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Heat and Mass Transfer Models and Measurements for Low-Temperature Storage of Biological Systems

  • Shahensha M. Shaik
  • Ram Devireddy
Reference work entry

Abstract

Living systems are routinely exposed to low temperatures, and their corresponding response has been of scientific and medical interest for centuries. The basic physicochemical phenomena that govern the response of the living systems to subzero temperatures are complex and interactively coupled so that the prediction and regulation of the associated events are often difficult. However, during the past several decades, significant progress has been made in devising useful techniques to further our understanding of biological systems at ultralow temperatures. This chapter presents some of the heat and mass transfer models and associated measurements in biological systems exposed to low temperatures specifically as applied to cryopreservation protocols.

Nomenclature

A

Cross-sectional area of the extracellular space

Ac

Effective membrane surface area available for water transport

Ao

Isotonic (initial) cell surface area

awi

Chemical potential of the intracellular fluid

awo

Chemical potential of extracellular fluid

B

Constant cooling rate

Bopt

Biophysically determined optimum cooling rate

c

Specific heat capacity

CPA

Cryoprotective agent

\( {C}_{CPA}^O \)

Concentration of the CPA outside the cell (extracellular space)

\( {C}_{CPA}^I \)

Concentration of the CPA inside the cell (intracellular space)

D

Diffusivity coefficient

ECM

Extracellular matrix

ELp

Apparent activation energy for Lpg

ELp[cpa]

Apparent activation energy for Lpg in the presence of CPAs

fec

Fractional mass of extracellular water in the medium

fic

Fractional mass of intracellular water in the medium

IIF

Intracellular ice formation

JCPA

Solute (CPA) flux across the cell membrane

Jv

Total flux across the cell membrane

k

Thermal conductivity

L

Latent heat of fusion per unit mass of solution

Lc

Length of the Krogh cylinder

Lp

Cell membrane hydraulic conductivity which measures the volume flow induced by a hydrostatic pressure difference

Lpg

Cell membrane hydraulic conductivity at TR

Lpg[cpa]

Cell membrane hydraulic conductivity TR in the presence of CPAs

ns

Number of moles of solutes in the cell as calculated from initial cell osmolarity

Ns

Number of water molecules

Nso

Initial (isotonic) number of water molecules

PIF

Probability of intracellular ice formation

r

Spatial coordinate

rv

Sinusoid (vascular) radius in the Krogh cylinder model

rvo

Initial sinusoid radius in the Krogh cylinder model

R

Universal gas constant

R2

Goodness of fit parameter

SCN

Surface-catalyzed ice nucleation

T

Temperature

TR

Reference temperature (273.15 K)

t

Time

V

Cell volume

VCN

Volume-catalyzed ice nucleation

Vo

Initial isotonic cell volume

Vb

Osmotically inactive cell volume

vcpa

Mole fraction of the CPA

ΔX

Distance between adjacent sinusoid centers in the Krogh cylinder model

Xwi

Mole fraction of the intracellular water

yi

Experimental data points

y(xi)

Model predicted data points

\( \overline{y} \)

Normal mean of the experimental data points

ΔCCPA

Concentration difference between the intracellular and extracellular space for solute (CPA)

ΔCCPAAVE

The “log mean” osmolality of the permeable solute

ΔCw

Concentration difference between the intracellular and extracellular space for water

ΔP

Hydrostatic pressure difference

\( \frac{\partial {C}_I}{\partial t} \)

Local unsteady CPA concentration term

\( \frac{\partial \left({vC}_I\right)}{\partial x} \)

Convection transport term

\( D\frac{\partial^2{C}_I}{\partial {x}^2} \)

Diffusion term, assuming constant D

Greek Symbols

v

Local convective velocity

νw

Partial molar volume of water

ω

Cell membrane permeability to cryoprotective agent (m3/N-s)

σ

Reflection coefficient or the “relative permeability” of the membrane to the solute

σi

Uncertainties in the experimental data points

ρ

Density

φs

Disassociation constant for sodium chloride

χ2

Residual variance

\( {\overline{\chi}}^2 \)

Total variance

Λ

Mass fraction of the medium which has not yet released latent heat during freezing

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringLouisiana State UniversityBaton RougeUSA

Section editors and affiliations

  • Ram Devireddy
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringLouisiana State UniversityBaton RougeUSA

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