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# A Simplified Procedure to Determine the Optimal Rate of Freezing Biological Systems

[+] Author and Article Information
Sreedhar Thirumala

Bioengineering Laboratory, Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803

Ram V. Devireddy1

Bioengineering Laboratory, Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803

1

Corresponding author. e-mail: devireddy@me.lsu.edu

J Biomech Eng 127(2), 295-300 (Dec 06, 2004) (6 pages) doi:10.1115/1.1865213 History: Received April 21, 2004; Revised October 19, 2004; Accepted December 06, 2004

## Abstract

The effect of several cell-level parameters on the predicted optimal cooling rate $Bopt$ of an arbitrary biological system has been studied using a well-defined water transport model. An extensive investigation of the water transport model revealed three key cell level parameters: reference permeability of the membrane to water $Lpg$, apparent activation energy $ELp$, and the ratio of the available surface area for water transport to the initial volume of intracellular water $(SA∕WV)$. We defined $Bopt$ as the “highest” cooling rate at which a predefined percent of the initial water volume is trapped inside the cell (values ranging from 5% to 80%) at a predefined end temperature (values ranging from $−5°C$ to $−40°C$). Irrespective of the choice of the percent of initial water volume trapped and the end temperature, an exact and linear relationship exists between $Lpg,SA∕WV$, and $Bopt$. However, a nonlinear and inverse relationship is found between $ELp$ and $Bopt$. Remarkably, for a variety of biological systems a comparison of the published experimentally determined values of $Bopt$ agreed quite closely with numerically predicted $Bopt$ values when the model assumed 5% of initial water is trapped inside the cell at a temperature of $−15°C$. This close agreement between the experimental and model predicted optimal cooling rates is used to develop a generic optimal cooling rate chart and a generic optimal cooling rate equation that greatly simplifies the prediction of the optimal rate of freezing of biological systems.

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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

The general variation of the predicted optimal cooling rate (Bopt,°C∕min) as a function of reference membrane permeability (Lpg,μm∕min-atm; ranging from 0 to 1.0μm∕min-atm). Bopt values are plotted by assuming SA∕WV=1.0μm−1 and at two different values of ELp (10.0 and 20.0Kcal∕mole). The Lpg values are shown along the x-axis and the Bopt values are shown along the y-axis.

Figure 2

Plot of the predicted optimal cooling rate (Bopt,°C∕min) as a function of activation energy (ELp,Kcal∕mole) assuming SA∕WV=1.0μm−1 and Lpg=1.0μm∕min-atm. The error bars represent the nonlinear variation of Bopt with ELp. In (a) the error bars represent variation for a positive 20% change in ELp, whereas in (b) the error bars represent negative 20% change in ELp. The ELp values are shown along the x-axis and the Bopt values are shown along the y-axis.

Figure 3

Plot of the predicted optimal cooling rate (Bopt,°C∕min) as a function of water volume/surface area (SA∕WV,μm−1) at different ELp values and at Lpg=1.0μm∕min-atm. The SA∕WV values are shown along the x-axis and Bopt values are shown along the y-axis in logarithmic scale.

Figure 4

Generic optimal cooling rate chart (GOCRC). Predicted GOCRC cooling rates are plotted as a function of activation energy (ELp,Kcal∕mole) at Lpg=1.0μm∕min-atm and SA∕WV=1μm−1. The dotted line represents a curve fitting used to generate a mathematical equation for BGOCRC value, BGOCRC=1009.5e−0.0546ELp (with a goodness of fit of R2=0.994). The ELp values are shown along x-axis and BGOCRC values are shown along y-axis in logarithmic scale.

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