M.V. Santos

Numerical simulation of cooling rates in vitrification systems used for oocyte cryopreservation

Oocyte cryopreservation is of key importance in the preservation and propagation of germplasm. Interest in oocyte cryopreservation has increased in recent years due to the application of assisted reproductive technologies in farm animals such as in vitro fertilization, nuclear transfer and the need for the establishment of ova/gene banks worldwide. However, the cryopreservation of the female gamete has been met with limited success mainly due to its small surface-area:volume ratio. In the past decade, several vitrification devices such as open pulled straws (OPS), fine and ultra fine pipette tips, nylon loops and polyethylene films have been introduced in order to manipulate minimal volumes and achieve high cooling rates. However, experimental comparison of cooling rates presents difficulties mainly because of the reduced size of these systems. To circumvent this limitation, a numerical simulation of cooling rates of various vitrification systems immersed in liquid nitrogen was conducted solving the non-stationary heat transfer partial differential equation using finite element method. Results indicate the nylon loop (Cryoloop®) is the most efficient heat transfer system analyzed, with a predicted cooling rate of 180,000°C/min for an external heat transfer coefficient h= 1000 W/m(2)K when cooling from 20 to -130°C; in contrast, the open pulled straw method (OPS) showed the lowest performance with a cooling rate of 5521°C/min considering the same value of external heat transfer coefficient. Predicted cooling rates of Miniflex® and Cryotop® (polyethylene film system) were 6164 and 37,500°C/min, respectively, for the same heat transfer coefficient.

Modeling heat transfer and inactivation of Escherichia coli O157:H7 in precooked meat products in Argentina using the finite element method

The presence of Escherichia coli is linked with sanitary deficiencies and undercooking of meat products. Recent studies have detected E. coli O157:H7 in black blood sausages. Minimum time-temperature specifications to kill the bacteria were obtained by numerical simulations of the microscopic heat conduction equation using the finite element method, and calculating the temperature profile of the sausage and the population of E. coli at the coldest point during heating. The model was validated by heating sausages in a water-bath. The effects of heat transfer coefficients and water temperatures on the required time to achieve an inactivation value (IV) of 12(log) are reported. Macroscopic heat balances were simultaneously solved to consider the temperature drop in the water batch as a function of the ratio between the mass of thermally treated sausage and the heat capacity of the system.

Comparison of heat transfer in liquid and slush nitrogen by numerical simulation of cooling rates for French straws used for sperm cryopreservation

Slush nitrogen (SN(2)) is a mixture of solid nitrogen and liquid nitrogen, with an average temperature of -207 °C. To investigate whether plunging a French plastic straw (commonly used for sperm cryopreservation) in SN(2) substantially increases cooling rates with respect to liquid nitrogen (LN(2)), a numerical simulation of the heat conduction equation with convective boundary condition was used to predict cooling rates. Calculations performed using heat transfer coefficients in the range of film boiling confirmed the main benefit of plunging a straw in slush over LN(2) did not arise from their temperature difference (-207 vs. -196 °C), but rather from an increase in the external heat transfer coefficient. Numerical simulations using high heat transfer (h) coefficients (assumed to prevail in SN(2)) suggested that plunging in SN(2) would increase cooling rates of French straw. This increase of cooling rates was attributed to a less or null film boiling responsible for low heat transfer coefficients in liquid nitrogen when the straw is placed in the solid-liquid mixture or slush. In addition, predicted cooling rates of French straws in SN(2) tended to level-off for high h values, suggesting heat transfer was dictated by heat conduction within the liquid filled plastic straw.

Implications of storage and handling conditions on glass transition and potential devitrification of oocytes and embryos

Devitrification, the process of crystallization of a formerly crystal-free, amorphous glass state, can lead to damage during the warming of cells. The objective of this study was to determine the glass transition temperature of a cryopreservation solution typically used in the vitrification, storage, and warming of mammalian oocytes and embryos using differential scanning calorimetry. A numerical model of the heat transfer process to analyze warming and devitrification thresholds for a common vitrification carrier (open-pulled straw) was conducted. The implications on specimen handling and storage inside the dewar in contact with nitrogen vapor phase at different temperatures were determined. The time required for initiation of devitrification of a vitrified sample was determined by mathematical modeling and compared with measured temperatures in the vapor phase of liquid nitrogen cryogenic dewars. Results indicated the glass transition ranged from -126 °C to -121 °C, and devitrification was initiated at -109 °C. Interestingly, samples entered rubbery state at -121 °C and therefore could potentially initiate devitrification above this value, with the consequent damaging effects to cell survival. Devitrification times were calculated considering an initial temperature of material immersed in liquid nitrogen (-196 °C), and two temperatures of liquid nitrogen vapors within the dewar (-50 °C and -70 °C) to which the sample could be exposed for a period of time, either during storage or upon its removal. The mathematical model indicated samples could reach glass transition temperatures and undergo devitrification in 30 seconds. Results of the present study indicate storage of vitrified oocytes and embryos in the liquid nitrogen vapor phase (as opposed to completely immersed in liquid nitrogen) poses the potential risk of devitrification. Because of the reduced time-handling period before samples reach critical rubbery and devitrification values, caution should be exercised when handling samples in vapor phase.

In-vitro development of vitrified–warmed bovine oocytes after activation may be predicted based on mathematical modelling of cooling and warming rates during vitrification, storage and sample removal

Heat transfer during cooling and warming is difficult to measure in cryo-devices; mathematical modelling is an alternative method that can describe these processes. In this study, we tested the validity of one such model by assessing in-vitro development of vitrified and warmed bovine oocytes after parthenogenetic activation and culture. The viability of oocytes vitrified in four different cryo-devices was assessed. Consistent with modelling predictions, oocytes vitrified using cryo-devices with the highest modelled cooling rates had significantly (P < 0.05) better cleavage and blastocyst formation rates. We then evaluated a two-step sample removal process, in which oocytes were held in nitrogen vapour for 15 s to simulate sample identification during clinical application, before being removed completely and warmed. Oocytes exposed to this procedure showed reduced developmental potential, according to the model, owing to thermodynamic instability and devitrification at relatively low temperatures. These findings suggest that cryo-device selection and handling, including method of removal from nitrogen storage, are critical to survival of vitrified oocytes. Limitations of the study include use of parthenogenetically activated rather than fertilized ova and lack of physical measurement of recrystallization. We suggest mathematical modelling could be used to predict the effect of critical steps in cryopreservation.

Convective heat transfer coefficients of open and closed Cryotop® systems under different warming conditions

The warming of cryopreserved samples supported by small volume devices is governed by heat transfer phenomena which are mathematically described by the solution of the transient heat conduction partial differential equations; the convective heat transfer coefficient (h) is an important parameter involved in the boundary condition which is related to the fluid dynamic behavior at the interface device-warming fluid (water, sucrose solution or air). Unfortunately, h values for small volume devices (i.e. Cryotop®) have not been experimentally determined. Moreover, heat transfer coefficients during warming of Cryotop® cannot be obtained through classical dimensionless correlations expressed in terms of Nusselt vs. Reynolds and Prandtl numbers that are available for regular geometries and single materials. It is the purpose of present work to determine the convective heat transfer coefficients (h) by numerically solving the heat transfer equation applying the finite element method. Numerical simulations allowed to predict time-temperature histories and warming rates under different protocols in Cryotop® system which were compared with literature warming rates reported for this device. The h values were calculated considering the heterogeneous structure of the domain (microdrop, plastic-support) and the irregular three-dimensional geometry. The warming conditions analyzed were: a) open system in contact with air and sucrose solution at 23 °C) and b) closed system in contact with air and water at 23 °C. The h values of the Cryotop® open system immersed in sucrose solution (23 °C), that according to literature achieved a survival in the order of 80%, are in the range of 1800-2200 W/m2K. The h values obtained in this work for warming conditions are critical parameters for cryobiologists when studying heat transfer rate in this small volume device.

Effect of Water Content on Thermo-Physical Properties and Freezing Times of Foods

For the prediction of temperature change in different foodstuffs during freezing and thawing processes, accurate estimation of the thermo-physical properties of the product is necessary, such as specific heat, density, freezable water content, enthalpy, and initial freezing temperature. These data allow the adequate design and optimization of equipment and processes. Water is a main component in all foods and greatly influences the behavior of these properties, depending on its concentration. During the freezing process, which involves the phase change of water into ice, the specific heat, thermal conductivity, and density undergo abrupt changes due to the latent heat release. This complex process does not have an analytical solution and it can be described as a highly nonlinear mathematical problem. Many difficulties arise when trying to numerically simulate the freezing process, especially when using the finite element method (FEM), which is especially useful when dealing with irregular-shaped foodstuffs. Several techniques have been applied to consider the large latent heat release when using FEM. One traditional method is the use of the apparent specific heat, where the sensible heat is merged with the latent heat to produce a specific heat curve with a large peak around the freezing point, which can be considered a quasi-delta-Dirac function with temperature (depending on the amount of water in the food product) (Pham 2008). However, this method usually destabilizes the numerical solution. Implementation of the enthalpy method, which can be obtained through the integration of the specific heat with temperature (Fikiin 1996; Comini et al. 1990; Pham 2008; Santos et al. 2010), and the Kirchhoff function, which is the integral of the thermal conductivity, allows the reformulation of the heat transfer differential equation into a transformed partial differential system with two mutually related dependent variables H (enthalpy) and E (Kirchhoff function) (Scheerlinck et al. 2001). These functions, H and E versus temperature, are smoother mathematical functions compared to the specific heat, thermal conductivity, and density versus temperature, avoiding inaccuracies and/or divergence of the numerical method. Even though it brings great advantage to the resolution of the problem, with the simultaneous enhancement of the computational speed of the program, this transformation of variables is not widely used in the literature. Unleavened dough and cooked minced meat were selected due to their significant difference in water content in order to explore the performance of the computational code written using the enthalpy-Kirchhoff formulation. Another important reason is because cooked minced meat and dough are both present in several ready-to-eat meals, therefore contributing valuable information to food processors interested in optimizing cooling and freezing operating conditions of semi- or fully processed goods. The objectives of this work are (1) to experimentally determine by differential scanning calorimetry (DSC) the thermo-physical properties of dough and cooked minced meat in the freezing range: specific heat as a function of temperature, bound water, heat of melting, initial freezing temperature, etc.; (2) to develop and validate a finite element algorithm to simulate the freezing process in regular and irregularly shaped foodstuffs; and (3) to introduce appropriate equations of the thermo-physical properties in the numerical program to assess the effect of total water content, bound water, and surface heat transfer coefficient on freezing times in an irregular food system.