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Finite Elements in Analysis and Design 44 (2008) 288297

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A finite element model for cryosurgery with coupled phase change andthermal stress aspects

Baohong Yanga, Richard G. Wana,, Ken B. Muldrew b, Bryan J. Donnellyc

aDepartment of Civil Engineering, University of Calgary, Calgary, Alta., Canada T2N 1N4bDepartment of Cell Biology and Anatomy, University of Calgary, Calgary, Alta., Canada T2N 4N1

cDepartment of Surgery, University of Calgary, Calgary, Alta., Canada T2N 4N1

Received 14 September 2007; accepted 4 November 2007

Available online 24 January 2008

Abstract

The process of freezing in a capillary porous medium is numerically simulated with regard to temperature and stress fields with consideration

of phase change in the context of cryosurgery applications. Using a transformation of the principal field variable into a new one, the so-

called freezing index, the bioheat transfer equation with a phase change is converted into a form that is readily amenable to a standard finite

element solution based on a fixed mesh, thereby circumventing the numerical difficulties associated with the moving boundary problem. The

temperature-dependent mechanical equilibrium equation is then solved in a quasi-static manner to analyze the resulting thermal stresses and

deformations. Using the developed finite element model, a multiprobe cryosurgery for prostate cancer treatment is simulated in detail based on

a real prostate geometry together with the actual cryoprobe placements and the freezing protocol used during surgery. The numerical results

are then discussed and provide both quantitative and graphical support to prostate cryosurgery planning.

2007 Elsevier B.V. All rights reserved.

Keywords: Multiprobe cryosurgery; Bioheat transfer; Phase change; Thermal stress; Freezing index; Finite element method

1. Introduction

Cryosurgery is a surgical technique that uses low-temperature

freezing to destroy abnormal tissues. Following the early

development of cryosurgical probes in the 1960s, modern

cryosurgery can now be applied to a slew of deep body con-

figurations such as the prostate, kidney, liver, breast and brain.

In a cryosurgical treatment, a single or multiprobe system is

placed in contact with the target tissue in order to extract heatquickly and inflict tissue injury in an effective manner. This is

accomplished by circulating a cryogen such as argon gas inside

the probes tip with the gas expanding from a high pressure to

a low pressure so as to induce a negative temperature change

according to the so-called JouleThomson effect. As the tem-

perature is lowered, the frozen domain propagates outward

Corresponding author.

E-mail address: [email protected] (R.G. Wan).

0168-874X/$- see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2007.11.014

from the probes into the tissue surrounding it. Therefore, the

volume of tissue affected by such a treatment is much larger

than the incisions made to insert the probes and the contacting

tissue.

As the tissue is subjected to a subzero temperature, it can

be injured due to various mechanisms, such as intracellular ice

formation, solution effects, cell membrane damage or delayed

injury [17]. Putting aside complex biophysics occurring at

the cellular level, tissue injury which characterizes the out-come of cryosurgery at the macroscopic level is controlled at

a local level by a number of parameters, such as the minimum

tissue temperature, cooling rate, duration of freezing exposure,

and mechanical stresses. From a tissue system viewpoint, tis-

sue volumes at various locations experience different thermal

and mechanical histories that impact on injury as the freezing

front propagates throughout the capillary porous medium. For

the purposes of cryosurgery planning, it is highly desirable to

compute the processes of ice propagation and induced ther-

mal stresses in the tissue system with some level of fidelity.
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B. Yang et al. / Finite Elements in Analysis and Design 44 (2008) 288 297 289

As such, one can engineer freezing strategies that will maxi-

mize tissue damage in the target region and minimize the dam-

age to the surrounding healthy tissue, and thereby enhance the

outcome of the surgery. There has been a limited number of

analytical solutions [8,9] to describe the ice formation around a

single cryosurgical probe in a regular shaped domain. In terms

of numerical modeling, most of the computations [10,11] areaimed at either a single probe or a multiprobe freezing in a do-

main with simple geometry such as a cylinder, although recent

works [12,13] attempt to look at the problem of multiprobe

placement in a given domain so as to achieve a targeted frozen

region through some optimization algorithm.

This paper presents finite element developments for simu-

lating both thermal and mechanical aspects of a multiprobe

prostate cryosurgery using a real clinical case study. To simulate

the freezing of tissues, transient bioheat equations with blood

perfusion induced heat generation as well as phase change are

solved, and the associated developed mechanical stresses are

subsequently investigated. In the thermal model, the moving

ice front is computed using a fixed finite element mesh with

a numerical strategy that involves a change in field variable,

notably, temperature into a so-called freezing index. In the me-

chanical model, the thermal strains due to the volumetric expan-

sion associated with tissue water phase change and the thermal

expansion of ice as a single phase are considered. The numeri-

cal modeling of a multiprobe prostate cryosurgery is performed

on a 3D finite element grid that has been accurately constructed

from a series of 2D ultrasound images of prostate cross-sections

captured from a patient before surgery. Details of cryoprobe

placements and thermal histories that correspond to an actual

cryosurgery are embedded into the analysis. The numerical

results give new perspectives in cryosurgery in that not onlylow temperatures are seen to contribute to tissue injury, but also

the combination of high cooling rates, extended time freezing

exposures, and high thermal stresses is very important.

2. Mathematical developments

It is hypothesized that tissue can be approximated as a cap-

illary porous medium penetrated by a biofluid (solution). By

considering surface tension effects through freezing point de-

pression as a function of tissue capillary size, one can ultimately

incorporate tissue microstructure in the modeling. However, as

a first approximation, the tissue thermal properties are herein

taken as the same as those of bulk water, given that the tissue

matrix contains approximately 78% water. Also, freezing point

depression as a result of solute and surface tension effects will

be ignored in this paper due to the high rate of cooling used in

cryosurgery.

2.1. Thermal model

The freezing of tissues is described at the continuum level

where bioheat equations [14], which include the effects of blood

perfusion and metabolic heat generation, are written with a

phase change. In such a problem, an internal moving boundary

(i.e. the ice front) with a jump condition is involved, on which

the latent heat has to be removed through heat conduction.

The ice front divides the whole domain into two regions:

the frozen region 1 and the unfrozen one 2. However, the

temperature, , over the whole domain is a continuous field

which satisfies the governing equations for bioheat conduction

in both phases, i.e.

(ki ) + bi (b ) = i cij

jt(1)

in which i , ci and ki are mass density, heat capacity and

thermal conductivity of the ith phase, respectively, where i = 1

for the frozen phase and i = 2 for the unfrozen phase. The term

bi (b ) describes the heat generation due to blood perfusion,

where b is the blood temperature and bi = mbi Cbi with mbibeing the blood perfusion rate and Cbi the specific heat of blood.

On the internal boundary, the ice front keeps an interfacial

temperature of 0 C while the latent heat has to be removed

through heat conduction, i.e.

(x, s(x)) = 0, (2)

(k22 k11) n + lV0 n = 0, (3)

where s(x) is the moment at which the ice front passes the field

point x, l = L with L being the latent heat per unit mass of

water or ice, and V0 is the velocity of the interface with a normal

n. It can be seen that the temperature gradient is discontinuous

across the interface and Eq. (3) is called the Stefan or jump

condition.

Turning to external boundary conditions, they are such that

either temperature, or constant heat flux, or linear heat flux

(Robin type) can be applied on the external boundary. These

are mathematically expressed by

= on 1, (4)

kij

jn= 0 on 2, (5)

kij

jn= qi hi on 3. (6)

The initial conditions are prescribed such that

(x, 0) = 0(x). (7)

Since the position of the interface is unknown priori, the

problem expounded in the above cannot be readily formulated

variationally over the whole domain. Herein, a transformation

which involves the introduction of a new variable v(x, t) called

the freezing index is used, i.e.

v(x, t) =

t

0 kj(x, ) d, ts(x),s(x)0 kj(x, ) d

+t

s(x)ki(x, ) d, t > s(x),

(8)

where i,j (i = j ) represent different phases and v(x, 0) = 0.

It can be proven that v is continuous over the whole domain

[15], i.e.

v1(x, s(x)) = v2(x, s(x)). (9)
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After the transformation, the governing equation for heat con-

duction over the entire domain can be rewritten in the freezing

index form, i.e.

v

t0

pi v d + Q + ijl + Cj0(x) = di v, (10)

where

i =

1 if x 1(t),

2 if x 2(t),j =

1 if x 1(0),

2 if x 2(0),

=

0 1

1 0

, Q =

t0

bib d,

pi = bi /ki , and di = i ci / ki .

The external boundary conditions and initial condition are

also transformed as follows:

v(x, t) = v(x, t ) = t

0ki (x, ) d on 1, (11)

jv

jn= 0 on 2, (12)

jv

jn= q

t0

ei v d on 3, (13)

where ei = hi / ki and q(x, t ) =t

0 qi (x, ) d. Eqs. (10)(13)

are the governing equations and boundary conditions in freez-

ing index form with the internal jump condition automatically

embedded into the formulation. The new governing equations

can be solved over the entire domain with a fixed mesh.

2.2. Mechanical model

The freezing of tissues may induce the development of

thermal stresses due to the volumetric expansion associated

with tissue water phase change and the thermal expansion or

contraction in a single phase. In this paper, the thermal stress

is studied using a quasi-static model. The soft tissues, both

in frozen and unfrozen regions, are considered as elastic ma-

terials with temperature-dependent properties. The physical

problem of solid mechanics involving the deformation of a

continuous body can be described mathematically by the

equilibrium equation, i.e.

LT + f = 0, (14)

where L denotes the differential operator, for a 3D case

LT =

j

jx0 0

j

jy0

j

jz

0j

jy0

j

jx

j

jz0

0 0j

jz0

j

jy

j

jx

,

= [x , y , z,xy ,yz ,zx ]T represents the stresses, and

f = [fx , fy , fz]T is a vector of body loads.

The boundary conditions can be either imposed values for

displacements on u or tractions on t, i.e.

u = u on u, (15)

1T = t on t, (16)

where u = [ux , uy , uz]T

represents the displacements,t = [tx , ty , tz]

T is a vector of surface tractions, and the matrix

1 is related to the unit normal vector n by

1T =

nx 0 0 ny 0 nz0 ny 0 nx nz 0

0 0 nz 0 ny nx

.

The prostate tissue material behavior is herein described by an

elastic constitutive tensor D such that

= D( th) (17)

with the total strain and the thermal strain th described by

= Lu, (18)

th =

R

() d +1

3eFwF ()

m, (19)

where R is a reference temperature at which there is no ther-

mal strain, () is a temperature-dependent thermal expansion

coefficient, e is the volumetric expansion associated with tissue

water phase change, Fw is the water fraction in prostate tissue,

F () is the fraction of ice in water, and m=[1, 1, 1, 0, 0, 0]T. As

such, Eq. (19) represents the thermal strain caused by isotropic

thermal expansion or contraction in a single phase, as well as

the volumetric expansion associated with water phase change.

3. Finite element implementation

3.1. Variational formulation

Applying variational principles to Eqs. (10) and (14), and

using boundary conditions, one arrives at a variational form of

the governing equations, i.e.

(di v)v d +

v v d

+

t

0

pi v d v d + 3

t

0

ei v d v d=

(ijl + Cj0)v d +

Qv d +

3

qv d,

(20)TD d =

TDth d +

uTfd +

t

uTt d.

(21)

The next step is to apply finite element approximations to freez-

ing index v and displacement u fields, i.e.

v(x, t ) = N(x) V(t), v(x, t ) = N(x) V(t ), (22)

u(x, t ) = M(x) U(t), u(x, t ) = M(x) U(t ) (23)

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in which, N and M denote shape functions, while V and U

denote time-dependent nodal freezing index and nodal displace-

ments, respectively. Noting that V and U are arbitrary we

obtain the following discrete problem, i.e.:

AV + BV + t

0

V d = R, (24)

KU = F, (25)

where

A =

di NTN d,

B =

jN

jx

T jN

jx

d,

=

pi NTN d +

3

ei NTN d,

R =

NT(ijl + Cj0) d +

NTQ d +

3

NTq d

(26)

and

K =

BTDB d,

F =

BTDth d +

MTfd +

t

MT t d (27)

with B = LM, and the prescribed values of v and u have to be

imposed on boundaries 1

and u

, respectively.

3.2. Time integration

The time derivative in Eq. (24) can be discretized using stan-

dard finite difference methods, i.e.

Vn+1 Vn

t= (1 )Vn + Vn+1, (28)

where 01, with = 12 for CrankNicholson scheme and

= 1 for backward-Euler scheme.

The time integration in Eq. (24) can be approximated by a

summation, i.e.t0V d

n

=

nk=0

k Vktk = n. (29)

Substituting Eqs. (28) and (29) into Eq. (24) leads to the

following non-linear equation system:

n+1Vn+1 = Pn+1 (30)

in which

n+1 = An+1 + tBn+1 + tn+1,

Pn+1 = Rn+1 Bn+1Vn (1 )tBn+1Vn n. (31)

The non-linearity arises from the dependence of the matrix

coefficients on the unknown variable Vn+1. An incremental

form is used here, i.e.

n +j

jV

n

Vn

Vn = Pn (32)

in which the higher order terms are neglected so as to linearize

the system of equations.

The physical parameters such as di for the two phases which

are discontinuous across the interface are mathematically asso-

ciated with a smoothed (regularized) jump function such that

di (v) =1 h(v)

2d1 +

1 + h(v)

2d2 (33)

in which

h(v) =exp(mv/ k2) exp(mv/ k1)

exp(mv/ k2) + exp(mv/ k1), (34)

where k1 and k2 are the heat conductivity constants, and theparameter m controls the transition width over the interface. In

fact, this transition zone physically corresponds to the mushy

zone.

4. Model verification

The developed thermal model has been validated by com-

paring the numerical results with experimental data and closed

form solutions, such as the Neumanns solution for a phase

change problem in a semi-infinite region. The mechanical

model has also been validated by comparing numerical results

with known closed form solutions.

4.1. Thermal model verification

An important analytical solution available for the phase

change problem without blood perfusion is due to Neumann

[16]. For a semi-infinite region x > 0 initially at a constant

temperature T0 (T0 > Tf and Tf is the freezing point of water)

Fig. 1. Comparison with Neumanns solution.

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Fig. 2. Calculated stresses and comparison with closed form solution: (a) stress x ; (b) comparison with closed form solution.

and the surface x = 0 subsequently maintained at a temperature

of T (T < Tf), Neumanns solution in terms of the error

function is written as follows:

T =

T +Tf T

erferf

x

2(1t )1/2

, 0 < x,

T0 T0 Tf

erfc[(1/2)1/2]

erfcx

2(2t)1/2

, < x < ,

(35)

where i = ki /(i ci ) is the thermal diffusion coefficient for

ice or water. is the transient position of the ice front, which

is = 2(1t )1/2. The non-dimensional parameter can be

determined from a transcendental equation as follows:

e2

erf

k21/21 (T0 Tf)e

21/2

k11/22 (Tf T) erfc[(1/2)

1/2]

=

L1/2

c1(Tf T) . (36)

In the finite element modeling, the semi-infinite region is ap-

proximated by a 1D finite domain, 0x15 mm. The compar-

ison of the numerical results with Neumanns solution is shown

in Fig. 1, from which it can be seen that the numerical results

are almost exactly identical with the analytical solution, except

that as the ice front goes further to the right, the numerical

solution becomes slightly higher than the analytical one. This

difference is due to the fact that the infinite boundary condition

cannot be exactly replicated in the finite element approxima-

tion; at the finite length of x = 15 mm, a temperature T = T0

has been prescribed.

4.2. Mechanical model verification

To verify the temperature-dependent mechanical model, a

long hollow circular cylinder with inner and outer radii of a

and b, respectively, is considered below. The cylinder is ini-

tially at a constant temperature of T0, and the inner and outer

surfaces are subsequently maintained at temperatures ofTa and

Tb, respectively. Both the inner and outer surfaces are subjected

to free traction. We assume that the coefficient of linear ther-mal expansion may be described by = 0(1 + 1T ), and that

Youngs modulus can be represented by E = E0 exp(E1T ).

The temperature distribution is symmetrical with respect to

the axis of the cylinder and does not vary in the axial direction.

At a sufficient distance from the end, the cross-sections of the

cylinder may be assumed to remain plane. Thus, the problem

can be reduced essentially to a plane strain problem. In the

finite element modeling, this problem is simulated in a quarter

plane. For a thermal steady state, expressions for temperature,

displacement and stress can be found in Ref. [17]. Herein, we

arbitrarily choose a =6cm, b =18cm, T0 = 25C, Ta = 350

C,

Tb = 300C, 0 = 8.33 10

6 C1, 1 = 0.00146C1, E0 =

2.05 105 MPa, E1 = 0.000433C1, and Poissons ratio

= 0.293. The calculated stress x is shown in Fig. 2(a), and

the radial stress r and circumferential stress compare very

well with the closed form solutions, as shown in Fig. 2(b).

5. Numerical simulation of a prostate cryosurgery

A typical multiprobe cryosurgery for prostate cancer is next

simulated in detail using the developed model. In such a surgery,

six cryoprobes each of 2mm in diameter and connected to

a cryomachine are typically placed at strategic locations into

the prostate to freeze the tissue to be destroyed. A urethral

warming tube circulating hot water is employed to protect the

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Fig. 3. Finite element mesh generation: (a) 2D contours; (b) 3D surface; (c) wireframe view; (d) FE geometry; (e) FE mesh.

Fig. 4. Freezing protocol.

urethral lining from being destroyed during cryosurgery. Details

of the surgery such as an actual prostate geometry, cryoprobe

placement layout and freezing protocol are embedded into this

numerical simulation.

5.1. Prostate geometry and FE mesh

A semi-automatic segmentation of 2D Ultrasound images

was performed before the surgery using an ultrasound imaging

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Fig. 5. Temperature, isotherm, cooling rate and freezing exposure index fields: (a) temperature (120s); (b) temperature (300s); (c) temperature (780s);

(d) 40 C isotherm (780 s); (e) cooling rate (10 s); (f) exposure index (780s).

system, which resulted in a series of prostate and urethra outline

contours. These 2D contours are then read into a C++ program

that has been developed in this work using visualization toolkit

(VTK) [18] to reconstruct a 3D surface, as shown in Figs. 3

(a)(c). Then, this prostate surface was used to generate a 3Dvolume by subtracting the urethra and six cryoprobes from the

prostate volume, as shown in Fig. 3(d). It can be seen that there

are two prostate surfaces, the inner surface is the real prostate

and the outer surface is an enlarged one which will be used as a

zero heat flux boundary in the simulation. Finally, a tetrahedral

finite element mesh, as shown in Fig. 3(e), was generated based

on the 3D geometry. A total number of 78, 728 nodes and

357, 225 tetrahedra was used in the numerical simulations.

5.2. Boundary conditions

A constant temperature of 37 C is applied throughout the

whole domain as the initial thermal condition. As far as the

boundary conditions are concerned, a zero flux is applied on

the outer surface, a constant temperature of 39.5 C is main-

tained on the urethral lining corresponding to hot water being

circulated inside it for its protection, and a Robin type bound-

ary condition is applied on the cryoprobe surfaces. The heattransfer coefficient in the Robin boundary condition reflects

the overall thermal resistance of the probe wall with respect to

its surroundings. As such, heat transfer coefficients for differ-

ent cooling powers (duty cycle) of the actual cryomachine, i.e.

25%, 50%, 75% and 100%, are back calculated from experi-

mentally measured temperatures on a cryoprobe which freezes

a water bath.

Fig. 4 illustrates the freezing protocol followed by the sur-

geon in the operating room. Six cryoprobes are each in turn ac-

tivated at a certain cooling power delivered by the cryomachine

following a sequence chosen by the surgeon based on experi-

ence. In the mechanical model, a zero displacement boundary

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Fig. 6. Displacement, principal stress and over stress ratio fields: (a) norm of displacements (780 s); (b) maximum principal stress 1 (780s); (c) minimum

principal stress 3 (780s); (d) over stress ratio (120 s); (e) over stress ratio (300s); (f) over stress ratio (780s).

condition is applied on probe surfaces due to the sticking effect

of frozen tissue.

5.3. Numerical results

Among the various numerical results obtained in this anal-

ysis, it is of interest to examine the space/time evolution of

the temperature field within the prostate when it is subjected

to freezing. Figs. 5(a)(c) show temperature contours at se-

lected times t= 120, 300 and 780 s in a representative transver-

sal section of the prostate whose outline appears in dark. As

the probes are activated, ice forms immediately around each

probe and begins to increase in size. Fig. 5(a) shows the tem-

perature contour at 120 s, when only two probes are activated

and the two ice balls are still separated. As the ice fronts prop-

agate, the ice balls touch each other and coalesce to invade the

remaining tissue domain. Fig. 5(b) shows the temperature con-

tour at 300 s, when four probes are activated and the ice balls

have coalesced. Fig. 5(c) shows the temperature contour at

780 s, when all the six probes are activated with the freezing

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being stopped at the next second. It is seen that the coolest

spots are in the periphery of the cryoprobes where the temper-

ature is in the neighborhood of 110 C. The temperature on

the prostate capsule is in the range of 20 C at the end of

cooling. An isosurface of40 C at 780 s is shown in Fig. 5(d),

which is often cited in the literature as the critical isotherm to

indicate cell death. However, it can be seen that the 40

Cisosurface cannot cover the whole target volume.

Fig. 5(e) shows the contours of cooling rate at 10 s which re-

veals that the rapid cooling, hence intracellular freezing injury,

is confined within a region near the probes. If rapid cooling

were to be considered as the only factor contributing to tissue

injury, the latter would be limited to tissue volumes near the

probes. However, the time exposure aspect of freezing needs to

be also investigated as mentioned earlier in the paper. A freez-

ing exposure index is hereby defined as

FI= t2

t1

(t ) dt,

where (t1) =20C and (t2) = 50

C. This index describes

the product of freezing temperature and time exposure that a

volume of tissue is held at for which solution (osmotic) effects

control the tissue injury. Fig. 5(f) shows the contours of freezing

exposure index at the end of freezing, i.e. t=780 s. It is seen that

the vicinity of the probes and areas in between them are sub-

jected to very high freezing exposure indices which cause cell

damage by solution effects. By contrast, it was seen in Fig. 5(e)

that regions close to the probes were subjected to fast cooling

rates, which hypothetically leads to intracellular ice formation.

The displacements (in mm) induced by freezing are given in

Fig. 6(a) which shows a significant tissue expansion at 780 s. Ifwe compare the displacements with temperature at certain times

during the freezing, it can be seen that the largest deformations

occur at the ice front due to the volumetric expansion associated

with phase change.

The stresses (in Pa) adjacent to the probe surfaces are much

higher than those in other regions, see Figs. 6(b) and (c) which

illustrate the maximum and minimum principal stress distri-

butions 1 and 3, respectively. This may be caused by the

zero displacement (sticking) boundary condition applied on the

probe surfaces. An over stress ratio (OSR), defined as the ratio

of the von Mises equivalent (deviatoric) stress over the com-

pressive yield stress, is calculated and its distribution is shownin Figs. 6(d)(f) for the selected times t = 120, 300 and 780 s.

In these figures, the zones of OSR 1 indicate areas of poten-

tial tissue damage by thermal stresses that can be high enough

to exceed the actual tissue compressive yield stress (y =

132 MPa, [19]), and thereby potentially cause cell membrane

damage at the microscopic scale. Based on an elastic analysis,

these stresses are higher than those obtained in elastoplastic

computations where plastic flow and stress redistribution are

considered.

It is also interesting to look at the time evolution of a few

controlling field variables such as temperature and von Mises

equivalent (deviatoric) stresses in order to understand the

interplay between temperature and stresses. Fig. 7 shows the

Fig. 7. Location of selected points.

Fig. 8. Temperature histories at six selected points.

Fig. 9. History of von Mises equivalent stress with time at selected points.

location of six selected points for which the time histories

of the above field variables are investigated. The temperature

history plots in Fig. 8 show that rapid cooling occurs when

the probe is just activated, in the region very close to that probe,

which was as well seen in Fig. 5(e). Turning to stress histo-

ries, Fig. 9 reveals that there is a peak in deviatoric stresses

that characteristically occurs at the instant the cryoprobes are

switched off (end of freezing), the magnitude of which is

higher at points (points 1, 2 and 3) close to the probe. This is

due to the sticking condition at the frozen tissue and cryoprobe

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interface which causes a constraint during ice contraction at

the end of freezing. This translates into both cracking and

crushing of ice, the extent of which depends on the thawing

rate (rate at which the freezing delivery is stopped). The peaks

in deviatoric stresses are more subdued for points 4, 5 and 6

away from the cryoprobes.

6. Conclusions

The freezing of tissues is simplified as a bioheat transfer

problem with a phase change and the ice front has been calcu-

lated using the freezing index method on a fixed finite element

mesh. The tissue deformations and stress distributions due to

the thermal expansion or contraction during freezing are cal-

culated based on a quasi-static mechanical model. Using the

developed finite element model, a typical multiprobe prostate

cryosurgery is simulated in detail based on an actual prostate

geometry, cryoprobe placement layout and freezing protocol.

The numerical results show that the tissue temperature, coolingrate, freezing exposure index and mechanical stresses can all

work effectively in concert to optimize cryoinjury. Refinements

to the current model are currently being made. These include

the consideration of plastic flow deformations with stress re-

distribution, as well as freezing point depression through solute

transport and surface tension arising from tissue microstructure

as capillaries. The freeze-thaw cycling as well as the thawing

rate aspects are also being investigated.

Acknowledgments

This work is supported by the Natural Science and Engineer-

ing Council of Canada (NSERC) and the Canadian Institutesof Health Research (CIHR).

References

[1] A.A. Gage, J. Baust, Mechanisms of tissue injury in cryosurgery,

Cryobiology 37 (1998) 171186.

[2] N.E. Hoffmann, J.C. Bischof, The cryobiology of cryosurgical injury,

Urology 60 (2002) 4049.

[3] P. Mazur, Kinetics of water loss from cells at subzero temperatures and

likelihood of intracellular freezing, J. Gen. Physiol. 47 (1963) 347369.

[4] P. Mazur, Freezing of living cells: mechanisms and implications, Am.

J. Physiol. 143 (1984) 125142.

[5] M. Toner, E.G. Cravalho, Thermodynamics and kinetics of intracellular

ice formation during freezing of biological cells, J. Appl. Phys. 67 (1990)15821593.

[6] J.P. Acker, J.A.W. Elliott, L.E. McGann, Intercellular ice propagation:

experimental evidence for ice growth through membrance pores,

Biophys. J. 81 (2001) 13891397.

[7] K. Muldrew, L.E. McGann, The osmotic rupture hypothesis of

intracellular freezing injury, Biophys. J. 66 (1994) 532541.

[8] T.E. Cooper, G.J. Trezek, Analytical prediction of the temperature field

emanating from a cryogenic surgical cannula, Cryobiology 7 (1970)

7987.

[9] J.C. Bischof, J. Bastacky, B. Rubinsky, An analytical study of cryosurgery

in lung, ASME J. Biomech. Eng. 114 (1992) 467472.

[10] J. Rewcastle, G. Sandison, L. Hahn, J. Saliken, J. McKinnon, B.

Donnelly, A model for the time-dependent thermal distribution within

an ice ball surrounding a cryoprobe, Phys. Med. Biol. 43 (1998)

35193534.[11] Y. Rabin, A. Shitzer, Numerical solution of the multidimensional freezing

problem during cryosurgery, ASME J. Biomech. Eng. 120 (1998) 3237.

[12] D.C. Lung, T.F. Stahovich, Y. Rabin, Computerized planning for

multiprobe cryosurgery using a force-field analogy, Comput. Methods

Biomech. Biomed. Eng. 7 (2004) 101110.

[13] D. Tanaka, K. Shimada, Y. Rabin, Two-phase computerized planning

of cryosurgery using bubble-packing and force-field analogy, ASME J.

Biomech. Eng. 128 (2006) 4958.

[14] H.H. Pennes, Analysis of tissue and arterial blood temperatures in the

resting human forearm, J. Appl. Physiol. 1 (1948) 93122.

[15] R. Wan, Z. Liu, K. Muldrew, J. Rewcastle, A finite element model for ice

ball evolution in a multi-probe cryosurgery, Comput. Methods Biomech.

Biomed. Eng. 6 (2003) 197208.

[16] H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, Oxford

University Press, London, 1959.[17] R.B. Hetnarski, Thermal Stresses, vol. 1, North-Holland, Amsterdam,

1986, pp. 409417 (Chapter 5).

[18] W. Schroeder, K. Martin, et al., The visualization toolkit, an object-

oriented approach to 3D graphics, Kitware, 2002 www.kitware.com.

[19] Y. Rabin, P.S. Steif, M.J. Taylor, T.B. Julian, N. Wolmark, An

experimental study of the mechanical response of frozen biological

tissues at cryogenic temperatures, Cryobiology 33 (1996) 472482.
http://www.kitware.com/http://www.kitware.com/http://www.kitware.com/

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