General tissue mass transfer model for cryopreservation applications

Abstract

Successful cryopreservation of complex specimens, such as tissues and organs, would greatly benefit both the medical and scientific research fields. Vitrification is one of the most promising techniques for complex specimen cryopreservation, but toxicity remains a major challenge because of the high concentration of cryoprotectants (CPAs) needed to vitrify. Our group has approached this problem using mathematical optimization to design less toxic CPA equilibration methods for cells. To extend this approach to tissues, an appropriate mass transfer model is required. Fick’s law is commonly used, but this simple modeling framework does not account for the complexity of mass transfer in tissues, such as the effects of fixed charges, tissue size changes, and the interplay between cell membrane transport and transport through the extracellular fluid. Here, we propose a general model for mass transfer in tissues that accounts for all of these phenomena. To create this model, we augmented a previously published acellular model of mass transfer in articular cartilage to account for the effects of cells. We show that the model can accurately predict changes in CPA concentration and tissue size for both articular cartilage and pancreatic islets, tissue types with vastly different properties.

Significance

Tissue cryopreservation involves exposure to concentrated solutions containing cryoprotectants, which causes a complex response that includes shrinkage and swelling of the tissue as well as the cells embedded in the tissue. Conventional diffusion models do not account for these phenomena. This study presents a general mass transfer model that accounts for various tissue-specific phenomena, including size changes caused by exposure to concentrated solutions. We show that this model can predict tissue size changes observed for articular cartilage and pancreatic islets, highlighting the generality of the modeling approach. This new model may enable the design of improved cryopreservation methods for various tissue types, which would have a broad impact in tissue transplantation and biomedical research.

Introduction

The successful cryopreservation of complex biological specimens such as tissues and organs would have immense benefits for the public health by improving the effectiveness of transplantations, accelerating medical research, and enabling emergency preparedness (1, 2, 3). However, these complex specimens have proven to be particularly challenging to cryopreserve (1,4).

Ice-free cryopreservation, known as vitrification, is a promising approach for complex specimens because it eliminates damage due to extracellular ice (5, 6, 7, 8). However, vitrification poses two new problems (i.e., osmotic damage and chemical toxicity) stemming from the high concentration of cryoprotectants (CPAs) required to suppress ice formation. Exposure to CPAs can induce excessive volume excursions (osmotic damage) and impart toxicity (4,9, 10, 11, 12). Osmotic damage can be prevented by incrementally adding and removing CPAs to reduce the extent of volume excursions (9,10,12, 13, 14), but avoiding toxicity is not as straightforward and remains a challenge (4,15, 16, 17).

Recent work has demonstrated the potential for using mathematical modeling to design less toxic methods for vitrification by manipulating the CPA composition and the time and temperature of CPA exposure (9,18, 19, 20, 21). These modeling approaches require an appropriate mass transfer model for predicting CPA transport in tissues. This is a difficult problem, as there are many tissue-based phenomena that need to be considered (22), including 1) mass transfer in the extracellular space, 2) coupling between extracellular and cell membrane mass transfer, 3) fixed electrical charges on the extracellular matrix, and 4) tissue volume changes. These phenomena are expected to be broadly applicable to tissues (22). For instance, tissue volume changes after exposure to CPA have been observed for various tissue types, including cartilage (23), pancreatic islets (24), ovarian tissue (25), and heart valves (26). Currently, there is no modeling paradigm that accounts for all of these phenomena.

In the literature, the most common way to model mass transfer is using Fick’s second law (6,27,28), but this approach does not account for cell membrane transport, fixed charges, nor tissue size changes. Others have presented more complicated models that address some of these phenomena (24,29, 30, 31, 32, 33, 34). In particular, the recent articular cartilage model presented by Abazari et al. (23,35) addresses nearly all of the phenomena, but it does not account for the coupling between extracellular and cell membrane mass transfer. This decoupling is appropriate in cell-sparse cartilage modeled by Abazari et al. (23), but its appropriateness for cell-dense tissues is questionable.

In this work, we present a general tissue mass transfer model that accounts for transport in the extracellular space, cell membrane transport, fixed charges, and tissue size changes. This general model was created by augmenting the articular cartilage model of Abazari et al. (23) to account for the coupling of extracellular mass transfer and cell membrane mass transfer. To show the generality of this approach, we applied our model to pancreatic islets and were able to match our previously published data. Overall, we have shown that we can model two very different tissues, articular cartilage and pancreatic islets, highlighting the potential for using this model to design improved cryopreservation procedures for many tissue types.

Materials and methods

Model description and definition of state variables

We consider one-dimensional transport in a thin slab of tissue, with a zero-flux condition on one end and the other end exposed to an infinite bath solution. Note that in the case of articular cartilage, this condition reflects that one end of the cartilage is exposed to the bath and the other to effectively impermeable bone. In general, the zero-flux condition can be considered a symmetry condition, in which both ends are in the bath and there is symmetry around the center of the tissue, the case considered below for pancreatic islets. The insulated side of the tissue is fixed in space, and the bath side is free to move as governed by the fluid fluxes and mechanical properties of the tissue. As such, this model is a classical one-dimensional moving boundary problem. Fig. 1 shows the general geometry of the model.

Figure 1.

Schematic illustrating modeling approach.

As shown in Fig. 1, the tissue is divided into four compartments: extracellular fluid, intracellular fluid, intracellular solids, and extracellular solids. Water, CPA, and salt (modeled as NaCl) can move through the extracellular space in the x-direction, and this can result in movement of solids in the x-direction as well. The cells are considered to be embedded in the extracellular solids and hence move with the solids. The intracellular fluid can exchange water and CPA with the extracellular fluid.

The variables used to describe the state of each of these compartments are depicted in Fig. 1, and subscripts are used to specify the chemical species and the compartment; iw, ic, in, s, n, w, and c represent intracellular water, intracellular CPA, intracellular salt, solids, extracellular salt, extracellular water, and extracellular CPA, respectively. All model variables are defined in the Nomenclature section in the Supporting materials and methods.

The mass concentration and volume fraction are defined per tissue volume and are used to write the continuity equations. The volume fraction of species k can be calculated from the mass concentration as follows:

$$\begin{array}{l}{\phi }_k=\frac{{\rho }_k}{{\overline{\rho }}_k},\hfill \end{array}$$ (1)

where ${\overline{\rho }}_k$ is the density of pure species k. The volume fractions of extracellular and intracellular salt were considered negligible, as done by Abazari et al. (23) for the extracellular salt.

The mole concentrations and mole fractions are used to represent driving forces for transport and are defined based on the fluid volume in which the chemical species is dissolved. For the extracellular fluid, the mole concentrations of water and CPA can be calculated from the mass concentrations:

$$\begin{array}{l}{C}_{\text{w}}=\frac{{\rho }_{\text{w}}}{M_{\text{w}}\left({\phi }_{\text{w}}+{\phi }_{\text{c}}\right)},\hfill \end{array}$$ (2a)

$$\begin{array}{l}{C}_{\text{C}}=\frac{{\rho }_{\text{c}}}{M_{\text{c}}\left({\phi }_{\text{w}}+{\phi }_{\text{c}}\right)}.\hfill \end{array}$$ (2b)

where M_k represents the molecular weight of species k, and where we have assumed that the volume fraction of the salt species is negligible. For the intracellular fluid, the mole concentrations of water and CPA can be expressed similarly:

$$\begin{array}{l}{C}_{\text{iw}}=\frac{{\rho }_{\text{iw}}}{M_{\text{w}}\left({\phi }_{\text{iw}}+{\phi }_{\text{ic}}\right)},\end{array}$$ (2c)

$$\begin{array}{l}{C}_{\text{ic}}=\frac{{\rho }_{\text{ic}}}{M_{\text{c}}\left({\phi }_{\text{iw}}+{\phi }_{\text{ic}}\right)}.\hfill \end{array}$$ (2d)

The extracellular matrix (solids species) is negatively charged and contains fixed positively charged sodium counterions that also contribute to the driving forces for transport. However, these fixed charges cannot move freely through the extracellular fluid and move with the matrix. The mole concentration of chloride ions can be calculated from the mass concentration:

$$\begin{array}{l}{C}_{\text{Cl}}=\frac{{\rho }_{\text{n}}}{M_{\text{n}}\left({\phi }_{\text{w}}+{\phi }_{\text{c}}\right)}.\hfill \end{array}$$ (2e)

The mole concentration of sodium ions includes the counterions for the chloride ions as well as the fixed charges:

$$\begin{array}{l}{C}_{\text{Na}}=C_{\text{Cl}}+C_{\text{fc}}.\hfill \end{array}$$ (3)

Thus, the sodium concentration is higher than the chloride concentration as a result of fixed charges. The total mole concentration of salt ions is then the sum of the sodium and chloride concentrations:

$$\begin{array}{l}{C}_{\text{n}}=C_{\text{Cl}}+C_{\text{Na}}.\hfill \end{array}$$ (4)

Whereas the concentration of fixed charges can change as the tissue shrinks and swells, the number of fixed charges per volume of solids does not change. This allows us to express the fixed charge concentration, C_fc, at any time in terms of a reference fixed charge concentration, $C_{\text{fc}}^{\text{o}}$ (see Supporting materials and methods for a derivation of Eq. 5a):

$$\begin{array}{l}{C}_{\text{fc}}=C_{\text{fc}}^{\text{o}}\left(\frac{{\phi }_{\text{w}}^{\text{o}}+{\phi }_{\text{c}}^{\text{o}}}{{\phi }_{\text{s}}^{\text{o}}}\right)\left(\frac{{\phi }_{\text{s}}}{{\phi }_{\text{w}}+{\phi }_{\text{c}}}\right).\hfill \end{array}$$ (5a)

The mole concentration of intracellular salt can be expressed in a similar manner. Because we consider salt to be impermeable to the cell membrane, the moles of intracellular salt per solids volume remains constant. This allows the intracellular salt concentration, C_in, at any time to be expressed in terms of a reference concentration, $C_{\text{in}}^{\text{o}}$:

$$\begin{array}{l}{C}_{\text{in}}=C_{\text{in}}^{\text{o}}\left(\frac{{\phi }_{\text{iw}}^{\text{o}}+{\phi }_{\text{ic}}^{\text{o}}}{{\phi }_{\text{s}}^{\text{o}}}\right)\left(\frac{{\phi }_{\text{s}}}{{\phi }_{\text{iw}}+{\phi }_{\text{ic}}}\right).\hfill \end{array}$$ (5b)

Finally, we can express the mole fraction, x, of species k in the extracellular fluid in terms of the mole concentrations:

$$\begin{array}{l}{x}_k=\frac{C_k}{C_{\text{w}}+C_{\text{c}}+C_{\text{n}}}.\hfill \end{array}$$ (6)

Transport equations

We begin by writing continuity equations based on the six mass concentrations (excluding intracellular salt) shown in Fig. 1. Movement of each species in the x-direction is expressed in terms of its average velocity, u, whereas exchange of water and CPA across cell membranes is expressed in terms of the two-parameter membrane transport model, as described below. This yields the continuity equations:

$$\begin{array}{l}\frac{\partial {\rho }_k}{\partial t}+\frac{\partial \left({\rho }_k{u}_k\right)}{\partial x}+{\chi }_k\frac{\partial V_{\text{i}k}}{\partial t}\eta {\phi }_{\text{s}}{\overline{\rho }}_k=0,\hfill \end{array}$$ (7)

where k = iw, ic, s, n, w, or c, and χ_k = −1 for k = iw and ic, χ_k = 0 for k = s and n, and χ_k = 1 for k = w and c. The variables V_iw and V_ic are the intracellular water and CPA volume per cell, respectively, and η is a constant term describing the cell density in the tissue (as defined in more detail below). These continuity equations follow from those given by Abazari et al. (23), with the addition of intracellular species and terms describing cell membrane transport.

The cell membrane transport terms are based on a classic two-parameter membrane transport model (36), which gives the rate of change of intracellular water volume and intracellular CPA volume for a single cell in terms of concentration driving forces:

$$\begin{array}{l}\frac{\partial V_{\text{iw}}}{\partial t}=L_{\text{p}}{A}_{\text{cell}}RT\left(C_{\text{ic}}+C_{\text{in}}-C_{\text{c}}-C_{\text{n}}\right),\hfill \end{array}$$ (8a)

$$\begin{array}{l}\frac{\partial V_{\text{ic}}}{\partial t}=P_{\text{c}}{A}_{\text{cell}}{\overline{\nu }}_{\text{c}}\left(C_{\text{c}}-C_{\text{ic}}\right).\hfill \end{array}$$ (8b)

In these equations, L_p is the hydraulic conductivity, P_c is the CPA permeability, A_cell is the (assumed constant) cell membrane surface area, ${\overline{\nu }}_{\text{c}}$ is the CPA molar volume, R is the universal gas constant, and T is temperature. The intracellular salt concentration, C_in, represents the total osmotic contributions of all intracellular salts. In the two-parameter formalism, salt is considered to be impermeable to the cell membrane (36). We used the two-parameter formalism for cell membrane transport because of its common use and ease of implementation. However, our approach for coupling extracellular and cell membrane transport is modular and can be used with any cell membrane transport model, including the recent model developed for nondilute solutions (37).

To use these membrane transport equations in the continuity equations, it was necessary to relate the volume changes for a single cell to the corresponding rate of transport between the intracellular fluid and extracellular fluid per tissue volume. This was accomplished by defining a constant term, η, to describe the cell density, which represents the number of cells per volume of solids. This term is considered constant because the cells are attached to the extracellular matrix and therefore move with the solids. The number of cells per tissue volume can then be calculated as $\eta {\phi }_s$. Also, η is used to interconvert between single cell volumes and tissue volume fractions as follows:

$$\begin{array}{l}{V}_{\text{iw}}=\frac{{\phi }_{\text{iw}}}{\eta {\phi }_{\text{s}}},\hfill \end{array}$$ (9a)

$$\begin{array}{l}{V}_{\text{ic}}=\frac{{\phi }_{\text{ic}}}{\eta {\phi }_{\text{s}}}.\hfill \end{array}$$ (9b)

Now, the only unknowns left from the continuity expressions are the four species velocities. To describe the velocity field, we relate the velocities to the chemical potential gradients, as described by Abazari et al. (23). The resulting multicomponent momentum balance is given below:

$$\begin{array}{l}-{\rho }_{\text{w}}\frac{\partial {\mu }_{\text{w}}}{\partial x}=f_{\text{cw}}\left(u_{\text{w}}-u_{\text{c}}\right)+f_{\text{ws}}\left(u_{\text{w}}-u_{\text{s}}\right)+f_{\text{nw}}\left(u_{\text{w}}-u_{\text{n}}\right),\hfill \end{array}$$ (10a)

$$\begin{array}{l}-{\rho }_{\text{c}}\frac{\partial {\mu }_{\text{c}}}{\partial x}=f_{\text{cw}}\left(u_{\text{c}}-u_{\text{w}}\right)+f_{\text{cs}}\left(u_{\text{c}}-u_{\text{s}}\right),\hfill \end{array}$$ (10b)

$$\begin{array}{l}-{\rho }_{\text{n}}\frac{\partial {\mu }_{\text{n}}}{\partial x}=f_{\text{nw}}\left(u_{\text{n}}-u_{\text{w}}\right),\hfill \end{array}$$ (10c)

where μ_k is the chemical potential of species k, and f_mn is the frictional coefficient between species m and n. A volume balance can be used to define a fourth relation (23):

$$\begin{array}{l}{u}_{\text{s}}\left({\phi }_{\text{s}}+{\phi }_{\text{iw}}+{\phi }_{\text{ic}}\right)+u_{\text{c}}{\phi }_{\text{c}}+u_{\text{w}}{\phi }_{\text{w}}=0.\hfill \end{array}$$ (11)

The frictional coefficients, as defined by Abazari et al. (23), are functions of the diffusivities of CPA and salt within water, D_cw and D_nw, respectively, as well as the permeabilities of CPA and water in the tissue, K_cs and K_ws, respectively. The frictional coefficients are defined as follows:

$$\begin{array}{l}{f}_{\text{cw}}=\frac{RT({\phi }_{\text{w}}+{\phi }_{\text{c}})C_{\text{c}}}{D_{\text{cw}}},\hfill \end{array}$$ (12a)

$$\begin{array}{l}{f}_{\text{nw}}=\frac{RT({\phi }_{\text{w}}+{\phi }_{\text{c}})C_{\text{Cl}}}{D_{\text{nw}}},\hfill \end{array}$$ (12b)

$$\begin{array}{l}{f}_{\text{cs}}=\frac{{\phi }_{\text{c}}^2}{K_{\text{cs}}},\hfill \end{array}$$ (12c)

$$\begin{array}{l}{f}_{\text{ws}}=\frac{{\phi }_{\text{w}}^2}{K_{\text{ws}}}.\hfill \end{array}$$ (12d)

The chemical potentials for water, CPA, and salt used by Abazari et al. (23) were derived in terms of mole fractions by Elliot et al. (38) and Elmoazzen et al. (37) and can be defined as follows:

$$\begin{array}{l}{\mu }_{\text{w}}={\mu }_{\text{w}}^{\ast }+\frac{P}{{\overline{\rho }}_{\text{w}}}-\frac{RT}{M_{\text{w}}}\left(1-x_{\text{w}}\right)\left(1+B_{\text{c}}{x}_{\text{c}}\right),\hfill \end{array}$$ (13a)

$$\begin{array}{l}{\mu }_{\text{c}}={\mu }_{\text{c}}^{\ast }+\frac{P}{{\overline{\rho }}_{\text{c}}}+\frac{RT}{M_{\text{c}}}\left(\ln \left(x_{\text{c}}\right)+0.5x_{\text{w}}^2-B_{\text{c}}{x}_{\text{w}}\left(1-x_{\text{c}}\right)\right),\hfill \end{array}$$ (13b)

$$\begin{array}{l}{\mu }_{\text{n}}={\mu }_{\text{n}}^{\ast }+\frac{RT}{M_{\text{n}}}\left(\ln \left(x_{\text{Na}}{x}_{\text{Cl}}\right)+x_{\text{w}}^2+2B_{\text{c}}{x}_{\text{w}}{x}_{\text{c}}\right).\hfill \end{array}$$ (13c)

In the above expressions, B_c is the constant second osmotic virial coefficient, and we do not need to define the reference chemical potentials, ${\mu }_k^{\ast }$, as chemical potentials only appear in differences or differentials.

The last unknown in the chemical potential expressions is the pressure P. This pressure is the gauge pressure defined relative to the pressure in the bath. The pressure, P, in the tissue can be expressed in terms of a reference gauge pressure, P^o, and an elastic pressure, P_elastic, that develops because of the tissue strain, as defined by Abazari et al. (23):

$$\begin{array}{l}P=P^{\text{o}}+P_{\text{elastic}}.\hfill \end{array}$$ (14)

Assuming a linear stress/strain relationship for compressive and tensile deformations yields

$$\begin{array}{l}{P}_{\text{elastic}}=H_{\text{A}}e,\hfill \end{array}$$ (15)

where the aggregate modulus of elasticity, H_A, is defined as a constant for both compressive and tensile deformations. Finally, the strain, e, can be related to the deviation in the solids volume fraction from the reference point:

$$\begin{array}{l}e=\frac{{\phi }_{\text{s}}^{\text{o}}}{{\phi }_{\text{s}}}-1.\hfill \end{array}$$ (16)

Variable transform to fix the size of the spatial domain

To solve the system of equations defined above, it is necessary to address the moving tissue boundary. Such a problem can be classified as a mass transfer analog to a classical one-dimensional Stefan problem in which several solution strategies have been proposed (39, 40, 41). We adopted a boundary immobilization method that fixes the domain size through coordinate transform (39). The transform is defined as follows:

$$\begin{array}{l}\alpha =\frac{x}{h},\hfill \end{array}$$ (17)

where h is the (time-dependent) location of the tissue/bath interface (see Fig. 1). There are only two unique spatial derivatives of first order: the chemical potential gradients and the flux gradients that can be transformed for the kth species:

$$\begin{array}{l}\frac{\partial {\mu }_k}{\partial x}=\frac{1}{h}\frac{\partial {\mu }_k}{\partial \alpha },\hfill \end{array}$$ (18a)

$$\begin{array}{l}\frac{\partial \left({\rho }_k{u}_k\right)}{\partial x}=\frac{1}{h}\frac{\partial \left({\rho }_k{u}_k\right)}{\partial \alpha }.\hfill \end{array}$$ (18b)

The temporal gradient follows from the definition of the total derivative in both x and α-space:

$$\begin{array}{l}{\frac{\partial {\rho }_k}{\partial t}}_x={\frac{\partial {\rho }_k}{\partial t}}_{\alpha }-\frac{\alpha }{h}\frac{dh}{dt}\frac{\partial {\rho }_k}{\partial \alpha }.\hfill \end{array}$$ (19)

It should be noted that the cell volume derivatives do not need to be transformed into the new domain because they are only defined through local concentrations (see Eqs. 8a and 8b).

Looking at the transformed equations, we see that we need an expression for the domain width, h, as well as its derivative dh/dt. The derivative (i.e., the Stefan condition (42) in classical problems) is governed by a mass balance at the boundary:

$$\begin{array}{l}\frac{dh}{dt}=u_{\text{s}}\left(\alpha =1\right).\hfill \end{array}$$ (20)

Eq. 20 can then be integrated to find h. However, we instead defined h using a mass balance on solids within the tissue:

$$\begin{array}{l}h=\frac{{\rho }_{\text{s}}^{\text{o}}{h}^{\text{o}}}{\int {\rho }_{\text{s}}\partial \alpha }.\hfill \end{array}$$ (21)

This mass balance form for the domain length produces the same result as the integrated form of Eq. 20 but has better numerical integration properties.

The physiological reference state

Typically, tissue properties are reported in the literature under normal physiological conditions. Thus, we define a physiological reference state in which the tissue is in equilibrium with a normal physiological solution and use this as a basis for the mass transfer simulations. From experimental data reported in the literature, we can estimate the reference tissue thickness, h^o, the fixed charge concentration, $C_{\text{fc}}^{\text{o}}$, the solids volume fraction, ${\phi }_{\text{s}}^{\text{o}}$, and the intracellular water volume fraction ${\phi }_{\text{iw}}^{\text{o}}$ (see Tables 1 and 2). Given these parameters, we can determine the state of the extracellular water, extracellular salt, and intracellular salt.

Table 1.

Parameters used in the cell-augmented cartilage model.

Parameter	Description	Value
Parameters from Abazari et al. (23)
ho	Cartilage thickness [m]	1e−3
$C_{\text{fc}}^{\text{o}}$	Fixed charge concentration [mol/m3]	200
${\phi }_{\text{s}}^{\text{o}}$	Solids volume fraction	0.2
Dnw	Diffusivity of salt in water [m2/s]	5e−10
Dcw	Diffusivity of DMSO in water [m2/s]a	2.25e−10
Kcs	Permeability of DMSO in cartilage [m4/N/s]a	4.7e−17
Kws	Permeability of water in cartilage [m4/N/s]a	4.98e−16
HA	Modulus of elasticity [Pa]a	2.4e6
New Parameters
ζ	Chondrocyte density of cartilage [cells/m3]	1e14 (43)
$V_{\text{it}}^{\text{o}}$	Chondrocyte isotonic volume [m3]	1e−15 (44)
Vb	Chondrocyte solids volume fraction	0.41 (44)
Lp	Chondrocyte hydraulic conductivity [m/Pa/s], DMSO, 21°C	2.68e−14 (45)
Pc	Chondrocyte CPA permeability [m/s], DMSO, 21°C	7.88e−8 (45)
Derived Parameters
${\phi }_{\text{iw}}^{\text{o}}$	Intracellular water volume fraction, $\zeta \ast V_{it}^o\ast (1-V_b)$	0.059
${\phi }_{\text{w}}^{\text{o}}$	Extracellular water volume fraction, $1-{\phi }_s^o-{\phi }_{iw}^o$	0.741
η	Chondrocytes per volume of solids [cells/m3], $\zeta /{\phi }_s^o$	5e14
Acell	Chondrocyte surface area [m2], $6^{2/3}\ast {\pi }^{1/3}\ast {V_{it}^o}^{2/3}$	4.84e−10
${\overline{\rho }}_{\text{s}}$	Pure species density of solids [kg/m3]b	1.13e3

^aBest-fit values as presented in Figs. 7, 8, 9, and 10 of Abazari et al. (23) for DMSO at 22°C.

^bCalculated based on the dry weight fraction presented by Abazari et al. (23) and the known volume fractions and pure species densities of all other species.

Table 2.

Parameters used for the pancreatic islet model.

Parameter	Description	Value
Literature Parameters
${\phi }_{\text{s}}^{\text{o}}$	Solids volume fraction	0.4 (24)
${\phi }_{\text{w}}^{\text{o}}$	Extracellular water volume fraction	0.2 (24)
$V_{\text{it}}^{\text{o}}$	Islet cell isotonic volume [m3]	970e−18 (24)
Vb	Islet cell solids volume fraction	0.4 (24)
Lp	Islet cell hydraulic conductivity [m/Pa/s]a, DMSO, 22°C	3.30e−14 (24,46)
Pc	Islet cell CPA permeability [m/s]a, DMSO, 22°C	1.86e−7 (24,46)
Acell	Islet cell surface area [m2]	408e−12 (24)
Ro	Islet radius [m]	82.2e−6 (24)
Dnw	Diffusivity of salt in water [m2/s]	5.18e−10 (24)
Dcw	Diffusivity of DMSO in water [m2/s]	5.31e−10 (24)
Derived Parameters
${\phi }_{\text{iw}}^{\text{o}}$	Intracellular water volume fraction, $1-{\phi }_s^o-{\phi }_w^o$	0.4
ζ	Islet cell density of islet [cells/m3], ${\phi }_{iw}^o/(V_{it}^o\ast \left(1-V_b\right))$	6.87e14
η	Islet cells per volume of solids [cells/m3], $\zeta /{\phi }_s^o$	1.72e15
${\overline{\rho }}_{\text{s}}$	Pure species density of solids [kg/m3]b	6.21e2
HA	Modulus of elasticity [Pa]	0
Fitted Parameters
$C_{\text{fc}}^{\text{o}}$	Fixed charge concentration [mol/m3]	59.7
Kcs	Permeability of DMSO in islets [m4/N/s]	6.35e−19
Kws	Permeability of water in islets [m4/N/s]	4.18e−17

^aThe membrane permeabilities of the two-parameter membrane transport model were fit from the three parameters of the Kedem-Katchalsky formalism that are reported by Benson et al. (24,46). A least-squares fitting approach was employed for an individual islet cell exposed to 1.5 mol/L DMSO at 22°C.

^bCalculated based on the islet dry weight fraction and the known volume fractions and pure species densities of all other species. The dry weight fraction was calculated by using the islet dry weight estimate of Parman (47) in conjunction with the initial islet size and water fraction of Benson et al. (24).

In particular, because the tissue and physiological bath are initially at equilibrium, we can equate the water and salt chemical potentials:

$$\begin{array}{l}{\mu }_{\text{w}}^{\text{tissue}}={\mu }_{\text{w}}^{\ast }+\frac{P^{\text{o}}}{{\overline{\rho }}_{\text{w}}}-\frac{RT}{M_{\text{w}}}\left(1-x_{\text{w}}^{\text{o}}\right)={\mu }_{\text{w}}^{\text{bath}},\hfill \end{array}$$ (22a)

$$\begin{array}{l}{\mu }_{\text{n}}^{\text{tissue}}={\mu }_{\text{n}}^{\ast }+\frac{RT}{M_{\text{n}}}\left(\ln \left(x_{\text{Na}}^{\text{o}}{x}_{\text{Cl}}^{\text{o}}\right)+{x_{\text{w}}^{\text{o}}}^2\right)={\mu }_{\text{n}}^{\text{bath}}.\hfill \end{array}$$ (22b)

We assume that the physiological bath is an aqueous solution with sodium and chloride concentrations equaling 0.15 mol/L each, rendering the bath chemical potentials known. At equilibrium, we can also equate the intracellular and extracellular salt concentrations, resulting in

$$\begin{array}{l}{C}_{\text{in}}^{\text{o}}=C_{\text{n}}^{\text{o}}=2C_{\text{Cl}}^{\text{o}}+C_{\text{fc}}^{\text{o}}.\hfill \end{array}$$ (22c)

Note that in the above equation, we have used (3), (4) to express the total extracellular salt concentration $C_{\text{n}}^{\text{o}}$ in terms of the chloride concentration and fixed charge concentration. These equations, when combined with Eqs. 1, (2a), (2b), (2c), (2d), (2e), 3, 4, (5a), (5b), and 6, can be solved to fully define the state of the tissue at the physiological reference state.

Initial conditions

We assume that the tissue is initially in equilibrium with a bath of known composition. In our simulations, we used a bath composition that was identical to the normal physiological solution but with a small amount of CPA to avoid complications involving the natural logarithm of CPA mole fraction (see Eq. 13b). Specifically, we used a bath composition with sodium and chloride concentrations again equaling 0.15 mol/L each and a CPA concentration of 10⁻⁴ mol/L. Below, we present a general strategy for determining the initial conditions within the tissue that can be applied for any initial bath composition.

Solving for the initial conditions is similar to defining the physiological reference state, but we must now consider CPA in addition to water and salt. At equilibrium, we can again equate the chemical potentials in the tissue and the bath:

$$\begin{array}{l}{\mu }_k^{\text{tissue}}={\mu }_k^{\text{bath}},k=w,c,n,\hfill \end{array}$$ (23a)

where the chemical potentials are defined as in Eqs. 13a, 13b, and 13c. Also, at equilibrium, we can equate the extracellular and intracellular concentrations of salt and CPA:

$$\begin{array}{l}{C}_{\text{in}}=C_{\text{n}}=2C_{\text{Cl}}+C_{\text{fc}},\hfill \end{array}$$ (23b)

$$\begin{array}{l}{C}_{\text{ic}}=C_{\text{c}}.\hfill \end{array}$$ (23c)

By combining Eqs. 23a, 23b, and 23c with Eqs. 1, (2a), (2b), (2c), (2d), (2e), 3, 4, (5a), (5b), 6, 14, 15, and 16, it is possible to solve for the state of the intracellular and extracellular solutions. Last, the initial tissue thickness can be calculated from Eq. 21.

Boundary conditions

Within the one-dimensional domain, we have two boundaries to consider: the insulated boundary and the bath boundary. At the insulated boundary, there is no flux of any species k across the boundary, and therefore, each velocity, u_k, is equal to zero (zero-flux condition).

For the bath boundary, we assumed that the boundary is always in equilibrium with the bath. As such, the boundary conditions can be found in exactly the same way as the initial conditions.

Numerical methods

The model was written and solved in MATLAB (R2020b; The MathWorks, Natick, MA). For temporal discretization, we employed MATLAB’s ODE 23 algorithm, an explicit variable-step Runge-Kutta method of third order accuracy. As a variable-step algorithm, ODE 23 adjusts the time step to keep the estimated error in any dependent variable within a given tolerance. For this analysis, default tolerances were used. We used the ODE 23 algorithm as it has the potential to provide greater efficiency and stability at cruder tolerances than higher order Runge-Kutta methods (48), which could be useful for future optimization work in CPA addition and removal protocol design.

For spatial discretization, we used central finite differencing with even spacing between nodes. To define the chemical potential gradient at the bath boundary and the flux gradient at the insulated boundary, we used backward differencing and forward differencing, respectively. Mass transfer Péclet number estimates were below two for our defined grids, ensuring central differencing is consistent with the physics of the problem (i.e., diffusion dominant) (49). Overall, the current numerical scheme can produce negative mass concentrations at very early times close to the bath boundary. To account for this, we adopted a negative value filtering procedure in the literature (50) and modified it by not distributing the negative values across positive nodes because of the relatively small magnitude of negative values calculated.

For grid convergence, we examined the average CPA concentration within the tissue using the classic grid convergence index (GCI) presented by Roache (51,52). We conducted three successive levels of grid refinement using a coarse, medium, and fine grid set at 26, 51, and 101 nodes, respectively, but all results are reported using the fine grid.

Given the transient nature of the problem, the GCI was calculated every minute for cartilage simulations and every 20 s for islet simulations. We found higher GCI values at early time points, with the GCI decreasing as the simulation progressed, as expected. For cartilage, GCI values higher than 5% were found for some simulation times ≤6 min, with high-end values <20%. No GCI values above 5% were found for the islet simulations. The use of the fine grid was deemed adequate for this study to maintain reasonable computation times.

Tissue-specific parameters

Articular cartilage

The introduction of coupled cell membrane and extracellular transport dynamics to the articular cartilage model of Abazari et al. (23) requires the inclusion of new parameters. We define the new parameters necessary for our augmented model in Table 1 as well as the original model parameters from Abazari et al. (23).

Pancreatic islets

Pancreatic islets can be modeled as radially symmetric spheres (24), so we again consider one-dimensional transport but in the radial direction r. This change in coordinate system requires several changes to the transport equations and the subsequent variable transform, as described in the Supporting materials and methods.

For the purposes of this study, we assume the modulus of elasticity to be zero, thus rendering P_elastic also equal to zero (53). The mechanical properties of islets are not very well studied, but we expect a 1000-fold reduction in their modulus when compared with cartilage (54). In cartilage simulations, we see no appreciable difference between volume predictions when the modulus is reduced 1000-fold and when the modulus is zero. Also, Benson et al. (24) reports an equilibrium islet volume relationship consistent with a negligible modulus.

Table 2 shows the parameters used in the islet model. The three fitted parameters were fit based on the islet size change data in the second panel of Fig. 3 of Benson et al. (24) using a least-squares approach and the coarse mesh of 26 nodes. A conventional least-squares objective function was written comparing model predictions and the data, and MATLAB’s fminsearch algorithm was called to minimize the objective function. The coarse mesh was used because of program runtime constraints. All simulation results are reported with the fine mesh of 101 nodes.

Results and discussion

Application of the model to articular cartilage

Abazari et al. (23) previously developed a transport model for articular cartilage and used the resulting concentration predictions to estimate chondrocyte volume changes within the tissue (35). This modeling approach neglects the coupling between extracellular and cell membrane mass transfer, which is reasonable for cartilage but becomes problematic for more cell-dense tissues. Here, we present a general tissue model, based on the model of Abazari et al. (23,35), that accounts for the coupling between extracellular and cell membrane mass transfer.

To validate our model, we compared the results of our model directly to the results given by Abazari et al. (23) for exposure of a cartilage slab to 6.5 mol/L DMSO at 22°C. In theory, our model without cells and that of Abazari et al. (23) should yield the same results, with any deviations most likely attributable to differences in numerical methods. As shown in Fig. 2, the model of Abazari et al. (23) and our model without cells yield similar predictions of normalized fluid weight and average DMSO concentration in the cartilage as a function of time. Fig. 2 also shows that accounting for cells in the model only has a modest effect on the predictions for articular cartilage, as expected, because cartilage has a relatively low cell density of around 10% by volume.

Figure 2.

Comparison between our current model with and without cells and that of Abazari et al. (23) for a slab of cartilage exposed to a 6.5 mol/L DMSO bath. All parameters are specified in Table 1. The top panel shows the normalized fluid weight (water and DMSO) of the cartilage, and the bottom panel shows the average DMSO concentration within the cartilage. To see this figure in color, go online.

Fig. 3 shows the equilibrium volume of articular cartilage after exposure to salt solutions of varying osmotic strength. Cartilage has been observed to shrink when exposed to hypertonic salt solutions, and model predictions are consistent with this trend. Once again, accounting for cells in the model only had a modest effect for articular cartilage.

Figure 3.

Equilibrium volume of articular cartilage after exposure to solutions with varying salt (NaCl) concentration. The data are for bovine articular cartilage and are presented as strain in Fig. 8 of Lai et al. (55), which we have converted to normalized volume through Eqs. 16 and 21. Model predictions with and without cells are also shown using the parameters for porcine articular cartilage in Table 1, except for changing the modulus of elasticity to 1e6 Pa to be consistent with the range of H_A values reported by Lai et al. (55) for bovine articular cartilage. To see this figure in color, go online.

Overall, the results shown in Figs. 2 and 3 demonstrate that our mathematical model can predict the measured trends for articular cartilage, including the transient shrink-swell response after exposure to CPA solution and the equilibrium response of tissue volume after exposure to solutions of varying salt concentration. However, the incorporation of cells into the model only has a modest effect because cartilage has a relatively low cell density.

Application of the model to pancreatic islets

Pancreatic islets have a cell density of around 67% by volume (see Table 2), nearly sevenfold higher than the cell density of cartilage. In this case, there is a clear need for a transport model that accounts for the effects of cells. As shown in Fig. 4, pancreatic islets have been observed to shrink and then swell after exposure to 1.5 mol/L DMSO (24). Our cell-augmented transport model accurately predicts this shrink-swell response and matches both the data and specialized islet model of Benson et al. (24). Unlike our proposed model, the model of Benson et al. (24) is not easily generalizable to other tissues, mainly because of its islet-specific geometric framework and the assumption that the ratio of intra- to extracellular space is fixed. Although this assumption appears to yield reasonable predictions for islets under the conditions tested, it is not based on the physics of the problem, and thus, its validity for other tissue types is questionable.

Figure 4.

Size change after exposing a pancreatic islet to 1.5 mol/L DMSO in phosphate-buffered saline (modeled as 0.15 mol/L sodium and 0.15 mol/L chloride in aqueous solution). Predictions for our current model were obtained using the parameters in Table 2. For the model with no cells, we added the intracellular water volume fraction to that of the solids. Model predictions and data from Benson et al. (24) are also shown. To see this figure in color, go online.

Most of the necessary model parameters for pancreatic islets were available in the literature (see Table 2), but three parameters were not found: fixed charge concentration, DMSO permeability, and water permeability. These parameters were found by fitting to experimental data in Fig. 4, resulting in the following best-fit parameters: $C_{\text{fc}}^{\text{o}}$ = 59.7 mol/m³, K_cs = 6.35e−19 m⁴/N/s, and K_ws = 4.18e−17 m⁴/N/s. The best-fit fixed charge concentration for pancreatic islets is lower than that of articular cartilage, which is not surprising because the glycosaminoglycan (GAG) content is higher in cartilage than most other tissue types (22). The ratio of GAG content to extracellular fluid volume can be used as a rough indicator of fixed charge concentration. The GAG content is ∼60-fold lower in pancreas than cartilage (56,57), and the extracellular fluid volume fraction is ∼4-fold lower in islets than cartilage. This results in a rough estimate for $C_{\text{fc}}^{\text{o}}$ in islets of 15-fold lower than $C_{\text{fc}}^{\text{o}}$ in cartilage. The best-fit $C_{\text{fc}}^{\text{o}}$ value for islets is ∼5 times higher than this rough estimate, but given the uncertainty of the estimate, the best-fit value is reasonable. The best-fit water and DMSO permeabilities for pancreatic islets were both lower than the corresponding permeability values in cartilage. This is consistent with the expected decrease in permeability as a result of the lower extracellular volume fraction in islets.

In Fig. 4, we also show predictions for our model in the acellular case. To generate an acellular model for pancreatic islets, we considered various approaches for dealing with the fraction of the islet volume occupied by cells, as discussed in the Supporting materials and methods. The most straightforward approach is to replace the volume occupied by cells with solids. The resulting acellular curve significantly underestimates the extent of islet shrinkage after CPA exposure as well as the extent of reswelling as the islet equilibrates in the CPA solution. As shown in Fig. S1, alternative approaches for defining the acellular case also result in predictions that are a relatively poor match to the data, which underscores the importance of accounting for cells in the model.

Pancreatic islets have also been observed to exhibit volume changes after exposure to anisotonic salt solutions. Fig. 5 shows measured equilibrium islet volumes over a range of solution concentrations, alongside the predictions of our cell-augmented islet model. The predictions for the model with cells are in good agreement with the data. In contrast, equilibrium predictions using any acellular version of the current model yields relatively poor replication of the data (see dotted line in Fig. 5 and Fig. S2), highlighting the need for a model with both intra- and extracellular compartments.

Figure 5.

Equilibrium volume of pancreatic islets after exposure to solutions with varying salt (NaCl) concentration. The data are from Benson et al. (58) with error bars representing the standard error of the mean. Model predictions were obtained using the parameters in Table 2. For the model with no cells, the intracellular water volume fraction was added to that of solids. To see this figure in color, go online.

Taken together, the results presented in Figs. 2, 3, 4, and 5 demonstrate that the cell-augmented model can be used to predict transient and equilibrium trends that have been observed in both cartilage and pancreatic islets, tissue types with vastly different properties. This underscores the general utility of this modeling approach for predicting transport in tissues.

Parametric analysis

Cell density, fixed charge concentration, and tissue mechanical properties can vary widely between different tissue types. Therefore, a parametric analysis was performed to examine the effects of these properties on the transport process. Fig. 6 illustrates the effects of cell density on the predicted tissue response after exposure to CPA. Overall, the increased presence of cells slows uptake of CPA into the tissue, as shown in the top panel of Fig. 6. This can be attributed to two different phenomena. First, cells decrease the volume fraction of extracellular space available for diffusion, which impacts the frictional coefficients in the model (see (12a), (12b), (12c), (12d)). Second, cells exchange CPA with the extracellular space, which slows down CPA transport into the tissue. This effect can be isolated by comparing the curves for 30% solids, 0% cells and 20% solids, 10% cells. In both cases, the extracellular fluid volume is the same, but the latter case (with 10% cells) exhibits slower CPA transport into the tissue. The bottom panel of Fig. 6 examines the effects of cell density on tissue volume changes. When solids density is held constant, the presence of cells reduces tissue shrinkage, increases the time for the tissue to reach its minimal volume, and slows the rate of swelling back to the original tissue volume.

Figure 6.

Effect of cell density on the CPA concentration and volume response of a tissue slab exposed to a 6.5 mol/L DMSO solution. Parameters are the same as in Table 1, unless otherwise noted. The cell percentages in the figure legends refer to the initial volume fraction of cell water. To see this figure in color, go online.

In Fig. 7, we examine the effects of fixed charge concentration, $C_{\text{fc}}^{\text{o}}$, and modulus of elasticity, H_A, for a tissue slab exposed to 6.5 mol/L DMSO. Fixed charge concentration has a small impact on the CPA concentration predictions, but it has a relatively large impact on the tissue volume response. In general, increasing fixed charge concentration results in more rapid initial tissue shrinkage as well as more rapid swelling back to the original tissue volume. On the other hand, the modulus of elasticity is a measure of the resistance to tissue size changes. Overall, there is an interplay between fixed charges and the modulus of elasticity that determines the extent of tissue size changes after CPA exposure.

Figure 7.

Effect of fixed charge concentration, $C_{\text{fc}}^{\text{o}}$, and modulus of elasticity, H_A, on the response of a tissue slab exposed to a 6.5 mol/L DMSO solution. The DMSO concentration in the top panels is the average DMSO concentration within the tissue. The panels with a high H_A indicate a modulus of 2.4e6 Pa (i.e., the modulus for cartilage), and the panels with a low H_A indicate a modulus of 2.4e4 Pa. The fixed charge concentration [mol/m³] was varied as indicated in each panel. Cell density was held constant at 0%. Parameters were the same as in Table 1, unless otherwise noted. To see this figure in color, go online.

The predictions shown in the bottom right panel of Fig. 7 indicate that for low fixed charge concentration and low modulus of elasticity, exposure to CPA can result in tissue shrinkage, followed by slower volume recovery over a much longer timescale. This phenomenon has been observed previously for ovarian tissue (25), which exhibited ∼20% shrinkage after exposure to CPA within ∼30 min, followed by a much slower rebound in tissue weight with a timescale of about a day. The similarities between experimental observations for ovarian tissue and model predictions suggest that the model may also be effective for predicting transport in ovarian tissue.

Implications for design of CPA addition and removal procedures

Mass transfer modeling can be used to design CPA addition and removal methods that minimize CPA toxicity and avoid mechanical damage due to osmotically driven volume changes. Mechanical damage may occur as a result of excessive tissue volume changes as well as excessive cell volume changes for the cells embedded within the tissue. CPA toxicity is expected to be dependent on the CPA concentration and exposure time, which varies spatially during equilibration of tissues with CPA. Therefore, to design CPA equilibration methods, it is useful to predict the volume response of the cells and the tissue as well as their CPA concentrations. Fig. 8 shows the tissue volume response after exposure to CPA for cartilage and pancreatic islets as well as the volume response for a surface cell (closest to the CPA bath) and an interior cell (farthest from the CPA bath). We also show the intracellular DMSO concentration for the surface and interior cell. Both the extracellular and intracellular concentrations are nearly identical in these simulations, with the intracellular concentration slightly lagging the extracellular concentration (not shown) because of the short timescale of CPA transport across the cell membrane.

Figure 8.

Left: intracellular DMSO concentration and volume predictions for a pancreatic islet exposed to 1.5 mol/L DMSO in phosphate-buffered saline (see Table 2 for parameters). Right: the same predictions as in the left panels but for a slab of cartilage exposed to 6.5 mol/L DMSO (see Table 1 for parameters). The surface cell is closest to the CPA bath, and the interior cell is furthest away. Cell volumes refer to the change in osmotically active volume, which excludes cell solids. Tissue volumes refer to the entire volume (including solids). To see this figure in color, go online.

One of the biggest differences between cartilage and islets is the timescale of transport, which is directly related to the difference in length scale between the two specimens. To wit, in the absence of cells and under Fick’s law, diffusion time scales with the square of length, and thus, we expect the cartilage to take on the order of 100 times longer to equilibrate than the islet, an approximation confirmed by our models (see Fig. 8). Our model, however, allows us to capture the cell response in the interior of the tissue, which can be valuable in predicting cell damage during cryopreservation protocols. In particular, in the case of our cell-augmented model, CPA takes a finite amount of time to move from the tissue surface to the interior, which can greatly influence the osmotic response of cells on the surface compared with those within the interior. The bottom panels of Fig. 8 show that cell volume changes are a greater function of tissue position in cartilage than in pancreatic islets, which is consistent with the longer timescale of transport in cartilage. These cell volume predictions are a key consideration when devising CPA addition and removal protocols, and Fig. 8 shows that our modeling approach can make these predictions in two vastly different tissues.

The predictions shown in Fig. 8 also have implications for the design of CPA equilibration methods that minimize CPA toxicity. In a previous work (9), we used a CPA toxicity cost function to design less toxic CPA equilibration methods for endothelial cells. Specifically, we were able to leverage the swelling induced by loading cells with CPA in hypotonic buffer to introduce more CPA in a shorter amount of time, resulting in a low toxicity novel protocol. As shown in Fig. 8, the tissue volume response after exposure to CPA is qualitatively similar to that of cells, which suggests that this toxicity reduction strategy may also extend to tissues. Further, the magnitude of tissue volume changes was substantially greater for islets than cartilage (see Figs. 3 and 5). This can be attributed to differing mechanical properties of the tissue and has been captured in the model with the modulus of elasticity. As such, manipulating the buffer tonicity during equilibration of an islet with CPA would be expected to have a greater impact on toxicity reduction when compared to cartilage.

Conclusions

In this work, we sought to develop a general model for mass transfer in tissues that addresses various tissue-based phenomena, including mass transfer in the extracellular space, coupling between extracellular and cell membrane mass transfer, fixed charges, and cell and tissue volume changes. To incorporate all phenomena in one model, we augmented the acellular articular cartilage model of Abazari et al. (23) to account for cells. To accomplish this, we incorporated the classic two-parameter membrane transport model into the overall model formalism. This new model is expected to enable mass transfer predictions in any tissue type, simply by changing tissue-specific parameters such as fixed charge density and modulus of elasticity. To show the general utility of the model, we compared predictions to experimental data for articular cartilage and pancreatic islets, demonstrating that the model can predict the observed changes in tissue size and CPA concentration for both of these tissue types. We also conducted a parametric analysis that showed that for tissues with a low modulus of elasticity and low fixed charge concentration, initial tissue shrinkage upon exposure to CPA is relatively rapid followed by much slower recovery of tissue volume, a trend that has previously been observed for ovarian tissue (25). Overall, the model shows promise for predicting mass transfer in various tissue types and lays the groundwork for future studies to better understand mass transfer in tissues and to develop improved cryopreservation methods.

This work can also serve as the basis for future refinements to improve the model. One limitation of the current model is that it uses the two-parameter membrane transport model, which assumes dilute solutions (36). More recent models of nondilute cell membrane transport (37) may provide more accurate predictions, especially for high CPA concentrations. Additional model refinements that could be addressed in future studies include the addition of cell-to-cell transport and consideration of the role of cell mechanics on the tissue mechanical response.

Author contributions

R.M.W. carried out the simulations, made the figures, and wrote the first draft of the manuscript. A.Z.H. designed the study. J.D.B. and R.S. contributed to model refinement and independently confirmed the model predictions. R.M.W., A.Z.H., J.D.B., R.S., and A.E. all contributed to writing and revision of the manuscript.

Editor: Guy Genin.