Dynamic Data-Driven Finite Element Models

for Laser Treatment of Cancer

J. T. Oden1, K. R. Diller2, C. Bajaj1, J. C. Browne1, J. Hazle4, I. Babuska1, J. Bass1, L. Biduat4,L. Demkowicz1, A. Elliott4, Y. Feng1, D. Fuentes1, S. Prudhomme1, M. N. Rylander3, R. J. Stafford4, and

Y. Zhang1

1 Institute for Computational Engineering and Sciences,2 Department of Biomedical Engineering,

The University of Texas at Austin, Austin TX 78712, USAoden,babuska,bass,leszek,feng,fuentes,serge,jessica@ices.utexas.edu,

kdiller@mail.utexas.edu, bajaj,browne@cs.utexas.edu,3 School of Biomedical Engineering and Sciences, Virginia Tech

Virginia Tech - Wake Forest University, Blacksburg, VA 24061, USAmnr@vt.edu,

4 University of Texas M.D. Anderson Cancer Center,Department of Diagnostic Radiology, Houston TX 77030, USA

jhazle,jstafford,Andrew.Elliott,Luc.Bidaut@di.mdacc.tmc.edu,Webpage: http://dddas.ices.utexas.edu

keywords: hyperthermia, real-time computing, medical imaging,

cancer treatment, optimization, goal-oriented error estimation

Abstract. Elevating the temperature of cancerous cells is known to increase their susceptibility tosubsequent radiation or chemotherapy treatments, and in the case in which a tumor exists as a well-defined region, higher intensity heat sources may be used to ablate the tissue. These facts are the basisfor hyperthermia based cancer treatments. Of the many available modalities for delivering the heatsource, the application of a laser heat source under the guidance of real-time treatment data has thepotential to provide unprecedented control over the outcome of the treatment process [7, 18]. The goalsof this work are to provide a precise mathematical framework for the real-time finite element solutionof the problems of calibration, optimal heat source control, and goal-oriented error estimation appliedto the equations of bioheat transfer and demonstrate that current finite element technology, parallelcomputer architecture, data transfer infrastructure, and thermal imaging modalities are capable ofinducing a precise computer controlled temperature field within the biological domain.

1 Introduction

Today, cancer is the second most common cause of death in the United States. In 2006, about 564,830Americans are expected to die of cancer. Among the most common forms of the disease are cancer ofthe prostate and breast, accounting for 10% and 15% of all cancer related deaths in men and women,respectively. Current detection technology has led to a 5-year relative survival rate of greater than 90% forprostate and breast cancer [4]. Given these survival rates, the quality of life of the patient post-treatment isa very important factor. Traditional medical treatment of early stage prostate and breast cancer, includingradical prostatectomy, pelvic lymphadenectomy, lumpectomy, and mastectomy are major surgical proceduresand are associated with many surgical complications. Ablation therapies delivered under various treatmentmodalities are a very promising, minimally invasive, alternative to standard treatment and show significantpotential as an effective cancer treatment to eradicate the disease while maintaining functionality of infectedorgans and minimizing complications and relapse.

2

The physical basis for thermal therapies is that exposing cells to temperatures outside their naturalenvironment for certain periods of time can damage and even destroy the cells. However, one of the limitingfactors in all forms of ablation therapy, including cryotherapy, microwave, radio-frequency, and laser, is theability to control the energy deposition to prevent damage to adjacent healthy tissue [19].

Calibration

Goal−oriented error estimation

Optimal control of laser

TransferData

TransferData MRI &

MRTI Scans

Image processingand Mesh generation

hp adaptive FEMcomputations

Houston: surgery/scans

Austin: simulations

Fig. 1. The Control Loop.

Recent advances in imaging technology allow the imaging of the geometry of tissue and an overlayingtemperature field using MRI and new MRTI technology. MRTI has the desirable properties of excellent soft-tissue contrast, the ability to provide fast, quantitative temperature imaging in a variety of tissues, and thecapability of providing biologically relevant information regarding the extent of injury immediately followingtherapy [5]. Image guidance [18, 20] has the potential to facilitate unprecedented control over bioheat transferby providing real time treatment monitoring through temperature feedback during treatment delivery. Asimilar idea using ultrasound guided cryotherapy has been studied and shows good results [19]. However, thehypothermic regime requires temperature differentials of 70o from normothermia to ablate the tissue. Tissue

3

ablation in the hyperthermic regime requires substantially less energy deposition, temperature differentialsof 10o, suggesting that MRTI guidance under laser therapy could provide superior control over the bioheattransfer.

The goal of this work is to deliver a computational model of bioheat transfer that employs real-time,patient specific data and provides real-time high fidelity predictions to be used concomitantly by the surgeonin the laser treatment process. The model employs an adaptive hp-finite element approximation of thenonlinear parabolic Pennes equation and uses adjoint-based algorithms for inverse analysis, model calibration,and adaptive control of cell damage. The target disease of this research are localized adenocarcinomas ofthe breast, prostate, cerebrum, and other tissues in which a well-defined tumor may form. The algorithmsdeveloped also provide a potentially viable option to treat other parts of the anatomy in patients withmore advanced and aggressive forms of cancer who have reached their limit of radiation and chemotherapytreatment.

The adaptive data driven computational control system for controlling laser treatment of prostate cancerdescribed is an example of a Dynamic-Data-Driven Applications System (DDDAS) in which simulationmodels interact with measurement devices and assimilate data over a computational grid for the purpose ofproducing high-fidelity predictions of physical events.

The development stages uses canines as the host of prostate and brain tumors. The actual laboratory atM.D. Anderson Cancer Center in Houston, TX is connected through a computational grid to the computingcenter in Austin, TX. Prior to treatment, MRI data is used to generate a finite element mesh of the patient-specific biological domain. The mesh is then optimized to a particular quantity of interest using goal-orientedestimation and adaption. The tool then proceeds to solve an optimal control problem, wherein the laserparameters (location of optical fiber, laser power, etc.) are controlled to eliminate/sensitize cancer cells,minimize damage to healthy cells, and control Heat Shock Protein (HSP) expression. Upon initiation of thetreatment process, real-time MRI data is employed to co-register the computational domain and MRTI datais used to calibrate the bioheat transfer model to the biological tissue values of the patient. As new datais given intermittently, computation is compared to the measurements of the real-time treatment and anappropriate course of action is chosen according to the differences seen. A parallel computing paradigm isused to meet the demands of rapid calibration and adapting the computational mesh and models to controlapproximation and modeling error. From a computational point of view, the orchestration of a successfullaser treatment is to solve the problems of co-registration, calibration, optimal control, and mesh refinementinvisibly to the surgeon, and merely provide the surgeon with an interface to the optimal laser parametersduring treatment. A schematic of the control loop is shown in Figure 1. The mathematical characterizationof so-called HSP expression is also described as are laboratory procedures for measuring tissue damage, meshgeneration, and the development of the computational infrastructure for calibration procedures, verificationand validation procedures, and inverse modeling and sensitivity analysis.

2 Mesh Generation

The first step in simulation is to construct a finite element mesh used for the governing bioheat transferequations. The mesh generation involves two steps: (1) construction, from MRI data, of a finite elementmesh to represent the geometry and (2) overlaying the MRTI temperature field onto the finite element mesh.

As an initial test of the imaging system when employed in experiments with laboratory animals, we firstconsider data pertaining to a mouse. A sample of the MRI data used to construct a mesh of the mouse andtumor is shown in Figure 3. In the case of prostate cancer, prostate tumor cells were inoculated in the hindlegs of a mouse and grown to a tumor burden of less than 1.0 cc, Figure 2. The hexahedral mesh of thetumor (blue) and tissue (yellow), shown in Figure 2, was created by a semi-automatic segmentation methodadapted to find the interface boundaries of tumor and tissue. Cubic spline and lofting methods are appliedto obtain smooth boundaries from the segmented MRI data [16, 21].

4

Fig. 2. Mouse and Mesh (from [16])

2.1 MRTI Data-Filtering

Spatio-temporal temperature distribution is measured during the laser treatment with update times lessthan 6 seconds per image and thickness between planes of 3.5 mm. The temperature field, measured in thebiological domain, is first filtered to eliminate any noise and then nodal temperature values are assigned tothe mesh by taking the interpolant of the MRTI temperature data.

Fig. 3. MRI Data of Mouse

3 HSP Characterization and Damage Model

Heat shock proteins are a latent defense mechanism built into all cells, including cancer cells, that provideenhanced viability of tumor tissue indirectly by preventing a damaged cell from going into apoptosis, resultingin the recurrence of cancer. Knowledge of temperature history versus time during treatment has been used topredict thermal necrosis in regions where damage is severe, but in regions where temperatures are insufficientto coagulate proteins, the results and subsequent effects have been difficult to predict. This is due, in part,to the expression of heat shock protein (HSP) in the regions of thermal stress. Consequently, knowledge ofthe thermal dose necessary to activate or de-activate HSP expression as a function of temperature and timein the affected tissue [16, 15] can be critical in planning and implementing an effective thermal treatment bylaser surgery.

5

Heat Shock Proteins are a family of gene products expressed in higher concentrations in the presence ofenvironmental stresses. The name vastly understates the HSP family’s astounding versatility. These proteinshave been identified as critical components of cell survival under adverse environmental conditions. Variouslevels of HSP species indicate the health or likelihood of cell proliferation or drug resistance. Also, measuresof cell damage as a function of temperature and time for a specific patient is a critical indication of the

Fig. 4. HSP Expression Model

effectiveness of thermo-therapies. Knowledge of thetemperatures necessary to elicit interaction in tumorresistance and cell damage is essential to effectivelyproduce a desired tissue response and surgical out-come. The work of Rylander [15] developed a modelfor HSP expression and cell damage based on an Ar-rhenius model for a mouse. Figure 4 shows the com-parison of the experimentally determined HSP ex-pression with the predicted values of (1). An empiri-cal formula for the concentration of HSP expression,H(u, t)

[

mgml

]

, at temperature, u, and time, t, basedon laboratory tests is [15]:

H(u, t) = H0eαt−βtγ

(1)

where α, β, and γ are time independent parametersthat may depend on temperature, with γ > 1.

Cellular damage is measured in terms of the dam-age fraction FD predicted by means of an Arrheniusintegral formulation [16].

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

∫ t

0

eEa

R u(x,τ) dτ (2)

where C0 is the initial concentration of healthy cells, Ct the concentration of healthy cells after heating attime t, A the pre-exponential scaling factor, Ea the activation energy of the injury process, R the universalgas constant, and u the absolute temperature.

4 Bioheat Transfer Model

Driving the prediction of the HSP and cellular damage is the temperature field produced by the well-knownPennes bioheat transfer model [13]. Liu [6] has shown Pennes model [13] to give good results for predictionof temperature field in the prostate. The formal statement of Pennes bioheat 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 Ω

given the Cauchy and Neumann boundary conditions

−k(u)∇u · n = h(u − u∞) on ∂ΩC

−k(u)∇u · n = G on ∂ΩN

and the initial conditionu(x, 0) = u0 in Ω

6

Fig. 5. Comparison of arctan fit to text bookvalues.

Here k[

Js·m·K

]

and ω[

kgs m3

]

are bounded functions of u,

determined empirically to be of the 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 tempera-ture, ρ is the density, and h is the coefficient of cooling. Thelaser source term Qlaser is a linear function of laser power andexhibits exponential decay with distance from the source.

Qlaser(x, t) = 3P (t)µaµtr

exp(−µeff‖x− x0‖)

4π‖x− x0‖

µtr = µa+µs(1 − γ) µeff =√

3µaµtr

P = is the laser power, µa, µs = are laser coefficients re-lated to laser wavelength and give probability of absorptionof photons by tissue, γ is the anisotropy factor, and x0 = isthe position of laser photon source. Tables 1 and 2 containconstitutive data from the work of Rylander [17].

Our weak form of the Pennes bioheat transfer model is asfollows:

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

Find u(x, t) ∈ V ≡ H1(

[0, T ], H1(Ω))

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), andlaser parameters, η = (P (t),x0, µa, µs), expressed as follows

B(u, β; v) =

∫ T

0

∫

Ω

[

ρcp

∂u

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

]

dxdt

+

∫ T

0

∫

∂ΩC

hu v dAdt +

∫

Ω

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

(3)

F (η; v) =

∫ T

0

∫

Ω

Qlaser(x, η)v dxdt +

∫ T

0

∫

∂ΩC

hu∞ v dAdt −

∫ T

0

∫

∂ΩN

G v dAdt +

∫

Ω

u0 v(x, 0) dx (4)

4.1 Calibration, Optimal Control

The control loop involves three major problems: calibration of the Pennes model to patient specific MRTIdata, optimal positioning and power supply of the laser heat source, and computing goal oriented errorestimates. Each of these problems is formally stated below.

7

Table 1. Model Coefficients (From Rylander [17])

Laser Wavelength 810 nm µs 14.74cm µa 0.046 cm ρtissue 1045 kg

m3

ua 310 K ρblood 1058 kg

m3 cbloodp 3840 J

kg·Kctissuep 3600 J

kg·K

Table 2. Model Coefficients

k(u) = k0 + k1 · atan(k2(u − k3)) ω(u) = ω0 + ω1 · atan(ω2(u − ω3))

k0 0.6489ˆ

J

s·m·K

˜

ω0 0.6267ˆ

kg

s m3

˜

k1 0.0427ˆ

J

s·m·K

˜

ω1 −0.137ˆ

kg

s m3

˜

k2 0.0252ˆ

1

K

˜

ω2 2.3589ˆ

1

K

˜

k3 315.314 [K] ω3 314.262 [K]

The problem of model calibration is to find the set of thermal conductivity parameters k0, k1, k3, k3,and blood perfusivity parameters ω0(x), ω1, ω3, ω3 that minimize the norm of the difference between thepredicted temperature field and the experimentally determined temperature field at each time instance ofthe experimental data. ω0(x) is allowed to vary over the spatial dimension as the blood perfusivity withinthe necrotic core of a cancerous tumor is expected to be significantly lower than the surrounding healthytissue. The calibration problem may be stated as follows:

Given a set of laser parameters, η0, and an experimentally determined temperature field at time in-stances tn, n = 1, 2, ...Nexp within the region Ωχ ⊂ Ω

uexp(x, tn) x ∈ Ωχ

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

∫

Ω

χ(x)(u∗(x, t) − uexp(x, t))2 dxdt +γ

2

∫

Ω

∇ω∗0(x) · ∇ω∗

0(x) dx

satisfies

Q(u∗(η0, β∗), β∗) = inf

β∈P

Q(u(η0, β), β)

P = β ∈ R4 × C1(Ω) × R

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

8

The problem of optimal laser heating control is to find the position, x0, power, P (t), and laser frequencyµa, µs that maximizes damage to the cancerous tissue while minimizing damage to healthy tissue in somemetric. The metric shown here is the L2

(

[0, T ];L2(Ω))

norm of the difference between the computed solutionand an ’ideal’ temperature field. Other metrics may be the L2(Ω) norm of the difference between thecomputed damage fraction, equation (2), and an ideal damage field within the biological domain at theend of the treatment process or the L2

(

[0, T ];L2(Ω))

norm of the difference between the computed HSPexpression field, equation (1), and an ideal HSP expression field.

‖FD(x, T )− F idealD (x)‖2

L2(Ω) ‖H − Hideal‖2L2([0,T ];L2(Ω))

Optimal control of the laser source term is stated as:

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

φ(x) ≡

37.0oC x ∈ ΩH

50.0oC x ∈ ΩC

where ΩH and ΩC are the domains of the healthy and cancerous tissue respectively, find the bestcombination of laser position and power, η∗ ∈ P, that produces the temperature field, u∗ ∈ V , suchthat

Q(u∗(η∗, β0), η∗) =

1

2‖u − φ‖2

L2([0,T ];L2(Ω)) =1

2

∫ T

0

∫

Ω

(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.

4.2 Mesh Refinement

Goal oriented error estimates [9] 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.• ‖uhp − u‖V ≤ ‖uh − u‖V where u, uh, uhp satisfy the Pennes model constraints

C(u; v) = 0 ∀v ∈ V

C(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 solution u ∈ V thanuh ∈ Vh ⊂ Vhp; likewise an arbitrary quantity of interest, Q, evaluated at Q(uhp) is a more accurate

9

representation of Q(u) than the quantity of interest evaluated at Q(uh). As an explicit example, consider thequantity of interest to be an averaged value of the temperature in a domain defined through the characteristicfunction, χ(x),

Q(u(η0, β0)) =1

2‖χ(x)u(x, t)‖2

L2([0,τ ];L2(Ω))

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 + u) − Q(u)]

δC(u; u; v) ≡ limθ→0

[C(u + u; 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

∫

Ω

[

ρcp

∂uh

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

h − ua) php

]

dxdt

−

∫ T

0

∫

∂ΩC

h(uh − u∞) php dAdt −

∫ T

0

∫

∂ΩN

G php dAdt +

∫ T

0

∫

Ω

Qlaser(x, η0)php dxdt

=

∫ T

0

Z(t)dt =

∫

Ω

X (x) dx +

∫

∂ΩC

R(x) dA +

∫

∂ΩN

N (x) dA

Where Z(t) is defined as

Z(t) = −

∫

Ω

[

ρcp

∂uh

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

h − ua) php

]

dx

−

∫

∂ΩC

h(uh − u∞) php dA −

∫

∂ΩN

G php dA +

∫

Ω

Qlaser(x, η0)php dx

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

X (x) =

∫ T

0

Qlaser(x, η0)php − ρcp

∂uh

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

h − ua) php dt

R(x) = −

∫ T

0

h(uh − u∞) php dt N (x) = −

∫ T

0

G php dt

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

10

4.3 Optimization Framework

The calibration and optimal control problems in the control loop may be posed as the following optimizationproblem

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 deriva-

tive, 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 by the adjoint method under the assumptions that the solution, objective function,and constraints admit the following Taylor series expansions.

u(q + θq) = u(q) + θu′(q; q) + θ2(...)

Q(u + θu, q + θq) = Q(u, q) + θδuQ(u, q; u) + θδqQ(u, q; q) + θ2(...)

C(u + θu, q + θq; v) = C(u, q; v) + θδuC(u, q; u; v) + θδqC(u, q; q; v) + θ2(...)

where the Gateaux derivatives are defined below.

u′(q; q) ≡ limθ→0

1

θ[u(q + θq) − u(q)]

δuQ(u, q; u) ≡ limθ→0

1

θ[Q(u + u, q) − Q(u, q)]

δqQ(u, q; q) ≡ limθ→0

1

θ[Q(u, q + q) − Q(u, q)]

δuC(u, q; u; v) ≡ limθ→0

[C(u + u, q; v) − C(u, q; v)]

δqC(u, q; q; v) ≡ limθ→0

[C(u, q + q; v) − C(u, q; v)]

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

11

Proof. The solution to the adjoint problem, p, implies that

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 expansion are used in computing the Gateaux derivatives of the variationalform 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+ θ2(...)

ω(u + θu, β + θβ) = ω(u, β) + θu∂ω

∂u+ θω0

∂ω

∂ω0+ θω1

∂ω

∂ω1+ θω2

∂ω

∂ω2+ θω3

∂ω

∂ω3+ θ2(...)

Notice that the variational form of the adjoint problem for the calibration, optimal control, and goal orientederror estimates are all the same:

δuC(u, β; u, p) = δuB(u, β; u, p) =

∫ T

0

∫

Ω

−ρcp

∂p

∂tu dxdt +

∫ T

0

∫

Ω

k(u, β)∇u · ∇p + u∂k

∂u∇u · ∇p dxdt

+

∫ T

0

∫

Ω

ω(u, β)up + u∂ω

∂u(u − ua)p dxdt +

∫

Ω

u(x, T )p(x, T ) dx +

∫ T

0

∫

∂Ω

hup dAdt

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

Given the spatially and temporally varying temperature field u(x, t) find the Lagrange multiplier 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 Ω

given the boundary conditions−k(u)∇p · n = h p on ∂ΩC

−k(u)∇p · n = 0 on ∂ΩN

and the terminal conditionp(x, T ) = 0 in Ω

12

When approximations of the temperature field, u(x, t), and the adjoint variable, p(x, t), are known, thefirst variation of the quantity of interest for the calibration problem may be obtained explicitly as follows.

δQ(β, β) =

−

∫ T

0

∫

Ω

∇u · ∇p k0 dxdt

−

∫ T

0

∫

Ω

atan(k2 (u − k3))∇u · ∇p k1 dxdt

−

∫ T

0

∫

Ω

k1 (u − k3)

1 + k22 (u − k3)2

∇u · ∇p k2 dxdt

−

∫ T

0

∫

Ω

−k1 k2

1 + k22 (u − k3)2

∇u · ∇p k3 dxdt

−

∫ T

0

∫

Ω

(u − ua)p ω0(x) dxdt +

∫

Ω

ω0(x) ω0(x) dx

−

∫ T

0

∫

Ω

atan(ω2(u − ω3))(u − ua)p ω1 dxdt

−

∫ T

0

∫

Ω

ω1 (u − ω3)

1 + ω22(u − ω3)2

(u − ua)p ω2 dxdt

−

∫ T

0

∫

Ω

−ω1 ω2

1 + ω22(u − ω3)2

(u − ua)p ω3 dxdt

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

δQ(η, η) =

∫ T

0

∫

Ω

3 µa µtr exp(−µeff‖x − x0‖)

4 π ‖x − x0‖p P (t)dxdt

∫ T

0

∫

Ω

3P (t)exp(−µeff‖x− x0‖)

4π‖x− x0‖

(

µtr + µa − µtr µa‖x− x0‖3(µa + µtr)

2µeff

)

p µadxdt

∫ T

0

∫

Ω

3P (t)exp(−µeff‖x− x0‖)

4 π ‖x − x0‖

(

µa − µtr µa‖x − x0‖3(µa(1 − γ))

2µeff

)

p µsdxdt

∫ T

0

∫

Ω

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

4 π‖x− x0‖3(x − x0) p x0dxdt

∫ T

0

∫

Ω

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

4 π‖x− x0‖3(y − y0) p y0dxdt

∫ T

0

∫

Ω

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

4 π‖x− x0‖3(z − z0) p z0dxdt

5 Results

Two main results are presented in this section. First, the overall computational feasibility of developing alaser treatment paradigm in which high performance computers control the bioheat data transferred from aremote site is demonstrated. Second, reliable computational predictions are shown when comparing resultsobtained using the Pennes model to the experimental MRTI data in canine cerebral tissue.

The typical time duration of a laser treatment is about five minutes. During a five minute span, one setof MRTI data is acquired every 6 seconds. The size of each set of MRTI data is ≈330kB (256x256x5 voxels)

13

and the bandwidth of a commercial T1 Internet connection is ≈300kB/s. Accounting for connection latency,each set of MRTI data can be transferred between Houston and Austin in under two seconds. A finiteelement mesh consisting of approximately 10000 degrees of freedom is sufficient to resolve the geometryof the biological domain. The execution time of a representative 10 second bioheat transfer simulation ispresented in Figure 6. The execution times represent 10 nonlinear state solves of Pennes model (10 onesecond time steps) combined with 10 linear adjoint solves to be used for calibration, optimal control, and/orerror estimates. Computations were done at the Texas Advanced Computing Center on Dual-Core LinuxCluster. Each node of the cluster contains two Xeon Intel Duo-Core 64-bit processors (4 cores in all) on asingle board, as an SMP unit. The core frequency is 2.66GHz and supports 4 floating-point operations perclock period. Each node contains 8GB of memory. The fastest time recorded is .67 seconds, meaning thatin a real time 10 second span Pennes model can predict out to more than two minutes! Equivalently, in a10 second times span roughly 14 corrections can be made to calibrate the model coefficients or optimize thelaser parameters.

Fig. 6. Executions times for a representative 10 second simulation (10 nonlinear state solve combined with 10 linearadjoint solve on 10000 dof system)

Preliminary computations comparing the predictions of Pennes model to experimental MRTI taken froma canine brain show very good agreement. A manual craniotomy of a canine skull was preformed to allowinsertion of an interstitial laser fiber. Thirty-six two dimensional 256x256 pixels MRI images, Figure 7, ofthe canine brain were acquired. The field view was 200mm x 200mm with each image spaced 1mm apart.A finite element mesh of the biological domain generated from the MRI data is shown in Figure 8. Themesh consists of 8820 linear elements with a total of 9872 degrees of freedom. MRTI thermal imaging datawas acquired in the form of five two dimensional 256x256 pixel images every six seconds for 120 time steps.The spacing between images was 3.5mm. The MRTI data was filtered then projected onto the finite elementmesh. Figure 9 shows a cutline comparison between the MRTI data and the predictions of Pennes model. It isobserved that the results delivered by the computational Pennes model slightly over diffuses the heat profile

14

peaks compared to measured values. However, at early times the maximum temperature value is within 5%of the MRTI value.

Fig. 7. Selected Slices of Canine MRI Brain Data and Iso-surface visualization of Canine MRI Brain Data

6 Conclusion

Results indicate that adaptive hp-finite element technology implemented on parallel computer architecturesand employing modern data transfer infrastructure and thermal imaging modalities are capable of predictingand guiding computer controlled temperature field within a biological domain at very good accuracies.Reliable finite element model simulations of laser therapies can be computed in a fraction of the timethat the actual therapy takes place. These prediction capabilities combined with an understanding of HSP

15

kinetics and damage mechanisms at the cellular and tissue levels due to thermal stress, together with recentadvances for controlling discretization and modeling errors through adaptive control strategies combinedwith the modern methods of inverse analysis, calibration, uncertainty quantification, sensitivity analysis, andoptimization [22, 12, 11, 14, 2, 8, 10, 9, 1, 3] provide a powerful methodology for planning and optimizing thedelivery of thermo-therapy for cancer treatments. Moreover, from a computational point of view, changingthe thermal delivery modality merely amounts to changing the source term in the governing PDE. Thistechnology has the potential to be extended to many areas of thermal treatment including RF, microwave,ultrasound, and even cryotherapy applicators.

Fig. 8. Finite Element Mesh of Canine MRI Brain Data

Acknowledgments. The support of this project by the National Science Foundation under grant CNS-0540033 is gratefully acknowledged.

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time = 51 s time = 71 s

time = 125 s time = 171 s

Fig. 9. Cutline Comparison of MRTI Data to Pennes model predictions