Analysis of Cryoprotectant Concentration During Cryopreservation in Articular Cartilage Sample Using Homogeneous and Porous Material Models

Abstract

Purpose

The study examines the concentration of cryoprotectant (CPA) in an articular cartilage sample during cryopreservation by computing the effective diffusion coefficient using different material models–homogeneous and porous.

Methods

The mass transfer phenomenon is coupled to the effective diffusion coefficient, which is determined by three different approaches. The first and second models, based on the Einstein-Stokes equation and the Arrhenius expression, respectively, treat the sample as a homogeneous material, whilst the third considers it as a porous medium. The effective diffusion coefficient is additionally weakly coupled to the heat transfer phenomenon described by the Fourier equation, and the third variant is also strongly coupled to the concentration of CPA.

Results

The final section of the article presents example calculations for the selected cryopreservation method, and the results are compared with the experimental results. Depending on the method applied to estimate the effective diffusion coefficient, the maximal relative errors in relation to experimental results are equal to 15.82%, 5.20%, and 24.96%, respectively.

Conclusion

A decrease in temperature and an increase in the concentration of dimethyl sulfoxide (DMSO) cause a reduction of the effective diffusion coefficient. Moreover, in the model considering the porosity of the sample, the lowest values of the effective diffusion coefficient were obtained. This study’s novelty lies in its comparative analysis of homogeneous and porous models, as well as its explicit coupling of temperature, concentration, and diffusion processes during cryopreservation.

Introduction

Biological tissues are often modelled as homogeneous materials. In reality, they consist of dispersed cells separated by pores that enable, for example, the transport of nutrients. Therefore, it is reasonable to treat them as a porous material consisting of a matrix and the pores filled with fluid, and characterized by porosity, which is defined as the ratio of the volume of the pores to the volume of the entire material. This structure of biological tissues affects transport phenomena such as heat and mass transfer. The thermal processes are caused by heat conduction in tissues or blood perfusion through tissue pores (blood convection), whereas mass transfer is related to changes in the concentration of cryoprotectant (CPA) in the sample domain, which occur due to diffusion [8].

The analysis of transport phenomena in biological tissues is applied in medical applications. For example, bioheat transfer is conducted when modelling treatments such as hyperthermia or photodynamic therapy [6, 32]. Mass transfer phenomena are crucial in the development of processes for drug delivery to appropriate cells or solvent transport in biodegradable scaffolds [8, 9]. Heat and mass transfer also occur during the cryopreservation process, the purpose of which is to store biological material at low temperatures. This process can lead to damage to the biological sample. To prevent undesirable changes in the sample, the cooling rate is properly adjusted and a chemical compound called a cryoprotectant (CPA), such as dimethyl sulfoxide (DMSO) or glycerol, is added. In the slow freezing method, the process of preservation is performed at a low cooling rate with a low concentration of CPA, whereas vitrification uses a high cooling rate combined with a high concentration of CPA [5, 7].

The subject of our study is a sample of articular cartilage. It is worthwhile to note that it is the tissue that covers the joint surfaces of bones, providing smooth movement and shock absorption, and does not contain blood vessels, lymphatics, or nerve connections. The articular cartilage is a tissue composed of water (about 80%), collagen, and proteoglycans. The diameter of its pores is expected to be between 2 nm and 6 nm according to hydraulic permeability measurements [15] and between 11 nm and 14 nm estimated by using thermoporometry [13].

The storage of articular cartilage is important in the clinical treatment of joint injuries [39]. Currently, tissue allografts and meniscal transplantation are being developed. Therefore, research on the methods for banking of allograft cartilage or tissue-engineered equivalents, which provide a time between procurement and surgical use, is crucial [19]. Cryopreservation of articular cartilage can be performed using conventional methods, as shown in [18, 19]. As indicated in [19], cryopreservation of articular cartilage by slow freezing is relatively simple, but the survival of chondrocytes (articular cartilage cells) is inadequate and variable, which makes it unsatisfactory for clinical applications. This effect is likely related to ice crystallization [17]. On the other hand, Song et al. [30, 31] cryopreserved the articular cartilage using vitrification, which resulted in higher chondrocyte survival. This method also depends on the warming rate, which may be technically difficult to control in the case of bulk samples [17]. Pegg et al. [17] proposed an alternative method called the liquidus-tracking (LT) protocol, which involves modulating the temperature and concentration of CPA of the bath of solution in which a sample is immersed. In this approach cooling and warming rates can be low. Moreover, it would allow avoiding ice crystallization and the use of high CPA concentrations, especially for temperatures above 0 °C, which ensures better effectiveness compared to other techniques.

In the literature, one can find examples in which cryopreservation is mathematically modelled to simulate, for example, the vitrification method [10, 16, 41] or the LT protocol [39]. The mathematical model describes transport phenomena, such as heat and mass transfer. Heat transfer processes are described by the Fourier equation [3, 25, 39], whilst mass transfer is defined by an equation based on Fick’s second law [10, 25, 39, 41].

Several of our previous works analysed the cryopreservation of articular cartilage samples by treating them as homogeneous materials [20, 27, 29]. As indicated in the mentioned papers, the results obtained differed from the experimental data from the literature [17]. It was also suggested that the discrepancies could be due to faults in the material model used.

This study presents a comparison of CPA concentration distribution in a sample by computing the effective diffusion coefficient using different material models—homogeneous (see Eqs. (2) and (3) in the Mathematical model section) and porous (cf. Eq. (4)). Moreover, mass transfer phenomena are coupled with heat transport processes (presented in Eqs. (1) and (11)). In addition, considering the porosity of the sample, the effective diffusion coefficient is strongly coupled with the concentration of CPA in the computational domain. The simulation results are validated by comparison with experimental data from the literature [17]. Unlike our previous works [20, 27–29], which treated articular cartilage mainly as a homogeneous medium and focussed on different methods of estimating the effective diffusion coefficient, the present study extends the analysis by incorporating a porous material model. Moreover, the coupling between diffusion, concentration, and temperature is explicitly considered. This approach provides new insights into the role of cartilage porosity during cryoprotectant permeation, which has not been addressed in our earlier publications. The CPA is assumed to be a solution composed of DMSO (dimethyl sulfoxide) and water, and the cryopreservation simulation is conducted using the LT protocol [28].

Although classical formulations of heat and mass transfer are well established, their direct application to biological tissues does not fully capture the influence of tissue heterogeneity and porous microstructure. The novelty of the present study lies in the comparative analysis of homogeneous and porous models of articular cartilage during cryopreservation, performed under the liquidus-tracking protocol. Unlike previous studies, this research explicitly considers the coupling between temperature, concentration, and the effective diffusion coefficient. This provides new insights into how the structural features of cartilage influence cryoprotectant permeation.

Materials and Methods

Mathematical Model

The distribution of the DMSO concentration in the considered computational domain is described by the relation [1, 27]:

$$\begin{array}{c}\frac{\partial c_d\left(r,z,t\right)}{\partial t}=\mathrm{\nabla }\left(D_{\text{eff}}\mathrm{\nabla }{c}_d\right),\end{array}$$ 1

where c_d [%(w/w) or mol·m⁻³] is the DMSO concentration in the sample, t is the time, r, z are the coordinate system, D_eff [m²·s⁻¹] is the effective diffusion coefficient, which in general depends on several factors, such us temperature, viscosity and structure of the medium, speed of diffusing molecules, and size of the fluid particles [1]. In our case, its value varies according to the assumed material model.

The rationale for introducing three variants of the effective diffusion coefficient is to evaluate the impact of different theoretical formulations on CPA concentration predictions in the tissue. Variants 1 and 2 are based on the Einstein–Stokes and Arrhenius relations and treat the sample as homogeneous. In contrast, variant 3 incorporates tissue porosity, tortuosity, and coupling between local DMSO concentration and diffusion properties. This better reflects the structural complexity of cartilage.

When the biological tissue is treated as a homogeneous medium, the effective diffusion coefficient can be determined from the Einstein–Stokes equation [11, 27]:

$$\begin{array}{c}{D}_{\text{eff}}=\frac{k_{B}T\left(r,z,t\right)}{6\pi r_s{\mu }_d},\end{array}$$ 2

where k_B [J·K⁻¹] is the Boltzmann constant (k_B = 1·38 × 10⁻²³ J·K⁻¹), T [K] is the temperature, r_s [m] is the radius of the spherical particle and μ_d [Pa·s] is the dynamic viscosity of DMSO.

A second method to determine the effective diffusion coefficient in a homogeneous material is based on the Arrhenius expression [24, 41]:

$$\begin{array}{c}{D}_{\text{eff}}=A\text{exp}\left(-\frac{E_a}{RT\left(r,z,t\right)}\right),\end{array}$$ 3

where A [m²·s⁻¹] is the prefactor, E_a [J·mol⁻¹] is the activation energy, R [J·mol⁻¹·K⁻¹] is the universal gas constant (R = 8.3145 J·mol⁻¹·K⁻¹).

For porous materials, the effective diffusion coefficient is defined as [38–40]:

$$\begin{array}{c}{D}_{\text{eff}}=D_{\text{dw}}\frac{\epsilon }{{\tau }^2},\end{array}$$ 4

where D_dw [m²·s⁻¹] is the diffusion coefficient of DMSO in water, ε is the water content in the sample and τ is the tortuosity factor, which represents the hindrance to flow diffusion caused by local boundaries or local viscosity [8]. The diffusion coefficient of DMSO in water is given by the expression:

$$\begin{array}{c}{D}_{\text{dw}}=D_0+{\mathrm{\Gamma }}^{0.5},\end{array}$$ 5

where D₀ [m²·s⁻¹] is the reference diffusion coefficient, and Γ is the thermodynamic factor that accounts for deviations from ideal solution behaviour and depends, amongst other variables, on temperature and the chemical potential of the system. It is defined as [38–40]:

$$\begin{array}{c}\mathrm{\Gamma }=1+x_d\left(\frac{\partial ln{\gamma }_d}{\partial x_d}\right),\end{array}$$ 6

where x_d is the mole fraction of DMSO and γ_d is the activity coefficient for DMSO calculated by the UNIFAC model [2]. The mole fraction of DMSO is coupled with its concentration in the computational domain.

The reference diffusion coefficient presented in Eq. (4) is the product of the diffusion coefficients at infinite dilution of DMSO in water (D_0,dw [m²·s⁻¹]) and water in DMSO (D_0,wd [m²·s⁻¹]) [26, 38–40]:

$$\begin{array}{c}{D}_0={\left(D_{0,\text{dw}}\right)}^{x_w}·{\left(D_{0,\text{wd}}\right)}^{x_d},\end{array}$$ 7

where

$$\begin{array}{c}{D}_{\text{0,dw}}=\left(2.98\times {10}^{-7}\right){\mu }_w^{-1.026}·{\left(0.285V_{c,d}^{1.048}\right)}^{-0.5473}·T,\end{array}$$ 8

$$\begin{array}{c}{D}_{0,\text{wd}}=\left(9.89\times {10}^{-8}\right){\mu }_d^{-0.907}·{\left(0.285V_{c,d}^{1,048}\right)}^{-0.45}·{\left(0.285V_{c,w}^{1.048}\right)}^{0.265}·T,\end{array}$$ 9

whilst x_w is the mole fraction of water, μ_w [Pa·s] is the dynamic viscosity of water, V_c,d [m³·mol⁻¹] and V_c,w [m³∙mol⁻¹] are the critical volumes of DMSO and water, respectively.

Furthermore, by introducing the three-parameter equation of the Vogel–Fulcher–Tammann–Hesse, the viscosity can be expressed as a variable depending on the temperature [26, 39]:

$$\begin{array}{c}ln\mu =P_1+\frac{P_2}{T-T_0},\end{array}$$ 10

where P₁, P₂ [K] and T₀ [K] are constants, and T [K] is the absolute temperature.

It should be noted that the effective diffusion coefficient depends on the temperature (see Eqs. (2, 3, 8, 9)). The temperature distribution in the computational domain was determined using the Fourier heat conduction equation [1, 27]:

$$\begin{array}{c}{c}_p\mathrm{\rho }\frac{\partial T\left(r,z,t\right)}{\partial t}=\mathrm{\nabla }\left(k\mathrm{\nabla }T\right),\end{array}$$ 11

where c_p [J·kg⁻¹·K⁻¹] is the specific heat capacity, ρ [kg·m⁻³] is the density, and k [W·m⁻³·K⁻¹] is the thermal conductivity.

The defined computational problem also requires determining the initial and boundary conditions. At time t = 0, the initial conditions are given by:

$$\begin{array}{c}T\left(r,z,t=0\right)=T_{\text{init}}\end{array}$$ 12

and

$$\begin{array}{c}{c}_d\left(r,z,t=0\right)=c_{\text{init}}\end{array}$$ 13

The boundary conditions are defined as follows (cf. Fig. 1) [27]:

$$\left\{\begin{array}{c}{\mathrm{\Gamma }}_1\text{and}{\mathrm{\Gamma }}_4:-nk·\mathrm{\nabla }T=\alpha \left[T\left(r,z,t\right)-T_{\mathit{bath}}\right]\text{and}-n{D}_{\mathit{eff}}·\mathrm{\nabla }{c}_d=0.9c_{\mathit{bath}},\\ {\mathrm{\Gamma }}_2\text{and}{\mathrm{\Gamma }}_3:-nk·\mathrm{\nabla }T=0\text{and}-n{D}_{\mathit{eff}}·\mathrm{\nabla }{c}_d=0,\\ \end{array}\right)$$ 14

where α is the natural convection heat transfer coefficient, n is the normal vector, T_bath and c_bath are the temperature and the DMSO concentration of bath solution, respectively. The coefficient 0.9 before c_bath simulates to the real mass exchange between the sample and the bathing solution (see [29, 39]).

Fig. 1.

Computational domain

Numerical Model

Numerical analysis was performed using the finite difference method (FDM), as described in detail in [12, 14, 27]. First, one defined a fixed time grid ($t^0<t^1<\dots <t^{f-1}<t^f<t^{f+1}<\dots <t^{\infty }$, where Δt = t ^f - t^{f −1} is the constant time step) and performed a discretization of the computational domain. The geometric mesh was prepared according to the concept called the five-point star [29]. It is important to note that the boundary nodes are not located directly on the boundary of the domain, but are positioned half a mesh step inward from it.

FDM approximates the differential equation by replacing derivatives with finite differences. At the central nodes, Eq. (1) takes the following discretised form:

$$\begin{array}{c}{\left(c_d\right)}_{i,j}^f={\left(c_d\right)}_{i,j}^{f-1}-\mathrm{\Delta }t\sum _{a=1}^4\frac{{\mathrm{\Phi }}_e}{W_e}\left({\left(c_d\right)}_e^{f-1}-{\left(c_d\right)}_{i,j}^{f-1}\right),\end{array}$$ 15

where i = 2, 3, …, n − 1, j = 2, 3, …, m – 1, with n and m denoting the number of nodes in the z- and r-directions, respectively. The index a corresponds to the set of neighbouring nodes e = {(i, j + 1); (i, j − 1); (i + 1, j); (i − 1, j)}, Φ_e [m⁻¹] denotes the shape function, and W_e [s∙ m⁻¹] represents the mass resistance. The shape function and the mass resistance can be expressed as:

$$\begin{array}{c}{\mathrm{\Phi }}_{i,j-1}=\frac{r_{i,j}-0.5h_1}{r_{i,j}{h}_1},{\mathrm{\Phi }}_{i,j+1}=\frac{r_{i,j}+0.5h_1}{r_{i,j}{h}_1},{\mathrm{\Phi }}_{i-1,j}={\mathrm{\Phi }}_{i+1,j}=\frac{1}{h_2},\end{array}$$ 16

$$\begin{array}{c}{W}_{i,j-1}=W_{i,j+1}=\frac{h_1}{D_{\text{eff}}},W_{i-1,j}=W_{i+1,j}=\frac{h_2}{D_{\text{eff}}}.\end{array}$$ 17

where r_i,j [m] is the radial coordinate of the node (i, j), h₁ [m] and h₂ [m] are the mesh steps in the r- and z-directions, respectively.

Analogously, using FDM, Eq. (11) for central nodes can be written as:

$$\begin{array}{c}{T}_{i,j}^f=T_{i,j}^{f-1}-\frac{\mathrm{\Delta }t}{c_p\rho }\sum _{a=1}^4\frac{{\mathrm{\Phi }}_e}{R_e}\left(T_e^{f-1}-T_{i,j}^{f-1}\right),\end{array}$$ 18

where R_e [m⁴∙K∙W⁻¹] is the thermal resistance, given by:

$$\begin{array}{c}{R}_{i,j-1}=R_{i,j+1}=\frac{h_1}{k},R_{i-1,j}=R_{i+1,j}=\frac{h_2}{k}.\end{array}$$ 19

As the explicit scheme is applied in the derivations, the numerical model must also satisfy the following stability condition:

$$\begin{array}{c}\mathrm{\Delta }t\le \sum _{a=1}^4\frac{{\mathrm{\Phi }}_e}{R_e}\text{and}\mathrm{\Delta }t\le \sum _{a=1}^4\frac{{\mathrm{\Phi }}_e}{W_e}.\end{array}$$ 20

It should be noted that Eqs. (1) and (11) must also be discretised for boundary nodes. As this derivation is provided in detail in [27], it is not repeated here.

Results

The article presents an analysis of heat and mass transfer in an articular cartilage sample during cryopreservation and the results are compared with experimental data. Pegg et al. [17] cryopreserved a sample obtained from a knee joint of a lamb using the LT protocol that they developed. In their experiment, the sample was immersed in a tube containing a CPTes2 solution proposed by Taylor and Hunt [33] and modified by Pegg et al. [17]. The CPTes2 solution mainly contains water and DMSO; therefore, the other components are neglected in our simulation.

An example of a device used for performing cryopreservation with the LT protocol is presented in the article by Wang et al. [36], where the test tubes containing the samples are placed in a temperature-controlled chamber. In addition, a set of pumps regulates the concentration of the cryoprotectant in the test tubes. The fact that the samples are placed in test tubes under controlled conditions has been taken into account in our analysis by modelling the appropriate boundary conditions.

Computational Example

Our calculations examined the effect of changing the effective diffusion coefficient on the concentration of DMSO in biological tissue, modelled both as a homogeneous and as a porous material. Table 1 presents the thermophysical and other parameters used to compute the effective diffusion coefficient. The initial values (when t = 0) are equal to T_init = 22 °C, c_init = 0%(w/w) [27], and the natural convection heat transfer coefficient α = 525 W·m⁻²·K⁻¹ [39]. The dimensions of the computational domain are R = 3 mm and H = 1 mm (cf. Fig. 1) [39]. The properties of the time grid and the geometric mesh are: ∆t = 0.005 s, h₁ = 0.1 mm and h₂ = 0.05 mm. It is assumed that the temperature (T_bath) and the concentration of DMSO (c_bath) of the bath solution in which the sample is immersed change according to the protocol named the liquidus-tracking method proposed by Pegg et al. [17]. Eight simulation steps were carried out, with the following values: T_bath ∈ {22; 22; − 5; − 8.5; − 16; − 23; − 35; − 48.5} ºC, c_bath ∈ {10; 20; 29; 38; 47; 56; 63; 72} %(w/w), with each step lasting t_bath ∈ {10; 10; 30; 30; 30; 30; 30; 30} s.

Table 1.

Parameters of the modelled process

Parameters		Values	References
cp	[J·kg−1·K−1]	3567.5	[4, 37, 39]
ρ	[kg·m−3]	1100	[4]
k	[W·m−3∙K−1]	0.518	[37]
ε	−	0.78	[39]
τ	−	1.4	[38, 39]
rs	[m]	2.541 × 10−10	[23]
μd	[Pa∙s]	1.996 × 10−3	[21]
μw	[Pa∙s]	0.8905 × 10−3	[21]
A	[m2∙s-1]	2.9895 × 10−7	[24]
Ea	[kcal∙mol−1]	3.9	[24]
Vc,d	[mL∙mol−1]	228	[39]
Vc,w	[mL∙mol−1]	56	[39]
P1 for DMSO	–	− 1.9528	[38, 39]
P2 for DMSO	[K]	331.8	[38, 39]
T0 for DMSO	[K]	172.41	[38, 39]
P1 for water	–	− 3.5418	[38, 39]
P2 for water	[K]	509.25	[38, 39]
T0 for water	[K]	149.68	[38, 39]

The computations were performed in an author’s programme prepared in MATLAB 2024b software, whilst the UNIFAC model calculations were conducted with the UNIFAC group contribution method activity calculator function available in MATLAB [22]. In general, the UNIFAC (UNIversal Functional Activity Coefficient) model is an activity coefficient model based on the local composition theory used in chemical thermodynamics. It assumes that the local environment of a molecule allows us to compute the probabilities of molecular configurations and activities in the solution. The UNIFAC model can be applied to estimate activity coefficients for arbitrary systems, provided that an interaction parameter exists between every pair of groups in the solution [34]. To determine and the activity coefficient for DMSO (γ_d), we need to specify the volume contributions (R_k), the surface area contributions (Q_k), the number of groups in each molecule (C), the mole fraction (x_i), the temperature of the system (T_react) and the interaction energy between every pair of groups (a_ij). In the case of DMSO, two groups –CH₃ and one group –SO₂, and for water, one group –H₂O are considered. The parameters R_k, Q_k, C, and a_ij were entered as matrices based on data from the literature [35]. Whereas x_i and T_react were introduced according to the temperature and concentration of DMSO in individual nodes of the computational domain.

Figure 2 presents the concentration of DMSO in the extracellular matrix of the sample as a function of time: (a) over the entire time simulation and (b) a zoomed-in view of the first 10 minutes of the initial step of the protocol. The graph shows the results at the point r = 0.050 mm, z = 0.475 mm of the computational domain. The plot contains three curves that correspond to three variants of the simulation. The first (blue dashed line) and second (red dash-dotted line) variants correspond to the sample modelled as a homogeneous material, where the effective diffusion coefficient is calculated using the Einstein-Stokes equation (Eq. (2)) for variant 1 and the Arrhenius expression (Eq. (3)) for variant 2. In the third variant (magenta solid line), the material is modelled as a porous medium (cf. Eqs. (4–9)). The plot also includes the concentration of the bath solution (black dotted line) and the experimental results (green points). As can be observed, the simulation results deviate from the experimental values, especially in the final steps of the process. The results for variant 2 are close to the experimental data, whereas those from variant 1 coincide with the concentration of DMSO in the bath solution. The values of the DMSO concentrations for variant 3 are lower than the experimental values.

Fig. 2.

Concentration of DMSO as a function of time: a over the entire simulation time and b during the first 10 minutes of the initial step of the protocol

Table 2 shows the effective diffusion coefficient values at the selected point in the computational domain (r = 0.050 mm, z = 0.475 mm) at the end of each step of the protocol. Figure 3 illustrates the change in the effective diffusion coefficient during the first 40 seconds of the third step of the protocol. The value of the effective diffusion coefficient decreases with decreasing temperature and, in the case of variant 3, with increasing concentration of DMSO. The highest values of the effective diffusion coefficient are observed for variant 1, whilst the lowest values are obtained for variant 3. It is worth remembering that in variants 1 and 2 the effective diffusion coefficient depends only on the temperature; therefore, once the temperature stabilises within the sample, the coefficient remains constant throughout the entire domain. In contrast, for variant 3, the effective diffusion coefficient depends on both temperature and concentration of DMSO; therefore, its value varies across the computational domain.

Table 2.

Effective diffusion coefficient in the selected point of the computational domain at the end of each step of protocol

Step	Effective diffusion coefficient, × 10−10 Deff [m2∙s-1]
	variant 1	variant 2	variant 3
1.	4.260	3.871	1.887
2.	4.260	3.871	1.831
3.	3.871	1.982	0.856
4.	3.820	1.799	0.780
5.	3.712	1.449	0.601
6.	3.611	1.170	0.450
7.	3.438	0.788	0.222
8.	3.243	0.480	0.067

Fig. 3.

Effective diffusion coefficient as a function of time during the first 40 seconds of the third step of the protocol

Validation

The experimental data used for validation were taken from Pegg et al. [17], who investigated the cryopreservation of bovine articular cartilage using the liquidus-tracking method. In this study, the DMSO concentration in the extracellular matrix was measured at various stages of the cooling protocol. These values, reported in [17], were extracted and used as reference data for the present validation. This dataset was selected due to the detailed description of the experimental protocol and the availability of concentration values at multiple time points, which enables a reliable comparison with the computational results. Table 3 presents the collation between the experimental results [17] and the simulated concentrations of DMSO in the extracellular matrix at the point r = 0.050 mm, z = 0.475 mm, and the calculated relative error for the data obtained at the end of each step of the established protocol. As can be seen, the smallest relative errors are obtained for variant 2, with values ranging from 0.55% to 5.20 %. In the case of variant 1, good agreement between the numerical and experimental results is observed during the initial stages of the process. However, at the end of the cooling process, the error increases significantly (up to 15.82%). The results for variant 3 have higher errors compared to the other models, ranging from 10.06% to 24.96%.

Table 3.

Comparison of the concentration of DMSO for simulation and experimental results

Step	Experimental data [17], cd [%(w/w)]	Simulation data (variant 1), cd [%(w/w)]	Relative error, δ [%]	Simulation data (variant 2), cd [%(w/w)]	Relative error, δ [%]	Simulation data (variant 3), cd [%(w/w)]	Relative error, δ [%]
1.	–	7.838	–	7.568	–	4.896	–
2.	16.3 ± 1.3	16.720	2.576	16.389	0.545	12.232	24.957
3.	24.5 ± 1.1	26.077	6.438	25.613	4.544	22.037	10.055
4.	34.2 ± 0.9	34.178	0.065	33.605	1.740	30.053	12.125
5.	41.7 ± 3.3	42.274	1.376	41.246	1.090	36.754	11.861
6.	47.8 ± 2.8	50.369	5.375	48.679	1.839	42.682	10.707
7.	52.2 ± 1.3	56.668	8.560	53.973	3.396	45.839	12.186
8.	55.9 ± 2.9	64.744	15.821	58.808	5.202	46.938	16.033

Discussion

This study goes beyond reproducing standard mass transfer equations by providing a structured comparison of different modelling assumptions for biological tissues. Examining both homogeneous and porous models, the analysis demonstrates how tissue microstructure and parameter coupling influence the accuracy of CPA transport predictions. This comparison enables a deeper understanding of the limitations of simplified approaches and highlights the necessity of considering structural parameters in predictive cryopreservation modelling.

The simulation results show that, counterintuitively, the porous model produced the largest relative errors compared to the experimental data, despite being theoretically the most realistic representation of articular cartilage. Several factors may explain this discrepancy. First, the porous model introduces additional coupling between concentration and temperature, which increases model sensitivity to uncertainties in thermophysical parameters such as viscosity and activity coefficients. Second, the structural parameters of the tissue (porosity, tortuosity, pore size distribution) are highly variable in real biological samples and difficult to determine accurately. Even small deviations in these parameters can significantly alter effective diffusion values. Third, the experimental data themselves may include variability due to the biological heterogeneity of the cartilage, which is not fully captured by the numerical representation. Taken together, these factors may account for the larger deviations observed in the porous case. Fourthly, the input parameters are taken from various sources in the literature and their values may differ slightly from those that characterized the sample used for the experiment.

Although the porous model differs most from the experimental data, it incorporates physical effects that are not present in homogeneous models. The higher error may result from the strong dependence of the porous structure on uncertain parameters, such as porosity, tortuosity, and pore size. In reality, these parameters are not constant and can vary across the tissue, thereby affecting local mass transport. Furthermore, these parameters are usually measured under static or near-equilibrium conditions, whereas cryopreservation involves strong, non-linear changes in temperature and concentration. Therefore, the discrepancies between the model and the experiment do not indicate that the porous medium approach is incorrect, but rather highlight the need for more accurate identification and validation of the parameters.

Future work will focus on performing sensitivity and uncertainty analyses to quantify the influence of variations in structural and thermophysical parameters on model predictions. Combining such analyses with fuzzy or stochastic approaches will provide a more robust framework for using porous models in biological cryopreservation. In this context, our earlier studies [27, 29] applied fuzzy and interval number theory to evaluate uncertainties in the mathematical model.

Conclusions

The article presents a simulation of the cooling process during cryopreservation of a cylindrical sample composed of articular cartilage. The novelty of this work lies in the comparative analysis of homogeneous and porous models of articular cartilage during cryopreservation, supported by validation with experimental data. Three different approaches were proposed to calculate the effective diffusion coefficient, where variants 1 and 2 assumed that the sample is a homogeneous material, whilst variant 3 treats the articular cartilage as a porous medium. The study analysed the effect of the effective diffusion coefficient on the distribution of DMSO concentration.

It is noteworthy that the effective diffusion coefficient is weakly coupled to the temperature in all cases, and in variant 3 it is additionally strongly coupled to the concentration of DMSO. A decrease in temperature and an increase in the concentration of DMSO cause a reduction in the effective diffusion coefficient. Moreover, in the model considering the porosity of the sample, the lowest values of the effective diffusion coefficient were obtained.

Differences in the effective diffusion coefficient directly affect the resulting concentration of DMSO. Using variant 1, the DMSO concentration at the end of each step closely matches that of the bath solution, and the relative errors range from 0.07% to 15.83%. For variant 2, the concentration of DMSO is closest to the experimental data amongst the other models, and the relative errors range from 0.55% to 5.20 %. For variant 3, the concentration of DMSO is lower than the experimental values and is associated with high relative error values (10.06% – 24.96%).

This demonstrates the limitations of simplified homogeneous models and highlights the necessity of including porosity effects in predictive simulations. Future work will focus on further development of the mathematical model, taking into account the porosity of the biological sample.

In summary, the study provides new insights into the transport mechanisms within articular cartilage during cryopreservation. The originality of this study lies in its comprehensive comparison of homogeneous and porous cartilage models during cryopreservation, supported by experimental validation. The coupling between diffusion, temperature, and concentration offers a new view of mass transport in biological tissues. These results provide a basis for improving porous models and for using fuzzy or stochastic methods to account for parameter uncertainties in biological systems.

Author Contributions

Conceptualization: Anna Skorupa and Alicja Piasecka-Belkhayat; Data curation: Anna Skorupa and Alicja Piasecka-Belkhayat; Formal analysis: Anna Skorupa and Alicja Piasecka-Belkhayat; Funding acquisition: Anna Skorupa; Investigation: Anna Skorupa; Methodology: Anna Skorupa and Alicja Piasecka-Belkhayat; Project administration: Anna Skorupa; Resources: Anna Skorupa; Software: Anna Skorupa; Supervision: Alicja Piasecka-Belkhayat; Validation: Anna Skorupa and Alicja Piasecka-Belkhayat; Visualization: Anna Skorupa; Writing – original draft: Anna Skorupa and Alicja Piasecka-Belkhayat; Writing – review and editing: Anna Skorupa and Alicja Piasecka-Belkhayat. All authors have read and agreed to the published version of the manuscript.

Data Availability

Data available on request.

Declarations

Conflict of interest

The authors declare that they have no competing interests.

Ethical Approval and Consent to Participate

Not applicable.

Consent for Publication

Not applicable.

Artificial Intelligence (AI)

The authors declare that they have not used AI-generated work in this manuscript.