremely difficult to measure experimentally, because histologic examination does not yield a distinction between microvessels that are perfusable and those that are not. The slope of the curve suggests that the model would be most accurate in predicting perfusable microvessel densities less than 500 lumens/mm 2. Although myocardial capillary density is near 2000 capillaries/mm 2 [11], most other native capillary beds have lower lumen densities, including skeletal muscle ( capillaries/mm 2 ) [12,13]. In addition, only recently have engineered tissues contained microvascular structures nearing 500 microvessels/mm 2 [10]. Therefore, the tissue inlet pressure or the effective permeability could be used to predict the microvessel density of perfused microvessels. A number of other parameters were shown to influence the pressure at the tissue inlet, so these must also be known to make an accurate prediction. These parameters include fluid viscosity, the fluid velocity at the tissue inlet, the ECM permeability, the degree of ECM anisotropy, and the average microvessel radius and its standard deviation, all of which are relatively straightforward to measure experimentally. Microvessel length must also be known if it approaches the dimension of the engineered tissue; equivalently, the dimension of the modeled tissue must be much greater than the length of a microvessel. Additionally, although the predicated tissue inlet pressure values are low, the modeled region represents only a small segment of a typical tissue sample; for example, the model would predict pressures in the range of mm Hg for a tissue 15 mm in length. If pressure at the tissue inlet cannot be measured precisely enough to make predictions, the residence time of a pulsed tracer within the tissue could be used. Theoretically, the tracer would reside in the tissue for a shorter time with a higher density of perfusable microvessels due to the faster velocities observed through the microvessels than in the ECM. However, this would require the extension of this model to include the species balance (convection and diffusion) equations for the tracer. Such an extension would also be useful in examining the oxygen distribution within the tissue, a prediction extremely relevant to tissue engineering. Despite being less accurate, the 2D FD model has many advantages over the FE model. The main advantage is computational speed: The FD model does not compute the flow field within microvessels (which is simply Poiseuille flow because of a negligible entrance length) or around the microvessels (whose effects become more negligible as the microvessel radius decreases); computing time for the FE model, which does compute these flow fields, would be prohibitively long when many microvessels were present. COMSOL also presented several software limitations, including the inability to handle permeability tensors for the study of ECM anisotropy, a limited range of bifurcation angles, and meshing limitations when using small dimensions. The 2D FD model also has the advantages of computational speed and coding simplicity over the 3D FD model. Extension to the experimentally Fig. 8 Schematic of a node (i) at which a microvessel begins. The microvessel is indicated by the solid straight line, and the arrow overlapping the microvessel indicates the fluid velocity in the microvessel (v t ). An interstitial velocity (v i ) was also defined at node i and is indicated by the second arrow. h represents the angle between the microvessel and the x-direction. relevant system, where the aligned microvessels can have out-ofplane orientation, would require using the 3D FD model. Although ideally the error between FD and FE models would be smaller, the vastly decreased computational time outweighs the difference in accuracy between the models. Although the FD model has been verified to be reasonably accurate and practically useful, it has limitations beyond its accuracy. One limitation is the restricted geometries: Microvessels can either be straight or have one bifurcation. Additional bifurcations could be included using the same principles to define pressures at bifurcation points, but this would add coding complexity. A more general model would allow for true networks of microvessels by connecting microvessels that were randomly placed near to each other; this would enable the examination of percolation behavior and is more similar to the way microvessel networks form in vitro. Acknowledgment The authors thank Victor Barocas for discussion leading to the model formulation, the Minnesota Supercomputer Institute for computing resources, and a predoctoral fellowship grant from the American Heart Association 11PRE (to KTM). Appendix The computational node (i) at a tube inlet is shown in Fig. 8. The surrounding nodes were labeled i 1 and i þ 1 in the x-direction and i n and i þ n in the y-direction, where n is the number of nodes spanning the tissue in the x-direction. The corresponding FD approximations ¼ v iþ1 v ¼ v iþn v ¼ P i 1 þ P ¼ P i n þ P i Dy At boundaries of higher coordinate value, backward FD approximations were applied for the velocity derivatives. References [1] Beard, D. A., and Bassingthwaighte, J. B., 2001, Modeling Advection and Diffusion of Oxygen in Complex Vascular Networks, Ann. Biomed. Eng., 29(4), pp / Vol. 137, MAY 2015 Transactions of the ASME

11 [2] Fry, B. C., Lee, J., Smith, N. P., and Secomb, T. W., 2012, Estimation of Blood Flow Rates in Large Microvascular Networks, Microcirculation, 19(6), pp [3] Goldman, D., and Popel, A. S., 2000, A Computational Study of the Effect of Capillary Network Anastomoses and Tortuosity on Oxygen Transport, J. Theor. Biol., 206(2), pp [4] Groebe, K., 1990, A Versatile Model of Steady State O2 Supply to Tissue. Application to Skeletal Muscle, Biophys. J., 57(3), pp [5] Secomb, T. W., Hsu, R., Park, E. Y., and Dewhirst, M. W., 2004, Green s Function Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks, Ann. Biomed. Eng., 32(11), pp [6] Helm, C. L., Fleury, M. E., Zisch, A. H., Boschetti, F., and Swartz, M. A., 2005, Synergy between Interstitial Flow and VEGF Directs Capillary Morphogenesis In Vitro Through a Gradient Amplification Mechanism, Proc. Natl. Acad. Sci. U. S. A, 102(44), pp [7] Lafleur, M. A., Handsley, M. M., Knauper, V., Murphy, G., and Edwards, D. R., 2002, Endothelial Tubulogenesis Within Fibrin Gels Specifically Requires the Activity of Membrane-Type-Matrix Metalloproteinases (Mt-Mmps), J. Cell Sci., 115(Pt. 17), pp [8] Rao, R. R., Peterson, A. W., Ceccarelli, J., Putnam, A. J., and Stegemann, J. P., 2012, Matrix Composition Regulates Three-Dimensional Network Formation by Endothelial Cells and Mesenchymal Stem Cells in Collagen/Fibrin Materials, Angiogenesis, 15(2), pp [9] Sieminski, A. L., Hebbel, R. P., and Gooch, K. J., 2004, The Relative Magnitudes of Endothelial Force Generation and Matrix Stiffness Modulate Capillary Morphogenesis In Vitro, Exp. Cell Res., 297(2), pp [10] Morin, K. T., Dries-Devlin, J. L., and Tranquillo, R. T., 2014, Engineered Microvessels With Strong Alignment and High Lumen Density via Cell- Induced Fibrin Gel Compaction and Interstitial Flow, Tissue Eng., Part A, 20(3 4), pp [11] Rakusan, K., 1971, Quantitative Morphology of Capillaries of the Heart. Number of Capillaries in Animal and Human Hearts Under Normal and Pathological Conditions, Methods Achiev. Exp. Pathol., 5, pp [12] Duey, W. J., Bassett, D. R., Jr., Torok, D. J., Howley, E. T., Bond, V., Mancuso, P., and Trudell, R., 1997, Skeletal Muscle Fibre Type and Capillary Density in College-Aged Blacks and Whites, Ann. Hum. Biol., 24(4), pp [13] Gavin, T. P., Stallings, H. W., III, Zwetsloot, K. A., Westerkamp, L. M., Ryan, N. A., Moore, R. A., Pofahl, W. E., and Hickner, R. C., 2005, Lower Capillary Density but No Difference in VEGF Expression in Obese Vs. Lean Young Skeletal Muscle in Humans, J. Appl. Physiol. (1985), 98(1), pp Journal of Biomechanical Engineering MAY 2015, Vol. 137 /