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Biomaterials. Author manuscript; available in PMC 2010 September 1.

Published in final edited form as:

Published online 2009 May 29. doi: 10.1016/j.biomaterials.2009.04.053

PMCID: PMC2726043

NIHMSID: NIHMS117103

Department of Biomedical Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA

The publisher's final edited version of this article is available at Biomaterials

See other articles in PMC that cite the published article.

This computational study analyzes how to design a drainage system for porous scaffolds so that the scaffolds can be vascularized and perfused without collapse of the vessel lumens. We postulate that vascular transmural pressure—the difference between lumenal and interstitial pressures—must exceed a threshold value to avoid collapse. Model geometries consisted of hexagonal arrays of open channels in an isotropic scaffold, in which a small subset of channels was selected for drainage. Fluid flow through the vessels and drainage channel, across the vascular wall, and through the scaffold were governed by Navier-Stokes equations, Starling’s Law of Filtration, and Darcy’s Law, respectively. We found that each drainage channel could maintain a threshold transmural pressure only in nearby vessels, with a radius-of-action dependent on vascular geometry and the hydraulic properties of the vascular wall and scaffold. We illustrate how these results can be applied to microvascular tissue engineering, and suggest that scaffolds be designed with both perfusion and drainage in mind.

Numerous materials have been developed to promote the formation of vascularized tissues in vitro and in vivo [1–3]. In many studies, these materials consisted of uniformly porous scaffolds (e.g., hydrogels, degradable polymer meshes) that contained vascular growth factors and/or cells [4–8]. More recently, scaffolds with pre-formed channels made by microlithography or other patterning techniques have been created, in the hope that pre-vascularizing these channels would accelerate perfusion upon transplantation *in vivo* [9–12]. The ability of scaffolds to yield a sufficiently large density of perfused vessels is often the primary concern, and many computational studies have attempted to design scaffolds with appropriate densities of channels to sustain a desired tissue metabolic rate [12–14]. In this work, we analyze the complementary issue of tissue drainage, specifically for scaffolds with pre-vascularized channels.

Why is drainage relevant? In vertebrates, nearly all organs contain specialized vessels—the lymphatics—to remove excess fluid, solutes, and cells that are transported across permeable blood vessel walls [15]. If fluid filters across blood vessels at a higher rate than the rate of drainage into lymphatics, then interstitial fluid accumulates and interstitial pressures rise [16, 17]. Increased interstitial fluid pressure, if not mirrored by increases in blood pressure, will lead to decreased transmural pressure (lumenal pressure minus interstitial pressure) across vascular walls, thereby lowering vascular diameter and hindering perfusion [18]. In particular, transmural pressure must remain above a certain threshold (the so-called “closing pressure”) to avoid vascular collapse [19–21]. Published values of closing pressures range from −5 to 25 cm H_{2}O [20–23]; in general, the greater the vascular tone, the greater the threshold value [22, 23]. Vascular constriction or collapse is thought to play a crucial role in pathological conditions where external compression of tissue greatly reduces blood flow, such as compartment syndrome [24–26].

We expect engineered tissues to be subject to the same constraints and possibly to be even more vulnerable to insufficient drainage. The highly cross-linked nature of many scaffolds implies that it is difficult for scaffolds to swell sufficiently to accommodate excess fluids. Moreover, engineered vessels can be more immature and permeable compared to native blood vessels [10]. As a first step towards understanding how to design drainage systems for engineered tissues, we used computational modeling to determine how the physical properties of a scaffold, and the organization of vessels contained within, influence the ability of the scaffold to be drained. We analyzed how drainage capacity affects pressure balance within a scaffold, and show that low densities of drainage channels can potentially lead to collapse of surrounding vessels.

In this study, we postulate that a minimum transmural pressure (analogous to the closing pressure *in vivo*) is required to maintain vascular patency. We show that—in the absence of drainage—all vascular networks will contain at least some vessel segments under negative transmural pressure at steady state. Under these conditions, it is likely that transmural pressures will fall below the minimum level required to avoid collapse and loss of perfusion. Thus, whether an engineered scaffold can be functionally perfused may depend not only on the total vascular density, but also on whether the scaffold has enough drainage channels to maintain vascular patency via a low interstitial pressure.

This study evaluates several candidate drainage systems, and determines which combination of scaffold properties and vascular geometries can maintain a given transmural pressure in all vessels. It extends the existing body of work in numerical and/or analytical investigations of microvascular fluid mechanics, which have predicted fluid flow and pressure within and immediately around capillaries, but which have not typically emphasized the role of drainage in determining flow and pressure [27–31]. Our study hypothesizes that the geometric relationship between the areas for filtration and drainage, and the hydraulic properties of the scaffold and endothelial walls, play complementary roles in regulating vascular transmural pressure.

The models considered here consist of arrays of parallel vessels, similar to those proposed by August Krogh in his model of oxygen transport [32]. In contrast to the standard Krogh model (in which all vessels are designated for perfusion), here we selected a subset for drainage. Given the potential similarity of our models to the Krogh model of capillary perfusion, we examined whether the concept of a radius-of-action or “Krogh radius”, which has greatly simplified the design of microvascular systems for perfusion [33, 34], could also be applied to the design of drainage systems.

We modeled pressures and flows in a slab of tissue comprised of a regular hexagonal array of perfusion vessels and drainage channels in a porous scaffold (Fig. 1). Here, “vessel” refers to a structure that contains an endothelial layer. “Channels” are barren and lack an endothelium. We do not refer to drainage channels as “lymphatics” to avoid confusion over the mechanism of drainage: In contrast to actual lymphatics *in vivo* [35], here the drainage channels are not endothelialized and are passively drained.

The geometries of our models followed those described by Vunjak-Novakovic and co-workers [14]. Vessels and drainage channels were cylinders of length *L* and diameter *D*, extended from one face of the slab to the other in a direction normal to the scaffold faces, and were separated by a center-to-center lattice spacing *h* (Fig. 1A). They were distributed within the hexagonal array such that drainage channels were separated by a distance 2*Nh*, where *N* is a positive integer.

The symmetry of this arrangement permitted reduction of the model tissue to an equivalent triangular wedge radiating from a single drainage channel (Fig. 1B; see below for appropriate boundary conditions). The simplified tissue contained *N* concentric layers of vessels.

Fluid flow through the scaffold obeyed Darcy’s law [36]:

$${\mathbf{v}}_{\mathit{scaffold}}=-K\nabla {P}_{\mathit{scaffold}}$$

(1)

Here, **v*** _{scaffold}* is the interstitial fluid velocity,

Fluid flow through the vessels and drainage channels obeyed steady-state Navier-Stokes equations:

$$\rho \phantom{\rule{0.16667em}{0ex}}({\mathbf{v}}_{\mathit{vessel}}\xb7\nabla )\phantom{\rule{0.16667em}{0ex}}{\mathbf{v}}_{\mathit{vessel}}=-\nabla {P}_{\mathit{vessel}}+\eta {\nabla}^{2}{\mathbf{v}}_{\mathit{vessel}}$$

(2)

$$\rho \phantom{\rule{0.16667em}{0ex}}({\mathbf{v}}_{\mathit{drain}}\xb7\nabla )\phantom{\rule{0.16667em}{0ex}}{\mathbf{v}}_{\mathit{drain}}=-\nabla {P}_{\mathit{drain}}+\eta {\nabla}^{2}{\mathbf{v}}_{\mathit{drain}}$$

(3)

where **v*** _{vessel}* and

To describe the perfusion of a scaffold with a defined medium, we imposed Starling s Law for filtration of protein-free perfusate at the vessel walls and on the front and back faces of the slab [39]:

$${v}_{n}={L}_{P}\left({P}_{\mathit{vessel}}-{P}_{\mathit{scaffold}}\right)$$

(4)

where *v _{n}* is the fluid filtration velocity normal to the wall, and

To test how the boundary conditions at the scaffold walls affected fluid pressures and flow, we modified our standard models in two ways: In one set of models, we imposed equation (4) at the drainage wall; these models effectively replace the drainage channel with a vessel, and are equivalent to models without drainage. In another set, we replaced equation (4) with fixed-pressure boundary conditions of *P _{scaffold}* =

Each design was completely defined by four geometric values (*N*, *D*, *h*, *L*), two hydraulic conductivities (*K*, *L _{P}*), three pressures (

For each model, we solved equations (1)–(4) using the finite element method (COMSOL Multiphysics ver. 3.4; Comsol, Inc.) with quadratic Lagrangian elements and the PARDISO solver algorithm. Each model yielded the transmural pressure *P _{t}*

To test whether hydraulic conductivity *K* was independent of *P _{scaffold}*, we measured

Since alginate gels are much more resistive than collagen gels are, we formed alginate gels in larger blocks (cross-sectional area = 4 cm^{2}, thickness = 1 mm). Alginate (4%; Sigma) was gelled with 60 mM CaCl_{2} at room temperature for 2 hours. The alginate slab was then carefully sandwiched between two plastic rings and sealed with silicone. The hydrostatic pressure differences applied across the gels were 3, 6, 10, or 13.5 cm H_{2}O; the average pressures in the gels were −5, 0, 5, 10, or 15 cm H_{2}O. Hydraulic conductivity was calculated with a formula analogous to the one described above.

Our objective was to analyze how the minimum transmural pressure *P _{t,min}* varies with the geometry of the scaffold (as described by

In all models, we assumed that *K* was independent of *P _{scaffold}*. To test this assumption, we measured

Figure 2A presents cross-sectional views of interstitial and lumenal pressures in a representative tissue construct with four layers of vessels per drainage channel (i.e., *N* = 4). Interstitial pressure *P _{scaffold}* increased with distance from the drainage channel (i.e., downwards in Fig. 2A), but decreased with distance along the vascular axis (i.e., left to right in Fig. 2A). The axial decrease in vessel lumenal pressure

(A) Interstitial and vascular pressures for a representative model (*N* = 4, *D* = 100 μm, *h* = 500 μm, *L* = 1 cm, *L*_{P} = 10^{−10} cm^{3}/dyn·s, *K* = 10^{−10} cm^{4}/dyn·s, *P*_{in} − *P*_{out} = 10 cm H_{2}O, *P*_{out} − *P*_{dr} **...**

In the absence of a drainage channel, negative transmural pressures emerged in some portions of the construct, while large segments of vascular walls effectively existed at *P _{t}* = 0 cm H

To obtain transmural pressures that were positive everywhere, it was necessary to have drainage channels *and* to have scaffolds whose walls exhibited hydraulic resistance (Fig. 2C). In the absence of this resistance (i.e., with constant-pressure boundary conditions), the interstitial fluid was not insulated from perfusion pressures, and all vascular inlets and outlets had *P _{t}* = 0 cm H

For a given model geometry and perfusion pressures, lowering the pressure *P _{dr}* at the ends of the drainage channel (or, equivalently, increasing

(A) Plot of minimum transmural pressure *P*_{t,min} versus scaffold hydraulic conductivity *K* (with *K* = 10^{−10} cm^{4}/dyn·s). (B) Plot of *P*_{t,min} versus vascular hydraulic conductivity *L*_{P} (with *L*_{P} = 10^{−10} cm^{3}/dyn·s). (C) Plot of normalized **...**

We note that for values of *L _{P}*/

Similarly, for values of *L _{P}*/

The scaffold geometry is given by the number *N* of vascular “shells” that surround each drainage channel and the diameter *D*, spacing *h*, and length *L* of each vessel. We varied the four parameters across ranges that we considered to be experimentally realizable for engineered scaffolds (Table 1). Based on the results of the previous section, we kept *P _{out}* −

Of the four geometric parameters, the number of vascular layers *N* had by far the strongest influence on transmural pressure (Fig. 4A). For instance, for a hydraulic conductivity ratio of *L _{P}*/

Varying the vessel diameter *D* demonstrated that transmural pressure exhibits a biphasic dependence on *D* (Fig. 5A). With diameters in the range of 50 to 200 μm, narrower vessels and drainage channels had a larger transmural pressure, all other parameters held constant. Since narrower vessels present a smaller surface area for filtration and hence a smaller overall hydraulic pathway, we plotted minimum transmural pressure versus a normalized, dimensionless hydraulic conductivity *DL _{P}*/

(A) Plot of minimum transmural pressure *P*_{t,min} versus diameter *D* of vessels and drainage channel. (B) Plot of *P*_{t,min} versus a normalized ratio of hydraulic conductivities *DL*_{P}/*K*. (C) Plot of *P*_{t,min} versus *DL*_{P}/*K* for models in which the diameter of the drainage **...**

In some cases (e.g., for *K* = 10^{−8} cm^{4}/dyn·s and *L _{P}* = 10

The effect of increasing vascular spacing *h* is to decrease the hydraulic conductance of the scaffold between the drainage channel and outermost vessel. Thus, we expected minimum transmural pressure to decrease as vessels and drainage were spaced further apart. Although *P _{t,min}* decreased with increases in

Changing the length *L* of vessels and the drainage channel proportionally alters the hydraulic conductance of the vascular wall and scaffold. Since the ratio of conductances does not change, we expected changes in *L* to have little effect on transmural pressures. Indeed, even for the most sensitive case of *L _{P}*/

To determine whether the driving pressure for perfusion *P _{in}* −

Our results indicate the following methods for increasing transmural pressure in a vascularized scaffold: 1) insulate the interior of the scaffold from perfusion pressures, 2) decrease the pressure at the ends of drainage channels, 3) decrease the pressure at vascular inlets, 4) reduce the hydraulic conductivity of vascular walls, 5) increase the hydraulic conductivity of the scaffold, 6) reduce the number of vessels per drainage channel, 7) decrease the diameter of vessels without changing the size of drainage channels, 8) decrease the spacing between vessels and drainage channel, and 9) increase the lengths of vessels and drainage channel (i.e., increase the thickness of the scaffold).

Transmural pressure is not equally sensitive to these changes, however. Whereas a two-fold increase in *P _{out}* −

To show how our results can be applied in microvascular tissue engineering, we consider the geometry proposed by Vunjak-Novakovic and co-workers for perfusing engineered cardiac tissue *in vitro* [14]. Here, *D* is 330 μm, *h* is 700 μm, and *L* is 2 mm. These dimensions were selected to create a scaffold that could effectively deliver oxygen to embedded cells. Flow velocities of 0.05–0.1 cm/s, which correspond to driving perfusion pressures of <0.1 cm were considered. We assume that the channels are vascularized, that all transmural H_{2}O, pressures must be positive to avoid collapse (e.g., by delamination of the vascular wall from the scaffold), and that ideally one wishes to obtain transmural pressures of at least 5 cm as a H_{2}Osafety margin.

How might such a vascularized scaffold be drained? As in all cases, the scaffold will need to be shielded from the perfusion pressures, either by directly cannulating each vessel or by modifying the scaffold so that it is covered by a layer of hydraulically resistive material (e.g., a monolayer of endothelial cells). Next, a subset of cylinders will need to be cannulated to form drainage channels. We have found that *L _{P}*/

To determine the accuracy of these predictions, we compared the extrapolated values with those obtained by direct numerical solution. We note that the vascular arrangement in [14] is essentially identical to that diagrammed in Figure 1, so direct solution is possible here. Numerical solution showed that, for a *P _{t,min}* of 5 cm H

The geometry in the previous example was simple enough that it could be directly solved, but more complex geometries of interest in microvascular tissue engineering (e.g., bifurcating or three-dimensional networks with various diameters) rapidly lead to computationally intractable models. To aid in analyzing complex geometries, it may be useful to borrow the concept of a radius-of-action or “Krogh radius” [32]. In the Krogh model of oxygenation, a capillary can supply oxygen to a cylindrical shell of tissue, whose thickness is a function of oxygen consumption rate per volume, the capillary wall oxygen tension, and the diffusion constant of oxygen. The ability to describe oxygenation by a single number (the Krogh radius) provides an intuitively simple design constraint on microvascular networks for perfusion [12].

To what extent does the same concept apply to drainage systems? Here, vessels “produce” interstitial pressure, and the drainage channel “removes” this pressure; the diffusion constant of interstitial pressure is proportional to the scaffold hydraulic conductivity *K* [52]. We expect that models in which interstitial pressure is “produced” at the same rate per volume will have the same interstitial pressure profile. For instance, an *N* = 6, *D* = 44 μm, *h* = 333 μm model with 100-μm-diameter drainage channel has the same dimensions and vascular surface area per volume as an *N* = 4, *D* = 100 μm, *h* = 500 μm model. We expect the two models to have nearly identical *P _{t,min}*, as is observed computationally: 8.74 cm H

For a vascular network of arbitrary geometry, we thus suggest the following procedure for designing a drainage network: First, one calculates the vascular surface area per volume for the given geometry. Second, one designs models with the same surface area per volume, but which consist of parallel arrays of vessels; these models should span a range of sizes. Computational solution of these models, or extrapolation from data in Figures 1–8, should yield the minimum transmural pressure as a function of distance between a drainage channel and the outermost vessel. From these numbers and a desired *P _{t,min}*, one obtains a “Krogh radius” of drainage, i.e., the maximum distance allowed between a drainage channel and vessel before drainage becomes insufficient.

We point out that mathematical correspondence with the standard Krogh model is not exact: In contrast with the oxygenation model, in which oxygen concentration profiles within the Krogh radius do not change when excess tissue is added outside the Krogh radius [32], here the interstitial pressure profiles always change when vessels are added (Fig. 4B). Nevertheless, the ability to convert a complex vascular network into a roughly equivalent parallel vascular array simplifies the design of drainage systems, especially for vascular geometries that are impractical to model computationally.

Our results are consistent with previous computational studies of drainage [16, 53]. These studies (in the area of tumor physiology) assumed that intra-tissue lymphatics were not functional, and that drainage occurred solely at the outer surface of the tissue volume. They determined that smaller tumors had lower interstitial pressures, which is consistent with our finding that a drainage channel can effectively drain only vessels in its vicinity.

Comparison with these studies also points the way to future enhancements of our model. First, we have discounted elastic coupling between vessel walls and the scaffold [18]. As a result, our model does not allow for a gradual degradation in perfusion rate as interstitial pressure rises and vessel diameter decreases. Second, we have assumed that the perfusate exerts no oncotic pressure, and that solute gradients do not exist in the scaffold. Both of these assumptions can be relaxed computationally (in the former, by using deformable meshes; in the latter, by modifying Starling’s Law and adding Fick’s Laws to the set of equations to be solved), but at considerable computational cost. We are currently exploring methods to realize these enhancements.

This work postulates that transmural pressure across a vessel must exceed a certain value to prevent vascular collapse. We used computational models to examine the implications of this postulate on the pressures and flows within a vascularized scaffold. We determined that the vascular geometry and the hydraulic properties of the vessel wall and the scaffold play complementary roles in determining transmural pressure.

An important principle demonstrated by our models is that, to maintain positive transmural pressure, the interstitial fluid must be permitted to flow into drainage channels and must be insulated from the pressures that drive vascular flow. We also found that the ratio of vascular to interstitial conductances will largely determine whether a scaffold can be effectively drained. In particular, if the hydraulic conductance of the scaffold is very low (as can occur in dense or large gels), then large portions of the capillary network will exist at approximately zero transmural pressure, a situation that does not favor vascular patency.

We found that a drainage channel—no matter what pressure it is set at—can only maintain a limited number of neighboring vessels above a threshold transmural pressure. A drainage channel thus has a limited range, which defines a tissue region for drainage similar in concept to the Krogh cylinder of oxygenation. The existence of this radius-of-action can potentially simplify the design of drainage systems for complex vascular networks. As our computational resources become more powerful, we intend to test whether vascular surface area per volume can accurately predict the spacing of channels needed to drain complex networks.

We note that the parallel geometry studied here can be realized experimentally by a variety of techniques [9, 10, 54–56]. Experimental tests of our computational predictions are thus possible, and will require the determination of closing pressures for engineered vessels.

We thank Dan Kamalic for assistance with the Whitaker Computational Facility in the Department of Biomedical Engineering at Boston University, and Paul Barbone, Keith Wong, and Celeste Nelson for helpful suggestions. This work is supported by the National Institute of Biomedical Imaging and Bioengineering (NIH award R01 EB005792).

*N*- Number of layers of vessels per drainage channel
*D*- Diameter of vessels
*h*- Inter-axial distance between vessels
*L*- Length of vessels, thickness of scaffold
*K*- Interstitial hydraulic conductivity
*L*_{P}- Hydraulic conductivity of vessel wall
*P*,_{in}*P*,_{out}*P*_{dr}- (Constant) hydrostatic pressures in vascular inlets, vascular outlets, and ends of drainage channel
*P*,_{vessel}*P*_{drain}- Hydrostatic pressures in vessels and drainage channel
*P*_{scaffold}- Interstitial pressure
*P*_{t}- Transmural pressure (=
*P*−_{vessel}*P*)_{scaffold} - v
, v_{vessel}_{drain} - Fluid velocities in vessels and drainage channel
- v
_{scaffold} - Velocity of interstitial fluid
*v*_{n}- Velocity of interstitial fluid normal to vessel wall (i.e., filtration velocity)

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