1

Cryogenic Simulations of Tissue Freezing

R. C. Dykhuizenl’* and A. Sambanis2

lThermal and Fluids Engineering,u Sandia National Laboratories, Albuquerque, NM87008-0836; ph. 505-844-9105, fax 505-844-8251, rcdykhu@sandia.gov

2Schoo1 of Chemical Engineering and Petit Institute for Bioengineering and Bioscience,Georgia Institute of Technology, Atlanta, GA 30332-0100* Author for correspondence

Abstract

This paper presents the first analysis of the cryopreservation of a large tissue. Preservation

of engineered tissue substitutes is of particular significance in ensuring the off-the-shelf

availability of these products. Freezing of living biological cells has been extensively stud-

ied, and models have been able to predict the associated chemical and thermal transients.

However, the progression from individual cells to the freezing of living tissues introduces

significant new problems due to their physical size and the associated mass and heat trans-

fer characteristics. This study has analytically examined the complexity of freezing living

tissue samples. For example, a specific freezing transient maybe specified for the surface

of the tissue, however, due to the limitations of the physics, the internal portions of the tis-

sue will experience a very different thermal transient. The high volume fraction of cells

within the tissue presents another challenge. When freezing a dilute suspension of cells,

one can treat the extra-cellular solutions as having an infinite capacitance. Therefore, any

osmotic flows through the cell membranes will not alter the extra-cellular chemical con-

centrations. However, tissues typically have a low extra-cellular volume fraction, so

osmotic flows will significantly alter both the cellular and extracellular chemical concen-

trations.

DISCLAIMER

This report was prepared as an account of work sponsoredby an agency of the United States Government. Neitherthe United States Government nor any agency thereof, norany of their employees, make any warranty, express orimplied, or assumes any legal liability or responsibility forthe accuracy, completeness, or usefulness of anyinformation, apparatus, product, or process disclosed, orrepresents that its use would not infringe privately ownedrights. Reference herein to any specific commercialproduct, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constituteor imply its endorsement, recommendation, or favoring bythe United States Government or any agency thereof. Theviews and opinions of authors expressed herein do notnecessarily state or reflect those of the United StatesGovernment or any agency thereof.

.T .-,, ./ .. !,. ., ..7-P., ,Ym-..e .,... 4-4. .,--? .—, . . . . ?—.. .--.-—. ——

DISCLAIMER

Portions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.

.--, , .— .7. - . . . . . . . . . . 7... .-,..- ,. . -,wm. .~.-, ,-... ,..,.,~ ~ . “\..-.-.-..s . a-q .= ,,% > .: r,., -,- ——-- -- -

, 1

Keywords: Tissue, freezing, modeling, simulation, cryopreservation

Introduction

As discussed by Carslaw and Jaeger, 1 an analytical solution for the freezing of pure

materials was published by J. Stephan in 1891. The freezing of biological materials is a

much more difficult problem due to the heterogeneous nature, the complex chemical

makeup, and the living cellular component of the material. There have been many recent

experimental studies on the freezing of biological materials that have concentrated on cells

or cell clusters, such as pancreatic islets.2*318’11‘20’22Current efforts are also considering

15719Modeling of biological cryopreservation has focused onsmall tissue samples.

4521individual cells and cell clusters. >‘

The freezing of living tissues is a significantly more difficult problem than the freezing of

single cells or small cell clusters due to the physical size of the material, which imposes

significant heat and mass transfer resistances. Natural and artificial tissues are composed

of cells (usually with several cell types) and extracellular components, including natural

and/or artificial matrices in complicated three-dimensional configurations that may include

vasculature. Each cell type may require a different freezing protocol. Nonetheless,

cryopreservation of tissues is of paramount significance in ensuring off-the-shelf

availability of these products.

This paper presents a first effort towards developing a framework for modeling the

cryogenic preservation of natural and artificial tissues. Simplifications have been made to

allow solutions at this time since little data are available about the complex processes

,

involved in tissue freezing. Even in this simple form, the model retains key characteristics

of the physical system and predicts significant differences between the freezing of a dilute

cell suspension and a tissue. Furthermore, such modeling studies can offer specific

suggestions regarding the architectural and other physical characteristics of tissues that are

needed in order to render them amenable to cryopreservation.RECX=tj/ ,-:, -

,,=-.z

Nov1;2oil’Tissue Model CM-rl

The model assumes that the tissue is composed of cells and extracellular space. The

extracellular space consists of a solid domain, which represents materials that do not

undergo solidification, such as extracelh.dar matrices and scaffolds, and a liquid domain,

which represents a solution of nutrients for the cells. Although ice may form both intra- and

extracellularly, at this point in the model development, we have ignored the nucleation of

ice in the intracellular domain. This avoids the complication of treating the cells in a

probabilistic manner, since some cells may nucleate earlier than others due to random

nucleation events. Absence of intracellular nucleation is not an unrealistic assumption, as

previous studies have shown for several cell types in the presence of a cryoprotective agent

to very low temperatures.

Even though nucleation may be unlikely due to a lack of heterogeneous nucleation sites,

the intracellular fluid may vitrify into a non-crystalline solid. Since the vitrification is not

accompanied by any significant latent heat, it is not required to track a vitrified volume

fraction within the cellular domain. To simulate vitrification, it is simply required to stop

osmotic exchanges with the extracellular domain.

-3-

- ?.7..- . . . ,-? ~. -- _T,,m . . . . .=, --7 Tdxitl. ,.,, . $ .,.. Crzvr- ..,-, . . ., . —-——- --—

,

Nucleation in the extracellular domain is assumed to occur immediately upon attainment

of the equilibration phase change temperature. This assumes that ice is nucleated artificially

by the operator, and the ice can grow throughout the assumed continuous liquid

extracellular space.

Two representative solutes are tracked within the two fluid domains. Salt is use to represent

solutes that do not permeate the cell membranes, but may migrate through the extra cellular

domain via diffusion. The model also tracks a cryo-protestant agent (CPA). Dimethyl

sulphoxide (Me2SO) is used in this study as a representative of CPA. Me2S0 may

permeate the cell membrane due to osmotically driven flows and diffusive processes. The

Me2S0 may also migrate through the extra cellular domain via diffusion in a similar

manner as the salt.

One-dimensional geometries (slab, cylindrical or spherical) maybe examined. Time-

dependent thermal and solute boundary conditions are imposed. The model tracks

transients as a function of the position within the tissue.

The physical sizes of the tissue cells are small, so diffusion of heat and solute within

individual cells is not resolved. However, the relatively slow transport of heat and solute

over the physical dimension of the tissue is modeled. This results in different thermal and

chemical transients at each radial location. A low osmotic coupling term typically results

in non-equilibrium distributions of solute between the two fluid domains.

The chemical makeup of living tissue is very complex, thus, simulations that include only

two solutes cannot completely describe the system. However, by simulating two solutes,

-4-

.. ._ee _,.---,7 ---, . . -. =-7.. . .. . ,, T.-,-;-.-R.”.-.-.W.T.<+,—.__.. ----,,.,.,, ..,-=.,,.-.- ...-< --,. ..........

I

one that permeates the cell membrane, and one that does not, the character of thermal

transients can be approximated. Pure ice begins to form at some subcooling below OC (the

amount of subcooling depends upon the solute content of the solution). The solute

concentrations in the liquid phase increases as ice forms. The cellular fluid volume

typically decreases due to osmotically driven flows caused by the introduction of CPA into

the extracellular domain and the concentration of both solutes due to freezing. However,

the cellular volume may increase later in some transients as Me2S0 invades the cell.

Mathematical Model

State Equations

In developing the state equations, we will allow for freezing in the intracellular domain.

However, due to the unavailability of nucleation parameters, in all calculations the state

variable for the intracellular ice volume fraction will be set to zero. Thus, the model will

only be applied to conditions where intracellular ice nucleation does not occur.

The total number of state variables at any radial location is 9 to completely describe the

thermal and chemical state of the two domains. These are the temperature, two ice volume

fractions, two liquid volume fractions, two Me2S0 contents, and two salt contents. A

nomenclature section at the end of the article shows the symbols used for state variables

and model parameters and provides the numerical values of the latter used in the

calculations.

These 9 state variables require 9 state equations which are presented here. The first state

equation is the energy equation. A single temperature is used for all domains:

-5-

q-- , .-,7 ... .. ... . ,‘~- .T.llfi. .t. ,- -,

,

aT= (Sc+ S-,)L+ V . (kVT)Pcp~ ( )

where S is the solidification rate of ice, and L is the latent heat of solidification. The last

term models the radial conduction of heat within the tissue.

The conservation equations for the intracellular and extracellular liquid volume fractions

are given below

act=

Tt = –Sc–e

a~x

z = –Sx+e

where El is the osmotic flow between the intracellular and extracellular domains.

Equation set 3 tracks the ice content of the intracellular and extracellular domains:

ticZ“s’ayx .

Z“s’

(2)

(3)

Examination of the above equation sets (2 and 3) reveals a basic assumption of the model:

the tissue sample does not change volume due to the various flow and phase change

processes. Whatever volume is lost by one domain, due to these processes, is gained by the

other domain. Thus, the model does not account for any tissue shrinkage or expansion that

may occur during the transient.

To track the motion of solutes through the system, a solute content variable is introduced.

Solute content is defined as the liquid volume fraction times the solute concentration. It was

found more convenient to use this definition, instead of solute concentration. The following

two state equations track the Me2S0 content in the extra- and intracellular liquid phases.

The solute mass balances in the extracellular domain are more complex due to radial

diffusion:

arc -JmA

z = V,n

arx J,nA uXDV2C~X .

x=vm+T

(4)

J,n represents the osmotic (volumetric) flux of Me2S0 through the cell membrane, and A

represents the cell membrane area per unit volume. In the results presented here, it is

assumed that osmotic flux occurs through the entire cell membrane. This is expected to be

the case when cells are embedded as single units in the tissue structure. However, in cases

where the cells form clusters, the area through which osmotic flux occurs should be reduced

to account for cell touching. The diffusion term in the extracellular solute equation is

modified by three terms. First, the diffusion term is multiplied by the extracellular liquid

volume fraction (UX). In this way, the diffusion naturally goes to zero as the amount of

extracellular liquid is decreased to this value. This modification is to account for the

reduction in the area available for diffusion in a multi-phase system.

The diffusion term in equation 4 is also divided by a tortuosity (~). The tortuosity is non-

dimensional, and greater than unity to simulate a tortuous path that the solute may take in

,

-7-

>..,.-,F. -,,,.3.--7mtY?:-.:... -T,??x.r:-.-w-w. x=---.-.’-?GT,w.r---------- “ ,. .=., -------’=lxssm..-:,t.. . ....-,+, b.. .W..a. . ... .. ...~,——.—., .,....—, iirr-=-w:.c ,“- * ., ., .-— -- - ““- -“”’ ... ..-

,

a typical tissue. We have used a value of 3 in the calculations presented in this paper, which

is expected to be representative of a natural tissue or of an engineered substitute in the final

stages of in vitro development. Engineered constructs are more permeable early on, i.e.,

shortly after seeding of cells onto a scaffold and before deposition of a significant amount

of matrix by the cells, so a lower value of tortuosity would be appropriate in this situation.

The salt content of the intracellular domain is constant since it is assumed that the salt does

not permeate the cell membrane. Thus, this state equation can be trivially replaced by an

algebraic statement that the intracellular salt content is constant. However, the salt content

of the extracellular domain must be tracked since the salt may diffuse within this domain

due to the changing concentrations that result during a transient. The conservation of the

salt content in the two fluid domains are given below:

PC = constant

(5)

The above lists nine state equation for the nine variables. However, since the volume

fractions must add to unity, one state equation can be replaced by an algebraic relation. The

current implementation of the model algebraically obtains the ice content in the

extracellular domain from the following volume balance equation:

(6)YC+I’X+%+%= 1–1

where L represents the solid volume fraction. These materials (such as cell membranes,

genetic material, scaffolds, etc.) do not participate in the phase change transients.

As indicated above, the above model does not account for any flow in the extracellular

domain. An outflow of fluid may result from the contraction of the tissue due to cell

dehydration. However, an engineered tissue with a rigid scaffold may limit tissue

contraction. Also, a small outflow may be induced due to a small increase in specific

volume that may accompany the formation of ice. It should be noted that when a tissue

sample is frozen from the outside in, the formation of a solid phase at the boundary may

limit any outflow. Thus, the thermal transient may result in significant thermally induced

stressesb. This model does not account for such thermal stresses.

Constitutive Eauations

The state equation set requires a number of constitutive equations. It has already been stated

that the fluid material properties will be determined by assuming a ternary sodium chloride-

Me2S0 solution in water. The first constitutive equations simply calculate solute

concentrations from the state variables:

c,C = pc/cxc

c.x = px/ctx

c mc = rc/~c.

c mc = rci~c

(7)

The salt content in the intracellular domain is.constant and set equal to the prescribed initial

content. However, since the volume fraction of the intracellular domain varies, the salt

concentration varies with time.

Second, the osmotic flows of water and Me2S0 are obtained from the solute

concentrations. The osmotic flow model is obtained from Kedem/Katchalsky equations. 10

The osmotic flow is assumed to be proportional to the difference in salt concentration:

0 = A(.l,n + J,v)

J,v = -P9iT[(c,c- C,r)+ CJ(c,nc- C,,,-r)] (8)

J m = v,n[C/l(l –CJ)J,v + 9iT@C,nc– C,nx)]

where Cl’ is the upstream concentration (intracellular or extracellular) depending upon the

direction of the osmotic flow J,V. This is a slight variation of a model currently in use by

many 12’13*14which employs the average of the two concentrations in place of Cl,. While

the original formulation allows a simpler equation set, which is easier to solve, it also

admits non-physical results. In certain situations the original model results in removal of

solute from regions where the solute concentration is zero.

The membrane permeabilities to water and Me2S0 are assumed to follow an Arrhenius

relationship with temperature.

P= Prefexp(-wj)

co=@refexp(-%(+-+))

(9)

The osmotic flows typically result in cell shrinkage during cooling. The slower the cooling,

the more shrinkage that will occur, for slow cooling allows more osmotic flow prior to

solidification. The cell membrane area per unit volume (A) is calculated below from the

cell diameter and the cell volume fraction assuming spherical cells:

A = 6V/d. (lo)

At this point the cell membrane area is fixed at it’s initial value. However, the formulation

can be easily modified in future versions resulting in a shrinking area as the cell volume is

reduced.

The solidification rates are assumed to be linear functions of the liquid subcooling and a

negative subcooling results in negative solidification (melting) rates:

s== qJTJcmc,CJ - 2-)

Sx= qx(z-,(cmx, C,x) - 7’) -(11)

This introduction of the above solidification rate constant (q) allows one to account for

non-equilibrium solidification by use of a relatively small value. However, the main reason

for including such a parameter is that the numerical model becomes easier to formulate,

with one differential equation for each state variable. Use of a large solidification rate

constant results in local equilibrium phase distributions. However, the solute

concentrations may be out of equilibrium. As mentioned above, the solidification rate

constant for the cellular domain is set to zero. This simulates conditions where the ice

nucleation in the cellular domain is unlikely, and/or the cellular fluid vitrifies. The

solidification rate constant for the extra-cellular phase is set to a large value so that

equilibrium is approached. It is assumed that the ice phase nucleation in this domain is

likely, or that the procedure includes an artificial nucleation of the ice phase in the

extracellular domain by an operator.

The solidification process in the extracellular domain continues until the total solute

concentration (Me2S0 and salt) reaches 45% by mass. This value is approximately

16 Peg: reports that vitrificationconstant through the eutectic trough as reported by Pegg.

often occurs at or prior to reaching this eutectic trough, so we will simply assume

vitrification at the eutectic trough concentration. A latent heat release does not accompany

the vitrification, and the model simply stops all osmotic flow once this state is reached.

Thus, the model simply reports a constant liquid volume fraction of constant concentration

to represent this glassy state. This simple phase change model allows for easy treatment of

thermal transients that involve both melting and solidification without requiring history

dependent parameters.

The thermal properties are determined from a volumetric average of the various

components that exist at any radial location. The thermal properties of the liquid phase are

currently treated as constants, equal to those of water. The thermal properties of the solid

18The two solute diffusivities areice phase are calculated as a function of temperature.

assumed to be linear functions of temperature with an intercept at O K.

Method of Lines

The spatial derivatives are put into a finite difference form. This procedure is often called

the Method of Lines. This allows use of standard ODE packages that automatically adjust

the time step to maintain a specified integration error. Grid convergence of the simulation

must still be done manually by the user. A diffuse suspension of cells can be simulated by

making the cell volume fraction very small.

17The package allowsThe SLATEC routine DDEBDF is used to integrate the equations.

use of error checks to control the time step during the transient. The user is still responsible

to determine if the node size is small enough to control numerical errors.

Initial and Boundarv Conditions

At the center of the tissue a symmetry boundary condition is imposed. At the outer

boundary, the solute concentration boundary condition is specified as a function of time. A

no flux boundary condition can also be applied to simulate conditions when the tissue is

removed from a liquid bath, or that the bath has frozen. The solute boundary condition can

be switched from a specified concentration to no flow at any time.

The outer boundary condition for temperature states that the rate of heat transfer between

the construct and the surroundings is equal to a heat transfer coefficient times the difference

of the surrounding temperature, prescribed as a function of time, and the temperature at the

surface construct.

q = I’z(z-b(t) -z-(l?)). (12)

By using an infinite heat transfer coefficient (h), this boundary condition can be used to

specify a prescribed boundary temperature on the edge of the domain.

The boundruy conditions at the initial time (O seconds) need not be equal to the initial

temperature condition imposed. This allows simulation of a step change in temperature or

concentration. Although a step change boundary condition may not be realistic, it is

important in validating the code using analytical solutions.

-13-

. -,r -L - .---.7-. ,. .,.- T.7,-.”Z-T . .. . , . . . .-?,-~,:mw-:s. . . . . . . .. ,. > —, =6m.:77*7.T.zcT% - .?<.7 -... , :.~..;?r’ X--K r ..--. -’- r>~ --=----- -- . — -

Code Verification

Test problems have been constructed to validate that the model reproduces one-

dimensional analytical solutions. The first validation problem set examined step changes in

the boundary temperature and solute concentration. Slab, cylindrical and spherical

geometries were used. Comparison with analytical resultsl were good, however, the

cylindrical and spherical geometries required more nodes for the same level of accuracy

since the numerical formulation is second order accurate in slab coordinates, but only first

order in cylindrical and spherical coordinates.

Other verification problem sets examined diffusion of a solute. This required setting the

osmotic flow parameter to zero. The solution for diffusion in the extra cellular domain is

independent of the extra cellular volume fraction, and the model simulations were in

agreement with this.

Finally, the code was used to simulate a classical Stefan problem. This required using a

phase diagram with little difference between the Iiquidous and solidous temperature to

simulate the freezing of a pure substance (isothermal solidification). A simple code

modification was required. The initial temperature was set to O C and the boundary

condition set to -10 C to allow use of the analytical solutions presented by Goodman.7 The

exact solution is approached as the solidous temperature approaches O C. For a solidous

temperature of- 1 C, the code predicted the freezing front to within 3%. For a solidous

temperature of-O. 1 C, the results were within 0.8% of the exact solution. These

comparisons are for internal nodes, for the error in the first couple of nodes was always

greater. This is because the finite difference representation of the step change in the

boundary condition can not simulate the infinite heat flux at zero time, which is a feature

of the exact solution. For all verification calculations and the calculations presented later in

this paper, the mesh convergence was determined. The level of accuracy of the code was

deemed adequate for the purpose of this study.

Time Scales of Tissue Processes

The numerical code described above allows examination of various thermal and chemical

transients within a tissue construct. However, prior to presenting the numerical results it is

best to examine the time constants that exist for the various processes. The ordering of the

time constants determine how the transient will progress. This information also

demonstrates that the maximum tissue thickness that can be supported by diffusion of

nutrients is comparable to the maximum thickness that can be presened when transport of

CPA is also by diffusion alone.

First, let us examine the thermal time constant. For simplicity we will assume that a tissue

can be represented by a constant property sphere of radius R. The finite size of the sphere

results in significant time lag between the central region and boundary layers. A thermal

transient solution is available for uniform temperature sphere exposed to a linear

temperature ramp at its surface. 1The governing equation is the well known Laplace

equation:

-15-

(13)

For a linear temperature ramp (B) imposed on the surface of the sphere, the temperature

at the center as a function of time is found analytically as follows:

[)pcPR2 2BpcpR2 mT(O, t)=B t–~ – z

(-l)n

[ )

–kn2x2t

xk(14)

n=l n2 ‘Xp R2PCP

Equation 14 reveals that the central temperature eventually approaches a linear response as

time gets large compared to the time constant of

R2pcP

kx2 “(15)

However, the central temperature lags the surface temperature by Bp cpR2/k. Using

water properties this results in a temperature lag of 2 K for a 2 mm diameter tissue and 100

K/rein. ramp rate. The time constant for this example is 0.6 seconds.

This temperature lag becomes significant when the tissue size is 2 cm in diameter, for then

the temperature lag is 200 K, and the time constant is 60 seconds. Numerical simulations

of actual tissue reveals that the thermal transient is much more complex due to the latent

heat effects making the heat capacitance effectively larger, and the time constants

effectively longer. However, this simple example illustrates that even relatively small

tissue samples will experience different thermal transients as a function of radial position.

The introduction of a CPA into a tissue is also difficult if one relies upon diffusion to

distribute the solute. If we ignore osmotic flows, which would only slowdown the diffusion

of CPA into the tissue, we can use the following equation to estimate the diffusive motion

of the CPA through the extracellular domain:

2

‘.YDa Cdx(1 -$&) = —— .

T 3r2(16)

In equation 16 we have assumed that the diffusive flux is reduced by the extracellular liquid

volume fraction, but the capacitance is only reduced by the solid volume fraction (for the

CPA must also enter the cellular domain). This simple assumption allows the use of

available analytical solutions.

concentration can be found: 1

For a step change in the boundary condition, the central

Cm-r(o,t).

[)–Daxn2n2t- 1+2 ~ (–l)”exp

C,nx(R, ij -n=l (1 - k)~R2 .

(17)

Equation 17 reveals that the central concentration lags the surface concentration with a time

constant of ( 1- k) R2~/axn2D. For a 2 cm diameter tissue this results in a time constant

of 8 days. Numerical simulations have confirmed this result. This demonstrates that relying

upon diffusion to distribute a CPA throughout a tissue will not be possible, just as diffusion

cannot be relied upon for transfer of nutrients and metabolizes in tissues of this thickness.

It has been determined that freezing of the tissue from the outside inward increases the

radially inward diffusive flux. This occurs due to the increased solute concentrations that

occur during the creation of pure ice. However, this mechanism has not been found to result

in a significant increase in the motion of a solute. Thus, one will have to rely upon

convective flows to distribute the solute.

Finally, we would like to examine the equilibration of the CPA between the intracellular

domain and the extracellular domain. The time constant for this process can be expressed

in the following form for a dilute suspension of cells. In this situation one can assume that

the concentration in the extra-cellular domain is constant. From equations 4,7 and 8, the

time constant can be estimated as:

d

6A9?To3 “(18)

Equation 18 only considers the motion of Me2S0 through the cell membrane. Using a

Me2S0 permeabilityof2x10-11 moles/N-see., and a cell size of 10 microns, the time

constant is approximately 40 seconds. Numerical simulations of dilute cell suspensions

have revealed a time constant approximately 1/2 this value due to osmotic water flow that

was ignored above. The time constant becomes even shorter for tissue simulations. Tissue

time constants are shorter since the extracellular domain is typically small. Therefore, the

concomitant concentration changes in the extracellular domain hasten the equilibration

process.

Numerical Solutions for Tissue Samples

In this section we present some numerical model results. As a sample problem, we used”a

spherical construct of 2 cm in diameter. This size was chosen simply to illustrate the large

effects of thermal lag, even in such a small tissue. Although the tissue very near the surface

experiences a thermal transient very close to that applied, central portions of the tissue

experience a very different thermal transient. We have assumed that initially the CPA is

uniformly distributed within the extracellular domain, but is absent from the intracellular

-18-

..), >7, , ---7’--!7 ,-fl . . .= . . ,, -. -.--,7-- - , , . ..m.- . .-, . ., ,&.k= ..- ,,W, .T ,,,

domain. While certainly an approximation, the mass transfer of CPA in the extracellular

domain is expected to be much faster than across the cell membranes, especially if

convective transport in the extracellular space exists. This is assumed to have been

accomplished via some convective mechanism since it is not possible for diffusion to

distribute the CPA evenly within this tissue within a reasonable time period. The surface of

the tissue is exposed to a 100 second transient from OC to -60 C, then held isothermally at

-60 C. No flux boundary conditions are used for the solute diffusion equations at the tissue

surface.

The nomenclature section presents the numerical value of all the parameters used in this

calculation. These values are representative of values others have used in simulating the

cryogenic behavior of actual tissues.

Figure 1 shows that the tissue surface (at the radial position of 1 cm) reasonably follows the

boundary condition transient. Due to the finite heat transfer coefficient (equation 12), the

boundary temperature is not reproduced exactly. Figure 1 also shows the significant

reduction in the temperature transient for the internal positions. The 0.6 C/second

temperature ramp rate is reduced to 0.015 C/second (prior to solidification) for the central

position. The reduced temperature transient is due to the latent heats associated with

freezing. After the freezing at any location is finished, the boundary temperature ramp rate

is approached.

Figure 2 shows the transient ice volume fraction. This represents extracellular ice, since the

current version of the model does not nucleate ice in the intracellular space. This plot shows

the increased delay in nucleation of ice as the depth into the tissue is increased. Due to the

-19-

., .,,.fi..,.r,~.+. -. m,. --., - .m>m.-m.. ,.,. -.Y .,-.-,,r.–.>.I-,.Zm. ......-. .m-.Y.s=ua7>,.--w>- .....=.. _y—, r,_. . ,. -.

.

insulating effect of the outer layers of the tissue, transients occur more slowly with depth

near the surface. However, the spherical geometry assumed results in a reversal of this

trend near the center. Due to the rapid reduction in mass near the center, later portions of

the transient occur more rapidly near the center. This is illustrated by the near instantaneous

growth of ice at the origin as shown in Figure 2 and the rapid decline in temperature after

freezing is completed as shown in Figure 1. This reversal in trend is not observed if a slab

geometry is assumed. In that case all portions of the transient occur slower with increasing

depth into the tissue.

Figure 3 reveals transient CPA concentration in the extracellular domain. The

concentration initially decreases due to the flow of water out of the cells and the flow of

CPA into the cells. The initial transient for all three locations shown is similar prior to

freezing. Only slight differences in temperature result in slightly different osmotic flow

rates. The initiation of freezing in the extracellular domain is denoted by the sudden rise in

the CPA concentration in Figure 3. The ice forms without any solute, resulting in an

increased solute concentration in the remaining extracellular liquid.

Figure 4 shows how the intracellular CPA concentration changes with time. Again, the

initial transient for all locations is approximately the same. Due to the different temperature

levels, equilibrium is approached at slightly different rates. However, when the

extracellular domain starts to form ice (Figure 2), the resulting increase in the extracellular

CPA concentration (Figure 3) results in an accelerated increase of the intracellular CPA

concentration. Once the intracellular fluid or the remaining fluid in the extracellular domain

vitrifies, all osmotic flow is stopped, and the intracellular concentrations remain constant.

-20-

. ...>.. ,.,,..<,.- ~. _ -~.~.-%., . —-mm.m n... .... . ... ..-.--.-.7? ..-,m.*m. , . ‘ . .—- ,..... --— a.n -- --,:-

Figure 5 shows that the final intracellular CPA concentration distribution is not a simple

function of position. The different times and temperature transients with position result in

a complex distribution of CPA within the intracellular domain. The base case results

presented here show that the lowest final intracellular CPA concentration occurs at the

center of the tissue with a maximum at a radial position of 0.7 cm. However, numerical

results using larger cell sizes or a reduced osmotic permeability have resulted in the final

CPA concentration in the intracellular domain being lowest at the surface of the tissue, with

a maximum near the radial position of 0.5 cm.

Figure 6 shows the transient cell diameter as a result of the osmotic flow. In general the cell

diameters initially decrease due to osmotic flow of water out of the cell. This initial

transient is similar at all locations since this is driven by the same chemical potential

differences. Only slight differences result from the different temperature levels at each

radial location. Examination of the central position shows that the cell size starts to increase

at 9 seconds and approaches its original value due to the CPA entering the cell. Then, at

125 seconds the extracellular domain starts to freeze, which causes a sudden increase in the

extracellular CPA and salt concentrations. This results in a renewed outward osmotic flow,

and further cell size reductions. Figure 6 reveals that the cells near the surface of the tissue

do not re-establish the original cell size prior to the predicted phase change in the

extracelh.dar domain. Again, the cell size transient is very sensitive to the initial cell size

and osmotic flow parameters.

7—--------

‘-i\ .\

s

‘\ ‘\\ s r=Ocm!‘\ \ \t‘\ ‘\ I ------- r= O.5cmII*\ ‘\ I\ ----- r = 1.0 cm

●\ i, \\●\. \. $\ ------ boundary

P-

0 100 200 300Time (seconds)

Figure 1: Temperature transients for three locations within the tissue. The boundarytemperature is included for reference.

0o 100 200 300

Time (seconds)

Figure 2: Extracellular ice volume fraction transient for three locations within the tissue.

-22-

.

0

/ .. -----— --- ———-_. _— _____/ , ---- —------------ ----------

/ /’

“’’””(“”CIt/1

100 200 300Time (seconds)

Figure 3: Extracellular domain CPA concentrations for three locations within the tissue.

Figure 4: Cell

o0

,i/’// It

:i :

II I —r=Ocm/’ 1’

//’ ------- r = 0.5 cm:

i/’ -----I r = 1.0 cm

/’ I//’ :

/’ /’/’ t’

100 200 300Time (seconds)

CPA concentration transients for three locations within the tissue.

-23-

0 0.2 0.4 0.6 0.8 1Radial position (cm)

Figure 5: Cellular CPA concentration distribution at the end of the transient

10

9

8

7

\ i\ 1

\II

I 1

\1I

I I

\

I r=OcmII II II ------- r = 0.5 cm\\\\

I

I1 —— —--1 r = 1.0 cmII

\ I

\I

\

tt

\ 1

-\ i\ \8‘-% ‘...

%. -------------------------------.- — — ______________________

o 100 200

Time (seconds)

Figure 6: Cell diameter transient for three locations within the tissue.

-24-

300

Conclusions

This paper presents the development of an initial modeling framework for simulation of

tissue cryopreservation processes. Successful cryopreservation of tissues is much more

difficult than the cryopreservation of a dilute suspension of cells due to the variety of

thermal transients experienced by cells at various locations within the tissue.

Many simplifying assumptions went into the creation of this model. Most of these involved

the treatment of the solidification event. It was assumed that the extracellular domain was

nucleated (naturally or artificially) near the equilibrium temperature, and that this

nucleation allowed growth of ice crystals throughout the extracelh.dar domain. When the

eutectic trough was reached in the extracellular domain, it was assumed that the remaining

liquid vitrified into a glassy state. The cellular domain was assumed to never nucleate, but

to vitrify when the neighboring extra-cellular domain vitrified.

The diffusion model is also simplistic. It is likely that pockets of high solute density can be

trapped within the extracellular space prior to the liquid fraction going to zero. In that case,

the diffusive flux might be modeled as going to zero faster than linearly with the change in

the liquid volume fraction. This suggests that the tortuosity should increase as the liquid

volume fraction is reduced.

Even with this simple model, some general conclusions were obtained. First, many time

constants govern the freezing transient of a tissue. It is not possible that the entire tissue

experience the same transient. Specifically, internal layers of the tissue will experience a

slower freezing transient. Second, it is shown that the final frozen state of tissue will not be

uniform. The solute concentrations and cell sizes will vary as a function of depth into the

-25-

. --.:,.~,--- - r ‘—---TKFZ.~)~,,, ,. ..., ,. .,.,,,,. ........... . . ,. .,---- -. . .

. I

tissue even if the solute is initially uniformly distributed. FFinally, it is demonstrated that

diffusion time constants are too slow to distribute CPA evwenly throughout any reasonably

sized tissue. Although such qualitative conclusions could b be reached on the basis of

physical considerations, our model can provide quantitative e aspects of these effects and can

generate the time and space profiles.

Development of a model allows rapid evaluation of differe~nt freezing protocols for natural

tissues to maximize the survival rate of the cells throughout the tissue. The model may

prove even more useful for the evaluation of the survival oEIf engineered artificial tissues. In

this case, various design options could also be evaluated.

To accomplish accurate quantitative predictions,

model structure will be necessary. Most of these

however. L refinements of the current

will requtiire experimental measurements

to determine model

Nomenclature

parameters.

Numerical values are provided to describe the sample calcm.dation presented in this paper.

A

CP

cd

Dc)E

h

J

k

L

P

R

cell membrane area per unit volume

heat capacitance (sensible heat)

solute concentration (initial conditions: C~C== C~X=0.3M; CdX= lM CdC=OM)

cell diameter (10 microns)

solute diffusion coefficient (1X10-9 m2/sec. fotir salt, 2x10-10 for Me2S0 at O

activation energy

heat transfer coefficient (1000 W/m2/K)

osmotic flux

thermal conductivity

latent heat

’14 m3/?/N/sec., at O C, with activationmembrane permeability to water (7x1Oenergy of 60670 J/mole)

Outer boundary of domain (1 cm)

,

xstT

v1)1

vaPr

Y

~ka

Pe

‘r

0)

Subscripts

b

c

m

ref

s

sat

x

universal gas constant

solidification rate

time

temperature (initial condition: O C)

Me2S0 specific volume(71 cm3/mole)

cell volume fraction (0.5)

liquid phase volume fraction (initial condition: UX=0.3, ctc=0.5)

salt content

Me2S0 content

solid ice phase volume fraction (initial condition: O)

solidification kinetics parameter (1 C-l)

solid volume fraction (0.2)

osmotic reflection coefficient (0.8)

material density

total osmotic flow between intracellular and extracellular domains

tortuosity for extracellular solute diffusion (3)

membrane permeability to Me2S0 (2x 10-11m3/N/sec., at OC, with activationenergy of 56500 J/mole)

boundary

intracellular

Me2S0

reference value

salt

solidification condition

extracellular

Acknowledgement

This work was supported in part by Sandia Corporation, a Lockheed Martin Company for

the U. S. Department of Energy under contract DE-AC04-94AL85000, and in part by the

ERC Program of the National Science Foundation under Award Number EEC-973 1643.

The authors also wish to thank Professor Z. F. Cui, Department of Engineering Science,

University of Oxford, for his valuable input.

-,-.,--7.7,’2-. -- .77.<J7-7 .7,-?.m-c.m c.-s,.T-m.,. ,, . . .. . . . .. ,-

.

References

1. Carslaw H. S. and Jaeger, J. C. Conduction of Heat in Solids, 2nd ed. Oxford UniversityPress, London, (1959).

2. Cattral, M. S., Lakey, J. R. T., Wamock, G. L., Kneteman, N. M. and Rajotte, R. V.Effect of cryopreservation on the survival and function of murine islet isografts andallografts, Cell Transplantation, 7,373-379 (1998).

3. Darr T. B. and Hubel, A. Freezing characteristics of isolated pig and human hepatocytes,Cell Transplantation, 6, 173-183 (1997).

4. deFreitas, R. C., Diner, K. R., Lakey, J. R. T. and Rajotte, R. V. Osmotic behavior andtransport properties of human islets in a dimethyl sulfoxide solution, Cryobiology, 35230-239 (1997).

5. Devireddy R. V., Barratt P. R., Storey K. B. and Bischof J.C. Liver freezing response ofthe freeze-tolerant wood frog, Rana sylvatica, in the presence and absence of glucose II.Mathematical modeling, Cryobiology, 38,327-338 (1999).

6. Diner, K. R. Modeling of Bioheat Transfer Processes at High and Low Temperatures,Advances in Heat Transfer, Editor, Cho, Y. I., 22, Academic Press, Inc. Boston, 157-358(1992).

7. Goodman, T. R. The Heat-Balance Integral and Its Application to Problems Involving aChange of Phase, ASME Transactions, 80,335-342, (1958).

8. Guyomard, C., Rialland, L., Fremond, B., Chesne, C. and Guillouzo, A. Influence ofalginate gel entrapment and cryopreservation on survival and xenobiotic metabolismcapacity of rat hepatocytes, Toxicology and Applied Pharmacology, 141,349-356 (1996).

9. Hubel, A., Toner, M., Cravalho, E. G., Yarmush,M. L. and Tompkins, R. G. Intracellularice formation during the freezing of hepatocytes cultured in a double collagen gel,Boitechnol. Prog. 7,554-559 (199 1).

10.Kedem, O. and Katchalsky, A. Thermodynamic analysis of the permeability ofbiological membranes to non-electrolytes, Biochimica et Biophysics Acts, 27, 143-179(1958).

11. Komiski, B., Darr, T. B. and Hubel, A. Subzero osmotic characteristics of intact anddisaggregate hepatocyte spheroids, Cryobiology, 38,339-352 (1999).

12.Levin, R. L., Cravelho, E. C. and Huggins, C. E. A membrane model describing theeffects of temperature on the water conductivity of erythrocyte membrane at subzerotemperature, Cryobiology, 13,415-429 (1976).

-28-

.-, .,T--,-.-T . . , -.-.= -Y-C--.,-.m.Tz-T-.--v- . . >T....V?::.-,..,-.;:.-m,-nsz=>:,m% .-rxm--T=i7—- ---- ------

. .

13.Mazaur, P. Kinetics of water loss from cells at subzero temperatures and the likelihoodof intercellular freezing, J. Gen. Physiol., 47,347-369 (1963).

14. McGrath, J. J. Quantitative measurement of cell membrane transport, Cryobiology, 34,315-334 (1997).

15. Muldrew, K., Novak, K., Yang, H., Zernicke, R., Schachar, N. S., and McGann, L. E.Cryobiology of Articular Cartilage: Ice Morphology and Recovery of Chondrocytes,Cryobiology, 40, 102-109 (2000).

16. Pegg, D. E. Equations for obtaining melting points and eutectic temperatures for theternary system dimethyl sulphoxide/sodium chloride/water, Cryo-Letters, 7,387-394(1986).

17.Shampine, L. F., and Watts, H. A. DEPAC - Design of a User Oriented Package of ODESolvers, SAND-79-2374, Sandia Laboratories, Albuquerque, NM, (1979).

18.Smith, D. J., Devireddy, R. V., and Bischof, J. C. Prediction of thermal history andinterface propagation during freezing in biological systems-latent heat and temperature-dependent property effects, Proceedings of the 5th ASME/JSi14E Joint ThermalEngineering Conference, San Diego, CA (March 15-19, 1999).

19.Song, Y. E., Khirabadi, B. S., Lightfoot, F., Brockbank, K. G. M., and Taylor, M. J.Vitreous cyropreservation maintains the function of vascular grafts, Nature Biotechnology,18,296-299 (2000).

20. Toner, M., and Cravalho, E. G. Thermodynamics and kinetics of intra-cellular iceformation during freezing of biological cells, J. Appl. Phys. 67, 1582-1592 (1990).

21. Toner, M., Cravalho, E. G. and Karel, M. Cellular response of mouse oocytes tofreezing stress: Prediction of intracellular ice formation, J. of Biomechanical Engineering,115, 169-174 (1993).

22. Watts, P., and Grant, M. H. Cryopreservation of rat hepatocyte monolayer cultures,Human and Experimental Toxicology. 15,30-37 (1996).