an efficient algorithm for phase change problem in tumor treatment using αfem

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International Journal of Thermal Sciences 49 (2010) 1954e1967

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

An efficient algorithm for phase change problem in tumor treatment using aFEM

Eric Li a,*, G.R. Liu a,b, Vincent Tan a, Z.C. He c

aCentre for Advanced Computations in Engineering Science (ACES), Department of Mechanical Engineering, National University of Singapore,9 Engineering Drive 1, Singapore 117576b Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576c State Key Lab. of Advanced Technology for Vehicle Body Design & Manufacture, Hunan University, Changsha 410082, PR China

a r t i c l e i n f o

Article history:Received 30 December 2009Received in revised form7 June 2010Accepted 7 June 2010Available online 13 July 2010

Keywords:Bioheat transferNumerical methodMeshfree methodFinite element method (FEM)Alpha finite element method (aFEM)

* Corresponding author. Tel.: þ65 94311537.E-mail address: [email protected] (E. Li).

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.06.003

a b s t r a c t

Cryosurgery is an effective treatment for killing tumor tissues. During the cryosurgery process, a phasetransformation occurs in the undesired tissue. Themost popular numericalmethod for simulation of phasechange problem is fixed grid methods where the latent heat is function of temperature. In this paper,a fixed grid method using the alpha finite element (aFEM) formulation is presented to simulate the phasetransformation and temperature field during the cryosurgery process for liver tumor treatment. The aFEMmodel is first established and tuned to have a close-to-exact stiffness compared with the standard finiteelement method (FEM). Three examples of liver tumor treatment including the single probe for regularshape of tumor, andmultiple probes for regular and irregular shape of tumor are presented. The numericalresults using alpha finite element method have demonstrated the effectiveness of the procedure.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

Traditional treatments for cancer include radiation therapy,chemotherapy and surgical removal [1]. However, the side effectsof these treatments may seriously weaken the patient. Currently,cryosurgery is extensively used in tumor treatments. The mainadvantage of cryosurgery over traditional kinds of surgery isminimalinvasion, less pain, little scarring, and low cost. In a cryosurgery,a continuous tip of probe directly contacts with the target tumortissue to introduce extremely low temperature resulting an irre-versible damage to the tumor cells. At present, the use of liquidnitrogen is considered as the most popular method of freezinglesions. Themechanismof the injury induced in the targeted tissue inthe cryosurgery has two factors: one is immediate effect related tocooling rate and the other is the after solidification knownas vascularinjury [2]. When the temperature falls into the freezing range, icecrystals form in the extracellular spaces. Extracellular crystallizationcell destruction occurswhen the temperature drops to in the range of�4 �C and�21 �C [1]. As the freezing continues, the ice crystals growresulting loss of liquid water, which causes the cell shrink andmembrane damage. These deleterious effects of cell dehydration arenot always lethal to tumor cells [2]. When the cooling rate is very

son SAS. All rights reserved.

high, extracellular ice crystallization has no sufficient time to form.In that case, the intracellular ice is present making destructionof membrane and organelles [3e7]. Vascular injury is also a key todestroy the tumor cell because vasculature is essential for tumorgrowth [1,4].

The important feature of cryosurgery process is that a phasechange process occurs in the treatment. The direction of ice growth,temperature distribution, and irregular shape of frozen regioncause difficulties in the cryosurgical simulation [8]. In particular,there is a big difference in freezing between the ideal materialhaving fixed freezing point and biological tissues. Phase change forbiological material occurs over a temperature range between �1 �Cand �8 �C, and lower limit due to the complex condition and inhomogeneity of tissues [8e11].

In the development of cryosurgery, a number of numericalmodels to solve the phase change problems of biological tissue havebeen proposed. Generally, the existing numerical method can becategorized into two groups: one is front tracking method; anotheris fixed grid method [12e19]. For the front tracking technique, theliquid and solid domains are two separated domains where thelatent heat is treated as a moving boundary condition. By applica-tion of the energy balance, the velocity of interface and positioncan be determined [20]. This requires deforming or altering grids,transformation or co-ordinates introduction of special algorithmsnear the phase change interface or chooses the space step or thetime step so that the interface coincides with the grids points [20].

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E. Li et al. / International Journal of Thermal Sciences 49 (2010) 1954e1967 1955

The advantage of front tracking methods is capable to provideaccurate prediction of solideliquid interface location, but thecomputer code is quite complicated. More importantly, the fronttracking methods are not suitable for material with a finite freezingrange [19]. Due to this reason, fixed grid method has become themost powerful method in simulation of phase change problem. Inthe fixed grid method, the solid and liquid parts are treated as onesingle domainwhere the position of interface needs to be specified.The fixed grid method not only has ease implementation, but alsocan handle complex multi-dimensional problem and finite freezingmaterial. However, oscillation may occur in the fixed grid methoddue to the discontinuity of effective heat capacity [13,19]. Anotherdisadvantage of fixed grid method is that the isothermal phasechange is difficult to handle because the effective heat capacitybecomes infinite at the freezing point [13]. In order to overcome thisproblem, a narrow temperature range is assumed in the simulation.

In the fixed grid method, there are several procedures to solvephase change problems such as finite difference method (FDM) andfinite elementmethod (FEM) depending on the discretization process.The main drawback for FDM is difficult to handle complex geometry.Due to this reason, FEM has gradually replaced FDM to solve heattransfer problem. In the FEM, the triangular element and tetrahedralelements are very popular because it is very easy to generate.However, FEMhas some inherent drawbacks due to its strong relianceon the element mesh. In order to overcome the shortcoming of FEM,meshfree methods have been developed to circumvent some of theproblems and have achieved remarkable progress [21e23]. Recently,strain smoothing techniques have been applied by Chen et al. [24] tostabilize the solutions of nodal integratedmeshfree methods and alsoin the natural element method [25]. Liu et al. have applied this tech-nique to formulate the linear interpolationmethod (LC-PIM) [26]. TheLC-PIM is formulated based on a small set of nodes in a local supportdomain by using the node-base strain smoothing technique and thepoint interpolation method (PIM) for field variable approximation.The unique feature of PIM is that the shape functions possess Deltafunction property, which allows straightforward imposition of pointessential boundary conditions. Due to the incompatible nature of thePIM shape functions, a generalization to the smoothing operation isneeded [27]. Instead of using compatible strains obtained from thestrainedisplacement relation, the generalized smoothing techniqueis adopted to smooth out the gradient over various types of thesmoothing domains in LC-PIM. The formulation of LC-PIM ensures thestability and convergence, providing softening effect to themodel andsignificantly improving the stability and accuracy [27]. Liu and Zhang[28] have provided an intuitive explanation and showed numericallythat when a reasonably fine mesh is used, LC-PIM can provideupper bound solution in energy norm for elasticity problems withhomogeneous essential boundary conditions. As the strain smoothingoperation is applied over the node-based domains, LC-PIM is alsotermed as node-based smoothed point interpolation method (or NS-PIM). Applying the strain smoothing technique to the finite elementmethod, the node-based smoothed finite element method (NS-FEM)has also been formulated. The NS-FEM can be viewed as a special caseof theNS-PIM, but the n-sided polygonal elementmeshes can be used.The NS-FEM always uses compatible displacement fields createdbased on elements, and has quite similar properties as NS-PIM thatallows incompatible displacement fields [28].

It is a fascinating idea to obtain exact solution in the energynorm using numerical method. For this purpose, a novel alphafinite element method (aFEM) using triangular elements in 2D andtetrahedral elements in 3D has been developed [29]. The aFEMmakes the best use of NS-FEM with upper bound property andFEM with lower bound property. The key point in the aFEM is tointroduce an a to establish a continuous function of strain energythat includes the contributions from the FEM and NS-FEM. When

a¼ 0, the aFEM is exactly the same as FEM, and the strain energy isunderestimated. When a ¼ 1, the aFEM becomes NS-FEM, and thestrain energy is overestimated. Furthermore, the aFEM is variationconsistence. However, using quadrilateral elements cannot providethe exact solution in certain norm.

Although different a value can achieve different solution in theaFEM, the aFEM is always stable and converges for any a value. Thisensures that nomatter what is the a value, the result is still reliable.At this moment, how to “tune” the a value in the aFEM is still anopen question. Our earlier work [29] has found for many cases thatan a˛½0:4;0:6� usually achieves much better results compared withthe standard FEM using the same node. Although this is not veryprecise, it is very simple to use. In this paper, the alpha value issimple fixed at 0.5.

This is the first time to extend and apply the aFEM in dynamicproblem with phase change. The paper is organized as follows:Section 2 briefly describes the theory of phase change problem incryosurgery. Section 3 introduces the finite element formulationof phase change problem. Section 4 outlines the tumor treatmentexamples. Finally, the conclusions from the numerical results arepresented in Section 5. The detail of numerical algorithm to formu-late the phase change equation using aFEM, including various timingschemes and lumped/distributedmassmatrix is shown in Appendix.

2. Theory of phase change problem

2.1. Model of cryosurgery

Many bioheat transfer models have been developed to simulatethe bioheat transfer in the living tissue [30]. The most popularmodel in the bioheat transfer was developed by Pennes [31].The main advantage of this model is only one variable (tempera-ture) involved in the simulation. As suggested by Pennes, thebioheat heat transfer is governed by the following equation [31]:

rcvTvt

¼ kV2T þ urcbðTbl � TÞ þ Qm þ Q (1)

Where r is density of tissue; k is the thermal conductivity; u isblood perfusion rate; c is the specific heat capacity of tissue; cb isthe specific heat capacity of blood; Tbl is the blood temperature; Qmis the volumetric heat source associated with the metabolism; Q isthe external volumetric heat source. The bioheat transfer in thehuman body is governed by heat conduction, blood perfusion,metabolism and external heat source. The blood perfusion rate andmetabolism are very important heat transfer processes to regulatethe body’s temperature. The net rate heat transfer between bloodand tissue is proportional to the product of the volumetric perfu-sion rate and the difference between the arterial blood temperatureand the local tissue temperature. In the cryosurgery, the extremelylow temperature is useful to destroy the malignant tissues. Thereare several assumptions made in the simulation of phase change ofcryosurgery.

1. Latent heat is constant.2. The thermal properties vary at the point of completely phase

change from liquid to solid.3. The density is constant at the solid and liquid region.4. At the interface, only heat conduction is involved in the heat

transfer.5. Blood flow rate is constant when the temperature is above the

lower phase-transition temperature.6. The metabolism is zero when the temperature is at the freezing

range.7. Liquid fraction is taken to be a function of temperature only.

H(T)

Ceff

E. Li et al. / International Journal of Thermal Sciences 49 (2010) 1954e19671956

2.2. Mathematical formulation of phase change problem

Let us consider a bounded region is divided into three regions,liquid part U1, solid region U3, and mushy part U2 as shown inFig. 1.Bioheat transfer in the solid:

rcsvTvt

¼ V$ðKsVTsÞ þ usrcsðTbl � TÞ þ Qm þ Q (2)

Bioheat transfer in the liquid:

rclvTvt

¼ V$ðKlVTlÞ þ ulrclðTbl � TÞ þ Qm þ Q (3)

Heat balance at the solid/liquid interface:

ks

�vTsvn

�� kl

�vTlvn

�¼ ðrlhl � rshsÞvx � rlhlvl (4)

where hs, hl are the enthalpies per unit mass of solid and liquid, vx,vl are the velocity of the interface and liquid at the interface.

Conservation of mass at the interface:

ðrl � rsÞvx ¼ rlvl (5)

vl ¼ �rs � rlrl

vx (6)

Substitute Eq. (6) into Eq. (4)

ksvTsvn

� klvTlvn

¼ rsLvx (7)

In this paper, the density is set constant at the whole domain,thus the bioheat transfer at the interface:

ks

�vTsvn

�� kl

�vTlvn

�¼ rL

vsvt

(8)

T1 ¼ T2 ¼ Tf for t > 0 and x ¼ sðtÞ (9)

where n is the unit normal on the phase interface (pointing into theliquid), s is the velocity of the interface and L is the latent heat perunit mass of solid. For a number of simple geometry and conditions,analytical and approximate solutions to the Stefan problem areavailable [32].

2.3. The enthalpy method

The essence of the enthalpy method is that the latent heat effectis incorporated into the heat capacity dependent on the tempera-ture. Thus, a newparameterH (enthalpy) is introduced. The enthalpy

Solid 1Ω

Moving

interface

Moving

interface

2Ω 3Ω

Mushy

Region

Liquid

Fig. 1. Domain of phase change.

is defined as the sum of the latent and sensible heat effect.Forisothermal freezing, the enthalpy H is defined as following:

HðTÞ ¼Z

rcsðTÞdT�T < Tf

�HðTÞ ¼

ZrcsðTÞdT þ rLþ

Zrcf ðTÞdT

�T � Tf

� (10)

In the biological material, there is no single freezing tempera-ture. Thus, a mush zone exists between the solid and liquid part.

HðTÞ ¼Z

rcsðTÞdT ðT < TsÞ

HðTÞ ¼Z

rcsðTÞdT þ rLþZ

rcf ðTÞdTþZrclðTÞdT ðTs � T � TlÞ

HðTÞ ¼Z

rcsðTÞdT þ rLþZ

rclðTÞdT ðT > TlÞ ð11Þ

where Ts is the solidus temperature, Tl is the liquidus temperature.With the above definition, the bioheat equation in Eq. (1) can berewritten in terms of enthalpy:

vHvt


where H is the enthalpy. Assume the linear release of latent heatover the mushy region, the variation of H(T) with temperature isshown in Fig. 2:

Take the derivative of H

vHvt

¼ vHvT

vTvt

(13)

The effective heat capacity is expressed as:

ceff ¼ dHdT

(14)

Thus, Eq. (1) becomes

rceffvTvt


Ts Tl

H(T)

Ceff

Fig. 2. Plot of enthalpy, effective heat capacity against temperature.

Tumor

ProbeT=37oC

103 mm

120 mm

Fig. 3. Geometry of investigated domain [44].

Table 1Thermal properties of liver tissues [10].

Unit Value

Thermal conductivity of the unfrozen tissue W/m �C 0.5Thermal conductivity of the frozen tissue W/m �C 2Heat capacity of the unfrozen tissue J/m3 �C 3.6 � 103

Heat capacity of frozen tissue J/m3 �C 1.8 � 103

Latent heat kJ/kg 4200Lower phase-transition temperature �C �8Upper phase-transition temperature �C �1Metabolic rate of the liver W/m3 4200Body core temperature �C 37Density of unfrozen tissue kg/m3 1000Density of frozen tissue kg/m3 1000Density of blood kg/m3 1000


The effective heat capacity, blood perfusion, metabolic heatgeneration and thermal conductivity for different phase region are:

ceff ¼ rcs ðT < TsÞceff ¼ rcf þ Q

Tl�TsðTs � T � TlÞ

ceff ¼ rcs ðT > TlÞ(16)

u ¼ 0 ðT < TsÞu ¼ 0 ðTs � T � TlÞu ¼ ub ðT > TlÞ

(17)

Qm ¼ 0 ðT < TsÞQm ¼ 0 ðTs � T � TlÞQm ¼ Qu ðT > TlÞ

(18)

k ¼ ks ðT < TsÞk ¼ ðklþksÞ

2 ðTs � T � TlÞk ¼ kl ðT > TlÞ

(19)

3. Finite element formulation for phase change problem

The discrete equations of the FEM can be obtained by multi-plying the governing Eq. (1) with a test function w in the entiredomain [33].ZU

w$rceffTtdU ¼ZU

w$kV2TdUþZU

w$ubcbðTb � TÞdU

þZU

w$QmdUþZU

w$QdU (20)

Using integration by parts:

rceff

ZU

wTtdUþ kZU

Vw$VTdUþ ubcb

ZU

wTdU

¼ ðubcbTb þ Qm þ QÞZU

wdU�ZG2

wqGdG

�ZG3

whðT � TaÞdG ð21Þ

In the FEM, the temperature is expressed as the following trial andtest function:

T ¼Xmi¼1

NiTi (22)

where Ni is the shape function, and Ti is the unknown nodaltemperature. In the Galerkin weak form, the weight function w isreplaced by shape function N, and the standard Galerkinweak formis expressed as:

rceff

ZU

NTtdUþ kZU

VN$VTdUþ ubcb

ZU

N$NTdU

¼ ðubcbTb þ Qm þ QÞZU

NdU�ZG2

NqGdG

� hZG3

N$NTdGþ hZG3

NTadG ð23Þ

The discretized system equation can be finally obtained andwrittenin the following matrix form:

½M�� _T�tþ½Kþ C�fTgt ¼ fFgt (24)

Therefore, the smoothed Galerkin weak form for Pennes’bioheat equation can be formulated.

K ¼ kZU

VN$VNdU The stiffness matrix (25)

M ¼ rceff

ZU

N$NdU The mass matrix (26)

C ¼ ubcb

ZU

N$NdUþ hZG3

N$NdG

The equivalent damping matrix ð27Þ

F ¼ðubcbTb þ Qm þ QÞZU

NdUþ hZG3

NTadG�ZG2

NqGdG

The force matrix ð28ÞEq. (24) is numerically solved via aFEM. The detail of numerical

algorithm of aFEM is presented in Appendix.

4. Numerical example

4.1. Case 1: single probe

The test problem considered is the freezing of a liver tumor. Forsmall tumor with regular shape, the single probe is sufficient tofreeze the tumor. For simplicity, the tumor is modeled as a circle of8 mm in diameter. The probe with 2 mm diameter is targeted at the

Fig. 4. Mesh for liver.


center of tumor. A schematic of the 2D geometry for the liver tumorcryosurgery is depicted in Fig. 3. The temperature at the outerboundary of liver is kept constant T¼ 37 �C. The initial temperaturein the tissue is set 37 �C.

The thermal property is listed in Table 1. Because the property oftumor is not available, the property of tumor is taken as the same asnormal tissue in the simulation. The internal working process of theprobe is very complicated, and it is assumed that the probe is

Fig. 5. Comparison for temperature contour at t ¼ 600 s.

working as a heat sink. The cooling rate is 2.4 � 10�7 W/m3 s until5 s. After 5 s, the heat sink keeps 1.2 � 10�8 W/m3 constant. Twosets of different mesh, coarse mesh with 291 nodes and fine meshwith 12 876 nodes are listed in Fig. 4. The reference model using12 876 nodes is sufficient to capture the physics of the problem,

0 100 200 300 400 500 600-250

-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture

αFEM with 291 nodesFEM with 291 nodesReference

Full scale distribution

0 5 10 15 20 25 30-120

-100

-80

-60

-40

-20

0

20

40

Time (s)

Tem

pera

ture


Zoomed-in distribution

Detail

b

a

Fig. 6. Temperature variation with time at the center of tumor. (a) Full scale distri-bution. (b) Zoomed-in distribution.


because the same numerical results are obtained using more densenodes (19 872 nodes). That implies that numerical results havealready converged using 12 876 nodes. In order to prevent therecurrence of cancer, the standard cryosurgical technique mustfreeze 10 mm beyond the tumor [1]. Thus, the mesh is dense in theregion surrounding the tumor in the coarse mesh.

Fig. 7. Size and location of ice ball.

It has been well known that final temperature is a key factorcontributing to freezing injury during cryosurgery [1,9e11].The destructive effects in cryosurgery can be classified as twotypes: one is immediate and another is delayed. The delayed effectis due to the restriction of blood flow. However, the vascular stasisshould be taken a few days. In this paper, only immediate effect(direct cell injury) is investigated. For the direct cell injury, theextracellular and intracellular ice crystallizations are serious effectto cell viability. In particular, the intracellular ice crystallization islethal to cells. The cell is collapsed due to the excessive stressresulting from volumetric expansion of water.

The temperature profiles at time t ¼ 600 s for the aFEM model,together with linear FEM and reference solutions are shown inFig. 5. It can be found that, the numerical solutions obtained usingthe present aFEM are in very good agreement with those ofthe reference ones compared with FEM. This validates our two-dimensional aFEM model for phase change problem in the bioheattransfer process. At the region surrounding the tumor, thetemperature changes very fast and phase change occurs. However,the FEMmodel using triangular element obtains very poor results inthis region. This is due to the “overly-stiff” phenomenon of a fully-compatible FEM model of assumed temperature based on theGalerkin weak form. As expected, the tissue temperature far awayfrom the tumor is almost unchanged compared the region near theprobe. In the cryosurgery, minimize the damage of healthy tissue is

Fig. 8. Comparison for temperature gradient.

Fig. 9. Geometry of liver.


crucial to determine a successful treatment. From Fig. 5, it is foundthat the healthy region of the liver tissue is not affected by thecooling too much. On the other hand, the temperature along theperiphery of the tumor is about �90 �C that is much lower than thecritical temperature (�40 �C) to kill the tumor.

Because the temperature at the center of tumor is the lowest, itis necessary to investigate the temperature variation with thetime at this point as shown in Fig. 6. It is found that there is a severeoscillation for FEM model using coarse mesh. The oscillation is dueto discontinuity in the effective heat capacity and thermal prop-erties. Additionally, the temperature variationwith time predicatedby FEM is much larger than reference. However, the aFEM modelcan avoid the oscillation in the simulation as shown in Fig. 6.Moreover, the numerical results from aFEM are more close to thereference compared with the FEM. This phenomenon is due to theuse of combination of FEM and NS-FEM which makes the modelclose-to-exact stiffness. The novel formulation of aFEM hasovercome the ‘overly-stiff’ of FEM and ‘overly-soft’ NS-FEM, thus itcan provide much more accurate results compared with the stan-dard FEM.

Fig. 7 presents the size and location of ice ball generated bythe freezing of probe at t ¼ 600 s. The isothermal surface T ¼ �1 �Cseparates the ice ball and unfrozen tissue. During the freezingprocess, the ice ball is developed by the probe. Based on the sizeand location of ice ball obtained from numerical simulation, theclinician can have a good understanding of the freezing necrosis fora specific probe. Therefore, the clinician is capable to select thecorrect probe parameters to achieve a desirable lesion size [1,9].As shown in Fig. 7, the ice ball for the FEM, aFEM and reference ispresented. The ice ball domain predicted by the FEM, aFEM andreference is listed as follow:

Fig. 10. Mesh information for

�39mm� x� 0mm;35mm� y� 75mm ðFEMmodelÞ�40mm� x� 2mm;35mm� y� 76mm ðaFEMmodelÞ�42mm� x� 3mm;32:5mm� y� 78mm ðreferencemodelÞ

It is easily noticed that the domain of ice ball developed by theFEM model is much smaller than reference model, however,the aFEM model can provide more close region compared with theFEM. In Fig. 7, it is found that the whole region developed by the iceball exceeds the region of tumor. The location and size of ice ball inmedical treatment is very beneficial to control the freezing tomaximize the damage of tumor cell. Further, these information canminimize the damage to healthy liver tissues to avoid an irrevers-ible injury to the neighboring liver tissue due to the over freezing.

The mushy zone (determined by the isothermal T ¼ �8 �C andT ¼ �1 �C) predicted by the aFEM, FEM and reference is alsodepicted in Fig. 7. As shown in Fig. 7, themushy zone is markedwithgreen color. It is obviously found that the location and size of mushyregion is developed by aFEM matches very well with the referencemodel compared with the FEM. The information of the mushy zoneis beneficial for cryosurgeon to plan a specific extent and size of thefreezing lesion [9]. In an addition, these critical isothermal linescould be used for surgical mapping [9].

The destructive mechanisms occurring after the phase transi-tion are also lethal to tumor cell in the solid state [2]. Themechanical stress is caused due to the temperature gradient duringthe cooling. Many experiments have shown that the stress due tothe constrained contraction of the frozen tissues can easily exceedthe yield strength of the frozen tissues, which results in plasticfracture or deformation [2]. As shown in Fig. 8, the temperaturegradients for the FEM, aFEM and reference model are presented.

regular shape of tumor.

Fig. 11. Mesh information for irregular shape of tumor.

Fig. 12. Comparison of temperature contour at time t ¼ 600 s.


It is found that the maximum temperature gradient occurs at theedge of probe. This is because the edge of probe is the boundary ofheat sink. It is also expected the temperature gradient at the regionfar away from the probe is quite small, which preserve the propertyof healthy tissue. From Fig. 8, it can be observed again that thecomputed results obtained using the aFEM are more accurate thanthat obtained from the linear FEM, and closer to reference one,especially in the high temperature gradient region. It is well knownthat triangular element is not suitable for FEM without additionaltreatment in some cases such high gradient problem due to the‘overly-stiff’ property. Such an “overly-stiff” behavior is responsiblefor the inaccuracy in temperature gradient solutions for triangularmesh. However, the aFEM works very well in triangular elementseven in coarse mesh in the numerical simulation. More impor-tantly, the triangular element is very easily generated in any meshgenerator. In an addition, the tissue in the human body is quitecomplicated, thus, the triangular element is a good candidate.

4.2. Case 2: multiple probes

It is very difficult to control the freezing process for large tumorsand irregular shape tumors with a single probe. Large tumorswith irregular shape are extremely tough to destroy due to difficultoptimization of cooling rate and location of probes. The use ofmultiple probes permits overlapping the required frozen areas inthe treatment of large tumors, and provides a method of destroyingthe tissue to the desired size and shape in complex tumor ablation[9]. Therefore, it is obvious that the use of multiple probes canshorten the time of treatment. However, the use of multipleprobes may limit the applicability of the thermocouple to measurethe tissue temperature, whichmakes some difficulty to monitor thefreezing effect. Thus, numerical simulation provides an effectivetool to track the tissue temperature responses and significantlyimprove the treatment. In an addition, the treatment parameterscan be optimized through numerical results before the tumoroperation [9].

Two examples of regular tumor with 12 mm diameter andirregular tumor as shown in Fig. 9 are presented in this section.Three identical probes with 2 mm diameter are targeted at tumor.In order to compare the numerical results of FEM and aFEM, foursets of mesh for regular and irregular shape of tumor are shown inFigs. 10 and 11 respectively. The cooling rates for regular andirregular shape of tumor are 1.4 � 10�7 W/m3 s and 1.44 � 10�7 W/m3 s respectively until 5 s. After 5 s, the heat sink is constant7�10�7W/m3 and 7.2�10�7W/m3 for regular and irregular tumorrespectively. Fig. 12 present the temperature contours at the timet ¼ 600 s for regular shape of tumor. From Fig. 12, it can be clearly

found that, compared with the numerical solution obtained fromstandard FEM using the same triangular mesh; the present aFEMsolution in temperature is much more accurate to the referenceresult. The temperature profile predicted by the FEM has a largedeviation from reference model. It is obviously shown that thetemperature surrounding the probe is much lower than the regionfar away from the probe in Fig. 12. This is one of attractive feature ofcryosurgery that the healthy region of the tissue is not damaged.

0 100 200 300 400 500 600

-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture


Fig. 13. Point A temperature with time for regular shape tumor. Fig. 15. Point C temperature with time for regular shape tumor.


In Section 4.1, it is mentioned the final temperature is a domi-nated factor to affect the destructive mechanism. The followingFigs. 13e15 check the numerical accuracy by plotting the timehistory of temperature at sample points A, B, C (center of eachprobe) in the regular shape of tumor (shown in Fig. 10), for presentaFEM and FEM using the same mesh as well as the referencesolutionwith 12 974 nodes. It can be clearly seen that the computedtemperatures of A, B, and C obtained from aFEM are closer to thereference results than those of linear FEM using the same linearmesh. The aFEM model so-constructed behaves ‘close-to-exact’stiffness, and hence can produce more accurate results.

Fig. 16 describes the temperature contours at the time t ¼ 600 sfor irregular shape of tumor. In Fig. 16, once again, it is clearly foundthat, the FEM results of temperature are much larger than thereference solutions at the tumor domain, while the aFEM solutionsare in good agreement with the reference solutions. For theirregular shape of tumor, the temperature far away from the probealso almost keeps unchanged, compared with temperature at thedomain surrounding the tumor. However, the temperature is more

0 100 200 300 400 500 600-250

-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture

α FEM with 352 nodesFEM with 352 nodesReference

Fig. 14. Point B temperature with time for regular shape tumor.

difficultly controlled in the irregular shape of tumor. Thus, accuratepredication of temperature distribution, precise treatment planningand optimization of cryosurgery process are very important tomakea successful treatment.

Fig. 16. Comparison of temperature contour at time t ¼ 600 s.

0 100 200 300 400 500 600

-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture


Fig. 17. Point D temperature with time for irregular shape tumor.

0 100 200 300 400 500 600-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture


Fig. 19. Point F temperature with time for irregular shape tumor.


Figs. 17, 18 and 19 compare the temperature variation with timeat sample points D, E, F (center of each probe) in the irregular shapeof tumor (shown in Fig. 11), for aFEM and FEM using 322 nodes aswell as the reference solution using 11 978 nodes. As expected, thenumerical results from aFEM are more close to the reference modelcompared with FEM model.

In the treatment planning, the reference results using very finemesh for both regular and irregular shape of tumor must be takenaround 25 h for Dell PC of Inter� Pentium (R) CPU 2.80 GHz, 2.00 GBof RAM. It is well known that accurate temperature prediction iscrucial in the treatment planning process. Although the computa-tion cost is lower for FEM model with coarse mesh (about 2 h for322 nodes), solution obtained from FEM has a large deviation withreference. For aFEM, the most important factor consuming moreCPU time is the less sparsity in the stiffness matrix due to morelocal nodes used in computing the smoothed strain fields. Based onthe analysis of Appendix, it is noted that the aFEM and FEM havethe same complexity, the computational time for aFEM usingcoarse mesh (322 nodes) is about 2.3 h. Thus, the aFEM is a goodway to simulate the phase problem in cryosurgery to reduce thecomputational time without losing the accuracy.

0 100 200 300 400 500 600-200

-150

-100

-50

0

50

Time (s)

Tem

pera

ture


Fig. 18. Point E temperature with time for irregular shape tumor.

The aFEM is found superior to the FEM in terms of computa-tional efficiency, accuracy, stable. This is because aFEM has over-come the drawback of NS-FEM with ‘overly-soft’ property and FEMwith ‘overly-stiff’ property. More importantly, the aFEM modelpossesses a “close-to-exact” stiffness and can produce exact solu-tions, in contrast to the “overly-stiff” FEM model that produceslower bound solutions, and NS-FEM that produces upper boundlower property [29].

5. Conclusion

In this paper, we have discussed the fixed grid method in phasechange problem. Additionally, the weak form formulation includingFEM and aFEM is also investigated in detail. Moreover, mass matrixselection, time discretization and integration of mass matrix havebeen analyzed. From the perspectives of accuracy, stability, conver-gence and efficiency, the aFEM formulation with lumped massmatrix is preferred. The alpha finite element method has combinedthe advantage of NS-FEM (upper bound property) and FEM (lowerbound property) to achieve ‘close-to-exact’ stiffness. In this paper,the regular and irregular shape of tumor with single and multipleprobes also have been discussed. The following conclusions can bederived as:

1. Through the numerical investigation, it is found that thenumerical results obtained from the aFEM are more accurate,stable compared to the FEMwith the same number of degree offreedom.

2. The same computational complexity is expected for both FEMand aFEM because there is no additional parameter or degreesof freedoms, and system matrices have the same dimensionwith FEM.

3. The aFEM is very suitable for analysis of phase change problemusing triangular element which can be automatically generatedin software. However, the numerical results obtained from FEMusing triangular element are poor, especially in the hightemperature gradient.

4. For the first time, the aFEM is applied in dynamic, non-linearproblem. The numerical results show that aFEM has overcomethe temporal instability induced by the excessive node-basedsmoothing operations of NS-FEM.


Appendix. Analysis of numerical algorithm for phase changeproblem

Briefing on the node-based finite element method (NS-FEM)

In this section, a brief formulation of NS-FEM is presented. In theNS-FEM, the domain discretization is the same as the standard FEM.However, the integration of stiffness matrix for NS-FEM in Eq. (25)is based on node instead of element, and the strain smoothingtechnique is applied.

To carry out the domain integration required in Eq. (25), a set ofnode-based non-overlapping sub-domains are created. The back-ground cells of three-node triangular elements are employed toconstruct the shape function. Based on the triangular mesh, theproblem domain U is further divided into N smoothing domainsassociated with nodes of the triangles such that U1WU2W.UN ¼U and UiXUj ¼ B; isj, where N is the total number of nodes. Asshown in Fig. A1, the smoothing domain Uk for node k is created byconnecting sequentially the mid-edge-point to the centroids of thesurrounding triangles of the node of interest. The boundary of thesmoothingdomainUk is labeledGkandtheunionofallUk formsexactlythe global domain U. As a result, each triangular element will bedivided into three quadrilaterals of equal area and eachquadrilateral isattachedwith nearest node. The cell associatedwith the node k is thencreated by combination of each nearest quadrilateral of all elementssurrounding the node k.

Using the node-based smoothing operation, the temperaturegradient is assumed to be the smoothed strain for node k defined by:

3k ¼ZUðkÞ

3ðxÞWðxÞdU ¼ZUðkÞ

VsTðxÞWðxÞdU (A1)

where W(x) is a given smoothing function that satisfies at leastunity property.ZUðkÞ

WðxÞdU ¼ 1 (A2)

Use the following constant smoothing function:

WkðxÞ ¼( 1SðkÞ

x˛UðkÞ

1 x;UðkÞ(A3)

Where S(k) is the area of cell expressed as:

SðkÞ ¼ZU

ðkÞ

dU ¼ 13

XNðkÞe

i¼1

SðiÞe (A4)

kΩ

kΓ

kCentroid of triangle

Field node

point-edge-Mid

in

Fig. A1. Background triangular cells and nodal smoothing domains for node k.

Where Ne is the number of elements around the node k, and Se(i) is

the area of the ith element around the node k.

3k ¼XI˛NðkÞ

n

BIðxkÞTI (A5)

Where Nn(k) is the number of nodes in the influence domain of node

k. When linear shape functions are used, it is the number of nodesthat is directly connected to node k in the triangular mesh.The BiðxkÞ is termed as the smoothed strain matrix on the cell:

BIðxkÞ ¼ 1SðkÞ

XNðkÞe

i¼1

13SðiÞe Bi (A6)

The entries Bi are constant due to the linear shape functionused. The assembly of stiffness matrix is similar to procedure in theFEM. The entries in sub-matrices of the stiffnessmatrixK in Eq. (25)are then expressed as:

K ¼XNk¼1

KðkÞ (A7)

where the KðkÞ is the smoothed stiffness matrix associate with thenode k. It can be calculated from:

KðkÞ ¼ZUk

BTkBdU ¼ BTkBUk

I Sk (A8)

It can be easily seen from Eq. (A8) that the resultant linearsystem is symmetric and banded (due to the compact supports ofFEM shape functions), which implies that the discretized systemequations can be solved efficiently.

The formulation of alpha finite element method

The aFEM makes a combination of NS-FEM and FEM by scalingan a˛½0;1�. The entries in sub-matrices of the system stiffnessmatrix KaFEM will be the assembly of the entries of those from boththe NS-FEM and the FEM. In the 2D NS-FEM, the area Se of eachtriangular element is divided into four parts with a scale factor a asshown in Fig. A2: three quadrilaterals scaled by (1 � a) at threecorners with equal area of (1/3)aVe, and the remaining Y-shapedpart in the middle of the element of area (1 � a)Ve. The NS-FEM isused to calculate three quadrilaterals, and the FEM is used tocalculate the remaining part (Y shape area). The entries in sub-matrices of the system stiffness matrix KaFEM will be the assemblyof the entries of those from both the NS-FEM and the FEM.The procedure for assembling the stiffness K is as follow:

KaFEMIJ ¼

XNn

l¼1

KNS�FEMIJðlÞ þ

XNn

k¼1

KFEMIJðkÞ (A9)

Centroid of triangleField node point-edge-Mid

+ =

FEM NS-FEM FEMα −

Fig. A2. Schematic of one partition of the node-based smoothing domain for node k asone vertex of four-node tetrahedral cell I (kek2ek3ek6).


where Ne is the number of total elements in the entire problemdomain:

KFEMIJðkÞ ¼

ZUðaÞ

e

BTI kBJdU (A10)

KNS�FEMIJðlÞ ¼

ZUðk;aÞ

�BðaÞI ðxkÞ

�Tk�BðaÞJ ðxkÞ

�dU (A11)

InwhichUðk;aÞe is the area associated the node k and bounded by the

boundary Gðk;aÞ as shown in Fig. A3; UðaÞe is the Y shape of area. The

smoothed strain BðaÞðxkÞ for Uðk;aÞe is determined by:

BðaÞI ðxkÞ ¼ 1

Sðk;aÞXNðkÞ

e

i¼1

13

�1� a2

�SðiÞe Bi ¼

1Sðk;aÞ

XNðkÞe

i¼1

13SðiÞe Bi ¼ BIðxkÞ

(A12)

The smoothed strain matrix is expressed as following:

Sðl;aÞ ¼Z

Uðk;aÞ

dU ¼XNðkÞ

e

i¼1

13

�1� a2

�SðiÞe ¼

�1� a2

�SðkÞ (A13)

Thus, the stiffness for NS-FEM is expressed as following:

KNS�FEMIJðkÞ ¼

�1� a2

�BTI kB

TJ S

ðkÞ (A14)

which implies that in the coding of aFEM, we can use the originalNS-FEM to calculate the stiffness matrix and then multiply (1� a2).

From the formulation of aFEM, it is seen that aFEM using a scalefactor a that controls the contribution of NS-FEM and FEM. Whena varies from 0 to 1, the solution of aFEM is continuous of that FEMand NS-FEM. Further, it is observed that the stiffness matrix of theaFEM has the same unknowns of only the temperature, the samebandwidth and sparsity as that of the standard FEM, and hence thesame computational complexity.

Assembly of mass matrix

The major challenge in the fixed grid method of phase changeproblem is to determine the mass matrix. Based on the expressionof mass matrix ðM ¼ rceff

RU

N$NdUÞ, it is obviously found that thelatent heat effect is incorporated into the mass matrix. If the shapefunctions used to describe stiffness over the element are the sameas mass distribution, then the corresponding mass matrix is

(k, )αΩ

( )αΩ

(k, )αΓ( )αΓ

Fig. A3. Cell associated with nodes for triangular elements in the aFEM.

regarded as the consistent mass matrix. In the lumped massmatrix, it is assumed that the mass is only concentrated on thenode. A lot of researchers have discussed the relative merits ofconsistent lumped and lumped mass matrix [33,34]. The expres-sion of consistent and lumped mass matrix is listed as following:

Mconsist ¼ rceffA3

264

12

14

14

14

12

14

14

14

12

375 Mlump ¼ rceffA

3

241 0 00 1 00 0 1

35(A15)

The A is the area of triangular element. Many authors claim thatthe lumpedmass matrix is a good way to simulate the phase changeproblem even for a constant specific heat capacity in terms of betterstability, accuracy and convergence [35e37]. In an addition, the useof a lumped mass matrix can reduce computational cost signifi-cantly. Thus, the lumped mass matrix is preferred in the simulation.

It is easy to integrate the mass matrix in completely liquid, solidormush region because they have a constant specific heat value. Thesimple approximation of specific heat to an element is based on theaverage temperature. However, this method only validates for alloywith very large finite freezing range [17]. There are some difficultiesto determine the mass matrix for the elements that contain two orthree phases. In this case, the effective heat capacity is not contin-uous function as shown in Fig. 2. The effective heat capacity may bemissed if the average temperature is not in the range of mushy.On the other hand, the effective heat capacity is overestimated ifthe average temperature is in the mushy region. A very popularsmoothing method is to split the moving front into a phase trans-formation region of width 23, which incorporating the latent heateffect into a equivalent heat capacity over the range 23 [17]. Thenumerical is sensitive to range 3. If the element is not dense or thetime step is large, the temperature maybe skips the phase changeinterval. Additionally, the latent heat of the associated volume maybe entirely missed if the interface jumps across one nodal point inless than one time step [17]. In order to tackle this problem, manyresearchers have proposed the ways to handle the discontinuity ofeffective heat capacity. Del Giudice [38] suggested the orientateddirection s of the temperature gradient:

ceff ¼

�vHvs

��vTvs

� ¼

��vHvx

�lsx þ

�vHvy

�lsy

�vTvs

� ; lsx ¼

�vTvx

��vxvs

�; lsy ¼

�vTvy

��vyvs

�(A16)

and vT=vs is defined:

vTvs

¼�vTvx

�2

þ�vTvy

�2

(A17)

Thus, the effective heat capacity is:

ceff ¼

"vHvx

vTvx

þ vHvy

vTvy

#"�

vTvx

�2

þ�vTvy

�2# (A18)

Lemmon [39] developed a newmethod to handle the effective heatcapacity:

ceff ¼8<:½ðvHvx

2þ�vHvy

�2ih�

vTvx

2þ�vTvy

�2i9=;

12

(A19)


Although the above two methods are good approximationsin evaluating the effective heat capacity, an additional variable(enthalpy) is introduced to increase the computational complexity.Further, the temperature gradient at each step must be evaluated inboth methods. Here, a simple and accurate method so called ‘nodalassembly’ of mass matrix to handle the effective heat capacity ispresented. In the FEM, the assembly of mass matrix is based onelement. The assembly of mass matrix based on the node guaranteesthat the phase change does not omitted. The principle of nodalassembly is the same as the node-based smoothed finite element(NS-FEM) as shown in Fig. A1. If one or two node’s temperature isin the mushy range, it is not necessary to calculate the averagetemperature. It implies the pre-process to judge the element inphaserange can be ignored. For numerical integration, the problemdomainU is partitioned into N integration domains Uk (k ¼ 1, 2, ., N) withone for each node based on the background triangular mesh.The integration domain Uk for node k is created by connectingsequentially themid-edge-points to the centroids of the surroundingtriangles of node k. According to the node’s temperature, we assigna specific effective heat capacity to a node and multiply the massmatrix. It is noted there is no smoothed technique involved in theassembly of mass matrix. If the effective heat capacity is continuousfunction or constant value, thenodal assembly is the same as elementassembly. The more accurate method is to find the interface usinginterpolation at each time step, which needs a lot of computationaltime and is very difficult applied in the computer code. Thus, thenodal assembly of mass matrix not only keeps the main feature offixed grid method, but also can capture the latent heat effect.

The time discretization

Although there are many time-stepping algorithms available, inthis paper only one-step methods and two-steps methods arediscussed. The stability and convergence are two important factorsin evaluating time-stepping scheme. The family of one-stepmethods is characterized as follows:

MðTnþ1 � TnÞDt

þ ðKþ CÞðqTnþ1 þ ð1� qÞTnÞ ¼ F 0 � q � 1

(A20)

The above time discretization is referred to as the q method. Inthe q method, q is a parameter that determines the time-steppingtechnique. When q ¼ 0, 1/2,1, the time-stepping scheme becomesEuler-forward, CrankeNicolson, and backward respectively.

If q > 1/2, the numerical results are unconditionally stablefor non-linear and linear problems. In the q method, only theCrankeNicolson scheme is second order accuracy in the time stepsize, and the rest are all first order accuracy. However, the qmethodpresents the oscillation in the temperature field although thenumerical results converge. The main reason is the discontinuity ofeffective heat capacity which results the oscillation to occur in theneighborhood of the freezing front. Thus, two-steps methods arepreferred in the time discretization for phase change problems. Thefirst two-time-level schemewas the Lees three-level technique [40]:

�Kn þ Cn�Tnþ1 þ Tn þ Tn�1

3

þMn

�Tnþ1 � Tn�1

2Dt

¼ Fn (A21)

Tnþ1 ¼�Kn þ Cn

3þ Mn

2Dt

�1

�� K

n þ Cn

3Tn � K

n þ Cn

3Tn�1 þ

Mn

2DtTn�1 þ Fn

(A22)

Lees three-level method is found stable, but the solutionexhibited strong oscillatory behavior [41]. This is because T definingin Eq. (A21) is based on the average T ¼ (Tnþ1 þ Tn þ Tn�1)/3[41].The most general two-steps time discretization is [41,42]:

MðTnþ1�TnÞ

DtþðKþCÞ

��12þa

�Tnþ1þ

�12�2a

�TnþaTn�1

�¼ F

(A23)

Tnþ1 ¼�MþDt�

�12þa

�ðKþCÞ

�1

�F�Dt� �

Dt� �12�2a

ðKþCÞ �M�Tn�Dt�a

�ðKþCÞTn�1

(A24)

Many researchers have found a ¼ 1/4 works very perfectly inpure solidification simulation [41]. The results found stable andvery accurate, and the thermal properties should be evaluated atthe time step tn [43]. Therefore, in this paper, we use Eq. (A24) witha ¼ 1/4 to discretize the time.

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an efficient algorithm for phase change problem in tumor treatment using αfem

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