Modeling the Simultaneous Transport of Multiple Cryoprotectants into Articular Cartilage Using a Triphasic Model

Abstract

Cryopreserving articular cartilage by vitrification can increase the availability of tissue for osteochondral allograft transplantation to treat cartilage defects. Developing well-optimized vitrification protocols can be supported by mathematical modeling to reduce the amount of trial-and-error experimentation needed. Fick’s law has been used to model cryoprotectant diffusion, but it assumes ideal, dilute solution behavior, neglects water movement, and assumes diffusion of each cryoprotectant is independent of the presence of other cryoprotectants. The modified triphasic model addresses some of these shortcomings by accounting for water movement and the nonideal, nondilute nature of cryoprotectant vitrification solutions. However, it currently only exists for solutions containing a single cryoprotectant. As such, we extend the modified triphasic model to include two permeating cryoprotectants so that simultaneous diffusion occurring in vitrification protocols can be more accurately modeled. Using previously published experimental data, we determine suitable values for the fitting parameters of the new model. We then model a successful vitrification protocol for particulated cartilage cubes by calculating concentration, freezing point, vitrifiability, and strain profiles at the end of each loading step. We observe that Fick’s law consistently underestimates cryoprotectant concentration throughout the cartilage compared to the modified triphasic model, leading to an underestimation of tissue vitrifiability. We additionally observe that simultaneous diffusion of cryoprotectants increases the permeation rate of each individual cryoprotectant, which Fick’s law fails to consider. This suggests that using the two-cryoprotectant modified triphasic model to develop vitrification protocols could reduce excess exposure to cryoprotectants and improve preserved tissue outcomes.

1. Introduction

Articular cartilage is a connective tissue responsible for reducing friction in joints during movement and transmitting loads to the subchondral bone. Injury to articular cartilage results in sustained defects because the avascular nature of articular cartilage limits its ability for self-repair.¹ This damage increases the risk of developing osteoarthritis,² which causes chronic pain, stiffness, and mobility problems.³ These symptoms can be debilitating and prevent those who face them from working and attending social events.³ Osteochondral allografting is an effective method for treating these cartilage defects.^4,5 This procedure involves transplanting a donor’s bone and articular cartilage into a recipient to restore tissue function. Although effective, osteochondral allografting is not always feasible since current hypothermic storage methods at 4 °C in culture media result in significantly reduced chondrocyte viability after 14 days.⁶ This time frame can be inadequate to complete the required infectious disease testing as well as size and contour matching.⁷ Cryopreserving articular cartilage by vitrification would allow donor tissue to remain viable long term,^8,9 thereby eliminating time-sensitive logistical obstacles and paving the way for improved clinical outcomes.

In the context of cryopreservation, vitrification is the process of forming an amorphous solid during rapid cooling of a cell or tissue in liquid nitrogen. Achieving this involves loading high concentrations of cryoprotectants into the cell or tissue prior to cooling to lower the freezing point and prevent damaging ice formation.¹⁰ At the same time, exposure to high concentrations of cryoprotectants can have cytotoxic effects detrimental to tissue viability.¹¹⁻¹³ As such, designing effective vitrification protocols involves ensuring that cryoprotectant permeation is sufficient to achieve vitrification, while also limiting cryoprotectant exposure to minimize toxic effects. One approach to mitigating toxicity is to load cryoprotectants in a stepwise manner at successively lower temperatures tracking the liquidus.^11,14 Multiple cryoprotectants are often loaded simultaneously in each step to achieve a high overall concentration while ensuring that the concentration of any one given cryoprotectant is not excessive.^13,15

Mathematical modeling can be used to guide stepwise protocol development and reduce the amount of trial-and-error experimentation needed.¹⁶ Fick’s law diffusion coefficients for cryoprotectants permeating into porcine articular cartilage have been determined from experimental data for several common cryoprotectants including dimethyl sulfoxide (DMSO), ethylene glycol (EG), propylene glycol, glycerol, and formamide.¹⁷⁻¹⁹ It was found that these coefficients provide suitable approximations of the permeation kinetics in human articular cartilage based on loading and efflux experiments.²⁰ The vitrifiability of a cryoprotectant solution can be determined using a model developed by Weiss et al.,²¹ and the freezing point of a cryoprotectant solution can be calculated using the multisolute osmotic virial equation.^22,23 Shardt et al.¹⁶ combined these models to predict the cryoprotectant concentration, freezing point, and vitrifiability profiles of articular cartilage at the end of each step of cryoprotectant loading. Starting from an existing 9.5 h cryoprotectant loading protocol,⁹ they optimized the procedure such that a similar level of vitrifiability was obtained within 7 h, thereby reducing cryoprotectant toxicity. Wu et al.²⁴ took a similar approach and designed two 2-step protocols for cryoprotectant loading into particulated articular cartilage cubes 1 mm³ in size. Table 1 shows their Protocol E-D, which involves the simultaneous loading of EG and DMSO. Protocol E-D was tested on porcine and human articular cartilage that was sliced into 1 mm³ cubes after dissection from healthy femoral condyles. After following the outlined cryoprotectant loading procedure, the cartilage cubes were placed into a sterile 1.8 mL Cryovial tube then plunged into liquid nitrogen and stored at −196 °C for either 0 or 180 days. Human chondrocyte assessment after storage, rewarming, and washing showed no statistically significant difference in membrane integrity at day 0 and day 180. The vitrification process maintained greater than 80% normalized chondrocyte membrane integrities, a high level of metabolic activity, unimpaired chondrocyte migration, and unchanged matrix productivity. Although this storage method was tested only for 6 months, the results for longer storage times are expected to be similar due to the lack of chemical and biological activity at −196 °C. As such, Protocol E-D provides evidence that mathematical modeling can support the development of successful vitrification protocols.

Table 1. Details of Protocol E-D Including Cryoprotectant Concentrations, Times, and Temperatures²⁴.

	concentrations	time	temperature
loading step 1	3 M EG	20 min	0 °C
loading step 2	4 M EG + 4 M DMSO	15 min	–5 °C

Although Fick’s law has been useful in modeling vitrification protocols, it has several limitations. First, Fick’s law assumes an ideal, dilute solution, which is not applicable to the highly concentrated cryoprotectant solutions used in vitrification protocols. Additionally, Fick’s law does not account for water movement during cryoprotectant loading that causes the cartilage to shrink and swell. In other contexts, this movement of water has been described by the biomechanical triphasic model of articular cartilage proposed by Lai et al.²⁵ Shaozi and Pegg²⁶ introduced this modeling approach to the context of cryobiology by including a cryoprotectant as an additional component. Although water movement is accounted for, their model still assumes that the cryoprotectant solution is ideal and dilute. Abazari et al.^27,28 addressed this limitation by accounting for the actual nonideality of cryoprotectant solutions by incorporating nonideal, nondilute chemical potential equations into the model. Compared to the results of this modified triphasic model, Fick’s law provides an underestimate of cryoprotectant concentration in cartilage.²⁹ Thus, protocols developed using Fick’s law likely load cryoprotectants in excess of the minimum amount needed for vitrifiability. Using the modified triphasic model instead would provide a more realistic estimate of cryoprotectant concentration and consequently vitrifiability. However, the modified triphasic model only accounts for one cryoprotectant, but loading protocols typically include multiple cryoprotectants diffusing simultaneously. As such, it is necessary to extend the model to include the simultaneous transport of multiple cryoprotectants.

The first objective of this paper is to extend the modified triphasic model to include two permeating cryoprotectants. The second objective is to determine suitable parameters for use in this model by fitting to experimental weight data reported by Abazari et al.²⁷ and concentration data reported by Jomha et al.^17,18 for DMSO and EG. The third objective is to use the developed model to determine the spatial profiles of cryoprotectant concentrations, freezing point, vitrifiability, and strain at the end of each loading step of Protocol E-D,²⁴ and compare with Fick’s law predictions. The fourth objective is to understand the behavior of two cryoprotectants diffusing simultaneously as opposed to single-cryoprotectant diffusion.

2. Governing Equations

2.1. Modified Triphasic Model for Two Cryoprotectants

The modified triphasic model is used to calculate the spatial and temporal distribution of cryoprotectant in cartilage as well as the change in cartilage thickness that occurs during cryoprotectant loading. This model, which involves the transport of a single cryoprotectant into articular cartilage, was developed by Abazari et al.²⁷ as an expansion upon previous work by Lai et al.²⁵ and Shaozi and Pegg²⁶ to account for the nonideal and nondilute nature of cryoprotectant solutions. Extending this model to describe the simultaneous transport of two cryoprotectants is necessary to increase its utility in modeling and optimizing vitrification cryoprotectant loading protocols. Equation derivation for the two-cryoprotectant version closely follows the derivation of the one-cryoprotectant version by Abazari et al.²⁷ It is described here in detail for clarity and completeness.

2.1.1. Continuity Equations

Continuity equations are used to describe the conservation of mass and continuous flow of each component in the model. Continuity takes the form of

1

2

3

4

5

for each component (w = water, n = mobile ions, c₁ = first cryoprotectant, c₂ = second cryoprotectant, s = solids), where ρ is density (kg/m³), v is velocity (m/s), and t is time (s). The fixed charges are included as part of the solid component of the cartilage.²⁷ The mobile ions include Na⁺ and Cl^–, which are related to one another by²⁵

6

where ρ_f denotes the density of the fixed charges (kg/m³). Equations 1–5 can be rewritten in terms of volume fraction φ by dividing by the component’s intrinsic density ρ̅ (kg/m³). Summing the resulting equations results in the expression

7

where φ = ρ/ρ̅. Volume changes due to the mixing of water, mobile ions, and cryoprotectants are assumed to be negligible. Recognizing that the time derivative term in eq 7 is zero because the sum of all volume fractions must equal one and that the volume fraction of mobile ions is negligible compared to the other components (i.e., φ_n ≈ 0) simplifies eq 7 to

8

Diffusion of water and cryoprotectant in articular cartilage occurs predominately in the direction perpendicular to the bone,³⁰ so the model can be reduced to one dimension. Under this condition, eq 8 can be used to show that³¹

9

2.1.2. Momentum Balance Equations

The component velocities were related to chemical potentials by Lai et al.²⁵ following nonequilibrium thermodynamics and momentum balance by the expression

10

where μ_α is the chemical potential of component α (J/kg) and f_αβ is the friction coefficient between components α and β (Ns/m⁴). Following classical mixture theories, Onsager reciprocity, and Lai et al.,²⁵ it is assumed that f_αβ = f_βα and that f_ns = 0. Following Abazari et al.,²⁷ the friction coefficients between the mobile ions and each cryoprotectant are neglected (i.e., f_nc1 = f_nc2 = 0). Unlike the model developed by Abazari et al.,^27,28 the version in this current paper involves two cryoprotectants rather than one. This introduces an additional friction coefficient f_c1c2, which is neglected (i.e., f_c1c2 = 0) in this work to avoid complicating the model with another parameter that must be found from experimental data. With these simplifying assumptions, the momentum balance equations can be written from eq 10 as

11

12

13

14

The friction coefficients are related to physically meaningful parameters, like diffusivity and permeability, by the following equations

15

16

17

18

19

20

where K_αβ is the permeability of component α in component β (m⁴/(Ns)), D_αβ is the diffusivity of component α in component β (m²/s), C_α is the concentration of component α (mol/m³), R is the universal gas constant, and T is the absolute temperature (K).

2.1.3. Chemical Potential Equations

The chemical potentials of the components are written based on the works of Elliott et al.²² and Elmoazzen et al.,³² which take a virial level approach and implement an arithmetic mixing rule for nonideal and nondilute solutions. In this case, the equations take the form

21

22

23

24

25

where μ_α^* is the reference state chemical potential of component α (J/kg), P is the absolute pressure (Pa), M_α is the molar mass of component α (kg/mol), x_α is the mole fraction of component α, and B_α is the mole fraction-based second osmotic virial coefficient of component α. Following Abazari et al.,²⁷ the effect of pressure on the chemical potential of the mobile ions is neglected due to their negligible volume fractions. Similarly, the mobile ions are dilute in the solution and are therefore assumed to behave ideally. Accordingly, their osmotic virial coefficients can be set to zero (i.e., B_Na⁺ = B_Cl^– = 0). Under this assumption and by acknowledging that the sum of all mole fractions equals one, the chemical potential equations can be simplified to

26

27

28

29

30

Since the mobile ions are treated as a single component in the continuity equations, it is necessary to combine the chemical potentials of Na⁺ and Cl^– as well. Following Lai et al.,²⁵eqs 27 and 28 can be combined using the argument M_n μ_n = M_Na⁺ μ_Na⁺ + M_Cl^– μ_Cl^– to give

31

where M_n is the molar mass of NaCl (kg/mol).

2.1.4. Domain Movement

Subjecting articular cartilage to highly concentrated cryoprotectant solutions induces initial shrinkage of the tissue due to water efflux driven by chemical potential differences between the tissue and treatment solutions. Consequently, the location of a particular point of cartilage tissue along the x-axis changes over time during treatment. Strain quantifies this displacement and is defined as

32

where l is the position at any given time during treatment (m) and l₀ is the initial position (m). For one-dimensional shrinkage, strain can be calculated in terms of the model variables by

33

where φ_s⁰ = 0.2 is the initial solid volume fraction.^27,28 Integrating strain with respect to x over the domain gives the overall change in cartilage thickness. Negative overall strain indicates shrinkage, whereas positive overall strain indicates swelling. Additionally, strain impacts the distribution of fixed charges on the tissue in the manner³¹

34

where C_f is the concentration of fixed charges at any given time during treatment (mol/m³) and C_f⁰ is the initial concentration of fixed charges (200 mol/m³).³³ The elastic stress on the cartilage is related to strain linearly by the aggregate compressive modulus, or cartilage stiffness, H_A (Pa) as

35

where P₀ is the initial elastic pressure in the cartilage (Pa), which is zero in this case due to a simplifying assumption that the concentrations of the bath and tissue at the boundary are equal, rather than the chemical potentials.

2.1.5. Conversion between Density, Concentration, and Mole Fraction

The continuity equations (eqs 1–5) are written in terms of component density, but it is necessary to convert these values to concentration for use in the friction coefficient equations (eqs 18–20) and to mole fraction for use in the chemical potential equations (eqs 26 and 29–31). First, concentration is related to density by

36

37

where the solid volume fraction is

38

since the volume fraction of the mobile ions was assumed to be negligible. Note that all concentrations are expressed in units of moles per m³ of solution volume and not per m³ of tissue volume. Equation 37 uses the molar mass of sodium to convert the density of the fixed charges to the corresponding concentration following Lai et al.²⁵ The concentration of sodium ions can also be calculated as

39

by understanding that the electroneutrality condition must be met in the tissue. The density of Cl^– ions can be calculated from the density of the combined mobile ions by

40

based on the definition of ρ_n given in eq 6 and using eqs 36, 37, and 39. The mole fraction of each component can then be calculated from its concentration as

41

where the total concentration is

42

2.1.6. Initial and Boundary Conditions

Figure 1 illustrates the geometry of a 1 mm³ cartilage cube that is the subject of theoretical modeling in this work. Numerical analysis is carried out in one dimension along the x-axis, with the center of the cube defined as x = 0 mm. A one-dimensional analysis provides a conservative estimate of cryoprotectant concentration and allows for direct comparison with previous works^16,27,28 that also assumed one-dimensional diffusion. The cartilage–solution boundary is initially at x = 0.5 mm but changes during the simulation as the half-thickness of the cartilage, h, changes.

Figure 1.

Schematic of a 1 mm³ cartilage cube immersed in a cryoprotectant solution with diffusion occurring in one dimension along the x-axis for the purpose of theoretical modeling.

At the center of the cube (x = 0 mm), the boundary condition is zero flux such that

43

At the cartilage–solution interface (x = h mm), the boundary condition is that the concentration of each component in the cartilage is equal to the concentration of that component in the bath solution. In terms of density, this condition is specified by solving eq 36 for ρ_α and specifying C_α as the bath concentration for each component (α = n, c₁, c₂). In this study, the concentration of mobile ions in the bath solution is taken to be 30 mol/m³, based on the results of Abazari et al.²⁷ The density of water at the boundary can be calculated using eq 38, the boundary densities of the mobile ions and cryoprotectants, and the relation φ = ρ/ρ̅. Note that this boundary condition is a simplifying assumption. A more precise boundary condition would be to assume that the chemical potential of each component in the cartilage is equal to the chemical potential of that component in the bath solution. However, using the chemical potential boundary condition makes it significantly more complex to solve the system of equations using available software. The slight decrease in predictive precision that could be expected using the concentration boundary condition was deemed worthwhile in exchange for greatly reducing the computational complexity.

At the beginning of the first step, there are no cryoprotectants in the cartilage cube. This corresponds to an initial condition of ρ_c1 = ρ_c2 = 0 across the domain at t = 0 s. However, the chemical potentials of cryoprotectant one and two (eqs 29 and 30) are mathematically undefined for this condition. Therefore, a small number close to zero is selected instead (i.e., ρ_c1 = ρ_c2 = 0.01 kg/m³). The initial concentration of mobile ions in the cartilage is 81 mol/m³ from the results of Abazari et al.²⁷ The initial density of water in the cartilage is determined based on the initial solid fraction as ρ_w⁰ = ρ̅_w (1 – φ_s). For the second step, the concentration profiles at the end of the first step are used as the initial condition.

2.1.7. Details of Numerical Solution

The system of equations was solved using code written in MATLAB R2021b (MathWorks, Natick, MA) linked to COMSOL Multiphysics 5.6 (COMSOL, Stockholm, Sweden). The problem was set up using the General Form PDE physics option in COMSOL for eqs 1–4, with ρ_w, ρ_n, ρ_c1, and ρ_c2 as the dependent variables. Equations 9 and 11–14 can be solved for v_w, v_n, v_c1, and v_c2, and the resulting expressions are coded as COMSOL variables alongside eqs 15–20, 26, 29–31, 33–36, and 38–42.

Table 2 lists all of the necessary constants for the model that are entered into COMSOL as global parameters. The model also requires cryoprotectant diffusion coefficients, cryoprotectant and water permeability coefficients, and cartilage stiffness modulus. These parameters must be determined by fitting the model to experimental data, which will be covered in Section 3.

Table 2. Constants Needed for the Modified Triphasic Model^a.

constant	value	reference
Mw	0.01802 kg/mol	(35)
MDMSO	0.07813 kg/mol	(35)
MEG	0.06207 kg/mol	(35)
Mn	0.05844 kg/mol	(36)
MCl–	0.03545 kg/mol	(37)
ρ̅w	1000 kg/m3	(35)
ρ̅DMSO	1101 kg/m3	(35)
ρ̅EG	1108 kg/m3	(35)
R	8.314 J/mol K
Cf0	200 mol/m3	(33)
φs0	0.2	(27, 28)
BDMSO	7.24	this work
BEG	3.41	(34)
Dnw	2.9 × 10–10 m2/s	(38)

^aThe osmotic virial coefficient for DMSO was obtained using a quadratic fit, following the methods of Zielinski et al.³⁴

The Moving Mesh physics option in COMSOL was used to model the changing cartilage thickness by prescribing a mesh displacement using an integral operator. The operator calculates the total strain across the domain to determine the change in the cartilage half-thickness as follows

44

A mesh refinement study was conducted to determine the mesh size that provides accurate results while minimizing computational intensity. Due to the nonlinear nature of the model, the mesh needs to be relatively fine for the solution to converge. The coarsest mesh that resulted in convergence of the model was “extra fine” (i.e., maximum element size of 0.01 mm). Refining the mesh to “extremely fine” (i.e., maximum element size of 0.005 mm) did not change the results. Therefore, it was concluded that “extra fine” was an adequately refined mesh for this problem.

The solution of the model was further verified by investigating the relative tolerance. COMSOL uses a default relative tolerance of 0.01. Reducing this tolerance by a factor of 1000 to 0.00001 did not change the solution. Therefore, the default relative tolerance of 0.01 is sufficient for this problem.

2.2. Fick’s Law of Diffusion

Fick’s law of diffusion is used to determine the spatial and temporal distribution of cryoprotectant in articular cartilage throughout loading. For one-dimensional diffusion, it takes the form

45

where C is the cryoprotectant concentration (mol/m³), D is the Fick’s law diffusion coefficient (m²/s), t is the time (s), and x is the position in the cartilage (m). Each cryoprotectant is considered individually and it is assumed that the diffusion of one cryoprotectant has no effect on the diffusion of another cryoprotectant. The diffusion coefficient is temperature-dependent and can be calculated from the Arrhenius equation

46

where A is the preexponential factor (m²/s) and E_a is the activation energy (kcal/mol). Table 3 lists the required parameters for the Arrhenius equation for DMSO and EG.

Table 3. Arrhenius Parameters for Use in Eq 46(17,18).

	A(m2/s)	Ea(kcal/mol)
DMSO	2.9895 × 10–7	3.9 ± 1.6
EG	1.833 × 10–7	3.8 ± 0.7

The geometry of the cube is shown in Figure 1. At the center of the cube (x = 0 mm), the boundary condition is no flow (i.e., ∂C/∂x = 0). At the cartilage–solution interface (x = 0.5 mm), the concentration of cryoprotectant on the surface of the cartilage is equal to the concentration of the solution. The initial condition for the first step is a cryoprotectant concentration of zero throughout the cartilage. The concentration profile at the end of the first step is used as the initial condition for the second step.

Equation 45 was solved with these initial and boundary conditions using the MATLAB built-in partial differential equation solver pdepe().

2.3. Freezing Point

The freezing point at each point along the cartilage thickness can be calculated from the cryoprotectant concentration profiles. Solving the modified triphasic model results in one set of concentration profiles and solving Fick’s law results in another set of concentration profiles. For both sets of concentration profiles, the freezing point across the cartilage is calculated using^22,23,34

47

where T_FP is the freezing point of the solution (K), T_FP⁰ is the freezing point of pure water (273.15 K), Δs₁ is the molar entropy difference between pure liquid water and pure solid water at T_FP⁰ (22.00 J/mol K), and π is the osmolality of the solution (osmol/kg solvent). The osmolality of the solution is calculated using the multisolute osmotic virial equation truncated to the second-order terms^22,23

48

where m_i is the molality (mol/kg solvent), k_i is the dissociation constant, and B_i^* is the second osmotic virial coefficient (molal^–1) of the (i – 1)th solute. The summations start from an index of 2 since the subscript 1 is reserved for water by convention. The index r signifies the total number of components in the solution. Table 4 includes the needed coefficients for eq 48 as calculated by Zielinski et al.,³⁴ who determined that second-order osmotic virial coefficients are adequate to describe this solution. Note that these osmotic virial coefficients are molality-based, whereas the coefficients used in the modified triphasic model are mole-fraction-based. The concentration of the solution can be used to calculate the molality as¹⁶

49

where ρ₁ is the density of water (998 kg/m³ at 22 °C)³⁹ and V_m,i is the molar volume of each cryoprotectant at 22 °C (L/mol), as listed in Table 4.

Table 4. Coefficients Needed for Freezing Point Calculation Including the Dissociation Constant,³⁴ Molality-Based Second Osmotic Virial Coefficient,³⁴ and Molar Volume³⁶ of each Cryoprotectant Used.

	k	B* (molal–1)	Vm(L/mol)
DMSO	1	0.108 ± 0.005	0.0709
EG	1	0.020 ± 0.001	0.0559
NaCl	1.678	0.044 ± 0.002

2.4. Vitrifiability

The vitrifiability of the cartilage along its thickness can be predicted based on the concentration distribution of cryoprotectants. The same calculation is performed for the concentration profiles determined from the modified triphasic model and those calculated using Fick’s law. The predictions are made using a statistical model of vitrifiability developed by Weiss et al.²¹ by calculating the vitrifiability score

50

where β_i is the coefficient for cryoprotectant i and β_ij is the interaction coefficient between cryoprotectants i and j, including self-interaction. Table 5 lists the coefficients for DMSO and EG. The model quantifies vitrifiability using thresholds α_n that correspond to ordinal scores of vitrifiability, which were assigned to experimental samples based on visual inspection. The current study is primarily concerned with the ordinal score that describes complete devitrification upon warming, which corresponds to α₁ = 167.6 ± 27.6. A vitrifiability score α greater than α₁ indicates that the solution is vitrifiable under the same or more favorable conditions than those used to develop the model (i.e., cooling rate greater or equal to 60 K/min and volume less than or equal to 5 mL).

Table 5. Minimum Threshold (α₁) and Coefficients (β) for Use in the Ordinal Model of Vitrifiability²¹.

parameter	estimate
α1	167.6 ± 29.2
βEG	39.9 ± 8.0
βDMSO	36.4 ± 6.7
βDMSO_EG	–3.7 ± 0.9
βEG_EG	–2.3 ± 0.6
βDMSO_DMSO	–1.4 ± 0.4

3. Fitting Methods and Results

Experimental data for average cryoprotectant concentration in porcine articular cartilage dowels after various treatment times along with the dowel weight change were used by Abazari et al.²⁷ to find suitable parameters for DMSO and EG for use in the single-cryoprotectant modified triphasic model. In that study, data for DMSO and EG were treated separately to determine values for cartilage stiffness H_A, water permeability K_ws, cryoprotectant diffusivity D_cw, and cryoprotectant permeability K_cs. However, theoretical understandings stipulate that cartilage stiffness and water permeability are independent of the cryoprotectant used.²⁷ Additionally, solving the two-cryoprotectant model requires that cartilage stiffness and water permeability each have a single value regardless of cryoprotectant since both cryoprotectants are present simultaneously. Therefore, the previously found parameters cannot be used directly in the extended model and new values will be found using the previously published data.

Figure 2 shows the geometry of the cartilage dowels used in the previously published experiments,^17,18,27 which is pertinent in determining the best-fit parameters. Each harvested porcine cartilage dowel is 10 mm in diameter and approximately 2 mm thick. Each cartilage dowel was removed from the bone base before treatment in either 6.5 M DMSO or 6.5 M EG for a specified time. Accordingly, cryoprotectant permeation was possible from the top, bottom, and circumferential sides of each dowel. However, the modified triphasic model is posed in one dimension, so the fitting procedure considers that permeation only occurs from the top and bottom of each dowel. This assumption is justified given that the combined surface area of the top and bottom of each dowel is approximately 2.5 times larger than that of the circumferential sides. Additionally, radial diffusivity in cartilage has been reported to be 2–10 times less than axial diffusivity.³⁰ The x-axis follows the axial dimension of the cartilage with x = 0 mm at the center and x = h at the cartilage–solution interface. The boundary condition at the center is no flow of any component. The boundary condition at the cartilage–solution interface is that the cryoprotectant concentration in the outermost edge of the cartilage is equal to the average cryoprotectant concentration in the cartilage after two hours of exposure to the cryoprotectant bath solution. At this point, the solution in the cartilage has equilibrated with the bath solution. Using the concentration after two hours of exposure as the boundary condition rather than the 6.5 M initial concentration of the cryoprotectant solution accounts for the fact that the volume of liquid the cartilage holds is not negligible compared to the volume of the bath solution. Therefore, modeling with the chosen boundary condition provides better agreement between the model predictions and experimental data.

Figure 2.

Schematic of a 2-mm-thick cartilage dowel immersed in a 6.5 M cryoprotectant solution with diffusion occurring in one dimension along the x-axis for the purpose of theoretical modeling.

Finding the best-fit values for the model parameters is computationally intensive due to the sheer number of possible combinations of the six parameters. To make the computation time reasonable, the data for DMSO and EG were initially treated separately so that the number of nested loops would be limited to four. On each iteration, the model was solved using one combination of parameters. The model solution was used to calculate the average cryoprotectant concentration C_c in the cartilage

51

and the normalized fluidized weight W_F of the cartilage

52

after a certain treatment time t_f. The combination of parameters that maximized the sum of R² values for the concentration data and normalized fluid weight data was chosen. The R² values can be calculated using

53

where n is the number of data points, y_i is the value of the ith data point, and ŷ_i is the predicted value of the ith data point calculated using the modified triphasic model. This equation was applied to the concentration data and normalized fluid weight data separately using eqs 51 and 52, respectively, to determine the ŷ_i values. The two resulting R² values were then summed together. Maximizing the sum of R² values was chosen over minimizing the sum of squared errors so that the fitting parameters could be optimized to both datasets despite the scales of the two datasets being drastically different. The search ranges for the parameters were selected such that they contained the fitting parameter results of Abazari et al.,²⁷ as the new values were expected to be similar to these. After completing this procedure for both DMSO and EG at 4 °C, the values found for cartilage stiffness and water permeability were averaged to obtain a single value for each. With cartilage stiffness and water permeability held constant, the fitting procedure was repeated for both DMSO and EG to fine-tune cryoprotectant diffusivity and cryoprotectant permeability. This method works well since predicting cryoprotectant concentration and normalized fluid weight is most sensitive to the values of cryoprotectant diffusivity and cryoprotectant permeability, respectively. A more detailed discussion about model parameter significance can be found in previous work by Abazari et al.²⁷ The entire procedure was repeated for 22 and 37 °C. Table 6 shows the best-fit parameters that were found using this process, and Figure 3 illustrates the fit to experimental data.

Table 6. Parameters for Use in the Two-Cryoprotectant Modified Triphasic Model.

parameter	4 °C	22 °C	37 °C
HA (Pa)	1.5 × 106	2.3 × 106	3.1 × 106
Kws(m4/(Ns))	4.3 × 10–16	4.5 × 10–16	5.0 × 10–16
Dcw for DMSO (m2/s)	1.6 × 10–10	2.3 × 10–10	3.9 × 10–10
Kcs for DMSO (m4/(Ns))	3.6 × 10–17	3.6 × 10–17	3.6 × 10–17
Dcw for EG (m2/s)	2.2 × 10–10	2.8 × 10–10	4.9 × 10–10
Kcs for EG (m4/(Ns))	2.0 × 10–17	2.1 × 10–17	2.7 × 10–17

Figure 3.

(a, b) Average concentration of cryoprotectant in a 2-mm-thick dowel of porcine articular cartilage dissected from the bone after various treatment times in a 6.5 M single-cryoprotectant solution. Experimental data are from Jomha et al.^17,18 The solid lines represent the best fit to the data calculated using the model solution for a single cryoprotectant and eq 51. (c, d) Normalized fluid weight of a 2-mm-thick dowel of porcine articular cartilage dissected from the bone after various treatment times in a 6.5 M single-cryoprotectant solution. Experimental data are from Abazari et al.²⁷ Solid lines represent the best fit to the data calculated using the model solution for a single cryoprotectant and eq 52.

The Arrhenius equation can be used to describe the temperature dependence of these fitting parameters.

54

The equation was linearized, and the activation energy E_a (kcal/mol) and prefactor A were found using linear regression. Table 7 lists the Arrhenius parameters and Figure 4 shows the Arrhenius plots illustrating the temperature dependence of the model parameters.

Table 7. Parameters for Use in Eq 54.

parameter	A	Ea(kcal/mol)
HA	1.40 × 109 Pa	3.76 ± 0.07
Kws	1.88 × 10–15 m4/(Ns)	0.8 ± 0.2
Dcw for DMSO	5.81 × 10–7 m2/s	4.5 ± 0.9
Kcs for DMSO	5.96 × 10–17 m4/(Ns)	0.28 ± 0.02
Dcw for EG	3.15 × 10–7 m2/s	4.0 ± 1.3
Kcs for EG	2.90 × 10–16 m4/(Ns)	1.5 ± 0.7

Figure 4.

Arrhenius plots showing the temperature dependence of the six fitting parameters. (a) Cartilage stiffness H_A (left y-axis; filled circles and solid line) and water permeability in cartilage K_ws (right y-axis; open triangles and dashed line). (b) Cryoprotectant diffusivity in cartilage D_cw for DMSO (dark blue) and EG (orange). (c) Cryoprotectant permeability in cartilage K_cs for DMSO (dark blue) and EG (orange).

4. Results and Discussion

4.1. Freezing Point, Vitrifiability, Strain, and Thickness of Cartilage for Protocol E-D

Figure 5 shows the calculated spatial profiles of freezing point, vitrifiability, and strain at the end of each loading step of Protocol E-D. Note that the Fick’s law results are shown from x = 0 to x = 0.5 mm, whereas the modified triphasic results are shown from x = 0 to x = h, where h is the predicted half-thickness of the cartilage cube.

Figure 5.

Modeling results for Protocol E-D (details listed in Table 1). Solid lines are the modified triphasic model predictions, and dashed lines are the Fick’s law model predictions. (a, b) Spatial distribution of cryoprotectant concentrations (orange is EG, dark blue is DMSO) from the center of a 1 mm³ articular cartilage cube (x = 0 mm) to the solution–cartilage interface (x = 0.5 mm for Fick’s law and x = h for the modified triphasic model) at the end of each loading step. (c, d) Freezing point profile at the end of each loading step. (e, f) Vitrifiability profile at the end of each loading step. The dotted horizontal line shows the minimum vitrifiability threshold corresponding to α₁. (g, h) Strain profile at the end of each loading step. Note that Fick’s law does not allow for strain.

The solid lines in Figure 5a,b show concentration distribution as calculated using the modified triphasic model (see Section 2.1.7 for details of numerical solution and Table 7 for Arrhenius parameters needed for eq 54). The dashed lines in Figure 5a,b show the concentration distribution as calculated using Fick’s law (eq 45 using diffusion coefficients calculated from eq 46 with the Arrhenius parameters from Table 3). At the end of the first step, the modified triphasic model predicts a slightly higher concentration of EG throughout the cartilage than Fick’s law predicts. However, the difference between the two models becomes more apparent at the end of the second step where the concentration of EG predicted by the modified triphasic model is significantly higher than that predicted by Fick’s law. In fact, at the end of the second step, the concentration of EG at the center of the cartilage cube has surpassed the concentration of EG in the bath solution. Although initially unintuitive, this result highlights the power of the two-cryoprotectant version of the modified triphasic model. Since there are three moving components, the concentration of cryoprotectant in the cartilage can increase both due to the movement of water out of the cartilage as well as the movement of cryoprotectant into the cartilage. Since water is a physically smaller molecule than EG and DMSO, it can move through the cartilage faster. At the start of the second step, the high osmolality of the bath solution drives water out of the cartilage. This results in the concentration of EG temporarily increasing above the equilibrium level by the end of the second step. Given more time, it would be expected that the solution in the cartilage would come to equilibrium with the bath solution. As anticipated, the modified triphasic model predicts that the concentration of EG and DMSO throughout the cartilage equilibrates with the bath solution when left an additional 45 min past the end of the second step. Simulations performed using the previously published single-cryoprotectant version of the modified triphasic model also showed that Fick’s law underestimates cryoprotectant concentration in articular cartilage throughout loading.²⁷⁻²⁹ As mentioned, our results for the two-cryoprotectant model are consistent with these findings. In addition, our model provides new insights into the transport behavior of a solution containing two cryoprotectants, which could not be found using the single-cryoprotectant version. It is important to note that all simulations were performed for a 1 mm³ cartilage cube. Performing the simulations for a different size of cube would change the numerical values of the computed concentration profiles. A larger cube would have a lower cryoprotectant concertation at its center compared to a 1 mm³ cube when using the same protocol. Conversely, a smaller cube would have a higher concentration. This is because it takes longer for the cryoprotectant to diffuse into the center when the cube is larger. Despite the difference in numerical values, the conclusions of this paper remain the same for all sizes of cartilage cubes.

Using the concentration distributions calculated using the modified triphasic model and Fick’s law, the freezing point throughout the cartilage, shown in Figure 5c,d, was calculated assuming nonideal solution behavior (eqs 47–49) using parameters from Table 4). At the end of the first step, the difference between the modified triphasic model and Fick’s law results is barely discernable. However, at the end of the second step, the freezing point predicted by the modified triphasic model is much lower. This is because the modified triphasic model predicts higher concentrations of cryoprotectant throughout the cartilage due to the transport behavior during the simultaneous diffusion of two cryoprotectants and water, which Fick’s law fails to take into account. This could be an important difference when designing multistep protocols that involve two cryoprotectants in the first step since it could allow the second step to be performed at a lower, less toxic temperature. In fact, it has been observed that loading cryoprotectants at lower temperatures greatly improves chondrocyte recovery, especially at higher cryoprotectant concentrations.¹¹ In addition, a more accurate prediction of the freezing point of the cryoprotectant solution would allow for better tracking of the liquidus during stepwise loading. This is important since the liquidus-tracking method has been shown to help prevent damaging ice crystallization during rapid cooling.¹⁴

The vitrifiability throughout the cartilage, shown in Figure 5e,f, was calculated using eq 50 and the parameters from Table 5. The horizontal dotted line illustrates the minimum threshold for vitrifiability as determined by Weiss et al.²¹ At the end of the first step, the cryoprotectant concentration throughout the cartilage is insufficient for vitrification. At the end of the second step, both the modified triphasic model and Fick’s law predict that the cartilage will be vitrifiable. However, the vitrifiability profile predicted by Fick’s law barely exceeds the minimum threshold, whereas the modified triphasic model predicts vitrifiability beyond what is minimally necessary. This suggests that it may be possible to further shorten vitrification protocols developed using Fick’s law to mitigate toxic effects while still achieving vitrifiability.

The strain throughout the cartilage, shown in Figure 5g,h, was obtained directly from the modified triphasic model solution. Fick’s law does not allow for strain, so the Fick’s law result is denoted as a horizontal line at zero strain. In both the first and second steps, the strain experienced in the cartilage is greatest at the center.

Figure 6 shows the predicted thickness of an initially 1 mm³ articular cartilage cube throughout Protocol E-D.²⁴ The solid line is the modified triphasic result calculated using eq 44. Note that Fick’s law assumes constant cartilage thickness so the prediction for Fick’s law is simply a horizontal line at the original cartilage thickness of 1 mm. Initially, the cartilage shrinks when suddenly immersed in a cryoprotectant solution since water quickly moves out of it due to difference in osmolalities. At the start of the second step, this change in thickness is more pronounced as the concentration of cryoprotectant in the bath solution is greater, thus providing a stronger driving force. After reaching a minimum thickness, the cartilage begins to expand back toward its original size. The modified triphasic model predicts that the thickness of the cartilage cube will return to 1 mm when left in the cryoprotectant bath solution an additional 45 min past the end of the second step.

Figure 6.

Predicted thickness of an initially 1 mm³ articular cartilage cube throughout the Protocol E-D loading procedure. The solid line shows the modified triphasic model predictions, and the dashed horizontal line represents Fick’s law. The vertical dotted line signifies the transition between the first and second steps of the loading procedure.

4.2. Comparison of Modeling Results for Cartilage Treated in Cryoprotectant Solutions with Equal Overall Concentration

Figure 7 shows the calculated spatial profiles of freezing point, vitrifiability, and strain at the end of 5 min of treatment at 0 °C in one of three cryoprotectant solutions each with an overall cryoprotectant concentration of 6 M (6 M DMSO, 6 M EG, or 3 M DMSO + 3 M EG). The profiles were calculated using the same equations as described for Figure 5. Note that the Fick’s law results are shown from x = 0 to x = 0.5 mm, whereas the modified triphasic results are shown from x = 0 to x = h, where h is the predicted half-thickness of the cartilage cube. For both treatments in 6 M DMSO and 6 M EG, Fick’s law underpredicts the concentration throughout the cartilage in comparison to the modified triphasic model. This discrepancy can be attributed to the fact that Fick’s law fails to account for the nonideal and nondilute nature of the cryoprotectant solution as well as water movement in and out of the cartilage. This underestimation of cryoprotectant concentration results in the predicted freezing point being higher for Fick’s law and the vitrifiability being lower. The rate of permeation of DMSO is greater than that of EG. Similarly, the strain the cartilage experiences when treated with 6 M DMSO is greater than that for the treatment with 6 M EG. The treatment in 3 M DMSO + 3 M EG achieved similar vitrifiability as each of the 6 M treatments, but with the benefit that the cartilage is subjected to less strain. Additionally, experiments have shown that solutions containing multiple cryoprotectants are less toxic to chondrocytes than those containing a single cryoprotectant at equal overall concentrations.¹³ Therefore, it is expected that the 3 M DMSO + 3 M EG solution would be less toxic than the 6 M DMSO or 6 M EG solutions while still reaching similar vitrifiability.

Figure 7.

Comparison of modeling results for a 1 mm³ articular cartilage cube after 5 min of treatment at 0 °C in one of three different cryoprotectant solutions with equal overall concentration (6 DMSO, 6 M EG, or 3 M DMSO + 3 M EG). The solid lines are the modified triphasic model predictions, and the dashed lines are the Fick’s law predictions. (a–c) Spatial distribution of cryoprotectant concentrations (dark blue is DMSO, orange is EG) from the center of a 1 mm³ articular cartilage cube (x = 0 mm) to the solution–cartilage interface (x = 0.5 mm for Fick’s law and x = h for the modified triphasic model) at the end of 5 min. (d–f) Freezing point profile at the end of 5 min. (g–i) Vitrifiability profile at the end of 5 min. The dotted horizontal line shows the minimum vitrifiability threshold corresponding to α₁. (j–l) Strain profile at the end of 5 min. Note that Fick’s law does not allow for strain.

Figure 8 shows the predicted thickness of an initially 1 mm³ articular cartilage cube throughout a 5-min treatment at 0 °C in one of three cryoprotectant solutions with equal overall concentration. The initial cartilage shrinkage is steepest for treatment in 6 M DMSO or 6 M EG. When the overall concentration is split between DMSO and EG, as is the case in the 3 M DMSO + 3 M EG treatment, the cartilage experiences a much less dramatic change in thickness. This can be explained by realizing that each cryoprotectant diffuses along its own concentration gradient. Therefore, the driving force for diffusion of each cryoprotectant is lower for the 3 M DMSO + 3 M EG solution in comparison to the 6 M single-cryoprotectant solutions. Consequently, the DMSO and EG move into the cartilage more slowly which also makes the water move out of the cartilage more slowly. Ultimately, this results in less extreme cartilage shrinkage.

Figure 8.

Predicted thickness of an initially 1 mm³ articular cartilage cube throughout a 5-min treatment at 0 °C in either 6 M DMSO (dashed line), 6 M EG (solid line), or 3 M DMSO + 3 M EG (dash-dotted line). The cartilage thickness was calculated using the modified triphasic model.

4.3. Comparison of Simultaneous and Individual Diffusion Predicted with the Modified Triphasic Model

Figure 9 compares the spatial concentration profile of cryoprotectant in a 1 mm³ articular cartilage cube after a 5 min treatment at 0 °C in 3 M DMSO, 3 M EG, or 3 M DMSO + 3 M EG as predicted by the modified triphasic model. Fick’s law modeling treats each cryoprotectant separately and assumes that the diffusion of one cryoprotectant has no impact on the diffusion of another cryoprotectant. In contrast, the modified triphasic model accounts for the presence of other cryoprotectants. When 3 M DMSO and 3 M EG diffuse simultaneously, it is predicted that the concentration of each cryoprotectant in the cartilage will be greater than when 3 M DMSO or 3 M EG diffuses individually. Since the overall concentration of the bath solution is greater when 3 M DMSO and 3 M EG diffuse simultaneously, the difference in osmolalities is higher and water rushes out of the cartilage at a greater rate than when the bath contains only 3 M DMSO or 3 M EG. This also causes the cartilage to shrink more during simultaneous diffusion. Fick’s law fails to account for water movement and so it cannot accurately model the simultaneous diffusion of multiple cryoprotectants and water.

Figure 9.

Comparison of the modified triphasic model predictions of spatial distribution of cryoprotectant concentrations (dark blue is DMSO, orange is EG) in a 1 mm³ articular cartilage cube after a 5 min treatment at 0 °C in 3 M DMSO (dash-dotted line), 3 M EG (dash-dotted line), or 3 M DMSO + 3 M EG (solid lines). The center of the cartilage cube is x = 0 mm and the cartilage–solution interface is initially at x = 0.5 mm. The lines end before x = 0.5 mm reflecting how much the cartilage has shrunken.

5. Conclusions

We have extended the modified triphasic model to include two permeating cryoprotectants so that vitrification protocols involving simultaneous diffusion of two cryoprotectants can be modeled in a way that accurately reflects nonideal, nondilute solution behavior and water movement. We have obtained Arrhenius parameters for cartilage stiffness, water permeability, diffusivity and permeability of DMSO, and diffusivity and permeability of EG from fitting the model to previously published experimental data. We have modeled Protocol E-D, a successful vitrification protocol for particulated cartilage,²⁴ using both the two-cryoprotectant modified triphasic model and Fick’s law to obtain concentration, freezing point, vitrifiability, and strain profiles at the end of each loading step. We have observed that Fick’s law consistently underestimates cryoprotectant concentration compared to the modified triphasic model because it fails to model water movement. This is consistent with findings for the previously published single-cryoprotectant version of the model.²⁷⁻²⁹ Therefore, vitrification protocols developed using Fick’s law can likely be further optimized using the modified triphasic model to reduce cryoprotectant exposure. We have shown that a bath solution containing lower concentrations of two cryoprotectants rather than a highly concentrated solution of one cryoprotectant can achieve similar vitrifiability while also minimizing the amount of cartilage shrinkage experienced during loading. Using our two-cryoprotectant modified triphasic model, we have seen that the diffusion of one cryoprotectant impacts the rate at which another cryoprotectant diffuses when both diffuse simultaneously. This phenomenon cannot be accounted for using Fick’s law. As such, the advancements made here in modeling the simultaneous transport of multiple cryoprotectants can support the future optimization of vitrification protocols to facilitate the long-term storage of articular cartilage.

The authors declare the following competing financial interest(s): N.M.J. and J.A.W.E. are co-inventors on the US (8 758 988) and Canada (2 788 202) patents for articular cartilage preservation: N.M.J., L. E. McGann., J.A.W.E., G. Law, F. Forbes, A. Abazari Torghabeh, B. Maghdoori, and A. Weiss University of Alberta, “Cryopreservation of articular cartilage”.