Thermal Analysis of Marginal Conditions to Facilitate Cryopreservation by Vitrification Using a Semi-Empirical Approach

Abstract

This study aims at the thermal analysis of marginal conditions leading to cryopreservation by vitrification, which appears to be the only alternative for indefinite preservation of large-size tissues and organs. The term “marginal conditions” here refers to cooling rates in close range with the so-called critical cooling rate, above which crystallization is avoided. The analysis of thermal effects associated with partial crystallization during vitrification is associated with the coupled phenomena of heat transfer and kinetics of crystallization. This study takes a practical, semi-empirical approach, where heat transfer is analyzed based on its underlying theoretical principles, while the thermal effects associated with partial crystallization are taken into account by means of empirical correlations. This study presents a computation framework to solve the coupled problem, while presenting a proof-of-concept for DP6 as a representative cryoprotective agent. The thermal effects associated with crystallization at various relevant cooling rates are measured in this study by means of differential scanning calorimetry. Results of this study demonstrate that, due to the thermal effects associated with partial crystallization, the cooling rate at the center of a large organ may lag behind the cooling rate in its surroundings under some scenarios, but may also exceed the surroundings cooling rate in other scenarios, leading to counter-intuitive effects associated with partial crystallization.

Introduction

Despite advances in medicine and biotechnology, and increased awareness about the benefits of organ donation and transplantation, there continues to be a gap between supply and demand of organs [24]. According to the Global Observatory on Donation and Transplantation, only 10% of the worldwide demand for organ transplantation is being met [13,21]. Organ cryopreservation is a promising technique which can successfully increase the availability of organs for transplant medicine by improving both logistics and outcomes of transplantation [13,22].

A typical cryopreservation protocol includes loading the specimen with a cryoprotective agent (CPA) solution, cooling to cryogenic temperatures, storage for an indefinite period of time, rewarming, and washing out the CPA solution, all while minimizing injury to cells and damage to the tissue structure. Vitrification is a cryopreservation protocol in which the formation of ice is avoided, and the organ is trapped in a glassy state [5,8,36], while classical methods of cryopreservation involve some degree of crystallization (vitreous in Latin means glassy). Cryopreservation of large vital organs such as kidney, heart and liver appear to require cryopreservation by vitrification [9,11] unlike smaller and less complex specimens, which can benefit from classical methods [3,5,8,36].

Formation of ice during cryopreservation is a path-dependent phenomenon, which is primarily dependent on the thermal history and availability of nucleators in the specimen. Cryoprotective agents are typically added to the cryopreserved biomaterial to suppress ice formation and growth, practically acting as glass promoting agents. Vitrification relies on the exponential increase in viscosity with the decreasing CPA temperature, where the crystallization potential is inversely dependent upon the viscosity of the material. When cooled fast enough with correlation to the viscosity, such that the cooling time is shorter than the typical time for crystallization, the specimen is eventually trapped in a glassy state. The temperature below which the viscosity is so high that the material can be considered solid on any practical time scale is known as the glass transition temperature (T_g).

While rapid cooling and rewarming rates promote vitrification, the resulting non-uniform cooling [4,16,17,18,33] and rewarming across the specimen can give rise to thermomechanical stresses, which can be harmful to the organ structure. Increasing the CPA solution concentration suppresses the critical cooling and rewarming rates, potentially decreasing thermomechanical stress effects. Unfortunately, CPAs are toxic [10], and the cryopreservation protocol must be designed such that the toxicity potential is minimized. Hence, reducing toxicity, promoting vitrification, and minimizing thermomechanical stress represent competing needs when selecting CPA solution concentration and when designing specific cryopreservation protocols [12].

A pervious study [4] presented an iterative, finite differences numerical method to predict the temperature and crystallization amount in selected one-dimensional cases—an infinite slab and a cylinder. There, predictions were based on enthalpy data measured by means of differential scanning calorimetry (DSC) and empirical equations to predict rate of ice crystallization. The current study focuses on thermal analysis of the conditions leading to partial vitrification in large specimens. Instead of solving the thermal field in the specimen in conjunction with the kinetics of crystallization, this study takes a practical, first-order approach, where the bulk properties of the material are considered. This approach is known as the enthalpy approach [25], which is now expanded to include the marginal conditions leading to phase change. To the best of our knowledge, the expanded enthalpy approach is presented here for the first time, along with a computational framework to solve the problem of partial vitrification in the context of cryopreservation. This computational framework is based on the finite elements analysis (FEA) commercial code ANSYS, while integrating recently developed physical properties of relevant CPA solutions [6,14,28,37].

Mathematical Formulation

Heat Transfer Model

The geometry analyzed in this study is of a cylindrical container, Fig. 1, consisting of two subdomains: CPA solution and container walls. No tissue sample is included in the current first-order analysis. In an actual process, the specimen would be permeated with the same concentration of CPA solution, substituting all biological solutions and occupying about three quarters of the specimen by volume. With this solution substitution in mind, the fact that the remaining portion of the organ structure does not undergo phase change, and the observation that the latent heat of CPA solution is an order of magnitude smaller than that of water—the key ingredient in biological solutions, the analysis of marginal freezing conditions presented here represents a first order approximation [6].

Figure 1:

Schematic illustration of (a) the CPA container and (b) cross section of the domain including boundary conditions.

Heat transfer in the vial and the CPA domain undergoing vitrification is assumed to be solely by conduction:

$$C\frac{dT}{dt}=\mathit{\nabla }\cdot (k\mathit{\nabla }T)$$ (1)

where C is the volumetric specific heat, T is temperature, t is time, and k is thermal conductivity. Continuity in temperature and heat flux is assumed on all interfaces. A previous study has shown significant natural convection in the CPA solution domain at high cryogenic temperatures (above −35°C) [12], where no crystallization is expected and in a range that is out of the focus of the current study, as discussed below. Hence, the intrinsic thermal conductivity of the CPA solution is used for the current analysis. The cryopreservation process is assumed to take place in the cooling chamber of a controlled-rate cooler [12]. A combined convection and radiation heat transfer coefficient is assumed between the outer surfaces of the container and the cooling chamber (Fig. 1(b)):

$$-k_j\frac{d{T}_j}{d\widehat{n}}=U\left(T_j-T_c\right)$$ (2)

where $\widehat{n}$ is a normal to the vial outer surface, U is the overall heat transfer coefficient, and the indexes j and c refer to the container and cooling chamber, respectively. A previous experimental study under similar conditions suggests U = 350 W/m²-°C for a typical cooling chamber, container size, and temperature range of interest in this study [12].

The thermal history consists of a uniform initial temperature, followed by constant cooling rate of the chamber temperature down to a storage temperature, and a constant temperature hold thereafter.

Physical Properties

Figure 2 displays selected apparent specific heat measurements for DP6 during cooling using DSC, as described below. The sudden increase in apparent specific heat in mid-range temperatures corresponds to crystallization, where the area under the curve during the corresponding temperature range equals to the enthalpy change during crystallization. The temperature range of crystallization widens while the peak effect of crystallization decreases with the increasing cooling rate, until such high cooling rate that crystallization effects are not observed anymore. This cooling rate threshold is known as the critical cooling rate. The critical cooling rate for DP6 based on this study is 5°C/min.

Figure 2:

Representative specific heat curves measured with DSC for the CPA cocktail DP6 at selected cooling rates.

The enthalpy approach [25,28,30] is used in this study in order to account for latent heat effects, where an effective specific heat represents the combined effects the intrinsic specific heat and the temperature-dependent effect of latent heat [20]:

$$h_{12}={\int }_{T_2}^{T_1}{C}_{eff}dT={\int }_{T_2}^{T_1}{C}_{p}dT+L_{12}$$ (3)

where, h₁₂ is the enthalpy change between the onset of crystallization at T₁ and termination of crystallization at T₂, C_eff is the effective specific heat, C_p is intrinsic specific heat, and L₁₂ is the latent heat released between the corresponding temperatures. Using the enthalpy approach to solve the heat transfer problem, the intrinsic specific heat in Eq. (1) is substituted with the effective specific heat from Eq. (3). Figure 3 illustrates the relationship between the various terms in Eq. (3), where the dashed line represents a simplified presentation of the effective specific heat used in the current study, which corresponds to an identical enthalpy changes measured with the DSC device. Note that the enthalpy approach is energy conserving by definition, which simplifies the numerical solution of Eq. (1) [25].

Figure 3:

Generic illustration of an effective specific heat, combining the intrinsic specific heat and the latent heat, where a family of similar curves were used in the current study to correlate phase transition with cooling rate in heat transfer simulations.

An ordinary solution using the enthalpy approach assumes cooling-rate independent effective specific heat, which resembles a quasi-steady phase change process. In contrast, it is not just that the effective specific heat is cooling-rate dependent in the current study, but also the cooling rate may vary during phase transition, as dictated by the underlying principles of heat conduction (Eq. (1)). This necessitates tracing the amount of local crystallization, where the local crystallization ratio is defined as:

$$R(t,x,y,z)=\frac{{\int }_T^{T_1}{C}_{eff}\left(\frac{\partial T}{\partial t}(t,x,y,z),T(t,x,y,z)\right)dT}{h_{12}}$$ (4)

Consistently, the thermal conductivity within the phase transition temperature range is calculated as a mixture of amorphous and crystallized materials [7]:

$$k=0.33\times (1-R)+R(1.20\times {10}^{-9}{T}^4+4.10\times {10}^{-7}{T}^3+3.11\times {10}^{-5}{T}^2-2.9\times {10}^{-3}T+4.29\times {10}^{-1})$$ (5)

where, the coefficients for the polynomial temperature approximation have been published recently [7].

Enthalpy Measurements and Modeling

DSC studies were performed with device model Q2000 of TA Instruments, Inc. (New Castle, DE). Crystallization effects were investigated below 10 °C/min. Each experimental run began with a temperature hold at 20 °C for 10 minutes, and then cooling at a constant rate to −120°C, at rates varying between experiments. Note that the glass transition temperature of DP6 is −115°C [37]. Each experiment was repeated three times for the same experimental conditions.

Experimental results indicate a critical cooling rate of 5°C/min and an average crystallization-onset temperature in the range of −50.7°C to −59.5°C over cooling rates ranging from 1°C/min to 5°C/min (average of repeated runs under the same conditions), with higher average temperature at lower cooling rates, Table 1. Figure 4 displays enthalpy changes of phase transition as a function of the cooling rate (L₁₂ in Eq. (3)). For comparison, the enthalpy change of pure water is 334 J/g—which is the key ingredient of the DP6 cocktail. It follows that the latent heat of DP6 crystallization under relevant conditions to cryopreservation is less than 15% of the latent heat of pure water. Experiments were also conducted to identify the melting point of DP6 in a quasi-steady process, where the peak melting temperature was found to be −28.3°C.

Table 1.

Measured cooling-rate dependent boundaries of the phase transition temperature range for DP6 using DSC (Fig. 2), where listed data represents the average of three runs, T₁ and T₂ are the upper and lower boundaries, respectively; this data set has been used as an empirical correlation for the simulation of partial crystallization, with a simplified version of it illustrated in Fig 3.

Cooling Rate (°C/min)	T1(°C)	T2(°C)
1	−50.7	−63.3
2	−52.1	−67
3	−55.1	−76
3.5	−56.4	−78.2
4	−59.5	−80.8
4.5	−58.0	Not Determined
8	Not Applicable	Not Applicable

Figure 4:

Latent heat, L, as a function of the cooling rate, H, for DP6 based on a differential scanning calorimetry study.

Table 2 lists the simplified physical properties used in the current study to describe the phase-change effect. Given the relatively low latent heat values, the narrow range of onset of crystallization when compared with the full range of cryogenic temperatures of interest, and the fact that the enthalpy approach employed here is energy conserving, the first order analysis in this study assumes the phase transition temperature range as well as the latent heat of fusion to be cooling rate independent. It is emphasized that the freezing process is analyzed as cooling-rate dependent, but the envelop conditions of freezing are approximated as temperature-independent. Nonetheless, as discussed below, the numerical framework could be expanded to account for cooling-rate variation in boundary conditions at the cost of computation runtime, which already comes at a high price.

Table 2:

Material properties used for thermal analysis in the current study for DP6 and acrylonitrile butadiene styrene (ABS) as the container material.

	Property	Value
DP6	Phase transition temperature range	T1=−50°C; T2=−75°C
	Glass transition temperature	−115°C
	Total latent heat	35000 J/kg
	Intrinsic specific heat	3000 J/kg-°C
ABS	Thermal conductivity	0.17 W/m-°C
	Thermal diffusivity	1.11×10−7 m2/s

Computational Framework

The solution to the heat transfer problem was performed using the FEA software ANSYS 19.1. To account for the cooling rate-dependent phase transition process, an ANSYS parametric design language (APDL) script was composed using a multiframe restart technique, following the flowcharts illustrated in Figs. 5–7. Time intervals for advancing through this framework were selected based on a convergence study described below. Figures 6 and 7 display flowcharts for the preprocessing and postprocessing stages, respectively.

Figure 5:

Computational framework for heat transfer simulations of CPA solution cooling which takes into account partial crystallization.

Figure 7:

Postprocessing detail for the computation framework displayed in Fig. 5, where h(t) is the accumulated enthalpy changes in the element due to phase change (it equals h₁₂ in Eq. 3 once crystallization is completed).

Figure 6:

Preprocessing detail for the computation framework displayed in Fig. 5

The selected ANSYS solver uses a nonlinear transient thermal solution based on the full Newton-Raphson method within each timestep. In order to increase the precision in cooling rate calculations (Fig. 6), it was calculated using a quadratic linear regression method based on the six most recent temperature data points in each element. Data analysis was performed with the commercial software Matlab.

The computation framework presented here is based on three simplifying assumptions: (i) the heterogeneous nucleation starts at T₁, (ii) the probability of crystallization in an element is independent of the amount of crystallization in adjacent elements, and (iii) the mass transport effect on the heat transfer process is negligible. These assumptions are associated with the high concentration of the CPA solution, where the viscosity is relatively high and the magnitude of heat diffusivity is orders of magnitude higher than that of the mass diffusivity. In practice, these assumptions lead to decoupling the analyses of heat transfer and mass transport, an approach which is consistent with the first order analysis presented in this study.

Results and Discussion

Unless otherwise specified for selected cases, the following properties and simulation parameters have been used: container diameter of 4.2 cm, height of 7 cm, and wall thickness of 1 mm; initial temperature of −25°C; storage temperature of −120°C; DP6 properties listed in Table 2 and presented in Fig. 4; and container (ABS) properties listed in Table 2. These geometrical parameters where selected since the resulting partial crystallization distribution is the most sensitive to the cooling boundary conditions, as displayed below. Doubling the size of the above container makes it consistent with the human kidney size [6], as discussed towards the end of the Results and Discussion. The ABS material is selected as a container material to reduce thermomechanical stress [29] (not simulated here), and the temperature range is relevant to DP6 and DP6 cocktails containing synthetic ice modulators [8]. The results discussed in this paper are relevant to commonly used materials for containers used to preserve large organs.

Convergence Study

Due to the potentially rapid changes in thermal properties during phase change, oscillatory instabilities may develop in the resulted cooling rate [35], which would affect the stability and convergence of the thermal solution. Convergence analysis considered variations in temperature and, independently, the amount of crystallization at the end of phase transition, as discussed below. The analysis was performed on (a) the number of elements, n, (b) the simulation time step size, Δt_s, and (c) the time-interval for multiframe cycles, Δt_c. The converged mesh had 56213 nodes and 15456 elements, a maximum simulation time-step of Δt_s=0.25 s, and a maximum multiframe time interval of Δt_c=0.5 s.

Figure 8 displays a CPA crystallization-volume histogram chart, which was used to determine convergence conditions, for four representative cooling rates at the surroundings of the container of 4.5°C/min, 7.5°C/min, 15°C/min, and 30°C/min. All the simulations began with a uniform temperature of −25°C and progressed to a storage temperature of −120°C. Low cooling rates resulted in a high portion of crystallization, while high cooling rates such as 15°C/min and 30°C/min resulted in a low portion of crystallization in some subregions of the domain, but crystallization always developed to some degree from the reasons discussed below. Of all the simulations studied, 7.5°C/min surroundings cooling rate resulted in the widest spread of partial crystallization, a case which was used for convergence analysis. Note that, for reasons explained below, partial crystallization within the domain may take place even if the surroundings cooling rate exceeds the critical cooling rate.

Figure 8:

Volume-crystallization histogram for selected cases, for varying surroundings cooling rate, where the critical cooling rate is 5°C/min. Note that the internal cooling rate distribution within the CPA cocktail domain deviates from the external rates listed due to the underlying principles of heat transfer. Further note that 7.5°C/min was taken as the boundary condition for a representative case for the convergence studies due to its wide distribution of crystallization within the domain.

Figure 9 displays the amount of crystallization and the temperature field at the end of phase transition temperature range, for a cooling rate of 7.5°C/min in a multiframe time-step convergence study. Here, all four simulations are subject to the same cooling protocol, the same simulation time-step of Δt_s=0.25 s, and the same mesh size as determined by the other convergence criteria. The multiframe restart script (Fig. 5) updates the local cooling rate and thereby the amount of crystallization in each element. Comparing the temperature fields with the corresponding crystallization fields in Fig. 9 illustrates how much more sensitive is the crystallization rate as a convergence criterion when compared with the temperature variations between successive runs. This could be explained by the relatively low latent heat of DP6 as a representative CPA cocktail, and also by the fact that the latent heat in general has a moderating effect on the localized temperature distribution. As discussed above, the numerical scheme is energy conserving, which means that the total amount of crystallization is essentially the same, regardless of the choice of Δt_c, but the distribution of the amount of crystallization may vary with the value of Δt_c.

Figure 9:

Representative results from a convergence study for multiframe time step, Δt_c, subject to cooling rate of 7.5C°/min, when the domain reaches the lower boundary of phase transition of −75°C: (a)-(d) display crystallization portion distributions, while (e)-(h) display the corresponding temperature fields.

A counter-intuitive result is observed in Figs. 9(a)–9(d), where the area with the highest percent of crystallization is not found at the center of the domain but at some intermediate radius in it. Here, it is commonly assumed by the cryobiology community that the cooling rate will decrease with the distance from the outer wall due to the transient nature of the process and the underlying principles of heat transfer. Hence, it is commonly assumed that the highest portion of crystallization would be found at the center of the domain, where the cooling rate is expected to be lower. While the above description sounds plausible for a process prevailing in a semi-infinite domain, the cross-section illustrated in Fig. 9 is of a finite domain—corresponding to an inward cooling process in a radial system subject to a constant cooling rate convective boundary condition. Such a system can be approximated as a semi-infinite domain only at the early stage of the cooling. However, as the thermal information propagates to the center of the domain, the corresponding thermal inertia potential diminishes, which results in cooling-rate acceleration there. The end result of this process is a maximum crystallization rate at some intermediate radius in the domain, as displayed in Fig 9. In general, this effect at various intensities can be expected in cylindrical and spherical systems undergoing cooling-rate dependent phase transition process.

The observations that the cooling rate may increase at the center of the domain and, hence, that the lowest cooling rate may be found at some intermediate distance between its outer surface and the center of the domain, are consistent with an earlier investigation into the thermal history in an infinite slab [33]. There, the analysis was based on a freezing front tracking method, the outcome was presented graphically, and the authors termed the above observations as the “centerline effect” [33]. A more recent study presented similar observations when analyzing partial crystallization in an infinite plate and an infinite cylinder configurations [4]. In terms of solution capabilities, the unique contribution of the current study is in presenting a practical and robust computation framework, which can be combined with any FAE commercial code, with the notable advantages of obtaining solutions for complex geometries and at a reasonable computation cost. Furthermore, the similarity in results between those earlier studies and the current study supports the simplifying assumptions that underly the computation framework, which essentially resulted in decoupling the problems of heat transfer and mass transport.

A crystallization-volume histogram is displayed in Fig. 10, where a maximum is observed at about 30% of the elements, when they reach about 35% of crystallization. At higher rates of crystallization, the percentage of elements experiencing higher percent of crystallization decreases, in a trend that becomes monotonic with the convergence of the solution (with the decrease of Δt_c). Conversely, a monotonic trend of increased percentage of elements displaying an increased percent of crystallization is observed below the same maximum.

Figure 10:

Results of a convergence study for Δt_c.

Cooling Rate Effect

Figure 11 displays the crystallization portion and the temperature field at the end of the phase transition temperature range (−75°C) for four representative cooling rates of 4.5°C/min, 7.5°C/min, 15°C/min, and 30°C/min, where only the first case is below the critical cooling rate of 5°C/min. Two unexpected observations can be made from Fig. 11: (i) crystallization may occur within the domain even when the surroundings cooling rate exceeds the critical cooling rate, and (ii) partial vitrification may occur at the center of the domain even when the surroundings cooling rate is lower than the critical cooling rate.

Figure 11:

Representative results for cooling rate study, where (a)-(d) display the crystallization portion distributions while (e)-(h) display the corresponding temperature fields when the highest temperature falls below the lower boundary for phase transition T₂; (a and e) 17.7min, (b and f) 19.3 min, (c and g) 21.8 min, (d and h) 27.3 min.

The observation that crystallization may occur at supra-critical cooling rates in the specimen’s surroundings has already been discussed in the context of the convergence study, as discussed above. For example, only 3% of the entire domain completely vitrifies when the external cooling rate is 7.5°C/min, Fig. 1(c), while 1% of the entire domain is fully crystallized. This means that 96% of the domain is partially vitrified to a varying degree. The implication to cryobiology applications is very significant, although some biological systems may sustain viability and functionality after vitrification at marginal conditions, as has been demonstrated in the case of VS55 and blood vessels [1]. However, the prediction of viability and functionality under marginal conditions must take into account more detailed analysis of the distribution of partial crystallization. The trend of crystallization described above is very difficult to trace from the temperature field displayed in the lower portion of Figs. 1(e)–1(h), which signifies the importance of the proposed detailed analysis.

The observation that only partial crystallization may be achieved at the center of the domain although the cooling rate at the surface is subcritical can be explained as follows. For the purpose of the current discussion, the solidus front is defined as the curved surface coinciding with the lower temperature boundary for phase change, T₂. The solidus front will propagate inwards with the decreasing surface temperature below T₂, where its velocity of propagation tends to increase with the increasing rate of cooling at the outer surface. At the same time, the same velocity tends to decrease with the increasing rate of latent heat release ahead of the solidus front, the magnitude of which is proportional to the velocity of the solidus front.

For the purpose of discussion, further consider the special case of a constant solidus front velocity in an infinite domain, characterized by temperature-independent thermophysical properties and uniform initial temperature. Under these conditions, the rate of latent heat release ahead of the solidus front must also be constant [31], which can be classified as a special case of the inverse Stefan problem [2,27,26]. In such a case, when the specific heat of the solid is much lower than the apparent specific heat during phase transition (a problem characterized by a low Stefan number [32]), the cooling rate at the surface and the velocity of the freezing front are linearly dependent [32,27].

By contrast, the same moderating effect of latent heat release ahead of the solidus front will decay as the solidus front gets closer to the volumetric center of the domain, and as the remaining portion of unfrozen material that can crystallize and, therefore, release heat declines. It is this decline that would give rise to the observed accelerated cooling rates at the center of the domain. In the example at hand, if the cooling rate at the outer surface is subcritical but not by a big amount, the effect described above may accelerate the cooling rate at the center in a sufficient amount to become a supra-acritical rate and hence arrest additional crystallization. Essentially, this is the reason for incomplete crystallization (i.e., partial vitrification) at the center of the domain displayed in Fig. 11(d), despite the subcritical cooling rate at the outer surface at all times.

Specifically for the case presented in Fig. 11(d), 73% of the elements are found to be 90–100% crystallized, but the core of the domain shows only 10–20% crystallization. Of course, the distribution of the portion of crystallization is not only dependent on the thermal history in the specimen’s surroundings, but also on the geometry of the container. One can even envision a scenario of complete vitrification at the center of the domain, and complete crystallization at the outer wall under special conditions. An intriguing aspect of that case from a cryobiology standpoint is that, if the specimen is small compared with the container and placed at its center, one may wrongly assume that the specimen has lost its viability based solely on visual observation.

Initial Temperature Effect

The initial temperature may also affect the cooling rate at different locations [33], which in turn affects in the rate of crystallization, as can observed from Fig. 12. Here, the two cases compared are identical with the exception of the initial temperature of 0°C and −25°C, where the cooling rate was kept the same at 7.5°C/min. Counter-intuitively, a higher initial temperature resulted in a larger portion of complete vitrification [6]. Specifically, elevating the initial temperature from −25°C to 0°C increased by 46% the volume of CPA solution that experienced less than 10% crystallization. Inversely, elevating the initial temperature from −25°C to 0°C also increased the by 10% the volume experiencing more than 90% crystallization. In other words, more distinct areas of vitrification and crystallization are developed in the medium when the initial temperature was elevated.

Figure 12:

Representative results for initial temperature study, where (a) and (b) display the crystallization portion distributions while (c)-(d) display the corresponding temperature fields when the highest temperature falls below the lower boundary for phase transition T₂; (a and c) 21.8 min, (b and d) 28.1 min.

This effect can be explained by taking the problem to hypothetically extreme conditions, which may not be relevant to cryobiology at first glance. Here, assume uniform thermophysical properties across the domain, a very high initial temperature, and a constant surrounding cooling rate thereafter. After a while, a quasi-steady temperature distribution will develop across the container that is the temperature difference between two points in time at every location in the container is the same. In other words, at a quasi-steady condition, the entire domain cools at the same rate, which equals to the surroundings cooling rate. Further assume that the initial temperature is so high that a quasi-steady condition is achieved before the point with coldest temperature in the domain passes into the phase transition temperature range. Now, if the surroundings cooling rate is already higher than the critical cooling rate, it would imply that the entire domain would vitrify in this hypothetical case.

Of course, in a real case the thermophysical properties are not exactly uniform and a very high initial temperature may not be a viable option for a biological material. The trend, however, would remain the same, meaning that a larger portion of the domain would vitrify with the increasing initial temperature, as long as the surrounding cooling rate remains constant. This explanation is consistent with the results presented in Fig. 12. This trend would not be affected by the three simplifying assumptions discussed in the computational framework section.

Storage Temperature Effect

Consider the same hypothetical case from above and a very low storage temperature (i.e., the minimum temperature that the surface of the container attains). As long as the difference between the storage temperature and the lower temperature boundary of phase transition is higher than the maximum difference within the quasi-steady temperature distribution, and as long as the surrounding constant cooling rate remains constant, complete vitrification can be expected from the same argument discussed above. Now, consider a new hypothetical storage temperature which equals to the lower boundary of phase transition. In the latter case the cooling rate at the center of the domain will decay once the outer surface reaches the storage temperature. Eventually in this case, the cooling rate is doomed to fall below the critical cooling rate and partial crystallization would follow.

In the case of DP6 for example, the lower boundary for significant crystallization under practical cooling conditions is about −80°C (Table 1), while the glass transition is −115°C [37]. For reasons not discussed in this study and which are associated with thermomechanical stress [19,20], it is often preferred to store the biological material around the glass transition temperature. This creates a size limit on the container size, where the maximum temperature difference in the container increases with the increasing container size, while the cooling rate decreases respectively.

As an example, a parametric study was performed on DP6 contained in a container size consistent with human kidney preservation [6], initial temperature of −25°C, and surrounding cooling rate of 7.5°C/min. Results of this study indicated that a storage temperature of −120°C would result in complete crystallization across the container, while a storage temperature of −180°C would result in partial vitrification at the center of the container. More broadly, results of this study suggest that the cooling rate distribution within the domain is a strong function of the storage temperature, which is consistent with previous studies [18] Unfortunately, this temperature may be impractical to prevent structural damage due to residual stresses [34]. While alternative thermal protocols could be explored and alternative CPA solutions can be investigated, such as DP6 mixed with synthetic ice modulators [8], the trend of decreasing storage temperature to compensate for an increasing specimen size in vitrification would remain the same.

Summary and Conclusions

A computation framework is proposed in the current study to analyze the conditions leading to partial vitrification in large specimens. Instead of solving the thermal field in the specimen in conjunction with the kinetics of crystallization, this study takes a practical first-order approach, where the bulk properties of the material are considered. The thermal analysis in this study is based on the enthalpy approach, which is now expanded to include the marginal conditions leading to phase change. DP6 is used in this study as a CPA model, using previously measured properties, combined with newly developed DSC data. A cylindrical polymeric container is used as a geometrical model for the analysis, when cooled at a constant rate within the chamber of a commercial controlled-rate cooler. This computation framework in this study is demonstrated on the FEA commercial software ANSYS.

Counter-intuitively, parametric studies indicate that (i) crystallization may occur within the domain even when the surroundings cooling rate exceeds the critical cooling rate; (ii) partial vitrification may occur at the center of the domain even when the surroundings cooling rate is lower than the critical cooling rate; and (iii) the extent of vitrification obtained in the specimen is dependent on the initial temperature as well as the storage temperature of the specimen, although the practical phase transition region is at neither of the extremes.

While the first-order analysis is presented here as a proof of concept, where the kinetics of crystallization is coupled with heat transfer analysis not by modeling but through an empirical correlation, this approach can be advanced to include a more realistic functional behavior of the effective specific heat and cooling-rate dependent boundaries of crystallization. Nonetheless, given the energy-conserving nature of the mathematical formulation, a higher-order formulation is expected to result in similar trends of crystallization, which is work in progress.