dynamic data-driven finite element models for laser ...oden/dr._oden... · the mouse and tumor is...

Dynamic Data-Driven Finite Element Models forLaser Treatment of CancerJ. T. Oden,1 K. R. Diller,2 C. Bajaj,1 J. C. Browne,1 J. Hazle,3 I. Babuška,1 J. Bass,1

L. Biduat,3 L. Demkowicz,1 A. Elliott,3 Y. Feng,1 D. Fuentes,1 S. Prudhomme,1

M. N. Rylander,4 R. J. Stafford,3 Y. Zhang1

1Institute for Computational Engineering and Sciences, University of Texas at Austin,Austin, Texas 78712

2Department of Biomedical Engineering, University of Texas at Austin, Austin,Texas 78712

3Department of Diagnostic Radiology, University of Texas M.D. Anderson CancerCenter, Houston, Texas 77030

4School of Biomedical Engineering and Sciences, Virginia Tech–Wake ForestUniversity, Blacksburg, Virginia 24061

Received 29 November 2006; accepted 31 January 2007Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20251

Elevating the temperature of cancerous cells is known to increase their susceptibility to subsequent radiationor chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher inten-sity heat sources may be used to ablate the tissue. These facts are the basis for hyperthermia based cancertreatments. Of the many available modalities for delivering the heat source, the application of a laser heatsource under the guidance of real-time treatment data has the potential to provide unprecedented control overthe outcome of the treatment process (McNichols et al., Lasers Surg Med 34 (2004), 48–55; Salomir et al.,Magn Reson Med 43 (2000), 342–347). The goals of this work are to provide a precise mathematical frame-work for the real-time finite element solution of the problems of calibration, optimal heat source control,and goal-oriented error estimation applied to the equations of bioheat transfer and demonstrate that currentfinite element technology, parallel computer architecture, data transfer infrastructure, and thermal imagingmodalities are capable of inducing a precise computer controlled temperature field within the biologicaldomain. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 904–922, 2007

Keywords: cancer treatment; goal-oriented error estimation; hyperthermia; medical imaging; optimization;real-time computing

Correspondence to: J. T. Oden, Institute for Computational Engineering and Sciences, University of Texas at Austin,Austin, TX 78712 (email: [email protected])Contract grant sponsor: National Science Foundation; contract grant number: CNS-0540033

© 2007 Wiley Periodicals, Inc.

DYNAMIC DATA-DRIVEN FINITE ELEMENT MODELS 905

I. INTRODUCTION

Today, cancer is the second most common cause of death in the United States. In 2006, about564,830 Americans were expected to die of cancer. Among the most common forms of the diseaseare cancer of the prostate and breast, accounting for 10 and 15% of all cancer related deaths inmen and women, respectively. Current detection technology has led to a 5-year relative survivalrate of greater than 90% for prostate and breast cancer [1]. Given these survival rates, the qualityof life of the patient posttreatment is a very important factor. Traditional medical treatment ofearly stage prostate and breast cancer, including radical prostatectomy, pelvic lymphadenectomy,lumpectomy, and mastectomy, are major surgical procedures and are associated with many sur-gical complications. Ablation therapies delivered under various treatment modalities are a verypromising, minimally invasive, alternative to standard treatment, and show significant potential asan effective cancer treatment to eradicate the disease while maintaining functionality of infectedorgans and minimizing complications and relapse.

The physical basis for thermal therapies is that exposing cells to temperatures outside theirnatural environment for certain periods of time can damage and even destroy the cells. However,one of the limiting factors in all forms of ablation therapy, including cryotherapy, microwave,radio-frequency, and laser, is the ability to control the energy deposition to prevent damage toadjacent healthy tissue [2].

Recent advances in imaging technology allow the imaging of the geometry of tissue and anoverlaying temperature field using MRI and new MRTI technology. MRTI has the desirable prop-erties of excellent soft-tissue contrast, the ability to provide fast, quantitative temperature imagingin a variety of tissues, and the capability of providing biologically relevant information regardingthe extent of injury immediately following therapy [3]. Image guidance [4,5] has the potential tofacilitate unprecedented control over bioheat transfer by providing real time treatment monitoringthrough temperature feedback during treatment delivery. A similar idea using ultrasound guidedcryotherapy has been studied and shows good results [2]. However, the hypothermic regimerequires temperature differentials of 70◦ from normothermia to ablate the tissue. Tissue ablationin the hyperthermic regime requires substantially less energy deposition, temperature differentialsof 10◦, suggesting that MRTI guidance under laser therapy could provide superior control overthe bioheat transfer.

The goal of this work is to deliver a computational model of bioheat transfer that employsreal-time, patient specific data and provides real-time high fidelity predictions to be used con-comitantly by the surgeon in the laser treatment process. The model employs an adaptive hp-finiteelement approximation of the nonlinear parabolic Pennes equation and uses adjoint-based algo-rithms for inverse analysis, model calibration, and adaptive control of cell damage. The targetdisease of this research are localized adenocarcinomas of the breast, prostate, cerebrum, andother tissues in which a well-defined tumor may form. The algorithms developed also providea potentially viable option to treat other parts of the anatomy in patients with more advancedand aggressive forms of cancer who have reached their limit of radiation and chemotherapytreatment.

The adaptive data driven computational control system for controlling laser treatment ofprostate cancer described is an example of a Dynamic-Data-Driven Applications System in whichsimulation models interact with measurement devices and assimilate data over a computationalgrid for the purpose of producing high-fidelity predictions of physical events.

The development stages uses canines as the host of prostate and brain tumors. The actuallaboratory at M.D. Anderson Cancer Center in Houston, TX, is connected through a compu-tational grid to the computing center in Austin, TX. Prior to treatment, MRI data is used to

Numerical Methods for Partial Differential Equations DOI 10.1002/num

906 ODEN ET AL.

FIG. 1. The control loop.

FIG. 2. Mouse and mesh (from [6]).



generate a finite element mesh of the patient-specific biological domain. The mesh is then opti-mized to a particular quantity of interest using goal-oriented estimation and adaption. The toolthen proceeds to solve an optimal control problem, wherein the laser parameters (location ofoptical fiber, laser power, etc.) are controlled to eliminate/sensitize cancer cells, minimize dam-age to healthy cells, and control Heat Shock Protein (HSP) expression. Upon initiation of thetreatment process, real-time MRI data is employed to coregister the computational domain andMRTI data is used to calibrate the bioheat transfer model to the biological tissue values of thepatient. As new data is given intermittently, computation is compared to the measurements ofthe real-time treatment and an appropriate course of action is chosen according to the differ-ences seen. A parallel computing paradigm is used to meet the demands of rapid calibrationand adapting the computational mesh and models to control approximation and modeling error.From a computational point of view, the orchestration of a successful laser treatment is to solvethe problems of coregistration, calibration, optimal control, and mesh refinement invisibly to thesurgeon, and merely provide the surgeon with an interface to the optimal laser parameters duringtreatment. A schematic of the control loop is shown in Fig. 1. The mathematical characteriza-tion of so-called HSP expression is also described as are laboratory procedures for measuringtissue damage, mesh generation, and the development of the computational infrastructure for cal-ibration procedures, verification and validation procedures, and inverse modeling and sensitivityanalysis.

II. MESH GENERATION

The first step in simulation is to construct a finite element mesh used for the governing bioheattransfer equations. The mesh generation involves two steps: (1) construction, from MRI data, ofa finite element mesh to represent the geometry and (2) overlaying the MRTI temperature fieldonto the finite element mesh.

As an initial test of the imaging system when employed in experiments with laboratory animals,we first consider data pertaining to a mouse. A sample of the MRI data used to construct a mesh ofthe mouse and tumor is shown in Fig. 3. In the case of prostate cancer, prostate tumor cells wereinoculated in the hind legs of a mouse and grown to a tumor volume of less than 1.0 cc (Fig. 2).The hexahedral mesh of the tumor (blue) and tissue (yellow), shown in Fig. 2, was created by asemiautomatic segmentation method adapted to find the interface boundaries of tumor and tissue.

FIG. 3. MRI data of mouse.


908 ODEN ET AL.

Cubic spline and lofting methods are applied to obtain smooth boundaries from the segmentedMRI data [6, 7].

A. MRTI Data-Filtering

Spatiotemporal temperature distribution is measured during the laser treatment with update timesless than 6 seconds per image and thickness between planes of 3.5 mm. The temperature field, mea-sured in the biological domain, is first filtered to eliminate any noise and then nodal temperaturevalues are assigned to the mesh by taking the interpolant of the MRTI temperature data.

III. HSP CHARACTERIZATION AND DAMAGE MODEL

Heat shock proteins are a latent defense mechanism built into all cells, including cancer cells, thatprovide enhanced viability of tumor tissue indirectly by preventing a damaged cell from going intoapoptosis, resulting in the recurrence of cancer. Knowledge of temperature history versus timeduring treatment has been used to predict thermal necrosis in regions where damage is severe, butin regions where temperatures are insufficient to coagulate proteins, the results and subsequenteffects have been difficult to predict. This is due, in part, to the expression of HSP in the regions ofthermal stress. Consequently, knowledge of the thermal dose necessary to activate or deactivateHSP expression as a function of temperature and time in the affected tissue [6, 8] can be criticalin planning and implementing an effective thermal treatment by laser surgery.

Heat Shock Proteins are a family of gene products expressed in higher concentrations in thepresence of environmental stresses. The name vastly understates the HSP family’s astounding ver-satility. These proteins have been identified as critical components of cell survival under adverseenvironmental conditions. Various levels of HSP species indicate the health or likelihood of cellproliferation or drug resistance. Also, measures of cell damage as a function of temperatureand time for a specific patient is a critical indication of the effectiveness of thermo-therapies.Knowledge of the temperatures necessary to elicit interaction in tumor resistance and cell damageis essential to effectively produce a desired tissue response and surgical outcome. The work ofRylander [8] developed a model for HSP expression and cell damage based on an Arrhenius modelfor a mouse. Figure 4 shows the comparison of the experimentally determined HSP expressionwith the predicted values of (1). An empirical formula for the concentration of HSP expression,H(u, t) [mg

ml ], at temperature, u, and time, t , based on laboratory tests is [8]:

H(u, t) = H0eαt−βtγ (1)

where α, β, and γ are time independent parameters that may depend on temperature, with γ > 1.Cellular damage is measured in terms of the damage fraction FD predicted by means of an

Arrhenius integral formulation [6].

FD(x, t) = 1 − e−�(x,t) �(x, t) = ln(C0/Ct) = A

∫ t

0e

EaRu(x,τ) dτ (2)

where C0 is the initial concentration of healthy cells, Ct the concentration of healthy cells afterheating at time t , A the pre-exponential scaling factor, Ea the activation energy of the injuryprocess, R the universal gas constant, and u the absolute temperature.



FIG. 4. HSP expression model.

IV. BIOHEAT TRANSFER MODEL

Driving the prediction of the HSP and cellular damage is the temperature field produced by thewell-known Pennes bioheat transfer model [9]. Liu [10] has shown Pennes model [9] to givegood results for prediction of temperature field in the prostate. The formal statement of Pennesbioheat model is as follows

Find the spatially and temporally varying temperature field u(x, t) such that

ρcp∂u

∂t− ∇ · (k(u)∇u) + ω(u)cblood(u − ua) = Qlaser(x, t) in �

given the Cauchy and Neumann boundary conditions

−k(u)∇u · n = h(u − u∞) on ∂�C

−k(u)∇u · n = G on ∂�N

and the initial condition

u(x, 0) = u0 in �


910 ODEN ET AL.

FIG. 5. Comparison of arctan fit to text book values.

Here k[ Js·m·K ] and ω[ kg

s m3 ] are bounded functions of u (Fig. 5), determined empirically to be ofthe form

k(u) = k0 + k1 atan(k2(u − k3))

ω(u) = ω0 + ω1 atan(ω2(u − ω3))

cp and cblood are the specific heats, ua the arterial temperature, ρ is the density, and h is the coef-ficient of cooling. The laser source term Qlaser is a linear function of laser power and exhibitsexponential decay with distance from the source.

Qlaser(x, t) = 3P(t)µaµtrexp(−µeff‖x − x0‖)

4π‖x − x0‖µtr = µa + µs(1 − γ ) µeff = √

3µaµtr



TABLE I. Model coefficients (from Rylander [11]).

Laser wavelength 810 nm µs 14.74 cm µa 0.046 cm ρtissue 1045 kgm3

ua 310 K ρblood 1058 kgm3 cblood

p 3840 Jkg·K ctissue

p 3600 Jkg·K

P is the laser power, µa, µs are laser coefficients related to laser wavelength and give probabilityof absorption of photons by tissue, γ is the anisotropy factor, and x0 is the position of laser photonsource. Tables I and II contain constitutive data from the work of Rylander [11].

Our weak form of the Pennes bioheat transfer model is as follows:

Given a set of model, β, and laser, η, parameters,

Find u(x, t) ∈ V ≡ H 1([0, T ], H 1(�)) s.t.

B(u, β; v) = F(η; v) ∀v ∈ V

where the explicit functional dependence on the model parameters, β = (k0, k1, k3, k3, ω0(x), ω1,ω3, ω3), and laser parameters, η = (P (t), x0, µa, µs), is expressed as follows

B(u, β; v) =∫ T

0

∫�

[ρcp

∂u

∂tv + k(u, β)∇u · ∇v + ω(u, β)cblood(u − ua)v

]dxdt

+∫ T

0

∫∂�C

huvdAdt +∫

�

u(x, 0)v(x, 0)dx (3)

F(η; v) =∫ T

0

∫�

Qlaser(x, η)v dxdt +∫ T

0

∫∂�C

hu∞ v dAdt

−∫ T

0

∫∂�N

G v dAdt +∫

�

u0 v(x, 0) dx (4)

TABLE II. Model coefficients.

k(u) = k0 + k1 · atan(k2(u − k3))

k0 0.6489[

Js·m·K

]k1 0.0427

[J

s·m·K]

k2 0.0252[

1K

]k3 315.314[K]ω(u) = ω0 + ω1 · atan(ω2(u − ω3))

ω0 0.6267[

kgs·m3

]ω1 −0.137

[kg

s·m3

]ω2 2.3589

[1K

]ω3 314.262 [K]


912 ODEN ET AL.

A. Calibration, Optimal Control

The control loop involves three major problems: calibration of the Pennes model to patient specificMRTI data, optimal positioning and power supply of the laser heat source, and computing goaloriented error estimates. Each of these problems is formally stated below.

The problem of model calibration is to find the set of thermal conductivity parame-ters k0, k1, k3, k3, and blood perfusivity parameters ω0(x), ω1, ω3, ω3 that minimize thenorm of the difference between the predicted temperature field and the experimentallydetermined temperature field. ω0(x) is allowed to vary over the spatial dimension as theblood perfusivity within the necrotic core of a cancerous tumor is expected to be signifi-cantly lower than the surrounding healthy tissue. The calibration problem may be stated asfollows:

Given a set of laser parameters, η0, and an experimentally determined temperature field at timeinstances tn, n = 1, 2, . . . , Nexp within the region �χ ⊂ �

uexp(x, tn) x ∈ �χ

find the best combination of model coefficients, β∗ ∈ P, that produces the temperature field,u∗ ∈ V , such that

Q(u∗(η0, β∗), β∗) = 1

2

∫ T

0

∫�

χ(x)(u∗(x, t) − uexp(x, t))2dxdt + γ

2

∫�

∇ω∗0(x) · ∇ω∗

0(x) dx

satisfies

Q(u∗(η0, β∗), β∗) = infβ∈P

Q(u(η0, β), β)

P = {β ∈ R4 × C1(�) × R

3 : ∃! u s.t. B(u, β; v) = F(η0, v) ∀v ∈ V}

The problem of optimal laser heating control is to find the position, x0, power, P(t), andlaser frequency µa, µs that maximize damage to the cancerous tissue while minimizing damageto healthy tissue in some metric. The metric shown here is the L2([0, T ]; L2(�)) norm of thedifference between the computed solution and an “ideal” temperature field. Other metrics maybe the L2(�) norm of the difference between the computed damage fraction, equation (2), andan ideal damage field within the biological domain at the end of the treatment process or theL2([0, T ]; L2(�)) norm of the difference between the computed HSP expression field, equation(1), and an ideal HSP expression field:

‖FD(x, T ) − F idealD (x)‖2

L2(�), ‖H − H ideal‖2

L2([0,T ];L2(�))

Optimal control of the laser source term is stated as:



Given a set of model coefficients, β0, and the ideal temperature field that maximizes damageto cancerous tissue while minimizing damage to healthy tissue

φ(x) ≡{

37.0◦C x ∈ �H

50.0◦C x ∈ �C

where �H and �C are the domains of the healthy and cancerous tissue respectively, find thebest combination of laser position and power, η∗ ∈ P, that produces the temperature field,u∗ ∈ V , such that

Q(u∗(η∗, β0), η∗) = 1

2‖u∗ − φ‖2

L2([0,T ];L2(�))= 1

2

∫ T

0

∫�

(u∗(x, t) − φ(x))2 dxdt (5)

satisfies

Q(u∗(η∗, β0), η∗) = inf

η∈P

Q(u(η, β0), η)

P = {η ∈ C0([0, T ]) × R5 : ∃! u s.t. B(u, β0; v) = F(η; v) v ∈ V}

where V is the appropriate space for the bioheat transfer model.

B. Mesh Refinement

Goal oriented error estimates [12] are calculated under the following framework:

• Fixing the parameters β0 and η0, denote

C(u; v) = B(u, β0; v) − F(η0; v)

• Let Vh and Vhp represent two finite dimensional spaces such that Vh ⊂ Vhp.• Assume that ‖uhp − u‖V ≤ ‖uh − u‖V where u, uh, uhp satisfy the Pennes model

constraints

C(u; v) = 0 ∀v ∈ VC(uhp; vhp) = 0 ∀vhp ∈ Vhp

C(uh; vh) = 0 ∀vh ∈ Vh

We presume that the solution uhp ∈ Vhp is a more accurate representation of the exact solutionu ∈ V than uh ∈ Vh ⊂ Vhp; likewise an arbitrary quantity of interest, Q, evaluated at Q(uhp)

is a more accurate representation of Q(u) than the quantity of interest evaluated at Q(uh). As anexplicit example, consider the quantity of interest to be an averaged value of the temperature ina domain defined through the characteristic function, χ(x),

Q(u(η0, β0)) = 1

2‖χ(x)u(x, t)‖2

L2([0,τ ];L2(�))


914 ODEN ET AL.

Using the following Taylor series approximations,

Q(uhp) ≈ Q(uh) + δQ(uh; uhp − uh)

C(uhp; vhp) ≈ C(uh; vhp) + δC(uh; uhp − uh, vhp)

with

δQ(u; u) ≡ limθ→0

1

θ[Q(u + θu) − Q(u)]

δC(u; u; v) ≡ limθ→0

[C(u + θu; v) − C(u; v)]

we may solve for the adjoint variable php ∈ Vhp such that

δQ(uh; u) = δC(uh; u, php) ∀u ∈ Vhp

Then

Q(uhp) − Q(uh) = −C(uh; php) = F(php) − B(uh; php)

= −∫ T

0

∫�

[ρcp

∂uh

∂tphp + k(uh, β0)∇uh · ∇php + ω(uh, β0)cblood(u

h − ua) php

]dxdt

−∫ T

0

∫∂�C

h(uh − u∞)phpdAdt −∫ T

0

∫∂�N

GphpdAdt +∫ T

0

∫�

Qlaser(x, η0)phpdxdt

=∫ T

0Z(t)dt =

∫�

X (x)dx +∫

∂�C

R(x)dA +∫

∂�N

N (x)dA

Where Z(t) is defined as

Z(t) = −∫

�

[ρcp

∂uh

∂tphp + k(uh, β0)∇uh · ∇php + ω(uh, β0)cblood(u

h − ua)php

]dx

−∫

∂�C

h(uh − u∞)phpdA −∫

∂�N

Gphp dA +∫

�

Qlaser(x, η0)phpdx

This quantity may be used to determine when greater time stepping accuracy is needed. X (x),R(x), and N (x), defined as

X (x) =∫ T

0Qlaser(x, η0)p

hp −ρcp∂uh

∂tphp −k(uh, β0)∇uh ·∇php −ω(uh, β0)cblood(u

h−ua)phpdt

R(x) = −∫ T

0h(uh − u∞)phpdt N (x) = −

∫ T

0Gphpdt

may be used to determine when spatial mesh refinement is needed.



C. Optimization Framework

The calibration and optimal control problems in the control loop may be posed as the followingoptimization problem

Find q∗ ∈ P s.t.

Q(u(q∗), q∗) = infq∈P

Q(u(q), q)

where q is a parameter in the control space P,

P ≡{

q such that∃!u ∈ V : C(u, q; v) = 0 ∀v ∈ V

}

and u is the state variable determined by a variational PDE of the form

C(u, q; v) = B(u, q; v) − F(q; v) = 0 ∀v ∈ V

in the appropriate Hilbert space, V .The demands of real-time computation will only permit the computation of the following

Gateaux derivative, δQ, of the objective function.

δQ(q, q) ≡ limθ→0

1

θ[Q(u(q + θq), q + θq) − Q(u(q), q)]

The gradient is computed using the adjoint problem under the assumptions that the solution,objective function, and constraints admit the following Taylor series expansions.

u(q + θq) = u(q) + θu′(q; q) + θ 2(. . .)

Q(u + θu, q + θq) = Q(u, q) + θδuQ(u, q; u) + θδqQ(u, q; q) + θ 2(. . .)

C(u + θu, q + θq; v) = C(u, q; v) + θδuC(u, q; u; v) + θδqC(u, q; q; v) + θ 2(. . .)

where the Gateaux derivatives are defined below.

u′(q; q) ≡ limθ→0

1

θ[u(q + θq) − u(q)] ∈ V ∀q

δuQ(u, q; u) ≡ limθ→0

1

θ[Q(u + θu, q) − Q(u, q)]

δqQ(u, q; q) ≡ limθ→0

1

θ[Q(u, q + θq) − Q(u, q)]

δuC(u, q; u; v) ≡ limθ→0

[C(u + θu, q; v) − C(u, q; v)]δqC(u, q; q; v) ≡ lim

θ→0[C(u, q + θq; v) − C(u, q; v)]


916 ODEN ET AL.

Theorem. Given q ∈ P, (and hence u ∈ V by definition), the derivative of the objectivefunction is

δQ(q, q) = δqQ(u(q), q; q) − δqC(u(q), q; q, p)

where p ∈ V is the solution to the adjoint problem.

Find p ∈ V :δuC(u(q), q; u; p) = δuQ(u(q), q; u) ∀u ∈ V

Proof. The solution to the adjoint problem, p, satisfies

δuC(u(q), q; u′(q; q); p) = δuQ(u(q), q; u′(q; q)) ∀q ∈ P.

From the first variation of the PDE constraints we have that

δuC(u(q), q; u′(q; q); p) = −δqC(u(q), q; q; p).

The solution follows

δQ(q, q) = δuQ(u(q), q; u′(q; q)) + δqQ(u(q), q; q)

= δqQ(u(q), q; q) − δqC(u(q), q; q; p).

The following Taylor series expansions are used in computing the Gateaux derivatives of thevariational form of Pennes model:

k(u + θu, β + θβ) = k(u, β) + θu∂k

∂u+ θk0

∂k

∂k0+ θk1

∂k

∂k1+ θk2

∂k

∂k2+ θk3

∂k

∂k3+ θ 2(. . .)

ω(u + θu, β + θβ) = ω(u, β) + θu∂ω

∂u+ θω0

∂ω

∂ω0+ θω1

∂ω

∂ω1+ θω2

∂ω

∂ω2+ θω3

∂ω

∂ω3+ θ 2(. . .)

Notice that the variational form of the adjoint problem for the calibration, optimal control, andgoal oriented error estimates are all the same:

δuC(u, β; u, p) = δuB(u, β; u, p)

=∫ T

0

∫�

−ρcp∂p

∂tu dxdt +

∫ T

0

∫�

k(u, β)∇u · ∇p + u∂k

∂u∇u · ∇p dxdt

+∫ T

0

∫�

ω(u, β)up + u∂ω

∂u(u − ua)p dxdt +

∫�

u(x, T )p(x, T ) dx +∫ T

0

∫∂�

hup dAdt



where the only difference is source term δuQ(u(q), q; u) which depends on the quantity of interest.The strong form of the adjoint formulation is as follows:

Given the spatially and temporally varying temperature field u(x, t), find the Lagrangemultiplier p(x, t) such that

−ρcp∂p

∂t− ∇ · (k(u)∇p) + ∂k

∂u(u, β)∇u · ∇p + ω(u, β)p

+ ∂ω

∂u(u, β)p(u − ua) =

u − uexp(x, t) (calibration)u − φ(x) (optimal control)

χ(x)u (error estimate)in �

given the boundary conditions

−k(u)∇p · n = hp on ∂�C

−k(u)∇p · n = 0 on ∂�N

and the terminal condition

p(x, T ) = 0 in �

When approximations of the temperature field, u(x, t), and the adjoint variable, p(x, t), areknown, the first variation of the quantity of interest for the calibration problem may be obtainedexplicitly as follows.

δQ(β, β) =

−∫ T

0

∫�

∇u · ∇pk0 dxdt

−∫ T

0

∫�

atan(k2(u − k3))∇u · ∇pk1 dxdt

−∫ T

0

∫�

k1 (u − k3)

1 + k22 (u − k3)2

∇u · ∇p k2 dxdt

−∫ T

0

∫�

−k1 k2

1 + k22 (u − k3)2

∇u · ∇p k3 dxdt

−∫ T

0

∫�

(u − ua)p ω0(x) dxdt +∫

�

ω0(x) ω0(x) dx

−∫ T

0

∫�

atan(ω2(u − ω3))(u − ua)p ω1 dxdt

−∫ T

0

∫�

ω1 (u − ω3)

1 + ω22(u − ω3)2

(u − ua)p ω2 dxdt

−∫ T

0

∫�

−ω1 ω2

1 + ω22(u − ω3)2

(u − ua)p ω3 dxdt


918 ODEN ET AL.

The first variation for the quantity of interest for optimal control is as follows:

δQ(η, η) =

∫ T

0

∫�

3 µa µtr exp(−µeff‖x − x0‖)4 π ‖x − x0‖ p P (t)dxdt

∫ T

0

∫�

3P(t)exp(−µeff‖x − x0‖)

4π‖x − x0‖(

µtr + µa − µtr µa‖x − x0‖3(µa + µtr)

2µeff

)p µadxdt

∫ T

0

∫�

3P(t)exp(−µeff‖x − x0‖)

4 π ‖x − x0‖(

µa − µtr µa‖x − x0‖3(µa(1 − γ ))

2µeff

)p µsdxdt

∫ T

0

∫�

3 P(t) µaµtr(µeff‖x − x0‖ + 1)exp(−µeff‖x − x0‖)

4 π‖x − x0‖3(x − x0)p x0dxdt

∫ T

0

∫�


4 π‖x − x0‖3(y − y0)p y0dxdt

∫ T

0

∫�


4 π‖x − x0‖3(z − z0)p z0dxdt

V. RESULTS

Two main results are presented in this section. First, the overall computational feasibility of devel-oping a laser treatment paradigm in which high performance computers control the bioheat datatransferred from a remote site is demonstrated. Second, reliable computational predictions areshown when comparing results obtained using the Pennes model to the experimental MRTI datain canine cerebral tissue.

The typical time duration of a laser treatment is about five minutes. During a five minute span,one set of MRTI data is acquired every 6 seconds. The size of each set of MRTI data is ≈330 kB

FIG. 6. Executions times for a representative 10 second simulation (10 nonlinear state solves combinedwith 10 linear adjoint solves on 10000 dof system).



(256 × 256 × 5 voxels) and the bandwidth of a commercial T1 Internet connection is ≈300 kB/s.Accounting for connection latency, each set of MRTI data can be transferred between Houstonand Austin in under two seconds. A finite element mesh consisting of approximately 10,000degrees of freedom is sufficient to resolve the geometry of the biological domain. The executiontime of a representative 10 second bioheat transfer simulation obtained using a Crank–Nicolsonscheme is presented in Fig. 6. The execution times represent 10 nonlinear state solves of Pennesmodel (10 one second time steps) combined with 10 linear adjoint solves to be used for cali-bration, optimal control, and/or error estimates. Computations were done at the Texas AdvancedComputing Center on Dual-Core Linux Cluster. Each node of the cluster contains two Xeon IntelDuo-Core 64-bit processors (4 cores in all) on a single board, as an SMP unit. The core fre-quency is 2.66GHz and supports 4 floating-point operations per clock period. Each node contains8GB of memory. The fastest time recorded is .67 seconds, meaning that in a real time 10 secondspan Pennes model can predict out to more than two minutes! Equivalently, in a 10 second timespan, roughly 14 corrections can be made to calibrate the model coefficients or optimize the laserparameters.

Preliminary computations comparing the predictions of Pennes model to experimental MRTItaken from a canine brain show very good agreement. A manual craniotomy of a canine skull waspreformed to allow insertion of an interstitial laser fiber. Thirty-six two dimensional 256×256 pix-els MRI images, Fig. 7, of the canine brain were acquired. The field view was 200 mm × 200 mmwith each image spaced 1 mm apart. A finite element mesh of the biological domain generatedfrom the MRI data is shown in Fig. 8. The mesh consists of 8820 linear elements with a total of9872 degrees of freedom. MRTI thermal imaging data was acquired in the form of five two dimen-sional 256 × 256 pixel images every six seconds for 120 time steps. The spacing between imageswas 3.5 mm. The MRTI data was filtered then projected onto the finite element mesh. Fig. 9 showsa cutline comparison between the MRTI data and the predictions of Pennes model. It is observedthat the results delivered by the computational Pennes model slightly over diffuses the heat profilepeaks compared to measured values. However, at early times the maximum temperature value iswithin 5% of the MRTI value.

VI. CONCLUSION

Results indicate that adaptive hp-finite element technology implemented on parallel computerarchitectures and employing modern data transfer infrastructure and thermal imaging modal-ities are capable of predicting and guiding computer controlled temperature field within abiological domain at very good accuracies. Reliable finite element model simulations of lasertherapies can be computed in a fraction of the time that the actual therapy takes place. Theseprediction capabilities combined with an understanding of HSP kinetics and damage mecha-nisms at the cellular and tissue levels due to thermal stress, together with recent advances forcontrolling discretization and modeling errors through adaptive control strategies combinedwith the modern methods of inverse analysis, calibration, uncertainty quantification, sensi-tivity analysis, and optimization [12–21] provide a powerful methodology for planning andoptimizing the delivery of thermo-therapy for cancer treatments. Moreover, from a compu-tational point of view, changing the thermal delivery modality merely amounts to changingthe source term in the governing PDE. This technology has the potential to be extended tomany areas of thermal treatment including RF, microwave, ultrasound, and even cryotherapyapplicators.


920 ODEN ET AL.

FIG. 7. Selected slices of canine MRI brain data and iso-surface visualization of canine MRI brain data.

FIG. 8. Finite element mesh of canine MRI brain data.



FIG. 9. Cutline comparison of MRTI data to Pennes model predictions.


922 ODEN ET AL.

References

1. American Cancer Society, Cancer facts and figures 2006, American Cancer Society, Atlanta, 2006.

2. K. Shinohara, Thermal ablation of prostate diseases: advantages and limitations, Int J Hyperthermia 20(2004), 679–697.

3. M. Kangasniemi, J. Stafford, R. Price, E. Jackson, and J. Hazle, Dynamic gadolinium uptake in thermallytreated canine brain tissue and experimental cerebral tumors, Invest Radiol 38 (2003), 102–107.

4. R. Salomir et al., Hyperthermia by MR-guided focused ultrasound: accurate temperature control basedon fast MRI and a physical model of local energy deposition and heat conduction, Magn Reson Med 43(2000), 342–347.

5. F. C. Vimeux et al., Real-time control of focused ultrasound heating based on rapid MR thermometry,Invest Radiol 34 (1999), 190–193.

6. M. N. Rylander, Y. Feng, J. Zhang, J. Bass, Stafford R. J., J. Hazle, and K. Diller, Optimizing Hspexpression in prostate cancer laser therapy through predictive computational models, J Biomed Optics11.4 (2005), 041113-1–041113-16.

7. Y. Zhang and C. Bajaj, Adaptive and quality quadrilateral/hexahedral meshing from volumetric data,Comput Methods Appl Mech Eng 195 (2006), 942–960.

8. M. N. Rylander, Y. Feng, and K. R. Diller, Thermally induced HSP 27, 60, and 70 expression kineticsand cell viability in normal and cancerous prostate cells, Cell Stress Chaperones, Preprint.

9. H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting forearm, J Appl Physiol1 (1948), 93–122.

10. J. Liu, L. Zhu, and L. Xu, Studies on the three-dimensional temperature transients in the canine prostateduring transurethral microwave thermal therapy, J Biomech Eng 122 (2000), 372–378.

11. M. N. Rylander, Design of hyperthermia protocols for inducing cardiac protection and tumor destructionby controlling heat shock protein expression, PhD thesis, The University of Texas at Austin, 2005.

12. J. T. Oden and S. Prudhomme, Estimation of modeling error in computational mechanics, J ComputPhys 182 (2002), 496–515.

13. T. I. Zohdi, J. Tinsley Oden, and G. J. Rodin, Hierarchical modeling of heterogeneous bodies, ComputMethods Appl Mech Eng 138 (1996), 273–298.

14. J. T. Oden and T. I. Zohdi, Analysis and adaptive modeling of highly heterogeneous elastic structures,Comput Methods Appl Mech Eng 148 (1997), 367–391.

15. J. T. Oden and K. Vemaganti, Adaptive hierarchical modeling of heterogeneous structures. Predictability:quantifying uncertainty in models of complex phenomena, Phys D 133 (1999), 404–415.

16. Serge Prudhomme and J. T. Oden, On goal-oriented error estimation for elliptic problems: applicationsto the control of pointwise errors, Comput Methods Appl Mech Eng 176 (1999), 313–331.

17. Roland Becker and Rolf Rannacher, An optimal control approach to a posteriori error estimation infinite element methods, Acta Numerica 10 (2001), 1–102.

18. J. T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptavity for the finite elementmethod, Comput Math Appl 41 (2001), 735–756.

19. J. T. Oden, S. Prudhomme, D. C. Hammerand, and M. S. Kuczma, Modeling error and adaptivity innonlinear continuum mechanics, Comput Meth Appl Mech Eng 190 (2001), 6663–6684.

20. W. Bangerth and R. Rannacher, Adaptive finite element methods for solving differential equations,Birkhäuser Verlag, Basel, 2003.

21. M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, MultiscaleModel Simul 1 (2003), 221–238.


dynamic data-driven finite element models for laser ...oden/dr._oden... · the mouse and tumor is...

Documents