An Inverse Problem Approach to Recovery of In Vivo Nanoparticle Concentrations from Thermal Image Monitoring of MR-Guided Laser Induced Thermal Therapy

Abstract

Quantification of local variations in the optical properties of tumor tissue introduced by the presence of gold-silica nanoparticles (NP) presents significant opportunities in monitoring and control of NP-mediated laser induced thermal therapy (LITT) procedures. Finite element methods of inverse parameter recovery constrained by a Pennes bioheat transfer model were applied to estimate the optical parameters. Magnetic resonance temperature imaging (MRTI) acquired during a NP-mediated LITT of a canine transmissible venereal tumor in brain was used in the presented statistical inverse problem formulation. The maximum likelihood (ML) value of the optical parameters illustrated a marked change in the periphery of the tumor corresponding with the expected location of nanoparticles and area of selective heating observed on MRTI. Parameter recovery information became increasingly difficult to infer in distal regions of tissue where photon fluence had been significantly attenuated. Finite element temperature predictions using the ML parameter values obtained from the solution of the inverse problem are able to reproduce the nanoparticles selective heating within 5°C of measured MRTI estimations along selected temperature profiles. Results indicate the maximum likelihood solution found is able to sufficiently reproduce the selectivity of the NP mediated laser induced heating and therefore the ML solution is likely to return useful optical parameters within the region of significant laser fluence.

1 Introduction

The introduction of commercially available systems integrating MRI guidance has renewed interest in the use of lasers for deep-seated tumor ablation. The image-guided laser ablation of tumors is a promising minimally-invasive technique currently being investigated as an alternative to conventional surgical interventions for non-surgical candidates [1, 2, 3, 4, 5]. Magnetic Resonance guided Laser Induced Thermal Therapy (MRgLITT) utilizing real-time temperature imaging feedback and dosimetry is currently being evaluated clinically in brain for treating focal cancerous lesions [6]. The advantages of replacing conventional surgical intervention with a minimally-invasive one are the potential reduction of the physical, emotional, and financial impacts of the procedure. The minimally invasive nature and non-ionizing radiation associated with thermal therapies translates to low impact, repeatable procedures ideally suited for non-surgical candidates and long term palliative management.

The challenges associated with delivery of this therapy paradigm in vivo remains the greatest barrier to successful clinical translation. Laser ablation uses small laser fibers, placed directly into the tumor, to induce heating and destroy the tumor tissue. While the region of damage is small and localized, the laser must first be carefully navigated through critical structures to the target site. Further, the laser induced heating isn’t confined exclusively to tumor tissue and there is a chance that nearby tissue can be damaged. Nanoparticle (NP) mediated laser ablation has been proposed in a variety of organs, including the brain, head and neck, liver, and prostate [7, 8, 9, 10, 11]. This minimally invasive approach to therapy may benefit from the use of emerging nanotechnology, such as gold-silica nanoshells, to facilitate a more conformal heating of the focal tumor using intravenously injected nanoparticles [8]. Such therapies would be highly synergistic with MR-guidance and temperature imaging as this is already used for guiding laser ablation in these environments.

Gold-silica nanoshells can be tuned to absorb in the near infrared (NIR) region of the optical spectrum and therefore may operate synergistically with the NIR lasers currently employed for MRgLITT applications. These particles may accumulate passively in the tumor tissue via the enhanced permeability and retention effect and facilitate a more conformal deposition of energy. However, the distribution of these particles throughout the tumor is often heterogeneous making it difficult to anticipate how a treatment may progress once the laser is turned on. Knowledge of the heterogeneous in vivo nanoparticle distribution is desirable to deliver a lethal thermal dose to the tumor while sparing healthy tissue and surrounding critical structures [12]. Determination of the patient specific local optical properties [13] in the region of the laser fiber could provide sufficient information for incorporating into a model of laser heating for assisting in monitoring [14] and controlling [15] devices used to deliver the procedure [16, 17] and, hence, provide a safer and more optimal approach to NP mediated LITT. The information could also be potentially useful in updating a treatment plan of the procedure.

Statistical inverse analysis techniques present a methodological way to capture physical parameters from measurements using appropriate models. Previously, inverse techniques have been used to locate the nanoparticles based on their observed magnetic properties [18]. Additionally, a Pennes bioheat transfer constrained optimization algorithm [19, 20] utilizing online intra-operative temperature monitoring has demonstrated potential to detect local variations in the constitutive properties relevant to the bioheat transfer. MRgLITT protocols use a non-damaging low power test pulse to confirm the proper location of the active tip prior to irradiating at higher powers for treatment. The MRTI estimated spatiotemporal temperature response from this low power pulse can simultaneously be used as a calibration pulse to recover the in vivo spatial distribution of optical parameters that manifest due to the presence of nanoparticles in the beam.

An algorithm for the inverse recovery of the spatially varying optical parameters is presented in this work. The inverse problem was applied retrospectively to thermometry measurements acquired during a NP mediated LITT procedure in a canine brain model. The emphasis of the technique is to demonstrate the feasibility of recovering heterogeneous optical parameters that may potentially be correlated with the presence of the in vivo nanoparticles distribution at the resolution of the MR temperature imaging.

High performance computing is required to solve the partial differential equation (PDE) constrained inverse problems involving thousands of optimization variables. The inverse framework is based on the notion of conjunction ¹ of probabilities and may be considered a generalization of conditional probabilities [21]. Within this framework, the noise or uncertainty of the thermal imaging data is seen to intuitively weight the cost function in the optimization process to find the maximum likelihood value. The characterizations of the model parameters may be used as the initial conditions of frontier algorithms in predictive simulation [22, 23] for uncertainty quantification within the PDE based computer models. A statistical characterization may ultimately be propagated to a quantity of clinical interest and provide information beyond single point deterministic predictions, such as expected values, variances, and confidence levels in the model prediction.

2 Materials and Methods

This study retrospectively analyzes data acquired during a recently closed animal protocol approved by the Institutional Animal Care and Use Committee of The University of Texas MD Anderson Cancer Center. Feasibility of the optical parameter recovery for the nanoparticle distribution was demonstrated using a transmissible venereal tumor (TVT) in canine brain tissue. The nanoparticle induced heating observed in MRTI was analyzed retrospectively. Details of the experimental setup are provided in [8]. Briefly, TVT cells were inoculated into the parietal lobe of the brain of immuno-suppressed canines. The tumors were allowed to grow over 6-8 weeks until they reached a nominal diameter of approximately 1cm. A $5.2\frac{\mathit{ml}}{\mathit{kg}}$ concentration of gold-silica nanoparticles (AuroShells®, Nanospectra Biosciences, Inc., Houston, TX) approximately 144-150nm in diameter with an optical absorption peak near 800nm was intravenously infused into the animal 24 hours prior to treatment. Details of the nanoparticle fabrication are provided in [8]. The weight of the animal was approximately 29 kg. On treatment day, the anesthetized animal was placed in a 1.5T clinical MRI scanner (Excite HD, GEHT, Waukesha, WI) and a water-cooled optical-fiber-based catheter (BioTex, Inc., Houston, TX) positioned adjacent to the tumor. Details of the laser system may be found in [16]. A test pulse of 3.0W for 30 seconds was first applied to verify the laser position of the 1cm isotropic diffusing fiber. A laser exposure of 3.5W of 808nm light was then delivered for 3 minutes and resulted in tumor specific heating due to the presence of the nanoparticles which absorbed much more energy than the brain tissue at 808 nm (Figure 1). MRTI utilized a multiplanar, data interleaved multi-shot gradient-echo, echo-planar imaging sequence for estimating the spatiotemporal distribution of temperature changes during heating. Multiplanar MRTI data was acquired every 6 seconds throughout the procedure. Post-treatment dynamic contrast enhanced (DCE) imaging was used to visualize the loss of perfusion in the affected areas and define the expected region of damage.

Figure 1.

A finite element model of in vivo TVT inoculated canine brain and the thermal imaging data set observed in MRgLITT. Registered anatomical and thermal imaging data of the canine procedure is shown. A single plane of MRTI during the heating is shown at the top right. The sagittal images shown intersect the axial direction of the fiber. Temperature is provided in °C. Preferential heating of the region with enhanced nanoparticle uptake is seen to the right of the illustrated fiber. The hexahedral finite element mesh consists of distinct regions for the catheter, diffusing tip, tumor burden, and healthy tissue. The mesh shown consists of N_dof =47,941 nodal degrees of freedom. The multi-planar MRTI data is projected onto the finite element mesh at each time step. The temperature along the applicator is held fixed with Dirichlet boundary conditions.

2.1 Statistical Inverse Computational Methods

Determination of the optical parameters was posed as a statistical inverse problem [21] defined on a probability space, Ω ≡ D × M, of the space of measured data, D, and the space of model parameters, M. The posterior probability state, σ : Ω → [0, 1], from which information on the optical parameters in the region of nanoparticle accumulation can be inferred is understood as the conjunction of the theoretical, Θ : Ω → [0, 1], and prior probability distribution functions, ρ : Ω → [0, 1].

$$\sigma (\overrightarrow{d},\overrightarrow{m})\propto \mathrm{\Theta }(\overrightarrow{d},\overrightarrow{m})\rho (\overrightarrow{d},\overrightarrow{m})(\overrightarrow{d},\overrightarrow{m})\in \mathrm{\Omega }$$

Under the assumptions of a delta functional for theoretical distribution, a Gaussian measurement model for MRTI measurements, u^{M RTI}, and an a priori uniform distribution for model parameters on M ⊂ M, the posteriori marginal distribution of the model parameters reduces to a non-Gaussian form restricted to the space of feasible parameters, M.

$${\sigma }_M(\overrightarrow{m})={\mathit{\int }}_{\mathrm{D}}d\overrightarrow{d}\sigma (\overrightarrow{d},\overrightarrow{m})\propto \{\begin{array}{rr}\exp \left(-\frac{{\parallel u(\overrightarrow{m})-u^{\mathit{M\; RT\; I}}\parallel }_D^2}{2}\right),& m\in M\\ 0,& m\notin M\end{array}$$ (1)

Here ∥ · ∥_D, (2), denotes the L₂ norm normalized by the zero-mean Gaussian noise, w, of the measurement model and $u(\overrightarrow{m})$ denotes the nonlinear mapping from parameter space to data space obtained as a solution to the Pennes bioheat transfer model [24].

$$\parallel u-u^{\mathit{M\; RT\; I}}{\parallel }_D^2={{\mathit{\int }}_{\mathrm{\Delta }T}{\mathit{\int }}_{\mathrm{\Omega }}\left(\frac{u(\overrightarrow{m})-u^{\mathit{M\; RT\; I}}}{w(x,t)}\right)}^2\mathit{dxdt}$$ (2)

In the present study, we are interested in the highest probability estimate of the model parameters. Over the probability space considered, the model parameter with the maximum likelihood, ${\overrightarrow{m}}_{\mathit{M\; L}}$, occurs at a maximum of the marginal probability distribution, ${\sigma }_M((\overrightarrow{m}))$. Equivalently, the maximum likelihood point, ${\overrightarrow{m}}_{\mathit{M\; L}}$, occurs at the minimum of the misfit function, $S(\overrightarrow{m})$.

$$S({\overrightarrow{m}}_{\mathit{M\; L}})=\underset{\overrightarrow{m}\in M}{\min }S(\overrightarrow{m})S(\overrightarrow{m})=\frac{\parallel u(\overrightarrow{m})-u^{\mathit{M\; RT\; I}}{\parallel }_D^2}{2}$$ (3)

The misfit function, $S(\overrightarrow{m})$, is interpreted as the distance between the bioheat transfer model prediction and measured thermal imaging data. Finding the maximum likelihood point is seen equivalent to solving a Pennes bioheat transfer constrained optimization problem over a physically feasible parameter space, M.

2.1.1 Theoretical Probability Density

The theoretical probability density, Θ, represents the mathematical model predicted relationship between the parameter space, M, and the data space, D. For a well defined mathematical model in which a unique solution exists, the theoretical probability density may be taken as a delta functional. This assumption explicitly ignores any potential modeling error and enforces the notion that the event of a temperature field not predicted by the theory has zero probability of occurring. The nonlinear mapping from parameter space to data space, $u(\overrightarrow{m})\in \mathrm{D}$, is obtained as a solution to the Pennes bioheat transfer model [24] coupled with the diffusion theory of light interaction with tissue [25].

$$\begin{array}{c}\rho c_p\frac{\partial u}{\partial t}-\nabla \cdot (k(\mathbf{x})\nabla u)+\omega (\mathbf{x})c_{\mathit{blood}}(u-u_a)={\mu }_a(\mathbf{x})\phi (m,\mathbf{x},t)\mathrm{in}\mathrm{\Omega }\\ -k(\mathbf{x})\nabla u·\mathbf{n}=0\mathrm{on}\partial \mathrm{\Omega }u(\mathbf{x},0)=u^0\mathrm{in}\mathrm{\Omega }\end{array}$$

The initial temperature field, u(x, 0) = u⁰, is taken as the measured baseline body temperature. The density of the continuum is denoted ρ, the specific heat of blood is denoted $c_b\left[\frac{J}{\mathit{kg}·K}\right]$, k(x) denotes the thermal conductivity, and ω(x) denotes perfusion. Zero heat flux Neumann data is assumed on the boundary, ∂Ω. Although likely to result in optical parameter estimation error due to the strong scattering assumptions [26], the classical spherically symmetric isotropic standard diffusion approximation (SDA) to the transport equation of light within a laser-irradiated tissue [25] is used as the kernel in modeling the laser source term. The SDA was used in this feasibility study for its computational simplicity in managing the algorithmic complexity of computing the gradient and Hessian of the misfit function, Section 2.2.1. The optical-thermal response to the laser source, q_laser = μ_a(x)φ(m, x, t), is modeled as

$$\begin{array}{c}\phi (m,\mathbf{x},t)={\mathit{\int }}_{U_{\mathit{tip}}}\frac{P(t)}{\mathrm{Vol}(U_{\mathit{tip}})}\frac{3{\mu }_{\mathit{tr}}\exp (-{\mu }_{\mathit{eff}}\parallel \mathbf{x}-\xi \parallel )}{4\pi \parallel \mathbf{x}-\xi \parallel }\mathit{d\xi }\\ \begin{array}{cc}& x\in U\backslash U_{\mathit{tip}}\\ {\mu }_{\mathit{tr}}={\mu }_a(\mathbf{x})+{\mu }_s(\mathbf{x})(1-g)& {\mu }_{\mathit{eff}}=\sqrt{3{\mu }_a(\mathbf{x}){\mu }_{\mathit{tr}}}\end{array}\end{array}$$

P(t) is the laser power as a function of time, μ_a and μ_s are laser coefficients related to laser wavelength and give probability of absorption and scattering of photons, respectively. The anisotropic factor is denoted g and ∥x − ξ∥ denotes the distance from the laser photon source, U_tip. Active cooling of the applicator is modeled by holding the temperature in this region fixed through Dirichlet boundary conditions at the ambient temperature [27].

Given an initial temperature field, u⁰, a set of model parameters, m ≡ (k(x), ω(x), μ_a(x), μ_s(x)) ∈ M, and the appropriate boundary condition, a model prediction of the temperature field, u(x, t), may be obtained. Discretizing the temperature with a Galerkin expansion at each time step, the data space is represented by the vector space of Galerkin coefficients for the thermal field. Similarly, a Galerkin discretization is used for the model parameters. The model space, M, is the space of model parameters for which a unique solution is well defined.

$$\begin{array}{ccc}u(x,t^k)=\sum _j^{N_{\mathit{dof}}}{d}_j^k{\varphi }_j(x)& k=1,2,..N_{\mathit{step}}& \mathrm{D}={ℝ}^{N_{\mathit{dof}\ast }{N}_{\mathit{step}}}\\ & m(x)=\sum _j^{N_{\mathit{model}}}{m}_j{\psi }_j(x)& \mathrm{M}\subset {ℝ}^{N_{\mathit{model}}}\end{array}$$

Here N_dof and N_model are the spatial temperature degrees of freedom and the parameter degrees of freedom, respectively. The number of time steps in the discretization is denoted N_step. In this application, an element of the data space, $\overrightarrow{d}\in \mathrm{D}$, is a set of Galerkin coefficients that may represent either an experimentally observed MRTI quantity or a quantity predicted by the bioheat transfer equation.

2.1.2 Prior Probability Density

The prior probability density, ρ, represents known information embodied in the experimentally obtained MRTI field, u^{M RT I}, and bioheat transfer parameters, m, from handbook data [28]. A Galerkin representation is used for the space time field obtained from the thermal images.

$$u^{\mathit{M\; RT\; I}}(x,t^k)=\sum _j^{N_{\mathit{dof}}}{\stackrel{\sim }{d}}_j^k{\varphi }_j(x){\overrightarrow{d}}^{\mathit{obs}}=\left[{\stackrel{\sim }{d}}_1^1,{\stackrel{\sim }{d}}_2^1,\dots \right]$$

The Galerkin coefficients, ${\stackrel{\sim }{d}}_j^k$, are the experimentally obtained pixel values of the thermal images. The temperature images and model parameters are assumed independent, $\rho (\overrightarrow{d},\overrightarrow{m})={\rho }_D(\overrightarrow{d}){\rho }_M(\overrightarrow{m})$. A Gaussian measurement model is assumed for ${\rho }_D(\overrightarrow{d})$

$${\rho }_D(\overrightarrow{d})=\mathrm{const}.\exp \frac{1}{2}\parallel \overrightarrow{d}-{\overrightarrow{d}}^{\mathit{obs}}{\parallel }_D=\frac{1}{{(2\pi )}^{(N_{\mathit{dof}\ast }{N}_{\mathit{step}})/2}|C_D{|}^{1/2}}\exp \frac{1}{2}{\left(\overrightarrow{d}-{\overrightarrow{d}}^{\mathit{obs}}\right)}^T{C}_D^{-1}\left(\overrightarrow{d}-{\overrightarrow{d}}^{\mathit{obs}}\right)$$

Here | · | is the matrix determinant. The norm over the data space ∥ · ∥_D (2) characterizes the inverse of the covariance matrix for the data space, $C_D^{-1}$ and is defined through the weighted space time norm. The MRTI provides a natural weighting function, w(x, t), that is inversely proportional to the amplitude of the signal to noise ratio (SNR) [29].

$$w(x,t)=\frac{1}{2\pi \alpha ·\gamma B_0·\mathrm{TE}}\frac{\sqrt{2}}{\mathrm{SNR}(x,t)}$$

Here α, $­0.0097\left[\frac{\mathit{ppm}}{°C}\right]$, is the temperature sensitivity coefficient, γ, 42.58 MHz/T, is the gyromagnetic ratio of water, B₀, 1.5T, is the static magnetic field strength, and TE, 20ms, is the echo time of the gradient echo pulse sequence. High signal to noise regions of the thermal images result in variance of the thermal signal within w(x, t) < 1°C; differences in the data will be weighted heavily in this region. Within low SNR regions, the variances are significantly larger and differences with the data are weighted less.

In general, $C_D^{-1}$ is not diagonal and has a sparsity dependent on the order of the shape functions and the quadrature rule used in the finite element approximation.

$$\begin{array}{lll}{\parallel u-u^{\mathit{M\; RT\; I}}\parallel }_D^2& =& \sum _k\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}{\left(\frac{d_i^k{\varphi }_i(x)-{\stackrel{\sim }{d}}_j^k{\varphi }_j(x)}{w(t^k,x)}\right)}^2\mathit{dx}\\ & =& {(\overrightarrow{d}-{\overrightarrow{d}}^{\mathit{obs}})}^T\left[\begin{array}{ccccc}M& 0& .& 0& 0\\ 0& M& 0& .& 0\\ .& 0& .& 0& .\\ 0& .& 0& .& 0\\ 0& 0& .& 0& M\end{array}\right](\overrightarrow{d}-{\overrightarrow{d}}^{\mathit{obs}})\end{array}M=\left[\begin{array}{ccc}.& .& .\\ .& {\mathit{\int }}_{\mathrm{\Omega }}\frac{{\varphi }_i(x){\varphi }_j(x)}{w(x)}\mathit{dx}& .\\ .& .& .\end{array}\right]$$

Physically realistic bounds for model parameters are obtained from tabulated handbook data [30, 25, 28] and are used to generate a uniform distribution over the set of feasible parameters, M ∈ M.

$${\rho }_M(\overrightarrow{m})=\{\begin{array}{cc}\mathrm{const}.\hfill & \overrightarrow{m}\in M\subset \mathrm{M}\hfill \\ \hfill 0& \overrightarrow{m}\notin M\hfill \end{array}$$

2.2 Discretization and Constitutive Data

The finite element model of the in vivo TVT inoculated canine MRgLITT is shown in Figure 1. The insertion of the applicator into the tissue was modeled using a mesh that consisted of distinct regions for the applicator, healthy tissue, and cancer burden. A quadrilateral mesh was extruded axially along the applicator to create the base of the hexahedral finite element mesh. The mesh for the tissue conforms to the boundary of the application and extends to the boundary of the brain to ensure that the boundary does not influence the heating. The entire mesh consists of N_dof =47,941 total nodes (grid points). At each time step of the Crank-Nicolson scheme used, the temperature across the applicator was held constant to model the effect of the room-temperature cooling fluid which protects the laser fiber during heating. The degrees of freedom across the applicator were held at 21°C by treating the corresponding degrees of freedom as Dirichlet boundary data. The bioheat transfer model temperature prediction is not seen to be very sensitive to the optical scattering [31]. Only optical absorption and thermal conductivity parameter estimation is considered in this work. The combined thermal conductivity and optical absorption estimation consisted of N_model =4,570 spatially varying parameters. The remaining input parameters are assumed deterministic. The laser power profile used during the therapy is shown in Figure 2. The test pulse, 3W for 30 seconds, and main therapeutic pulse, 3.5W for 3min, are shown. The amount, duration, and magnitude of heating data needed to accurately recover the optical parameters during the “test” pulse of the experimental data and subsequent critical evaluation of accuracy of the calibrated computer model prediction during the therapeutic delivery is considered outside the scope of this work. The current work seeks to demonstrate the potential of parameter estimation, hence, parameter estimation for verifying uptake and potentially planning the procedure is considered for the time interval of the strongest fluence signal for the data acquired. Data acquired during the time window of the therapeutic heating and subsequent cooling, δt = [204, 504], N_step = 300 seen in Figure 2, is used to demonstrate the feasibility of recovering the 3D spatially varying optical distribution. The Crank-Nicolson scheme was used to propagate the temperature field at one second time steps. Table 1 summarizes the constitutive data used.

Figure 2.

Power profile. The laser power profile used during the nanoparticle mediated MRgLITT procedure is shown. A 3W test pulse is applied for 30 seconds to verify the applicator position. The test pulse is followed by the main therapeutic pulse, 3.5W for 3min. The time window of the therapeutic pulse, δt_heating, and the cooling, δt_cooling, provides the maximum fluence and is used in the parameter recovery in this work. The time-temperature history of the finite element method (FEM) predicted and MRTI measured heating is shown on the right axis at two spatial locations. The spatial locations are illustrated in Figure 3(c) as (1) and (2). The history at the two locations illustrates the temperature differences between the selective and non-selective heating regions. Point (2) was chosen within the nanoparticle selective heating region. Point (1) was chosen on the opposite side of the fiber, outside of the region of expected nanoparticle accumulation, and shows significantly less heating.

Table 1.

Constitutive Data [28, 25, 30]

$\left[k^{\mathit{lb}},k^{\mathit{ub}}\right]\frac{W}{m·K}$	$\omega \frac{kg}{m^{3}s}$	g	${\mu }_s\frac{1}{m}$	$\left[{\mu }_a^{\mathit{lb}},{\mu }_a^{\mathit{ub}}\right]\frac{1}{m}$	$\rho \frac{\mathit{kg}}{m^3}$	$c_{\mathit{blood}}\frac{J}{\mathit{kg}·K}$	$c_p\frac{J}{\mathit{kg}·K}$
[0.1, 0.7]	9.0	0.88	34.0e3	[60.0, 600.0]	1045	3840	3600

The field was optimized in a 1.5cm diameter spherical ROI about the laser applicator centroid.

2.2.1 Gradient and Hessian Evaluation

The bound constrained optimization suite of TAO [32] was used in finding the maximum likelihood value of (3). Bounds were applied by restricting the search space to the feasible parameters, M. A trust-region Newton-conjugate gradient (Newton-CG) method ² was used to solve the optimization problem. The method allows search directions of negative curvature. Both gradient and Hessian information was used. A BFGS preconditioner and Eisenstat-walker convergence criteria was used for the inexact Newton solve.

A Crank-Nicolson time discretization scheme is used to obtain a finite element solution of the Pennes bioheat transfer model. Gradient and Hessian information from the misfit function is computed consistently with the discretization. For computational efficiency, the Hessian is not explicitly computed and only Hessian-vector products are considered [33]. An adjoint variable p(x, t) is needed to compute the gradient of the misfit function with respect to thermal conductivity and optical absorption.

$$\frac{\partial S(u,m)}{\partial {\widehat{k}}_i}=\sum _{k=1}^{N_{\mathit{step}}}\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}-\nabla u^{k-\frac{1}{2}}·\nabla p^k{\psi }_i\mathit{dx}\frac{\partial S(u,m)}{\partial {\widehat{\mu }}_{a_i}}=\sum _{k=1}^{N_{\mathit{step}}}\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}\frac{\partial }{\partial {\mu }_a}{q}_{\mathit{laser}}{p}^k{\psi }_i\mathit{dx}$$

Notice that the gradient of the misfit with respect to the optical absorption impose a natural regularization that is weighted by the fluence, φ(x, m)

$$\frac{\partial S(u,m)}{\partial {\widehat{\mu }}_{a_i}}=\sum _{k=1}^{N_{\mathit{step}}}\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}{p}^k{\psi }_i{\mathit{\int }}_{{\mathrm{\Omega }}_{\mathrm{tip}}}\frac{{\mu }_{\mathit{tr}}+{\mu }_a-{\mu }_{\mathit{tr}}{\mu }_a\parallel x-\xi \parallel \frac{3({\mu }_a+{\mu }_{\mathit{tr}})}{2{\mu }_{\mathit{eff}}}}{{\mu }_{\mathit{tr}}{\mu }_a}\phi (\mathrm{x},m)\mathit{d\xi dx}$$ (4)

and the gradient is zero when either the power is off or when the fluence is zero far away from the applicator. Physically, this means that the inverse problem technique cannot recover model parameters in regions where there is no fluence as expected.

A second adjoint variable, p̃(x, t), is needed to compute the product of the Hessian with an arbitrary model parameter, $\overrightarrow{q}$, representing a linear combination of sensitivities, ${\sum }_j^{N_{\mathit{model}}}\frac{\partial u^{k-\frac{1}{2}}}{\partial m_j}{q}_j$.

$$\begin{array}{lll}\sum _{j=1}^{N_{\mathit{model}}}\frac{{\partial }^{2}S(u,m)}{\partial k_i\partial m_j}{q}_j& =& \sum _{k=1}^{N_{\mathit{step}}}\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}\left[\sum _{j=1}^{N_{\mathit{model}}}-\nabla \frac{\partial u^{k-\frac{1}{2}}}{\partial m_j}{q}_j·\nabla p^k-\nabla u^{k-\frac{1}{2}}·\nabla {\stackrel{\sim }{p}}^k\right]{\psi }_i\mathit{dx}\\ \sum _{j=1}^{N_{\mathit{model}}}\frac{{\partial }^{2}S(u,m)}{\partial {({\mu }_a)}_i\partial m_j}{q}_j& =& \sum _{k=1}^{N_{\mathit{step}}}\mathrm{\Delta }t{\mathit{\int }}_{\mathrm{\Omega }}\left[\sum _{j=1}^{N_{\mathit{model}}}\frac{{\partial }^2}{\partial {\mu }_a^2}{q}_{\mathit{laser}}{p}^k{\psi }_j{q}_j-\frac{\partial }{\partial {\mu }_a}{q}_{\mathit{laser}}{\stackrel{\sim }{p}}^k\right]{\psi }_i\mathit{dx}\end{array}$$

Matrix shell operations provided by PETSc [34] were implemented to compute the Hessian vector products.

3 Results

The results for a recovered maximum likelihood solution are shown in Figures 3 and 4. A comparison of the maximum over time MRTI data to the finite element model is shown in Figure 3(b) and (c). The nanoparticle selective heating is seen to be reproduced by the finite element modeling. The time-temperature history of the two points labeled (1) and (2) in Figure 3(c) is shown in Figure 2. Point (2) plots the predicted and measured temperature in the nanoparticles mediated heating region and point (1) shows the measured and predicted temperature in the normal tissue on the contralateral side of the applicator. A distance versus maximum temperature profiles is shown in Figure 4. The location of the upper and lower profiles is illustrated in Figure 3(b). Pointwise agreement between the model prediction and MR temperature imaging values is seen to be < 5°C depending on location along this profile.

Figure 3.

Maximum likelihood finite element prediction. (a) The edema ring seen in the anatomical imaging is the signature of the extent of the thermal dose delivered. The solutions are overlaid onto the post treatment contrast enhanced anatomical imaging of the lesion within the ROI illustrated. In the image shown, the active tip is placed to the side of the tumor such that the tumor is on the right side of the laser and normal tissue is to the left of the laser. (b) The finite element prediction of the maximum temperature distribution corresponding to the maximum likelihood value is shown. (c) The corresponding maximum MR thermal imaging over time is projected onto the finite element mesh and provided for comparison. Temperature is shown in °C. The temperature history at spatial locations denoted (1) and (2) is provided in Figure 2 and illustrates the temperature differences between the selective and non-selective heating regions as a result of the nanoparticles being passively taken up in the tumor vasculature and not in normal brain. (d) The optical parameter map is provided in [m⁻¹]. (e) The map of the thermal conductivity is provided in $\left[\frac{W}{m·K}\right]$. The combined thermal conductivity and optical absorption recovery, consisted of N_model =4,570 parameters. The tissue properties near the applicator were allowed to vary in the region where selective heating is expected; this region is cropped from the full 3D simulation and highlighted in (d), (e), and (f). (f) The unit power fluence distribution is provided as a reference in $\left[\frac{W}{m^2}\right]$. The optical absorption map shows high optical properties on the periphery of the tumor and near the laser. Information about the optical parameters further away from the high fluence region is not provided in the technique. Spatial profiles of the optical absorption, thermal conductivity, temperature, and fluence are illustrated in (b)-(f) and plotted in Figure 4.

Figure 4.

Spatial profiles of the maximum likelihood solution. The location of the two profiles shown is provided in Figure 3. All profiles are plotted with respect to the same distance (mm) from the applicator tip to quantitatively compare the solutions. The solid lines correspond to the lower profile, “lo”, shown in Figure 3. The dotted lines correspond to the upper profile, “hi”, shown in Figure 3. The spatial variation of the thermal conductivity and optical absorption is shown at the bottom. The maximum temperature seen in the finite element method (FEM) prediction and MRTI measurements along the profile is shown at the top. Good agreement is seen between the measured and predicted temperature. The unit power fluence, obtained with the spatially varying optical absorption parameters, is also provided as a reference for the laser source at the top.

The corresponding maximum likelihood parameter values for the optical absorption and thermal conductivity are shown in Figure 3(d) and (e). Contrast enhanced imaging of the perfusion deficit and surrounding edema resulting from the heating is used as an anatomical reference to overlay the maximum likelihood solution. The largest optical absorption values are seen close to the active diffusing tip where the fluence is the largest. The large optical absorption values observed at larger distances from the diffusing tip are artifacts of the optimization solvers in the presence of low fluence. The computed fluence for unit power, 1W, with the inverse problem recovered optical parameters is provided in Figure 3(f).

4 Discussion

Results indicate the maximum likelihood solution found, ${\overrightarrow{m}}_{\mathit{M\; L}}$, reproduces the selectivity and magnitude of the NP mediated laser induced heating and therefore the ML solution is likely to return useful parameters. Fast MRTI techniques combined with advanced, high performance inverse analysis algorithms offer a methodology to recover the in vivo spatial distribution of optical parameters due to presence of treatment modifiers, such as gold-silica nanoshells, near the laser. Reported measured correlations between optical absorption and nanoparticle concentration are provided in Table 2 as a reference for the optical absorption recovery in Figure 3. As seen in Figure 3, recovery of the map of the optical absorption is limited in the current technique to regions of significant penetration of photon fluence. Accurate μ_a(x) estimation outside this region is not possible as the fluence induced heating is the fundamental mechanism by which the absorption parameters is recovered. The gradient (4), and subsequent changes in μ_a(x), is zero when the power is off and decays with the fluence as distance increases from the tip. The changes in parameters observed in these regions are likely caused by the ill-posed formulation in this area and are an artifact of the optimizer. The optical parameters far from the laser source do not affect the heating near the applicator.

Table 2.

Concentration Correlations, Provided by Nanospectra Biosciences, Inc., Houston, TX

μa[m−1]	conc [NP/ml]
215	3.58e9
362	6.02e9
474	7.88e9

The current technique determined the maximum likelihood solution by jointly optimizing biothermal and optical parameters. Although the maximum temperature distribution is seen to reproduce the nanoparticle selective heating, further work is needed in decoupling and validating the biothermal parameter estimation from the optical parameter estimation. Validation of the optical parameter estimation may be compared to experimental methods such as neutron activation analysis. Perfusion imaging techniques may also provide additional information for correlating the nanoparticle uptake [35]. Intuitively, a physics motivated optimization process should recover biothermal properties during the cooling regime of the procedure followed by optical parameter recovery, while holding the biothermal parameters fixed, during the heating regime of the procedure [36]. Further, absorption parameters during the initial stages of heating may be approximated during the initial linear temperature rise before thermal conductivity has influenced the heating. Recovering the biothermal parameters independently of optical parameters at different stages of the procedure decouples the inverse solution and could improve computational efficiency by reducing the time period and, thus, computational expense of the cost function evaluation per iteration [19]. This would eliminate the potential for changes in thermal conductivity to compensate for changes in optical parameters and vice-versa, seen in Figure 3. For example, inaccuracies in the cooling flow of the applicator could lead the optimization process to attempt to correct for this by adjusting the thermal conductivity parameters accordingly. Subsequently, the recovery of the optical parameters assuming incorrect thermal conductivity parameters will be skewed.

Ultimately, the number of iterations and computational cost per iterations needed to converge to a maximum likelihood solution using the joint and decoupled approach should be critically evaluated for clinical timeliness. Depending on the presence of local minima and parameter constraints, the joint and independent solution techniques may also give different quality solutions. For real-time application, at least an order of magnitude speedup is needed over initial codes used in this feasibility study, the runtime to converge to a maximum likelihood value on the 84 cores used was approximately 6-8 hours. All computations were performed on resources allocated at Texas Advanced Computing Center (TACC) on the lonestar system. The solvers were run on seven compute nodes, each with two Xeon Intel®Hexa-Core 64-bit 3.33GHz Westmere processors (12 cores in all) and 24GB per node. The development of computationally inexpensive methods for providing a high quality initial guess and preconditioner [37] as well as GPU-based FEM solvers [38] are under consideration for improving computational efficiently. Further computational efficiency may also be gained through full space Newton methods [39, 40] that optimize the state space and parameter space jointly.

Future studies will incorporate more accurate laser models [41] that numerically solve the governing equations of radiation transport to account for the path-wise dependence of the optical attenuation. For this coupled system of radiation and bioheat transfer equations, the algorithmic complexity of computing the gradient and Hessian of the misfit function, Section 2.2.1, increases significantly. Higher accuracy δP models [26] may provide an increased penetration of the underlying fluence model for the inverse recovery techniques. δP models [26] that relax the high scattering assumptions of the SDA are also expected to be more appropriate for the nanoparticle environment.

Information pertaining to the variance of the parameters about the maximum likelihood point would provide further clarity as to the sensitivity of the solution. Further, techniques that propagate the statistical characterization of the posteriori state of the model parameters (1) would provide valuable information as to the confidence in a therapy simulation based on the model parameters recovered. Unfortunately, the posteriori state (1) is of a non-Gaussian form and is very computationally expensive to infer this level of statistical information. Frontier techniques in uncertainty propagation, including Markov Chain Monte Carlo [42, 43] and stochastic Galerkin [22, 44] methods are under consideration. Bayesian inference ideas of model ranking and averaging of model predictions [45, 46, 47, 48] may also be explored. One possible computational alternative may be to assume that the model space prior ρ_M is Gaussian. Under this assumption, a second order Taylor series expansion of the misfit function reveals that the posteriori probability distribution σ_M may be approximated as Gaussian near the maximum likelihood point. Further, the Hessian may be shown to be the inverse of the covariance matrix of the posterior distribution.

$$\begin{array}{c}S(\mathbf{m})\approx S({\mathbf{m}}_{\mathit{M\; L}})+\mathrm{g}({\mathbf{m}}_{\mathit{M\; L}})(\mathbf{m}-{\mathbf{m}}_{\mathit{M\; L}})+\frac{1}{2}{(\mathbf{m}-{\mathbf{m}}_{\mathit{M\; L}})}^{T}H({\mathbf{m}}_{\mathit{M\; L}})(\mathbf{m}-{\mathbf{m}}_{\mathit{M\; L}})\\ \mathrm{g}({\mathbf{m}}_{\mathit{M\; L}})=\frac{\partial S}{\partial \mathbf{m}}({\mathbf{m}}_{\mathit{M\; L}})=\overrightarrow{0}\Rightarrow {\sigma }_M(\mathbf{m})\approx \mathrm{const}.\exp \left\{-\frac{1}{2}{(\mathbf{m}-{\mathbf{m}}_{\mathit{M\; L}})}^{T}H({\mathbf{m}}_{\mathit{M\; L}})(\mathbf{m}-{\mathbf{m}}_{\mathit{M\; L}})\right\}\end{array}$$

Here the maximum likelihood point, m_{M L}, is a critical point of the misfit function. Notice that the second derivative test of optimization theory implies that the Hessian is positive definite and its inverse exists. Effcient estimates of the diagonal of the covariance matrix [49] as the variance in the model parameters would have significant practical influence.

5 Conclusion

A finite element based inverse recovery technique has been employed to recover the relevant optical parameters needed for modeling nanoparticle mediated thermal therapy and agreement of predicted heating with MRTI measurements demonstrated. Despite the use of an inferior fluence model, the maximum likelihood approach produced a model of the nanoshell selective heating observed within the tumor with a high degree of accuracy. Further work developing the statistically oriented algorithm presented here into clinically reproducible techniques of quantitatively validating nanoparticle concentrations appears promising. The combined technique with validated and accurate computational tools for MRgLITT planning, monitoring, and procedure optimization represents a suitable cancer treatment option for inoperable cancerous lesions.

The presented concepts for inverse recovery of the nanoparticle distribution are not restricted to laser induction alone and could be readily extended to an energy delivery approach that is able to adequately deliver a nanoparticle mediated thermal dose throughout the target lesion of interest. Further mechanisms include external electromagnetic field approaches [50], radio-frequency [51] and photo-acoustic heating [52, 53]. Noninvasive knowledge of the patient specific nanoparticle uptake can be exploited to help accurately plan and statistically optimize the outcome of thermal therapy delivery techniques and even enhance radiation therapy dose [54].