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Models and Optimal Control in Freezing andThawing of Living Cells and Tissues

Karl-Heinz Hoffmann, Nikolai D. Botkin and Varvara L. Turova

Abstract. This paper outlines results obtained by the authors in theframework of the DFG Priority Program “Optimization with partialdifferential equations” (SPP 1253). The intention of the authors wasrelated to the application of the theory of partial differential equationsand optimal control techniques to the minimization of damaging factorsin cryopreservation of living cells and tissues in order to increase thesurvival rate of frozen and subsequently thawed out cells. The paperpresents mathematical models of the processes of freezing and thawingand describes the application of optimal control theory to the design ofoptimal cooling and warming protocols which reduce damaging effectsand improve the survival rate of cells.

Mathematics Subject Classification (2010). Primary 92B05, 35Q92, 35F21,65M06; Secondary 49L20.

Keywords. Freezing and thawing of biological cells, damaging factors,mathematical model, control system, Hamilton-Jacobi equations, gridmethods, optimal cooling and warming rates.

1. Introduction

The authors of patent [1] have found that certain tooth follicles contain theso called pluripotent (able to develop into multiple types) stem cells. Thepatent also outlines the application of such stem cells in tissue engineering,gene therapy, and in identifying, assaying or screening with respect to cell-cellinteractions.

These technologies involve freezing and thawing out of small tissue sam-ples in such a manner that the cells preserve their functional properties. Opti-mization and control are necessary here because of several competitive effectsof cooling. Slow cooling causes slow freezing of the extracellular fluid, whichresults in an increase in the concentration of salt in the remaining unfrozenpart of the extracellular solution. Since the intracellular liquid remains un-frozen relatively long, the osmotic effect leads to the cellular dehydration and

2 Karl-Heinz Hoffmann, Nikolai D. Botkin and Varvara L. Turova

shrinkage. Another effect of this process is a large stress which can damagethe integrity of cell membranes. If cooling is rapid, the water inside the cellsforms small, irregularly-shaped ice crystals (dendrites) that are relatively un-stable. If frozen cells are subsequently thawed out too slowly, dendrites willaggregate to form larger, more stable crystals that may cause damage. Max-imum viability is obtained by cooling at a rate in a transition zone in whichthe combined effect of both these mechanisms is minimized. Thus, an opti-mization problem can be formulated for mathematical models describing theprocesses of freezing and thawing. Moreover, some general arguments and ourfreezing experiments show that better results can be obtained if the ambienttemperature falls not monotonically in time, especially in the temperaturerange where the latent heat is released. Thus, time dependent optimal con-trols (optimal cooling protocols) are reasonable. Our numerical simulationsand experiments show that positive effects can be achieved by creating tem-perature gradients in the freezing area or by forcing ice nucleation throughmechanical vibration or some temperature shocks localized in a small area(seeding). Another control tool is related to cryoprotective agents which varyeutectic properties of solutions.

Additional difficulties arise when preserving structured solid tissues. Sig-nificant problems are associated with the difficulty of controlling heat transferin a large object with a complex internal structure. The presence of differ-ent cell types, each with its own requirements for optimal cryopreservation,limits cell survival when a single thermal protocol is imposed on all of thecells. Extracellular ice can cause damage of the structural integrity of the tis-sue. Mechanical stresses caused by delayed freezing of the intracellular waterare dangerous for cell membranes. Each of these is an additional source ofdamage, over and above those that are already known from studies of cellsin suspension. Therefore, optimization and optimal control are necessary inthis case.

2. Mathematical Models

Usually, tissue samples are being frozen using special plants e.g. of the Ice-Cube family developed by SY-LAB, Gerate GmbH (Austria), see Figure 1.The main part of such plants is a freezing chamber supplied by a coolingsystem. The plant is controlled by a computer that allows the user to pre-scribe a cooling protocol to be tracked. Tissue samples are put into plasticampoules containing solutions. The ampoules are placed into a rack locatedin the freezing chamber.

The main objective of this report is to outline mathematical modelsdescribing processes running in the ampoules during freezing and to sketchthe application of optimal control theory to the design of improved coolingprotocols that reduce damaging effects caused by the release of the latentheat and by stresses arising due to delayed freezing of the intracellular water.

Such models can be classified as follows.

3

• The first model utilizes mean values of thermodynamical parameters todescribe the mean boundary temperature of the ampoule. The controlhere is the temperature regime in the freezing chamber. A version ofsuch a model including the design of improved cooling protocols is im-plemented in IceCube plants and tested in experiments on freezing ofdental tissues.

• The second model of freezing deals with spatially distributed parametersand describes ice formation in the liquid surrounding the tissue sample.This approach is based on the so-called phase-field models described bypartial differential equations. They have been introduced by Caginalp(see [2]) and studied by many scientists. We base the study here on theresults of [3] where an optimal control problem for a phase-field modelis considered and investigated both from the mathematical and algo-rithmic points of view. The design of optimal controls utilizes gradientdescent methods and techniques of adjoint equations.

• The third model should describe ice formation on the cellular level. Thisincludes modeling of phase changes in the extracellular liquid confinedinside of small pores of the extracellular matrix and computing of me-chanical stresses exerted on cell boundaries due to delayed freezing ofthe intracellular water. Another effect is dehydration of cells due to theosmotic outflow caused by the increase of the salt concentration in theextracellular liquid during its freezing. The opposite effect, rehydration,occurs during thawing. Formation of dendrites that can aggregate intolagre sharp ice crystals should also be accounted for in this model.

The basic tools here are the theory of ice formation in porous mediaand Stefan type models, see [4, 5, 6].

3. Mean Value Temperature Response Model

A preliminary approach to the control of global (averaged) thermodynam-ical parameters was elaborated and verified using a Freezer IceCube 15M(SY-LAB, Gerate GmbH, Austria). IceCube 15M is developed for controlledfreezing of small tissue samples put into plastic ampoules (see Figure 1).

The main parts of the plant are a freezing chamber containing a cooling sys-tem based on gas nitrogen, a rack for placing ampoules, and two temperaturesensors that measure the chamber and sample temperatures, respectively. Theplant is equipped with a computer that allows the user to input a coolingprotocol either manually or as a file prepared in off-line regime. The computercontrols the cooling system and forces the chamber temperature to track theprescribed cooling protocol.

Freezing experiments show an irregular behavior of the temperaturenear the freezing point because of the release of the latent heat and thecrystallization. A typical response of the object to the cooling with a constantrate is shown in Figure 2.


Figure 1. Outlook of the IceCube plant (to the left) andits freezing chamber with two temperature sensors and am-poules containing tissue samples (to the right).

f

SD

L

chamber temperaturesample temperature

time

T

Figure 2. Idealized typical temperature response of thesample when the chamber temperature falls linearly in time.Three dangerous processes are pointed out.

The supercooling (S) is not too dangerous itself but in combination withthe latent heat release (L). This yields the creation of damaging dendrites: thelonger runs the latent heat release (L), the more dendrites appear. A suddendrop of the temperature (D) causes a temperature shock to cells. Therefore,it would be preferable to reduce both the duration of the latent heat release(L) and the temperature drop (D).

The following simple thermodynamical model is based on averaged val-ues of parameters (see [7] and [8]):

d

dtH = −α

(

T (H)− Te

)

(3.1)

Te = u, |u| ≤ µ. (3.2)

5

HereH and T are the averaged enthalpy density and the temperature, respec-tively; Te the chamber temperature; u the chamber temperature rate whosemaximal absolute value µ is about 20C/s; α = |S||V |−1h, where S, V , andh are the surface area, the volume, and the overall heat transfer coefficientof the ampoule, respectively. The main part of the model is the constitutivelaw T = T (H), which is specific for the sample. Nevertheless such a relationis robust for a series of similar samples. The function T (H) can be obtainedapproximatively by recovering the mean surface temperature T (t) of the am-poule from measurements for a given temperature profile Te(t). Substitutingthe functions T (t) and Te(t) into (3.1) and computing H(t) yields the pairT (t), H(t), which defines implicitly the desired function.

The aim of the control is to smooth the temperature response of thesample during the production of the latent heat (see Figure 2). Formally, thisis expressed through the minimization of the following performance index:

J =

t2∫

t1

(

ddtT (H(t))− θ0

)2

dt ≡

t2∫

t1

(

α∂T

∂H(H) · (T (H)− Te) + θ0

)2

dt, (3.3)

where θ0 is a desirable slope of the temperature curve. Ordinary differentialequations (3.1), (3.2) and functional (3.3) form a controlled system, wherethe control variable u is the rate of the chamber temperature. This modelpossesses the following nice property: the functional J does not depend (up toa positive multiplier) on the choice of α, whenever the constitutive law T (H)is restored using (3.1), and H(t) is found from (3.1) with the same value ofα. Therefore, the value of α can be simply chosen as α = 1, which savesthe trouble of measuring the physical parameters |V |, |S|, and h. Applicationof optimal control theory allows us to find cooling profiles which essentiallyimprove the temperature response of samples.

A very important question is the robustness of optimal control if thechamber and sample temperatures are measured with an error. This has beeninvestigated using the theory of optimal conflict control. The disturbanceswere considered as controls of an opposite player maximizing the objectivefunctional. New numerical methods for solving Hamilton-Jacobi equationsarising from conflict control problems with state constraints were (see [9, 10,11]) applied, and the robustness of solutions was validated.

The above sketched optimization techniques are implemented in Ice-Cube freezers. With the graphical interface of an IceCube plant, users canchoose an optimization option to compute the optimizing cooling impulse.The corresponding optimized temperature response is much better than anon-optimized one (see [7]).

4. Distributed Modeling of Ice Formation in the Liquid Milieu

Nowadays, phase field techniques for modeling of solidification and freezingprocesses become very popular. They are based on the consideration of theGibbs free energy which depends on an order parameter that assumes values


from -1 (solid) to 1 (liquid) and changes sharply but smoothly over the so-lidification front so that the sharp liquid/solid interface becomes smoothed.The rate of smoothing is controlled by a small parameter, which enables toreach arbitrary approximation of the sharp interface.

We use a phase field model (see [2],[3],[5], and [12]) to describe phasechanges in the milieu containing a tissue. The control parameter is the tem-perature in the chamber. Note that the heat flux on the boundary of theampoule is proportional to the temperature jump on the boundary. For sim-plicity, we do not include the solid part (i.e. the tissue and the walls of theampoule) into the following description bearing in mind that they are ac-counted for in numerical simulations presented in [8]. Therefore, Ω being theinterior of the ampoule, Γ the boundary of Ω. The equations read as follows:

ut +ℓ

2φt −K∆u = 0, x ∈ Ω, (4.1)

τφt − ξ2∆φ− 1

2(φ − φ3)− 2u = 0, x ∈ Ω, (4.2)

−K∂u

∂n= h(u − ue(t)− g),

∂φ

∂n= 0, x ∈ Γ, (4.3)

u|t=0 = u0 ≡ const > 0, φ|t=0 = φ0 ≡ −1. (4.4)

Here, u is the scaled distribution of the temperature; φ the phase function:φ = 1 for the frozen state and φ = −1 for the liquid state; ℓ the scaledlatent heat; K the scaled heat conductivity coefficient; h the scaled overallheat conductivity; g the boundary control (add-on to the nominal coolingprotocol ue(t) = u0 + θ0t, where θ0 < 0 is a given slope). In contrast to thework [3], we do not assume C2 regularity of Γ. It is supposed that Γ is ofthe class C0,1, i.e. Lipschitz continuous. Such an assumption covers varioustechnical designs of ampoules and permits a direct extension of the result tothe case where a solid part (tissue) immersed into the fluid is present.

The regularity of solutions is investigated in [13] and [14]. In particular,the existence and continuity of solutions in time under discontinuous initialdates, φ0 ∈ L2(Ω), is proved.

First, the following functional that estimates the mean quadratic devi-ation from the nominal cooling protocol ue(t) was considered:

J = 1

2

∫ tf

t0

∫

Ω

(u− ue(t))2dxdt. (4.5)

The adjoint system is derived as in [3] with some modifications related to theboundary method of control.

−pt −K∆p− 2q = h(u − ue(t)), x ∈ Ω,

−τqt −ℓ

2pt − ξ2∆q − 1

2(1 − 3φ2)q = 0, x ∈ Ω,

−K∂p

∂n= hp,

∂q

∂n= 0, x ∈ Γ,

p(tf ) = 0, q(tf ) = 0.

(4.6)

7

Here, p is the adjoint variable corresponding to u; q the adjoint variablerelated to φ. The appropriate regularity of solutions of (4.6) was established(see [7]) so that the derivative of the functional J with respect to the controlg is defined by the formula:

J ′(u, φ)(δg) =

∫ tf

t0

∫

Γ

δg(t, s)p(t, s)dsdt, for all δg ∈ L2(

(t0, tf )× Γ)

.

(4.7)Therefore, we can identify J ′(u, φ) with p|(0,tf )×Γ. The method of conjugategradients looks as follows. Consider the n th step. Assume that the controlgn is already known. Compute then the states un, φn, and the adjoint statespn and qn. Compute gn+1 as gn+1 = gn+αndn, where αn is an approximatesolution of the line search problem αn → minα J(gn+αdn), and the conjugatedirection dn is found from the relation (see [15] for finite dimensional case)

dn = −pn + βndn−1, βn =

[∫ t2

t1

∫

Γ

(pn−1)2dsdt

]−1 ∫ t2

t1

∫

Γ

(pn)2dsdt.

The numerical results obtained by minimizing the functional (4.5) showoscillations around the nominal protocol ue(t) (see [8]), which makes thisfunctional practically unusable. To avoid such effects, it is necessary to includethe time derivative of the temperature into the functional. Several functionalsof such a type have been considered and rejected because of essential technicaldifficulties carefully discussed in [8].

An appropriate solution is the use of the following functional that es-timates the deviation of the slope of the mean temperature from a givenslope:

J = 1

2

∫ tf

0

([u]t − θ0)2dt, where [u] = 1

|Ω|

∫

Ω

udx.

The idea is to express [u]t through values that do not contain time derivatives.Integrating the model equations (4.1) and (4.2) over Ω yields:

[u]t − θ0 = γu,φ,g(t) :=

− 1

|Ω|

[

h

∫

Γ

[u− ue(t)− g]ds+ 1

4τ

∫

Ω

(φ− φ3)dx+ 1

τ

∫

Ω

udx

]

− θ0.(4.8)

It was shown that the corresponding adjoint system is well-posed, and thederivative of the last functional J is correctly defined. Nevertheless, the im-plementation of such method is not simple. First, the computation of anoptimized cooling protocol is time consuming. Second, optimized controlsobtained have a complex structure on the boundary, which can hardly beimplemented technically. Therefore, it is reasonable to look for a control thatis constant on the boundary so that g is a function of t only. Such a controlg(t) can be computed from the condition γu,φ,g(t) = 0, i.e. [u]t − θ0 = 0 (see


relation (4.8)). This yields the following feedback rule:

g(t) = 1

h|Γ|

[

h

∫

Γ

[u− ue(t)]ds +1

4τ

∫

Ω

(φ − φ3)dx + 1

τ

∫

Ω

udx

]

+|Ω|h|Γ|θ0.

(4.9)The result of such a heuristic rule seems to be comparable with that obtainedby using adjoint equations, but the run time is sufficiently shorter (see [7]).

4.1. Improvement of the Heat Conductivity of Ampoule Walls

The authors of patent [16] have shown that the temperature gradient insidethe ampoule during freezing is favorable for the homogeneous ice nucleationand avoiding the dendritic crystal growth. The idea of the invention [16] is tocover the bottom of ampoules with a metal, say copper, and to place the tissuesample at a certain distance from the bottom. Computations performed withthe use of the above-described phase field model (4.1)–(4.4) confirm this ideaand indirectly show the improvement of the phase transition process (see [7]).Paper [17] confirms the idea to improve the heat conductivity of the ampoulewalls by placing of micro rods on its surfaces.

4.2. Ice Formation in the Extracellular Matrix

The description of ice formation in the extracellular matrix of a tissue is basedon the thermodynamics of porous media (see e.g. [4]). The main feature ofporous or soil media is that the unfrozen water content is a function of thetemperature only. Such a function can be considered as material property thatdepends on the pore size distribution and the material of the solid matrix. Theinteraction of the liquid with the solid matrix causes as a rule the homogenousnucleation of the liquid and, therefore, avoids it from supercooling. Thus,the unfrozen water content is not defined from any equation but is a givenfunction of the temperature. In freezing soil science, it is measured directlyby NMR (Nuclear Magnetic Resonance). There are a lot of theoretical workson derivation of this function (see e.g. [4] and [18]).

The model looks as follows:

ρC∂θ

∂t+ ρL

∂βℓ

∂t−K∆θ = 0, −K

∂θ

∂n= λ(θ − θ0t− g). (4.10)

Here, ρ is the density (assume that the densities of the liquid and iceare equal), L is the phase change latent heat, C is the specific heat capacity(assume they are equal for the liquid and ice), βℓ is the liquid water volumefraction (unfrozen water content), and g is the boundary control (add-onto the nominal cooling protocol θ0t, where θ0 < 0 being a given slope).The function βℓ was recovered from data obtained in experiments with tissuesamples on an IceCube plant (see [7, 8] for βℓ, simulations with system (4.10),and optimization problems).

It is well known that the increase of the volume during the water toice phase change is rather large. If the phase change occurs in a pore of aporous material, a very large stress can be exerted on the pore walls. Thecomputation of stresses in porous media is based on the theory of linear

9

elasticity, homogenization theory, and the model of ice formation from theprevious section. Simulations show that the stress inside porous media canrise up to 2Bar. The stress exerted on a sole cell located in a pore withfreezing water is about 1Bar (see [7]).

5. Consideration of Damaging Processes on Cellular Level

Processes of freezing and thawing inside of pores of the extracellular matrixplay an imported role in cryopreservation. One of the objectives of our studywas the development of mathematical models of such processes and elabora-tion of control procedures to reduce damaging factors arising during freezingand thawing.

Three injuring factors are considered. One of them is a large stressexerted on cell membranes. Another factor is excessive shrinkage and swellingdue to osmotic dehydration and rehydration occurring during the freezingand warming phases, respectively. The third effect is related to the growth ofdendrites that appear during freezing and can aggregate into large crystalsduring thawing. The growth of dendritic seeds occurs at rapid cooling rates.

Large stresses exerted on cell membranes occur at slow cooling be-cause of non-simultaneous freezing of extracellular and intracellular fluids.The use of rapid cooling rates is limited by dendritic growth. Therefore, itis reasonable to apply control theory to provide simultaneous freezing of ex-tracellular and intracellular fluids even for slow cooling rates. To this end,mathematical models containing control variables and optimization criteriawere formulated. We have started with spatially distributed models describ-ing the dynamics of phases in each spatial point (see e.g. [2], [4], and [3]), andthen an averaging was applied to reduce partial differential equations to afew ordinary differential equations with control parameters and uncertainties(see [19]). These equations contain nonlinear dependencies given by tabulardata, which complicates the application of traditional control design methodsbased on Pontryagin’s maximum principle. Nevertheless, dynamic program-ming methods related to Hamilton-Jacobi-Bellman-Isaacs (HJBI) equationsare suitable. Stable grid procedures that enable to design optimized controls(cooling protocols) for ODE systems describing competitive ice formation in-side and outside of living cells have been developed (see [9, 10], and [8]). Itshould be noticed that the formation of dendritic seeds are also included intoour models by utilizing of the corresponding thermodynamical relations.

Cell dehydration and rehydration occur because of abnormal watertransport across cell membranes. This effect is caused by the osmotic pres-sure arising because of different salt concentrations in intra- and extracellularfluids. Conventional models of cell dehydration during freezing (see e.g. [20]and [21]) describe the change of the cell volume. The cell shape is supposedto be spherical or cylindrical. However, as it is reported by biologists (see e.g.[22]), controlling cell shape is also important for the survival rate of cells. Forthis reason, mathematical models concerned with the evolution of cell shape


depending on the temperature distribution and the amount of intra- andextracellular ice have been developed in our project. The study was basedon the theory of ice formation in porous media (see [4]) and Stefan-typemodels (see [5]) describing the motion of the cell membrane due to osmoticflow into and out of the cell. The evolution of cell shape was described withHamilton-Jacobi type equations solved using both finite-difference schemesand reachable set methods (see [23] and [24]).

5.1. Balance of Ice Formation in Intra- and Extracellular Liquid

The cells of a tissue are surrounded by an extracellular liquid confined in smallvessels of an extracellular matrix. Biologists suppose that cells may commu-nicate through these vessels. Since the mechanism and the role of cell-cellinteractions are still not well understood (see e.g. [25]), we have considereda simplified model which does not take into account possible cell-cell inter-actions. For this reason, the extracellular liquid is assumed to be confined insmall non-communicating cavities or pores of the extracellular matrix (see asketch in the left part of Figure 3). Each cell has a membrane that provides aphysical separation between the intra- and extracellular environments, whichmay cause delayed freezing of intracellular liquid. This effect results in a verylarge stress exerted on the cell membrane. The magnitude of this effect canbe approximately estimated as follows: p ≈ Eice æ · (1 − βℓ), where p is thepressure, Eice is the elastic modulus of ice, æ is the ratio of volume expan-sion due to the water-to-ice phase transition, and βℓ is the unfrozen waterfraction so that 1− βℓ is the ice content. A rough estimate yields p ≈ 1Bar,which may be dangerous for cell membranes. To reduce this effect, the fluidsinside and outside the cell must freeze simultaneously. This can be achievedby lowering the freezing point of the extracellular fluid using a cryoprotector,say dimethyl sulfoxide, and optimizing cooling protocols.

Some averaging technique described in [8, 19, 24] yields the followingODE model:

x = −α1[Θ1(x)−Θ2(y)]− λ[Θ1(x)− z] + v1,

y = −α2[Θ2(y)−Θ1(x)] + v2,

z = u.

(5.1)

Here, x is the averaged density of the internal energy of the extracellularliquid, y is the same for the intracellular fluid, z is the temperature outsidethe pore (chamber temperature), u is the cooling rate, and v1, v2 are dis-turbances interpreted as data errors. The control variable u (cooling rate)is restricted by |u| ≤ µ, the disturbances v1, v2 are bounded by |v1| ≤ ν,|v2| ≤ ν. The functions Θ1(x) and Θ2(y) are the inverse to the functionsx = ρ Cθ1 + ρLβ1

ℓ (θ1) and y = ρ Cθ2 + ρLβ2ℓ (θ2), respectively. The functions

β1ℓ and β2

ℓ express the unfrozen liquid fractions for extra- and intracellularfluids, respectively, ρ is density, C the specific heat capacity, and L the specificlatent heat. Thus, Θ1(x) and Θ2(y) express the temperatures in the extra-

11

and intracellular liquids through the internal energies x and y, respectively.See also Figure 3 for the further explanations.

z

x

y

θ

βℓ(θ)

e

θ

Figure 3. To the left: two-dimensional sketch of a cell lo-cated inside a pore of the extracellular matrix. The pore isfilled with an extracellular fluid, whereas the cell containsintracellular liquid. At the center: a typical form of the func-tion defining the fraction of unfrozen liquid. The graphs ofthese functions can be shifted to the left or right accordingto the freezing points of the extra- and intracellular liquids.To the right: the inverse to the function e = ρ Cθ + ρLβℓ(θ)(expression of the temperature through the internal energy).

According to the meaning of the functions βiℓ, i = 1, 2, exact simul-

taneous freezing of the extra- and intracellular liquids can be expressed asvanishing of the following functional:

J=

∫ tf

0

∣

∣β1ℓ

(

Θ1(x(t)))

− β2ℓ

(

Θ2(y(t)))∣

∣

2dt (5.2)

that estimates the difference of the ice fractions in the extra- and intracellularregions.

Differential game (5.1) and (5.2) assumes that the objective of the con-trol u is to minimize the functional (5.2), whereas the objective of the dis-turbance is opposite. Moreover, the trajectories should remain in a stateconstraint set represented in the form of inequalities defined on trajectoriesof (5.1).

The value function of differential game (5.1) and (5.2) has been com-puted as a viscosity solution (see [26]) to the corresponding HJBI equationusing upwind grid methods developed in [9, 10, 11]. The optimal feedbackcontrol was designed by applying the procedure of extremal aiming (see [27]).The computations have been performed on a Linux computer admitting 64gbmemory and 32 threads. The coefficient of parallelization was equal to 0.7 perthread (23 times speedup totally). The grid size was equal 3003, the numberof time steps equaled 30000. The runtime is approximately 60 min.

The simulation presented in Figure 4 shows the case of different freezingpoints for the pore, θ1s, and the cell, θ2s, with θ1s − θ2s = −13C. Thus, thefreezing point of the extracellular fluid is lowered, e.g. by adding a cryopro-tector. This enables us to freeze the intracellular fluid using temperatures


laying above the freezing point of the extracellular liquid, which makes pos-sible simultaneous freezing.

The effect of supercooling of the intracellular fluid can be accounted forby introducing a kink into the dependence of the temperature on the internalenergy at the freezing point (see [28] and [8] for the exact definition and thecorresponding simulations).

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

Graph of the value function

at t = 0, z = 0

Ice fractions in the extracellular

space and in the cell versus time

β2ι (t)

β1ι (t)

V(0,x,y, z)|z=0

x0,y0

Figure 4. Almost simultaneous freezing of extra- and in-tracellular liquids.

5.2. Balance of Ice Formation with Accounting for Dendrite Growth

Accounting for the growth of dendrite seeds is done by the following modifica-tion of the internal energy inside the cell: y = ρCθ2+ρLβ2

ℓ (θ2)+ρDκβ2ℓ (θ2),

where the last term is treated as a dendrite generation energy (D is thespecific latent heat of dendrite growth). The value κ is computed as fol-lows: κ = κ0(y)

− = −κ0α2(θ2 − θ1)+, where (a)− = min a, 0 and

(a)+ = max a, 0. The objective of the control is to minimize the gener-ation of dendrite seeds along with the balance of ice formation so that thefollowing extended functional is considered:

J =

tf∫

0

∣

∣β1ℓ

(

Θ1

)

− β2ℓ

(

Θ2

)∣

∣

2dτ + κ0ρDα2

tf∫

0

(

Θ2 −Θ1

)+β2ℓ

(

Θ2

)

dτ. (5.3)

Here Θ1 = Θ1(x(t)) and Θ2 = Θ2(x(t), y(t)). Additionally, the state con-straint z ≤ 2C (remember that z is the temperature outside the pore) isimposed.

It was observed in simulations (see [8]) that 30% less dendritic seeds areformed if functional (5.3) is used instead of functional (5.2).

13

5.3. Simulation of Cell Thawing with Optimization of the Osmotic Inflowand Accounting for the Dendrite Growth

Optimization of the Osmotic Inflow. First note that, compared to the freezingprocedure, the dependence of the unfrozen water fraction on the temperatureis modified (the graph of the function is shifted to the right to account forthe delay of thawing). The functional to minimize expresses the amount ofliquid moving into the cell, which is proportional to the difference betweenthe concentration of the physiological salt solution and salt concentration inthe cell

J = α

tf∫

0

∣

∣

∣c0 − g

( m0

W0 β2ℓ (Θ2(y(τ)))

)∣

∣

∣dτ. (5.4)

Here, c0 is the concentration of the physiologic salt solution, m0 the saltamount in the cell, W0 the initial cell volume, g a function defining thesaturation of salt concentration in the cell. Thus, W0β

2ℓ is the unfrozen water

volume inside the cell, and g(m0/(W0β2ℓ )) the salt concentration in the cell.

Additionally, the state constraints

−50C ≤ z ≤ 40C and Θ2(y) ≤ 20C (5.5)

are imposed to prevent excessive warming. Optimization results and opti-mized warming protocols are presented in [8].Accounting for the Dendrite Growth. Dendrite seeds formed during freezing,can form large ice crystals at the thawing stage, which may be dangerous forcells. To take into account the growth of dendrites, the following functionalis considered:

J = α

tf∫

0

∣

∣

∣c0−g

( m0

W0 β2ℓ

(

Θ2(y(τ)))

)∣

∣

∣dτ+

maxt∈[0,tf ]

[(

1− β2ℓ (Θ2(y(t)))

)

p(t)β2ℓ (Θ2(y(t)))

]

.

(5.6)

The first term in (5.6) coincides with (5.4), and the second one expressesthe amount of ice formed due to aggregation of dendrite seeds according tothe Poisson law (see [29])

p(t) = 1− e−λ(θ,∆E,D(θ))t.

The nucleation rate λ is a function of the temperature, the activation energy,and the diffusive mobility of dendritic seeds, respectively. In the simulation,the nucleation rate λ is supposed to be constant.

Since both terms in the functional are non-antagonistic, it is clear thatrapid thawing is preferable. To prevent excessive warming, state constraints(5.5) were applied.

Two simulations of system (5.1) with state constraints (5.5) were per-formed: the first one with functional (5.4), and the second one with functional(5.6). The comparison of the simulation results shows that 7% less dendritesare formed in the second case, see [8].


5.4. Mathematical Models of Dehydration and Rehydration of Cells

Each biological cell is located inside a pore or channel filled with a salinesolution called extracellular fluid. The cell interior is separated from the outerliquid by a cell membrane whose structure ensures a very good permeabilityfor water. This leads to abnormal water transport across the cell membranein the presence of osmotic pressure caused by different salt concentrations inintra- and extracellular fluids.

5.4.1. Dehydration of Cells. In the freezing phase, the mechanism of the os-motic effect is the following. Ice formation occurs initially in the extracellularsolution. Since ice is practically free of salt, the water-to-ice phase changeresults in the increase of the salt concentration (cout) in the remaining extra-cellular liquid. The osmotic pressure forces the outflow of water from the cellto balance the intracellular (cin) and extracellular (cout) salt concentrations.Modeling of cell shrinkage is based on free boundary problem techniques. Themain relation here is the so-called Stefan condition: V = α(cout − cin), whereV is the normal velocity of the cell boundary (directed to the cell interior),and the right-hand-side represents the osmotic flux that is proportional tothe difference of the salt concentrations. The coefficient α is the product ofthe Boltzmann constant, the temperature, and the hydraulic conductivity ofthe membrane (see e.g. [21]). Note that α is practically constant in our case.The extracellular salt concentration cout depends on the unfrozen fractionβℓ(θ) of the extracellular liquid.

The intracellular and extracellular salt concentrations are estimated us-ing the mass conservation law as follows:

cin = c0inW0c /Wc(t), cout = c0outW

0/W (t), W (t) =

∫

W 0

βℓ(θ(t, x))dx,

where W 0c and Wc(t) are the initial and current cell volumes, respectively,

W 0 and W (t) are the initial and current volumes of the unfrozen part of thepore. The distribution of the temperature θ(t, x) is found from the phase fieldmodel (4.10).

The cell region Σ(t) is searched as the level set of a function Ψ(t, x), i.e.

Σ(t) = x : Ψ(t, x) ≤ 1, x ∈ R3 (or R2).

Assuming that the cell boundary propagates with the normal velocityV yields the following Hamilton-Jacobi equation for the function Ψ(t, x):

Ψt − α(cout − cin)|∇Ψ| = 0, Ψ(0, x) = infλ > 0: x ∈ λ · Σ(0). (5.7)

Here |∇Ψ| denotes the Euclidean norm of the gradient. Examples of thecorresponding computer simulations can be found in [23, 24, 8].

5.4.2. Rehydration of Cells. In the thawing phase, the osmotic effect resultsin the inflow of water into cells and therefore causes cell swelling. We use the

15

following mass conservation law for the salt content:

W 0c c0in = Ws c

0in+Wℓ cin, Wc = Ws+Wℓ, Wℓ(t) ≈

∫

W 0c

βℓ (θ(t, x))dx, (5.8)

where W 0c is the initial volume of the frozen cell, Wc the current volume of

the cell, Ws and Wℓ are volumes of the frozen and unfrozen parts of the cell,respectively, c0in and cin the salt concentrations in the frozen and unfrozenparts of the cell, respectively. Relation (5.8) yields:

cin = c0in(

1 + (W 0c −Wc)/Wℓ

)

,

where Wc is calculated from the current cell shape. The salt concentrationcout outside the cell is supposed to be a constant, and finally we arrive at anequation of the form of (5.7). The corresponding computer simulation can befound in [8].

5.4.3. Accounting for the Membrane Tension Using Reachable Set Approach.In reality, the deformation of the cell membrane depends on the membranetension which is a function of the curvature. Therefore, a more realistic ex-pression for the normal velocity of the cell boundary would be:

V(t, x) = α(cout(t)− cin(t)) + γσ(x),

where σ(x) is the angular curvature at the current point x of the cell boundary(see [8]), and γ is a constant. The resulting equation reads

Ψt −(

α(cout − cin) + γσ(x))

|∇Ψ| = 0, Ψ(0, x) = infλ > 0: x ∈ λ · Σ(0).(5.9)

Note that accounting for the curvature can alter the convexity/concavitystructure of the Hamiltonian depending on the state x.

The problem was treated using the method of reachable sets (see [27,30]). Examples of the application of reachable sets method to (5.9) in R2 canbe found in [23, 24, 8]).

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Karl-Heinz HoffmannZentrum MathematikTechnische Universitat MunchenBoltzmannstr. 285716 MunchenGermanye-mail: [email protected]

Nikolai D. BotkinZentrum MathematikTechnische Universitat MunchenBoltzmannstr. 285716 MunchenGermanye-mail: [email protected]

Varvara L. TurovaZentrum MathematikTechnische Universitat MunchenBoltzmannstr. 285716 MunchenGermanye-mail: [email protected]

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