Insights into Interstitial Flow, Shear Stress, and Mass Transport Effects on ECM heterogeneity in Bioreactor-Cultivated Engineered Cartilage Hydrogels

Tony Chen, Mark Buckley, Itai Cohen, Lawrence Bonassar, Hani A. Awad

Supplemental Information

1. Biphasic theory solution of unidirectional flow over a permeable hydrated mixtureReproduced from Hou et al. (1989)

Hou et al. (1989) reported a theoretical solution to the classical problem of Poiseuille flow over a

porous mixture (Figure A1) using the biphasic theory. Here we reproduce the solution in

dimensionless form for reference.

Nomenclature:ϕ s Volume fraction of the solid phase in an immiscible biphasic mixture

ϕ f Volume fraction of the fluid phase in an immiscible biphasic mixture

v f Flow velocity

μf Free fluid viscosity (in the flow channel)

μa Apparent fluid viscosity in the biphasic layer

κ Darcian permeability

k Hydraulic permeability k¿κ / μa K Interstitial drag coefficient in the biphasic layer, which is inversely proportional

to the Darcian permeability κ=ϕ f 2 μ f / K

∂ p/∂ y Pressure drop driving the flow in the channelh1 Height of flow channel

h2 Thickness of biphasic layer

Figure S1. Schematic representation of the problem of Poiseuille flow over a biphasic layer.Reproduced with modification from Hou et al. (1989).

In setting up the problem, the flow of an incompressible Newtonian fluid (e.g. viscous) is

assumed to be driven by a unidirectional pressure gradient along the flow channel and over a

biphasic layer, which in turn is assumed to be an immiscible mixture of a linearly elastic, and

isotropic solid phase ( ϕs ) and an incompressible, viscous fluid phase ( ϕf ) such that ϕ s+ϕ f =1. The

boundary conditions at the interface between the free fluid and the biphasic layer surface were

derived by considering the balance laws for mass, linear momentum, and energy.

The force balance for the fluid in the channel and biphasic layer yields the following governing

equations, respectively:

−∂ p∂ y

+μ f∂2v f

∂ z2 =00 ≤ z ≤ h1(S1)

−ϕ f ∂ p∂ y

+μa∂2 v f

∂ z2 −K v f =0−h2 ≤ z ≤ 0(S2)

subject to the “pseudo no-slip” and the momentum “jump” boundary conditions at the biphasic

layer in contact with the free fluid ( z=0 )

ϕ f vf|z=0−¿= v f|z=0+¿(S3) ¿¿

μa

ϕf∂ v f

∂ z |z=0−¿=μf∂ v f

∂ z |z=0+¿(S4) ¿

¿

and the no-slip boundary condition at the rigid impermeable walls ( z=h1 ,−h2 )

v f=0 (S5)

The governing equations (S1 and S2) can be written in nondimensional form as

1+ 1ϕf

∂2 Qf

∂ ξ2 =0 0≤ ξ ≤ λ (S6)

δ 2 ∂2 Qf

∂ξ2 −Q f=ηδ 2−1≤ ξ ≤ 0 (S7)

Where

Qf =ϕf μ f

h22( ∂ p

∂ y )v f

is the nondimensional fluid flux

ξ= zh2

is a nondimensional (thickness) coordinate

λ=h1

h2 is the ratio of the flow channel height to the biphasic layer thickness

δ 2=μa

K h22 is the ratio of viscous drag of the outside fluid to the interstitial drag

of the fluid within the porous biphasic layer

η=ϕf 2 μ f

μa is a weighted viscosity ratio

The pseudo no-slip condition and momentum jump boundary condition at the channel-biphasic

layer interface (ξ=0), respectively, reduce to the following nondimensional form:

ϕ f Q f|ξ=0−¿=Q f|ξ=0+¿(S8) ¿¿

ϕf

η∂ Qf

∂ξ |ξ=0−¿=

∂Q f

∂ ξ |ξ=0+¿(S9) ¿

¿

The no slip boundary conditions at the rigid impermeable walls (ξ= λ ,−1 )

Qf =0 (S10)

The solution to the nondimensional governing equations subject to the aforementioned boundary

conditions can be expressed in a compact form as

Qf =[ −ϕf

2(ξ−λ )(ξ+ηδ

2 δ ∙ [cosh (1/δ )−1 ]+λ ∙ sinh (1 /δ )λ ∙ cosh (1 /δ )+ηδ ∙ sinh (1/δ ) )0 ≤ ξ ≤ λ

¿η δ 2(1−e−ξ+1

δ )+ηδ 2 δ (ηδ−λ )+( λ2−2 η δ 2) ∙ e1/ δ

−ηδ+ λ+(ηδ +λ ) ∙ e2 /δ ∙ sinh( ξ+1δ )−1≤ ξ ≤ 0

(S 11)

Therefore, the interstitial flux depends upon δ , λ, and η, whose effects have been discussed in detail by

Hou et al. (1989). In our setup, it is reasonable to assume that η is constant since the free fluid and

interstitial fluid are identical and thus have the same viscosity value. The experimentally measured flow

velocity values were fitted using least square nonlinear regression (fminsearch function in MATLAB)

to the equation v f=Q f h22( ∂ p

∂ y )/ϕ f μ f , where Qf is the piecewise biphasic solution expressed in equation

S11, to determine the values of δ and λ. While the nominal dimensions of the chamber and hydrogel

were known, we chose to include λ as an optimization variable to eliminate any experimental

uncertainty in determining the z-coordinate of the hydrogel surface. The drag coefficient and the

apparent permeability where then determined from the optimized value of δ as discussed in the

manuscript.

2. Computational fluid dynamics modeling

Since the theoretical solution described in section 1 was derived for Poiseuille flow over an infinite slab

of a porous material which does not account for the finite geometry of the bioreactor system and the

end conditions that might give rise to two-directional, reversible flow, a 2-D computational fluid

dynamics (CFD) model of the parallel plate bioreactor system was created using the geometry and

dimensions of the scaled-down bioreactor used in the FRAP-based flow visualization experiments

(Table 1). CFD modeling was performed using COMSOL Multiphysics v4.0a (Comsol, Stockholm,

Sweden). The bioreactor was discretized into 3 regions comprising the flow media channel, TEC

agarose hydrogel, and the anchoring cell-free agarose hydrogel (Figure 2a). To optimize the

computational time without sacrificing the accuracy of the numerical solution, the mesh within the TEC

hydrogel region, which comprised 111,864 triangular elements, was refined to be biased towards the

interfaces with the free fluid channel and the anchoring agarose layer (Figure S2). The remainder of

the geometry was meshed using 105,064 uniformly distributed triangular elements. “No-slip”

boundary conditions were enforced at the walls. The viscous fluid in the media flow channel was

modeled using the Navier-Stokes equations, while the hydrogel regions were modeled using the

Brinkman’s equation. The TEC hydrogel permeability was set to the value determined for each flow

rate (Table 2). The permeability of the cell-free agarose hydrogel was scaled as a function of the

agarose concentration as described by Gu et al (2003). The CFD model was solved using the PARDISO

static solver. Convergence of the model was confirmed using mesh refinement.

The fluid velocity profiles and flow streamlines (Figure S3) were examined at the midline of the

bioreactor and locations 1 mm, either upstream or downstream, from the midline. These measurements

were done to verify that the flow is mostly unidirectional at the bioreactor midline, where the FRAP

measurements were taken. The model was also used to compute the shear stress profile at midline

through the depth of the bioreactor (Figure 5).

Figure S2. (a) The bioreactor computational grid showing the different regions comprising the CFD model including the flow channel, the TEC hydrogel, and the anchoring agarose hydrogel layer. (b) Magnification of a region at the interface between the flow channel and the TEC hydrogel showing the biased grid density with increased elements at the interface.

Flow Channel

TEC Hydrogel

TEC Hydrogel

e

Figure S3. Flow velocity profiles within the flow chamber and the TEC hydrogels were estimated from a CFD model of the scaled-down parallel plate bioreactor for the case of 0.14 ml/min at the midline of the bioreactor and locations 1 mm, either upstream or downstream, from the midline. Panels a and b depict the predicted parallel (y-velocity) and perpendicular (z-velocity) components of the flow velocity, respectively. Note the difference in the scale of the two velocity components, which suggests that the assumption that the flow is mostly unidirectional is valid. Panels c and d show the depth dependent interstitial flow y-velocity and z-velocity profiles, respectively, within the TEC hydrogel

region. Panel e depicts representative streamline plots showing the predicted flow paths within the different bioreactor regions.