A Poroelastic Model Describing Nutrient Transport and Cell Stresses Within a Cyclically Strained Collagen Hydrogel

Abstract

In the creation of engineered tissue constructs, the successful transport of nutrients and oxygen to the contained cells is a significant challenge. In highly porous scaffolds subject to cyclic strain, the mechanical deformations can induce substantial fluid pressure gradients, which affect the transport of solutes. In this article, we describe a poroelastic model to predict the solid and fluid mechanics of a highly porous hydrogel subject to cyclic strain. The model was validated by matching the predicted penetration of a bead into the hydrogel from the model with experimental observations and provides insight into nutrient transport. Additionally, the model provides estimates of the wall-shear stresses experienced by the cells embedded within the scaffold. These results provide insight into the mechanics of and convective nutrient transport within a cyclically strained hydrogel, which could lead to the improved design of engineered tissues.

Introduction

One of the primary challenges facing the successful creation of engineered tissue constructs is the transport of oxygen and nutrients to cells within the scaffold. Multiple advances have been made in the vascularization of scaffolds, but even this process requires sufficient transport to the endothelial cells forming the networks (1). However, many tissue-engineering scaffolds can be highly porous, especially protein-based hydrogel biomaterials. For instance, a 2-mg/mL collagen type-I hydrogel is 99.6% porous (2). Highly porous scaffolds do not create much of a barrier to solute transport (3), but these gels are often used for applications that involve substantial mechanical deformation (2,4–7). Previous studies have indicated that cyclic strain induces substantial fluid pressure gradients within porous materials (8,9), which affects transport of solutes (10–12). These studies suggest that transport in a cyclically strained porous scaffold has both a convective and diffusive component. This study describes a poroelastic model that incorporates convection to predict the wall-shear stress on cells embedded in a cyclically strained collagen hydrogel within a flexible polydimethylsiloxane (PDMS) well. The model is validated by tracking the position of 1–5 μm beads during application of cyclic strain to the hydrogel.

Poroelastic theory has proven to be a useful model for describing the mechanics and transport of both actual tissue and various biomaterials. In its most basic form, the constitutive equations for solid and fluid stress are coupled through the fluid pressure term, which results in a formulation that can predict macroscopic quantities like fluid velocity and solid-phase stress. First formulated by Biot (13) for transport through soil, poroelastic formulations have because been used to describe the coupled fluid and solid mechanics of bone (14), cartilage (15), and brain tissue (16). Experimental validation of these models is difficult in vivo, though recent advances have been made for measuring strain-induced fluid flow in bone (17) and transport within soft tissue (18). Solute transport has been incorporated into poroelastic models in the past, both for bending porous cantilever beams (19–21) and for intervertebral disks (22) and brain tissue (23). These models predicted that the concentration gradient was strongly dependent upon the strain-induced fluid transport. Considering the porosity of collagen hydrogels compared to in vivo tissue, it is intuitive that the fluid flow environment is even more crucial to solute transport.

The poroelastic description of the flow/strain apparatus described by our model is built upon the classic Mandel problem (24). The Mandel problem is defined as the response of a long rectangular, isotropic poroelastic strip that is confined by two impermeable, rigid, parallel plates subject to a Heaviside step load. It was first proposed and solved analytically by Mandel (24) and is a basic benchmark problem in the mechanics of poroelastic materials. Analytical solutions to the Mandel problem have been studied for different material properties such as transversely isotropic poroelastic materials (25,26), and poroviscoelastic materials (27), both of which were subject to Heaviside step loads. Additional studies have been done for anisotropic materials (28,29), material with negative Poisson’s ratios (28,29), variability in the porosity and permeability (30), and allowing for compressibility within the pores of different media (26). An analytic solution to the Mandel problem subject to cyclical loading was given by Kameo et al. (8) for isotropic poroelastic materials and by Hoang and Abousleiman (31) for poroelastic and poroviscoelastic materials subject to general time-dependent loads.

The innovation of this model is the application of Mandel’s problem to specific boundary conditions that describe a flow/strain apparatus used to mechanically stimulate collagen hydrogels (32). The device is essentially a flexible well that has an open surface at one end, which introduces an asymmetry to the problem. Moreover, the apparatus is able to apply cyclic strain alone and in combination with an imposed cross flow, both of which are incorporated into the boundary conditions. The model is then used to predict fluid pressure and velocity within the deforming gel for the different testing conditions specific to each experiment. To validate the model, both the transient response and transport of micrometer-scale beads were built into a numerical model complementing the analytical model. A quantity of 1–5-μm diameter beads was chosen to minimize diffusive transport, because the pore size of collagen is approximately the same scale. The results of this model are validated experimentally by applying strain cycles with different amplitude and frequency to an acellular collagen hydrogel and measuring the bead penetration to match with the numerical predictions. Because three-dimensional collagen hydrogels are often used as rudimentary models of tissues in experiments where both mechanics and mass transfer are important, it is important to have validated theoretical and experimental models that provide understanding on the influence of the strain and its frequency on solute transport and shear stresses.

Experimental Methods

Collagen hydrogels

A quantity of 2.0 mg/mL (0.2% w/v) Type-1 collagen hydrogels was polymerized by combining acid-solubilized collagen to 0.1 M NaOH, 5× DMEM, and fetal bovine serum and then incubating the mixture at 37°C, 100% humidity, and 5% CO₂ for 30 min. Mixtures were polymerized in flexible PDMS rectangular wells. The PDMS wells had the following dimensions: 8 mm height × 10 mm length × 15 mm width. For gel polymerization, the PDMS wells were etched in sulfuric acid using a previously described protocol to facilitate gel attachment (32).

Parameter measurement

Assuming linear strain and an isotropic medium, only elastic modulus and Poisson’s ratio were required to describe the gel solid mechanics. These parameters were measured using unconfined compression in a previous study (32). To complete the poroelastic description, viscosity of the fluid in the hydrogel as well as the permeability of the medium were needed. The viscosity is assumed to be that of water at 25°C, and the permeability was measured using a PDMS well equipped with an inlet for flow (32). A syringe pump was used to deliver a constant volume flow rate to the well, and the observed pressure drop was measured to calculate Darcy permeability.

Particle tracing validation

PDMS wells were connected to an axial mechanical testing apparatus using nylon grips, as described in Galie et al. (32). The surrounding bath was filled with 180 mL of PBS containing 15 mg of 1–5-μm blue fluorescent spheres. Cyclic strain of varying amplitude and frequency was applied for 3600 cycles. After the cyclic stretch, the PDMS wells were removed from the grips and visualized by a confocal microscope. The gel surface was marked by the presence of a large concentration of beads. Using a built-in stepper motor, the focal plane was moved upward by 50-μm steps until there were no beads in focus. The objective was then moved downward in 5-μm steps until at least one bead came into focus. This process was repeated in two different locations of each gel, and each amplitude/frequency combination was tested three times, resulting in six total measurements for each cycle combination. Fig. 1 illustrates the process to determine penetration distance.

Figure 1.

Model validation: gels polymerized in PDMS wells were cycled at varying strain magnitude and frequency in a solution containing fluorescently labeled beads. Confocal microscopy was then used to determine the penetration depth of the beads after 3600 cycles.

Mathematical Model

We model the collagen hydrogels as a Biot poroelastic medium (13) coupled with Darcy’s law. Assuming an intrinsically incompressible solid matrix and an infinitely incompressible fluid, the u−p formulation is

$${\mu }_s^{\ast }{\nabla }^2{\underset{¯}{u}}^{\ast }+\left({\lambda }^{\ast }+{\mu }_s^{\ast }\right)\nabla \left(\nabla ·{\underset{¯}{u}}^{\ast }\right)=\nabla p^{\ast },$$ (1)

$$\frac{\partial }{\partial t^{\ast }}\left(\nabla ·{\underset{¯}{u}}^{\ast }\right)=k^{\ast }{\nabla }^2{p}^{\ast },$$ (2)

where ${\underset{¯}{u}}^{\ast }={\left(u_1^{\ast },u_2^{\ast }\right)}^T$ is the solid displacement, p^∗ is the pressure, μ_s^∗ and λ^∗ are the solid Lamé constants, k^∗ = κ^∗/μ^∗f is the hydraulic conductivity, κ^∗ is the permeability, and μ_f^∗ is the fluid viscosity. Using Darcy’s law, we relate the relative fluid velocity, ${\underset{¯}{v}}^{\ast }={\left(v_1^{\ast },v_2^{\ast }\right)}^T$, to the pressure by the relation

$${\underset{¯}{v}}^{\ast }=-\frac{k^{\ast }}{\phi }\nabla p^{\ast },$$ (3)

where φ is the porosity of the hydrogel.

The gel resides in a two-dimensional rectangular domain of length L^∗ and height H^∗, where the wall at x^∗ = 0 is an oscillating wall with frequency ω and amplitude A^∗. Because we will examine two different domain configurations, we shall discuss boundary conditions specific to each configuration separately.

We nondimensionalize Eqs. 1–3 by assuming the scales ${\underset{¯}{u}}^{\ast }=H^{\ast }\underset{¯}{u}$, ${\underset{¯}{x}}^{\ast }=H^{\ast }\underset{¯}{x}$, t^∗ = ω⁻¹t, p^∗ = H^∗2ωk⁻¹p, and ${\underset{¯}{v}}^{\ast }=\omega H^{\ast }\underset{¯}{v}$ to obtain the nondimensional u – p equations

$${\mu }_s{\nabla }^2\underset{¯}{u}+\left(\lambda +{\mu }_s\right)\nabla \left(\nabla ·\underset{¯}{u}\right)=\nabla p,$$ (4)

$$\frac{\partial }{\partial t}\left(\nabla ·\underset{¯}{u}\right)={\nabla }^{2}p,$$ (5)

along with the nondimensional Darcy’s law

$$\underset{¯}{v}=-\frac{\nabla p}{\phi }$$ (6)

for 0 ≤ x ≤ L and 0 ≤ y ≤ L. Here, L = L^∗/H^∗, μ_s = μ^∗_sk^∗/ωH^∗2, and λ = λ^∗k^∗/ωH^∗2.

Domain Configuration I—No Crossflow

In this configuration, the gel is encased by three impermeable solid walls with one boundary open to a bulk fluid that allows for fluid to enter or exit the hydrogel, as shown in Fig. 2. The bottom and right walls are fixed while the left wall oscillates at a prescribed strain and frequency. Here, the components of the dimensional elements of the stress tensor are

$${\sigma }_{xx}^{\ast }=\left({\lambda }^{\ast }+2{\mu }_s^{\ast }\right)\frac{\partial u_1^{\ast }}{\partial x^{\ast }}+{\lambda }^{\ast }\frac{\partial u_2^{\ast }}{\partial y^{\ast }}-p^{\ast },$$ (7)

$${\sigma }_{yy}^{\ast }=\left({\lambda }^{\ast }+2{\mu }_s^{\ast }\right)\frac{\partial u_2^{\ast }}{\partial y^{\ast }}+{\lambda }^{\ast }\frac{\partial u_1^{\ast }}{\partial x^{\ast }}-p^{\ast },$$ (8)

$${\sigma }_{xy}^{\ast }={\mu }_s^{\ast }\left(\frac{\partial u_1^{\ast }}{\partial y^{\ast }}+\frac{\partial u_2^{\ast }}{\partial x^{\ast }}\right).$$ (9)

The hydrogel and the pore fluid are assumed, initially, to be in the reference configuration, so we enforce the initial conditions u^∗₁(x^∗,y^∗,0) = u^∗₂(x^∗,y^∗,0) = p^∗(x^∗,y^∗,0) = 0 throughout the domain.

Figure 2.

Domain configuration and boundary conditions with a single open boundary.

Solving the governing equations (see Appendix A), the dimensionless solutions for the displacements and the pore pressure are

$$u_1\left(x,t\right)=A\left(1-\frac{x}{L}\right)\sin \left(t\right),$$ (10)

$$u_2\left(y,t\right)=\frac{A}{L}\left[y\sin \left(t\right)+\frac{{\mu }_s}{\lambda +2{\mu }_s}\left(C_L\left(y,t\right)+C_T\left(y,t\right)\right)\right],$$ (11)

$$p\left(y,t\right)={\mu }_s\frac{A}{L}\left[2\sin \left(t\right)+\frac{\partial C_L}{\partial y}+\frac{\partial C_T}{\partial y}\right],$$ (12)

where

$$C_L\left(y,t\right)=\left(\frac{1+i}{2}\frac{\sinh \left(\nu \left(1+i\right)y\right)}{\nu \cosh \left(\nu \left(1+i\right)\right)}{e}^{it}+\frac{1-i}{2}\frac{\sinh \left(\nu \left(1-i\right)y\right)}{\nu \cosh \left(\nu \left(1-i\right)\right)}{e}^{-it}\right)$$ (13)

and

$$C_T\left(y,t\right)=4\left(\lambda +2{\mu }_s\right)\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{1+s_n^2}\sin \left(\left(2n-1\right)\frac{\pi }{2}y\right)e^{-s_{n}t}.$$ (14)

The pore velocity is

$$v_2\left(y,t\right)=\frac{{\mu }_s}{\phi }\frac{A}{L}\left(\frac{{\partial }^2{C}_L}{\partial y^2}+\frac{{\partial }^2{C}_T}{\partial y^2}\right),$$ (15)

where ν = (2(λ + 2μ_s))^−1/2,

$$\frac{{\partial }^2{C}_L}{\partial y^2}=-\nu \left(\left(1-i\right)\frac{\sinh \left(\nu \left(1+i\right)y\right)}{\cosh \left(\nu \left(1+i\right)\right)}{e}^{it}+\left(1+i\right)\frac{\sinh \left(\nu \left(1-i\right)y\right)}{\cosh \left(\nu \left(1-i\right)\right)}{e}^{-it}\right)$$ (16)

and

$$\frac{{\partial }^2{C}_T}{\partial y^2}=\left(\lambda +2{\mu }_s\right)\frac{A}{L}\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{1+s_n^2}{\left(2n-1\right)}^2{\pi }^2\sin \left(\left(2n-1\right)\frac{\pi }{2}y\right)e^{-s_{n}t}.$$ (17)

As t → ∞, the transients’ term, C_T, rapidly goes to zero and we are left with a purely oscillating solution for the displacements, pore pressure, and relative fluid velocity.

In Figs. 3–6, we plot the solid displacements, pore pressures, and relative fluid velocity over a single cycle for parameters relevant to the experiment design (see Appendix C for definition of parameters). We can see that the purely horizontal cyclic strain causes a purely vertical pressure field and relative fluid velocity. Examining the displacements, we can see that there is no poroelastic response in the horizontal displacement, where u₁ exhibits a linear profile that cycles in phase with the strain, and a weak poroelastic response in the vertical displacement, where the poroelastic response is dwarfed by the linear profile that cycles in phase with the prescribed cyclic strain. This weak poroelastic response yields a small pressure field that induces a relative fluid flow near the open boundary (y = 1). Note that for these parameters, the transients decay quickly within a single cycle and for high cycle counts, we can neglect the transient terms.

Figure 3.

Nondimensional horizontal solid displacement in the x direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. The horizontal solid displacement is linear in space and purely sinusoidal in time.

Figure 4.

Nondimensional vertical solid displacement in the y direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. The vertical solid displacement is dominated by the y sin(t) term where the extra poroelastic effects are smaller in magnitude and isolated to a region near the open boundary. These smaller effects cause the relative fluid flow.

Figure 5.

Nondimensional pore pressure in the y direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. The pressure is dominated by a spatially constant sinusoidal term but at the open boundary, there are significant changes in the pore pressure. Note that the transient terms have decayed to zero in approximately one cycle and future cycling yields the limit cycle solution.

Figure 6.

Nondimensional relative pore fluid velocity in the y direction over a single cycle using the parameters given in Appendix C with γ = 10% and ω = 2π radians/s. The relative fluid flow is isolated to the open boundary end with no significant relative fluid flow in the interior of the gel or near the solid boundary. This implies that the cells nearest to the open boundary will experience greater fluid shear stresses than the rest of cells in the interior.

Domain Configuration II—Crossflow

In this configuration, the gel is encased by two impermeable solid walls with two parallel boundaries open to a bulk fluid, along with a pressure gradient between the two open boundaries generating a crossflow through the domain. In the absence of a crossflow, this is the cyclic Mandel problem, whose solution has been given previously by Kameo et al. (8) and by Hoang and Abousleiman (31). Once again, the right wall is a fixed impermeable wall whereas the left wall is an impermeable wall that is driven by a prescribed displacement, as shown in Fig. 7. We scale as before and use the dimensionless driving pressures P₁ = ωH^∗2P^∗₁/k^∗ and P₂ = ωH^∗2P^∗₂/k^∗. Solving the dimensionless equations for these boundary conditions (see Appendix B), the solution for u₂ is

$$u_2\left(y,t\right)=\frac{A}{L}(\frac{1}{\lambda +2{\mu }_s}\left(\lambda y-{\mu }_s\right)\sin \left(t\right)+8{\mu }_s\sum _{n=1}^{\infty }\frac{1}{{\nu }_n\left({\nu }_n^2+1\right)}\left[{\nu }_n\left(\cos \left(t\right)-e^{-{\nu }_{n}t}\right)+\sin \left(t\right)\right]\cos \left(\left(2n-1\right)\pi y\right)),$$ (18)

where ν_n(λ + 2μ_s)(2n – 1)² π². Once again, the displacement is governed by a transient solution and a limit cycle solution. The pressure is

$$p\left(y,t\right)=P_1-\left(P_1-P_2\right)y+p_o\left(y,t\right),$$ (19)

where

$$P_0\left(y,t\right)=-8\frac{A}{L}{\mu }_s\left(\lambda +2{\mu }_s\right)\sum _{n=1}^{\infty }\frac{\left(2n-1\right)\pi }{{\nu }_n\left({\nu }_2^{\ast }+1\right)}\left[{\nu }_n\left(\cos \left(t\right)-e^{-{\nu }_{n}t}\right)+\sin \left(t\right)\right]\sin \left(\left(2n-1\right)\pi y\right).$$ (20)

Substituting the pore pressure into Darcy’s law and including the crossflow solution, the relative fluid velocity is

$$v_2=\frac{1}{\phi }\left[\left(P_2-P_1\right)+8\frac{A}{L}{\mu }_s\sum _{n=1}^{\infty }\frac{1}{{\nu }_n^2+1}\left[{\nu }_n\left(\cos \left(t\right)-e^{{\nu }_{n}t}\right)+\sin \left(t\right)\right]\cos \left(\left(2n-1\right)\pi y\right)\right],$$ (21)

with no flow in the horizontal direction.

Figure 7.

Domain configuration and boundary conditions with a crossflow through two open boundaries.

Figs. 8–10 show the nondimensional poroelastic response in the vertical displacement, (u₂ – A/L sin(t)), the nondimensional pore pressure without the crossflow gradient, and the nondimensional relative pore fluid velocity minus the fluid velocity for γ = 10% and ω = 2π radians/s (1 Hz). We can see that the solutions are antisymmetric about the center point where u^∗₂ is fixed. If we neglect the crossflow, the pore pressure is constant in the center of the hydrogel and there is a relative fluid flow near both open boundaries. Once again, the transients decay rapidly for these parameters, so we can neglect the transient terms after the initial cycle.

Figure 8.

Nondimensional vertical solid displacement in the crossflow case in the y direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. The solution has been translated to ensure that the center of the hydrogel is fixed. The displacement is antisymmetric about the center of the hydrogel and deviates from the linear profile rapidly as we approach the open boundary.

Figure 9.

Nondimensional pore pressure for the crossflow case in the y direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. Here, the linear pressure term from the crossflow has been removed to show the poroelastic response. The pressure is constant in the center of the hydrogel and rapidly approaches the ambient pressures as we approach the open boundaries.

Figure 10.

Nondimensional relative pore fluid velocity for the crossflow case in the y direction over a single cycle using parameters given in Appendix C with γ = 10% and ω = 2π radians/s. Here, the constant crossflow velocity has been neglected to show the poroelastic response. The relative fluid flow is confined to the regions near the open boundaries with the largest velocity located at either y = 0 or y = 1.

Trajectories of Lagrangian Particles in the Hydrogel

To validate the experimental and analytical results, we model the position of Lagrangian particles embedded in the hydrogel and compare the predicted penetration of these particles with the penetration depth observed in the experiment described above. We chose a particle diameter on the same scale as the pore size of 2 mg/mL collagen (1–5 μm), to minimize transport by diffusion. The Stokes number (the ratio of the characteristic timescale for the particles to the characteristic timescale for the fluid) for a small spherical particle in our model is

$$St=\frac{{\rho }_p{d}_p^2\omega }{18{\mu }_f},$$ (22)

where ρ_p = 1 g/cm³ is the particle density, d_p = 1–5 μm is the particle diameter, ω = π to 4π Hz is the frequency of the oscillating wall, and μ_f = 1 cP. For these parameters, the Stokes number ranges from St = 10⁻⁴ to St = 10⁻⁶. Therefore, we assume that the velocity of the particle depends solely on the relative velocity of the pore fluid at that particle’s position, which we obtain using the analytic solution from the first domain configuration (Fig. 2). Due to the small drag forces on the particles, we neglect inertia forces and assume there is no relative motion between the particle and fluid. Hence, the position of the particle at time t is governed by the equation

$$\dot{Y}\left(t\right)=v_2\left(Y\left(t\right),t\right),$$ (23)

where Y(t) is the particle position, and $\dot{Y}\left(t\right)$ is the particle velocity, which is the relative fluid velocity, v₂(Y(t),t), given by Eq. 44.

The position of the Lagrangian particle is computed numerically using adaptive backward Euler discretization in time, employing Richardson extrapolation to select the time step size. Fig. 11 shows the trajectory of a particle initially at Y(0) = 0.999 using the parameters in Appendix C with γ = 10% and ω = 2π radians/s (1 Hz). Note that the particle has a net penetration into the hydrogel even though the driving oscillations would yield no net penetration after a full cycle in the bulk fluid. Fig. 12 shows the computed relative penetration depth versus frequency for various amplitudes with the results from the benchtop experiments. We can see that the analytical solution matches well with the experiments and we recover the increased penetration depth as frequency decreases or the amplitude increases. This result is similar to prediction of a decreased skin or penetration depth of a poroelastic diffusion wave as the frequency is increased. The overestimation of the bead penetration depth is expected because in the mathematical model, we place the bead is initially in hydrogel whereas in the experiment, the beads initially reside in the bulk fluid. There is a short delay between when the cyclic strain is started and when the bead enters the hydrogel, leading to a smaller penetration depth in the experiments. These results lead us to believe the mathematical model sufficiently models the behavior of the hydrogel and the pore fluid for these domain configurations, and will yield accurate predictions of the relative fluid velocity and pore pressure as well as the displacements and stresses in the solid matrix.

Figure 11.

Particle-penetration depth versus number of cycles for γ = 10% and ω = 2π radians/s (1 Hz) over 3600 cycles using the parameters given in Appendix C. While the particle oscillates, there is a net displacement of the particle into the hydrogel.

Figure 12.

Relative particle-penetration depth versus frequency in Hertz for various strain rates after 3600 cycles using the parameters given in Appendix C. (Lines) Theoretical results; (points) results from the experiments. The penetration distance decreases as the frequency is increased with significant penetration for small frequencies and it increases as the strain rate is increased. This qualitative behavior is seen in the results from both the mathematical model and the experiments. This result is similar to prediction of a decreased skin or penetration depth of a poroelastic diffusion wave as the frequency is increased. The analytic results show a slightly larger penetration depth that the experiments, which is expected because it is assumed that the bead is initially in the hydrogel in the mathematical model, but the bead resides initially in the bulk fluid in the experiments.

Shear Stress due to Relative Fluid Flow on Cells Embedded in the Hydrogel

Due to the relative fluid flow induced by the cyclic strain, any cells embedded within the hydrogel will experience wall-shear stresses due to this flow. We estimate the wall-shear stress using the results from Wang and Tarbell (33), where they estimate the average wall-shear stress on the surfaces of spherical cells in a square or triangular array to be

$$\tau =\frac{4}{\pi }\sqrt{\frac{-{\mu }_f{u}_0\text{Δ}p}{L_a}}F\left(C\right),$$ (24)

where μ_f is the viscosity of the pore fluid, u₀ is the fluid velocity across the cell array, Δp is the fluid pressure drop across the cell array, L_a is the length of the cell array, and F(C) depends on the type of cell array (square or triangular) and the volume fraction of cells, C. For our purposes, we assume that F(C) = 1 and Δp/L_a → ∂p/∂y for small cell arrays (L_a << 1), so the average wall-shear stress becomes

$${\tau }^{\ast }\left(y^{\ast },t^{\ast }\right)=\frac{4}{\pi }\sqrt{-{\mu }_f^{\ast }{v}_2^{\ast }\frac{\partial p^{\ast }}{\partial y^{\ast }}}=\frac{4}{\pi }\sqrt{\frac{{\mu }_f^{\ast }\phi }{k^{\ast }}}{v}_2^{\ast }\left(y^{\ast },t^{\ast }\right).$$ (25)

Because the average wall-shear stress is proportional to the relative fluid velocity, the shear stress on a cell embedded within the hydrogel is largest near the open boundaries, whereas the cells far from the open boundary experience little or no wall-shear stress. The maximum wall-shear stress, for the parameters given in Appendix C, is shown for various values of γ and ω in Fig. 13. The average wall-shear stress is largest for higher cyclic strain rates and lower frequencies. For a cycle frequency of 1 Hz with a 5% cyclic strain, the maximum average wall-shear stress is ∼6 × 10⁻⁴ N/m² (6 × 10⁻³ dynes/cm²).

Figure 13.

Dimensional maximum average wall-shear stress on cells embedded in the hydrogel for the parameters given in Appendix C versus cyclic frequency for various values of the strain rate. The maximum stress increases as the strain rate is increased and as the cyclic frequency is decreased. Cells located on the open boundary always experience the largest average wall-shear stress.

Conclusions

In this article, we develop and solve a mathematical model of a constrained cyclically strained hydrogel using Biot poroelastic theory (13). The model is able to predict the pore pressure and relative pore-fluid velocity within the deforming gel for the different testing conditions. The results of this model are validated by applying strain cycles with different amplitude and frequency to an acellular collagen hydrogel and measuring the bead penetration, which matches the predictions from numerical simulations of the motion of the beads due to the relative pore fluid velocity. Additionally, we are able to make predictions of the average wall-shear stress due to the relative fluid velocity on cells embedded within the hydrogel using the analytical solution of the mathematical model.

In addition to predicting wall-shear stress on cells embedded in the previously described flow/strain apparatus, these results have relevance in predicting the rate of convective transport of solutes through tissues. In particular, our model incorporates cyclic strain to simulate the mechanical deformations to which many tissues are exposed, and which affects the rate of convective flow in the interstitium. The model therefore provides insight into nutrient transport in dynamically strained tissues. In this study, we used well-defined engineered tissue analogs composed of collagen. However, our findings may also be extended to native tissues by appropriately altering the relevant parameters. The model also allows prediction of the shear stress to which cells are exposed, and therefore provides insight into the effects of physical forces on cells in three-dimensional tissues. Taken together, our findings could be used to rationally design engineered tissues as well as to understand the physical and chemical environment in normal and diseased tissues.

Additional complexity can be added to the model to account for the anisotropy present in many in vivo tissues and in vitro models. For example, depending upon the orientation of constraints, cells embedded in collagen hydrogels can compact the gel and alter its isotropic organization. Such anisotropy would not change the constitutive equations, but the Lamé and permeability constants would then change with direction. However, the constitutive equation for the Biot poroelastic formulation would be affected if the model were to account for large deformations, which may be relevant for different tissues in the body that undergo high magnitudes of strain. Greater levels of strain would also likely increase the viscoelastic contribution of the collagen fibers within the hydrogel, necessitating the addition of a viscoelastic constitutive equation to our poroelastic formulation. The study of these additional complexities is the subject of future research.

Appendix A: Analytic Solution for Domain Configuration I—No Crossflow

From the boundary conditions, we can see that u₁ = A(1 − x/L) sin(t), u₂ = u₂(y,t), and p = p(y,t), where

$$\left(\lambda +2{\mu }_s\right)\frac{{\partial }^2{u}_2}{\partial y^2}=\frac{\partial p}{\partial y},$$ (26)

$$\frac{{\partial }^2{u}_2}{\partial y\partial t}=\frac{{\partial }^{2}p}{\partial y^2}+\frac{A}{L}\cos \left(t\right).$$ (27)

We integrate Eq. 26 with respect to y and use the condition p(x,1,t) = 0 to obtain that

$$p\left(y,t\right)=\left(\lambda +2{\mu }_s\right)\left(\frac{\partial u_2}{\partial y}-{\frac{\partial u_2}{\partial y}|}_{y=1}\right).$$ (28)

It follows that Eq. 27 becomes

$$\frac{{\partial }^2{u}_2}{\partial y\partial t}=\left(\lambda +2{\mu }_s\right)\frac{{\partial }^3{u}_2}{\partial y^3}+\frac{A}{L}\cos \left(t\right)$$ (29)

with the boundary conditions

$$u_2\left(0,t\right)={\frac{{\partial }^2{u}_2}{\partial y^2}|}_{y=0}=0,$$ (30)

$${\frac{\partial u_2}{\partial y}|}_{y=1}=\frac{\lambda A}{L\left(\lambda +2{\mu }_s\right)}\sin \left(t\right),$$ (31)

and the initial condition u₂(y,0) = 0.

We integrate Eq. 29 with respect to y to obtain

$$\frac{\partial u_2}{\partial t}=\left(\lambda +2{\mu }_s\right)\frac{{\partial }^2{u}_2}{\partial y^2}+\frac{A}{L}y\cos \left(t\right)+g\left(t\right).$$ (32)

Applying the boundary conditions, we get that g(t) ≡ 0 and we let u₂(y,t) = (A/L) y sin(t) + c(y,t) to obtain

$$\frac{\partial c}{\partial t}=\left(\lambda +2{\mu }_s\right)\frac{{\partial }^{2}c}{\partial y^2}$$ (33)

with the conditions c(y,0) = 0, c(0,t) = 0, and

$${\frac{\partial c}{\partial y}|}_{y=1}=-\frac{A}{L}\frac{2{\mu }_s}{\lambda +2{\mu }_s}\sin \left(t\right).$$ (34)

We take the Laplace transform of Eq. 33 with respect to t,

$$s\tilde{c}=\left(\lambda +2{\mu }_s\right)\frac{d^2\tilde{c}}{d{y}^2},$$ (35)

where $\tilde{c}\left(y,s\right)=\mathrm{L}\left\{c\left(y,t\right)\right\}={\int }_0^{\infty }{e}^{-st}c\left(y,t\right)d{t}^{\ast }$ with the boundary conditions $\tilde{c}\left(0,s\right)=0$ and

$${\frac{\partial \tilde{c}}{\partial y}|}_{y=1}=-\frac{2{\mu }_s}{\lambda +2{\mu }_s}\frac{A}{L}\frac{1}{1+s^2}.$$ (36)

Solving Eq. 35 for $\tilde{c}\left(y,s\right)$ and applying the boundary conditions, we obtain

$$\tilde{c}\left(y,s\right)=-\frac{2{\mu }_s}{\lambda +2{\mu }_s}\frac{A}{L}\frac{1}{1+s^2}\frac{\sinh \left(\nu y\sqrt{2s}\right)}{\sqrt{2s}\nu \cosh \left(\nu \sqrt{2s}\right)},$$ (37)

where ν = (2(λ + 2μ_s))^−1/2. We compute the inverse Laplace transform of Eq. 37 using complex contour integration to obtain the solution in terms of a limit-cycle solution and a transient solution:

$$c\left(y,t\right)=\frac{A}{L}\frac{{\mu }_s}{\lambda +2{\mu }_s}\left(C_L\left(y,t\right)+C_T\left(y,t\right)\right).$$ (38)

The limit-cycle solution that arises from the poles at s = ±i is

$$C_L\left(y,t\right)=\left(\frac{1+i}{2}\frac{\sinh \left(\nu \left(1+i\right)y\right)}{\nu \cosh \left(\nu \left(1+i\right)\right)}{e}^{it}+\frac{1-i}{2}\frac{\sinh \left(\nu \left(1-i\right)y\right)}{\nu \cosh \left(\nu \left(1-i\right)\right)}{e}^{-it}\right).$$ (39)

The transient solution arises from the poles at s = s_n = (λ + 2μ_s)(2n − 1)²(π/2)² and is

$$C_T\left(y,t\right)=4\left(\lambda +2{\mu }_s\right)\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n}{1+s_n^2}\sin \left(\left(2n-1\right)\frac{\pi }{2}y\right)e^{-s_{n}t}.$$ (40)

Hence, the solution for the displacements and the pore pressure is

$$u_1\left(x,t\right)=A\left(1-\frac{x}{L}\right)\sin \left(t\right),$$ (41)

$$u_2\left(y,t\right)=\frac{A}{L}\left[y\sin \left(t\right)+\frac{{\mu }_s}{\lambda +2{\mu }_s}\left(C_L\left(y,t\right)+C_T\left(y,t\right)\right)\right],$$ (42)

$$p\left(y,t\right)={\mu }_s\frac{A}{L}\left[2\sin \left(t\right)+\frac{\partial C_L}{\partial y}+\frac{\partial C_T}{\partial y}\right].$$ (43)

Using the nondimensional Darcy’s law, Eq. 6, there is no flow in the horizontal direction and the relative fluid velocity in the vertical direction is

$$v_2\left(y,t\right)=\frac{{\mu }_s}{\phi }\frac{A}{L}\left(\frac{{\partial }^2{C}_L}{\partial y^2}+\frac{{\partial }^2{C}_T}{\partial y^2}\right),$$ (44)

where

$$\frac{{\partial }^2{C}_L}{\partial y^2}=-\nu \left(\left(1-i\right)\frac{\sinh \left(\nu \left(1+i\right)y\right)}{\cosh \left(\nu \left(1+i\right)\right)}{e}^{it}+\left(1+i\right)\frac{\sinh \left(\nu \left(1-i\right)y\right)}{\cosh \left(\nu \left(1-i\right)\right)}{e}^{-it}\right)$$ (45)

and

$$\frac{{\partial }^2{C}_T}{\partial y^2}=\left(\lambda +2{\mu }_s\right)\frac{A}{L}\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{1+s_n^2}{\left(2n-1\right)}^2{\pi }^2\sin \left(\left(2n-1\right)\frac{\pi }{2}y\right)e^{-s_{n}t}.$$ (46)

Appendix B: Analytic Solution Domain Configuration II—Crossflow

We begin by removing the inhomogeneous pressure boundary conditions separating the crossflow solution from the solution that is due to the oscillating wall:

$$p\left(x,y,t\right)=P_1-\left(P_1-P_2\right)y+p_o\left(x,y,t\right).$$ (47)

Note that the crossflow solution does induce any deformation in the medium.

We now proceed as before, by letting

$$u_1\left(x,t\right)=A\left(1-\frac{x}{L}\right)\sin \left(t\right),$$ (48)

and assuming the vertical displacement and the pore pressure depend only on y and t (u₂ = u₂(y,t) and p = p(y,t)), we obtain the following equations:

$$\left(\lambda +2{\mu }_s\right)\frac{{\partial }^2{u}_2}{\partial y^2}=\frac{\partial p}{\partial y},$$ (49)

$$\frac{{\partial }^2{u}_2}{\partial y\partial t}=\frac{{\partial }^{2}p}{\partial y}+\frac{A}{L}\cos \left(t\right),$$ (50)

with the boundary conditions p(0,t) = p(1,t) = 0 and

$${\frac{\partial u_2}{\partial y}|}_{y^{\ast }=0}={\frac{\partial u_2}{\partial y}|}_{y=1}=\frac{A}{L}\frac{\lambda }{\lambda +2{\mu }_s}\sin \left(t\right),$$ (51)

along with the initial conditions p(y,0) = u₂(y,0) = 0.

Integrating Eq. 49 with respect to y and applying the boundary conditions yields

$$p\left(y,t\right)=\left(\lambda +2{\mu }_s\right)\frac{\partial u_2}{\partial y}-\frac{A}{L}\lambda \sin \left(t\right),$$ (52)

and Eq. 50 becomes

$$\frac{{\partial }^2{u}_2}{\partial y\partial t}=k\left(\lambda +2{\mu }_s\right)\frac{{\partial }^3{u}_2}{\partial y^3}+\frac{A}{L}\cos \left(t\right).$$ (53)

Integrating Eq. 53 with respect to y and applying the boundary conditions, we obtain

$$\frac{\partial u_2}{\partial t}=k\left(\lambda +2{\mu }_s\right)\frac{{\partial }^2{u}_2}{\partial y^2}+\frac{A}{L}y\cos \left(t\right)+\frac{dg}{dt},$$ (54)

where g(t) corresponds to a translation of the medium in time. Hence, we set g(t) to fix u₂ at y = 1/2. We eliminate $\dot{g}\left(t\right)$ and the inhomogeneous term in the boundary conditions by setting

$$u_2\left(y,t\right)=\frac{A}{L}\frac{\lambda }{\lambda +2{\mu }_s}y\sin \left(t\right)+C\left(y,t\right)+g\left(t\right)$$ (55)

to obtain the equation

$$\frac{\partial C}{\partial t}=\left(\lambda +2{\mu }_s\right)\frac{{\partial }^{2}C}{\partial y^2}-\frac{A}{L}\frac{2{\mu }_s}{\lambda +2{\mu }_s}y\cos \left(t\right)$$ (56)

with the boundary conditions

$${\frac{\partial C}{\partial y}|}_{y=0}={\frac{\partial C}{\partial y}|}_{y=1}=0,$$ (57)

and the initial condition C(y,0) = 0. We expand in a Fourier cosine series by multiplying Eq. 56 by 2cos(nπy) and integrate over the domain with respect to y for n = 0, 1, 2, … to obtain

$$\frac{d{C}_n}{dt}=-a^2{n}^2{\pi }^2{C}_n-b\cos \left(t\right)\underset{0}{\overset{1}{\int }}y\cos \left(n\pi y\right)dy,$$ (58)

where

$$C_n\left(t\right)=2\underset{0}{\overset{1}{\int }}C\left(y,t\right)\cos \left(n\pi y\right)dy$$

with

$$a^2=\left({\lambda }^{\ast }+2{\mu }_s\right)\text{and}b=2A{\mu }_s/L\left(\lambda +2{\mu }_s\right).$$

Hence,

$$C\left(y,t\right)=\frac{1}{2}{C}_0\left(t\right)+\sum _{n=1}^{\infty }{C}_n\left(t\right)\cos \left(n\pi y\right).$$ (59)

Note that

$$2\underset{0}{\overset{1}{\int }}y\cos \left(n\pi y\right)dy=\{\begin{array}{cc}1& \text{for}n=0\\ 0& \text{for}n>0,\text{even}\\ -\frac{4}{n^2{\pi }^2}& \text{for}n\text{odd}\text{.}\end{array}$$ (60)

For n = 0, we have C₀(t) = b sin(t) and, for n > 0, even, we have C_n(t) ≡ 0. For n odd, Eq. 58 is

$$\frac{d{C}_n}{dt}=-a^2{n}^2{\pi }^2{C}_n+\frac{4b}{n^2{\pi }^2}\cos \left(t\right),$$ (61)

with the initial condition C_n(0) = 0.

We now take the Laplace transform of Eq. 61 with respect to t and use the initial condition to obtain

$$\widetilde{C_n}=\frac{4b}{n^2{\pi }^2}\frac{s}{\left(1+s^2\right)\left(s+{\left(an\pi \right)}^2\right)},$$ (62)

where

$$\widetilde{C_n}\left(s\right)=\mathcal{L}\left\{C_n\left(t\right)\right\}.$$

We compute the inverse Laplace transform of $\widetilde{C_n}$ by taking the partial fraction expansion of Eq. 62,

$$\widetilde{C_n}=\frac{4b}{{\left(n\pi \right)}^2\left({\left(an\pi \right)}^4+1\right)}\left[{\left(an\pi \right)}^2\left(\frac{s}{1+s^2}-\frac{1}{s+{\left(an\pi \right)}^2}\right)+\frac{1}{1+s^2}\right],$$ (63)

and taking the inverse Laplace transform of each term to obtain

$$C_n\left(t\right)=\frac{4b}{{\left(n\pi \right)}^2\left({\left(an\pi \right)}^4+1\right)}\left[{\left(an\pi \right)}^2\left(\text{cos}\left(t\right)-e^{-{\left(an\pi \right)}^{2}t}\right)+\sin \left(t\right)\right].$$ (64)

Combining Eqs. 59 and 64, we obtain

$$C\left(y,t\right)=-\frac{b}{2}\sin \left(t\right)+\sum _{n=1}^{\infty }\frac{4a^{2}b}{{\nu }_n^2\left({\nu }_n^2+1\right)}\left[{\nu }_n\left(\cos \left(t\right)-e^{-v_{n}t}\right)+\sin \left(t\right)\right]\cos \left(\left(2n-1\right)\pi y\right),$$ (65)

where ν_n = a²(2n − 1)² = (λ+2μ_s)(2n − 1)² π². The final solution for u₂ is

$$u_2\left(y,t\right)=\frac{A}{L}(\frac{1}{\lambda +2{\mu }_s}\left(\lambda y-{\mu }_s\right)\sin \left(t\right)+8{\mu }_s\sum _{n=1}^{\infty }\frac{1}{{\nu }_n\left({\nu }_n^2+1\right)}\left[{\nu }_n\left(\cos \left(t\right)-e^{-{\nu }_{n}t}\right)+\sin \left(t\right)\right]\cos \left(\left(2n-1\right)\pi y\right)).$$ (66)

Once again, the displacement is governed by a transient solution and a limit cycle solution. The pressure, without the crossflow terms, is

$$p\left(y,t\right)=-8\frac{A}{L}{\mu }_s\left(\lambda +2{\mu }_s\right)\sum _{n=1}^{\infty }\frac{\left(2n-1\right)\pi }{{\nu }_n\left({\nu }_n^2+1\right)}\left[{\nu }_n\left(\cos \left(t\right)-e^{-{\nu }_{n}t}\right)+\sin \left(t\right)\right]\sin \left(\left(2n-1\right)\pi y\right).$$ (67)

Substituting the pore pressure into Darcy’s law and including the crossflow solution, the relative fluid velocity is

$$v_2=\frac{1}{\phi }\left[\left(P_2-P_1\right)+8\frac{A}{L}{\mu }_s\sum _{n=1}^{\infty }\frac{1}{{\nu }_n^2+1}\left[{\nu }_n\left(\cos \left(t\right)-e^{{\nu }_{n}t}\right)+\sin \left(t\right)\right]\cos \left(\left(2n-1\right)\pi y\right)\right],$$ (68)

with no flow in the horizontal direction.

Appendix C: Dimensionless Parameters

$$\begin{array}{l}A=\gamma L,\\ H^{\ast }=8\text{mm,}\\ k=1.43\times {10}^{-11}{\text{m}}^4\text{/Ns,}\\ L^{\ast }=10\text{mm,}\\ {\mu }_s^{\ast }=5680.8\text{poise,}\\ {\lambda }^{\ast }=11,027.5\text{poise,}\\ \gamma =5-20\%,\\ \phi =0.996,\\ \omega =\pi \text{to}4\pi \text{rad/s}\text{.}\end{array}$$