Introduction. Keywords: bioheat transfer; biomagnetism; finite elements; simulation; thermal damage; thermal medicine.

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1 Biomed. Eng.-Biomed. Tech. 2015; 60(5): Frederik Soetaert, Luc Dupré, Robert Ivkov and Guillaume Crevecoeur* Computational evaluation of amplitude modulation for enhanced magnetic nanoparticle hyperthermia DOI /bmt Received March 8, 2015; accepted July 17, 2015; online first August 26, 2015 Abstract: Magnetic nanoparticles (MNPs) can interact with alternating magnetic fields (AMFs) to deposit localized energy for hyperthermia treatment of cancer. Hyperthermia is useful in the context of multimodality treatments with radiation or chemotherapy to enhance disease control without increased toxicity. The unique attributes of heat deposition and transfer with MNPs have generated considerable attention and have been the focus of extensive investigations to elucidate mechanisms and optimize performance. Three-dimensional (3D) simulations are often conducted with the finite element method (FEM) using the Pennes bioheat equation. In the current study, the Pennes equation was modified to include a thermal damage-dependent perfusion profile to improve model predictions with respect to known physiological responses to tissue heating. A normal distribution of MNPs in a model liver tumor was combined with empirical *Corresponding author: Guillaume Crevecoeur, Department of Electrical Energy, Systems and Automation, Ghent University, Technology Park 913, B-9052 Zwijnaarde, Belgium, Phone: , Fax: , guillaume.crevecoeur@ugent.be Frederik Soetaert: Department of Electrical Energy, Systems and Automation, Ghent University, Technology Park 913, B-9052 Zwijnaarde, Belgium; and Department of Radiation Oncology and Molecular Radiation Sciences, The Johns Hopkins University School of Medicine, Baltimore, MD 21231, USA Luc Dupré: Department of Electrical Energy, Systems and Automation, Ghent University, Technology Park 913, B-9052 Zwijnaarde, Belgium Robert Ivkov: Department of Radiation Oncology and Molecular Radiation Sciences, The Johns Hopkins University School of Medicine, Baltimore, MD 21231, USA; NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD , USA; Department of Oncology, The Johns Hopkins University School of Medicine, Baltimore, MD 21231, USA; Institute for NanoBioTechnology, The Johns Hopkins University, Baltimore, MD 21218, USA; and Department of Materials Science and Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA nanoparticle heating data to calculate tumor temperature distributions and resulting survival fraction of cancer cells. In addition, calculated spatiotemporal temperature changes were compared among magnetic field amplitude modulations of a base 150-kHz sinusoidal waveform, specifically, no modulation, sinusoidal, rectangular, and triangular modulation. Complex relationships were observed between nanoparticle heating and cancer tissue damage when amplitude modulation and damage-related perfusion profiles were varied. These results are tantalizing and motivate further exploration of amplitude modulation as a means to enhance efficiency of and overcome technical challenges associated with magnetic nanoparticle hyperthermia (MNH). Keywords: bioheat transfer; biomagnetism; finite elements; simulation; thermal damage; thermal medicine. Introduction Hyperthermia cancer therapy is a modality used to treat cancer by selectively raising the local tissue temperature to inflict damage to cancer cells [22]. A dose-dependent cytotoxic effect results when temperature is raised to a range between 41 C and 46 C and maintained for a period of time [47]. Hyperthermia has proven to be an effective component of multimodality treatments for managing local and recurrent disease when it is combined with radiation and chemotherapy because the biological and physiological effects of tissue heating complement DNA- and tissue-damaging effects of many standard therapies [30, 35, 47]. Despite the compelling preponderance of clinical and biological data, clinical applications of hyperthermia are typically confined to treating superficial tumors because technical challenges associated with energy delivery and control remain for treating tumors located deep within the body (e.g. liver). The success of treatment with hyperthermia depends upon the precision of energy delivery and control to achieve a prescribed thermal dose within a tumor and its margins in adjacent normal tissue,

2 492 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH while minimizing off-target heating elsewhere. Stated another way, it is the combination of anatomical complexities, and physics and engineering challenges associated with precise heat delivery and control, that have impeded widespread clinical implementation of hyperthermia cancer therapy. Magnetic nanoparticles (MNPs) have emerged as a promising platform to address the technical challenges presented by heating tissue. MNPs can generate significant heat when they are exposed to alternating magnetic fields (AMFs), which are not attenuated by tissue [8, 9, 11, 14, 17, 29, 36, 37, 43]. Furthermore, remote activation of the nanoparticles enables non-invasive control of energy deposition to the tumor and its surroundings, enabling specific treatment algorithms that exploit the unique properties of intra-tissue heat generation and transfer [2, 4]. While magnetic nanoparticle hyperthermia (MNH) is an auspicious hyperthermia technique, new challenges to balance the nanoparticle material properties with AMF conditions become evident. Principally, MNPs need to be sufficiently biocompatible and optimized to generate therapeutic heating to achieve the desired clinical benefits at acceptable (tolerable) tissue concentration; and with AMF frequency and amplitude combinations appropriate for both patient and clinical personnel. Although it is recognized that efficiency of heating by MNPs depends upon general magnetic properties of the nanoparticle and AMF frequency and amplitude, the details of these interactions continue to generate considerable debate [6, 8, 9, 11, 14, 17, 21, 31, 33, 43]. Nevertheless, a consensus has emerged that the interactions are complex and that the internal magnetic structure of the nanoparticles [14], as well as interparticle magnetic dipole-dipole interactions [6], can have considerable influence [11]. Likewise, interaction of AMFs with biological tissue can deposit non-specific heating due to induced eddy currents, and this power deposition can initiate systemic thermoregulatory responses in mammals creating complex thermal gradients throughout the body [1, 3, 32, 45]. Thus, although it is recognized that considerable clinical potential exists to effectively treat malignant solid tumors by heating MNPs with AMFs, it is also evident that this platform technology harbors complexities demanding additional study to enhance clinical implementation. An outstanding challenge identified in the first demonstration of MNP cancer hyperthermia is the need to balance nanoparticle heating performance with AMF frequency and amplitude [24, 31]. The heating performance of a given sample of MNPs, often reported as specific loss power (SLP, or equivalent) with unit watts per gram of magnetic material (W/g magnetic material), depends upon amplitude when measured at fixed AMF frequency [5]. Generally, loss power (and thus heating rate) increases with increasing AMF amplitude to a value approximating magnetization saturation of the sample [11]. Although a detailed and robust description of the physical mechanism(s) governing the amplitude-dependent SLP remains to be elucidated, it is recognized that empirical description of this relationship provides useful information for energy calculations relevant to therapy as demonstrated in mouse model studies [10]. Similarly, modulating AMF power and amplitude have been explored to elucidate effects on tissue heating and thermoregulatory response (e.g. perfusion), although much remains to be studied [3]. The consensus opinion is held that exposing a large volume of a patient (i.e. torso, ~30 cm diameter) to AMF limits frequency and amplitude combinations to approximately H f A/(m s), where H is the field amplitude (A/m) and f is the frequency (Hz) [3]. Clinical application thus imposes a difficult performance specification for the MNPs: to generate heat with AMF, a process that depends upon AMF amplitude, while minimizing the AMF amplitude to avoid off-target heating. Experimental and computational efforts have been devoted to address this challenge and to develop an understanding of the interplay between nanoparticle-amf interactions and the heat generated within tissues to extract principles useful to optimize implementation [4, 7, 34, 39]. Certainly, progress has been realized, but often, simplifications, such as neglecting temperature-dependent tissue response (i.e. thermoregulation via perfusion), are implemented to enhance computational efficiency. Furthermore, little attention has been devoted to study the potential effects of modulating AMF amplitude as a means to enhance the efficiency of nanoparticle heating. We hypothesized that modulating the amplitude of a constant-frequency AMF may significantly enhance heat generated by nanoparticles having a non-linear amplitude-dependent SLP (i.e. SLP(H)); and that active thermoregulation in response to this heating, i.e. temperature- and thermal damage-dependent perfusion provides a critical component to assess resulting temperaturedependent tissue damage. To test this hypothesis, we calculated temperature in model tumor tissue that results from heat produced by model MNPs using experimental SLP(H) values reported for bionized nanoferrite (BNF) MNPs [5]. Amplitude-modulated MNH was tested by applying different amplitude modulations of a 150-kHz sinusoidal waveform, creating time-dependent variation of the SLP [i.e. SLP(H(t))]. Consequently, volumetric heating power

3 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 493 also becomes time-dependent, providing a basis to evaluate whether amplitude modulation enhances the overall estimated heating efficiency. The Pennes bioheat equation, describing macroscopic heat transfer in biological tissues, was solved using the finite element method (FEM). The tumor-tissue model used in the present study comprised concentric spheres (tumor within larger tissue/organ) having thermophysical parameters of liver and tumor obtained from the literature. Corresponding blood and blood perfusion values were also obtained from literature. Blood perfusion, a physiological thermoregulatory process and a cooling term in the model, is defined as a constant value in the classic Pennes equation. We compare the classic form with one containing modifications to include measured thermal damage-dependent response. The MNP distribution is defined as a Gaussian about tumor center and is assumed to be time-independent. Change of tissue temperature resulting from nanoparticle heating was calculated, and the generated data were used to calculate thermal damage using established timeat-temperature models that describe cell/tissue thermal damage by an Arrhenius function (i.e. survival fraction). Comparison of the thermal damage among the different modulations shows significant differences that are highly dependent on the studied location (center or boundary of the tumor). This opens perspectives, because amplitude modulation might be a useful method to enhance the thermal damage due to MNPs without the price of increasing eddy current heating. Materials and methods Model and Pennes heat transfer calculations Macroscopic heat transfer in biological systems is typically modeled using the Pennes bioheat equation [42]: T( r, t ) ρc = [ k T( r, t)]- ω( r, t) ρ c [ T( r, t)-t ] + P. b b b MNP t The left-hand side of this energy balance equation describes the change of temperature T ( C) in time (t) at a certain point in space ( r ) with a prefactor consisting of the mass density ρ (kg/m 3 ) and specific heat capacity c (J/[kg K]) of the biological tissue. Meanwhile, the right-hand side of Eq. 1 contains the heat sources and sinks causing the temperature change. The first term is the classic heat conduction term containing the thermal conductivity k (W/[m K]). The Pennes bioheat equation also incorporates the heat sink effect of blood perfusion. This effect is proportional to the blood perfusion rate ω (s -1 ), the blood mass density ρ b, the specific heat capacity of blood c b, and the difference between the tissue temperature, T, and the blood temperature (T b = 37 C). Finally, P MNP (W/m 3 ) is the (1) Figure 1: Schematic representation of the geometry of the numerical model. A 1-cm 3 spherical tumor (d tumor = cm) is placed in the center of a sphere with healthy liver tissue (d liver = 12 cm). Body temperature (37 C) is enforced at the outer boundary of the healthy liver tissue (Dirichlet boundary condition). volumetric power dissipation of the MNPs in an AMF. We assume in this contribution that there is no power dissipation originating from eddy currents that may be induced by the AMF. In case one wishes to incorporate this effect, an additional electromagnetic induction term in the right-hand side of Eq. 1 needs to be included. In the present study, the Pennes bioheat equation is solved in a model liver. We geometrically approximate the liver as a sphere with a diameter of 12 cm. The outer diameter needs to be chosen carefully to ensure that the applied Dirichlet boundary condition T V = 37 C is a good approximation, as in [4]. A spherical tumor of 1 cm 3 is present in the center of the liver (d = cm). At the interface between healthy and tumor tissue, continuity conditions for the temperature and heat flux are present. A schematic representation of the simulation environment is given in Figure 1. The relevant thermophysical properties of healthy liver tissue, liver tumors, and blood are summarized in Table 1 [27]. Other simplified geometrical models for the liver are cylinder shapes [27]. Although concentric spheres are extremely useful as a simple tool to deduce general trends for clinical procedures, the use of patient-specific 3D imaging data would be encouraged. Candeo and Dughiero [7], for instance, use computed tomography data for computational studies of MNH. Calculation of thermal damage to tissue Thermal damage in biological systems exposed to elevated temperature depends upon both temperature and time at temperature. Survival studies of mammalian cells in culture exposed to elevated

4 494 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH Table 1: Thermophysical properties of healthy liver tissue, liver tumors, and blood, as used in this computational study [27]. Tissue ρ (kg/m 3 ) c (J/[kg K]) k (W/[m K]) Healthy liver tissue Liver tumor Blood N/A temperature have demonstrated that the cytotoxic effects of heat obey an Arrhenius relationship [15, 16, 40, 41]. Therefore, we use the survival fraction based on a kinetic model with the underlying assumption of a single transition state energy barrier with an irreversible molecular flux [18, 40, 41]. This Arrhenius damage index Ω can mathematically be modeled by the following differential equation: Ω( r, t) E = A t RT( r, t ) a exp -, where A is the frequency factor, ΔE a is the activation energy barrier, and R is the universal gas constant (8.314 J/[mol K]) [18]. The survival fraction α can then be calculated as [18, 40] (2) α ( r, t) = exp(- Ω( r, t)). (3) This survival fraction can be interpreted as the ratio of the concentration of living cells at the chosen end-point and the initial concentration. To calculate the survival fraction α, we use the following values: A = s -1 and ΔE a = J/mol [18]. Modifications of bioheat equation to include thermal damage-dependent perfusion and the hyperemic effect in the vicinity of these damaged regions [28, 44]. This profile is believed to be a more realistic representation of the physiological perfusion responses due to hyperthermia. MNP heating model The magnitude of nanoparticle power deposition, P MNP (W/m 3 ) in the Pennes bioheat equation (Eq. 1), was assumed to vary with applied magnetic field amplitude, depending upon measured SLP(H). The SLP is defined as the power generated per mass unit of Fe (W/g Fe). Following Attaluri et al. [4], a tissue MNP concentration corresponding to a tissue iron concentration of 5 mg Fe/cm 3 tumor was used. For a 1-cm 3 tumor, the total injected iron mass thus is m Fe = 5 mg Fe. The concentration of MNPs or the m Fe constitutes an important parameter for hyperthermia applications. The product of m Fe and the SLP thus equals the power deposited by the MNPs, where SLP is a function of both frequency (f) and magnetic field amplitude (H) [5]. Frequency was fixed, f = 150 khz, similar to conditions reported by Bordelon et al. [5] for the BNF-Starch particles (micromod Partikeltechnologie GmbH, Rostock, Germany). The magnetic heating properties, structure, and synthesis of these particles have already been extensively characterized [5, 12, 13, 25, 26]. The BNF-Starch particles have a very high and non-linear SLP at mid to high amplitudes [5]. We have listed the experimental SLP values of [5] for different applied magnetic field amplitudes in Table 2 of a 150-kHz sinusoidal waveform with no modulation. Because we need a continuous set of SLP values, we have fitted a sixth-order polynomial approximation to the data of Table 2. The resulting SLP(H) is described by SLP( H) = e H e H e H H H H The approximated values of the SLP, using Eq. 6, are listed in Table 2 and are in close correspondence with the reported experimental values. While approximating the SLP (ΔSLP = difference (6) In the classic Pennes bioheat equation, blood perfusion rate, ω, is constant and equal to the baseline blood perfusion rate, ω 0, during a hyperthermia treatment [42]. Local blood perfusion response to temperature, however, is complex and can increase or decrease depending upon both temperature and time at temperature [28]. In addition, thermal damage introduces a non-linear correlation with the degree of tissue damage, as observed in renal tissues [28]. This empirical non-linear correlation can be separated into four linear approximations that describe the relative perfusion rate in terms of the degree of vascular stasis DS [44]: ω [ 30 DS( r, t) + 1] if 0.00 DS ω [ -13 DS( r, t) ] if 0.02< DS ω( r, t) = ω [ DS( r, t) ] if 0.08< DS 0.97 (4) 0 ω [ DS( r, t) ] if 0.97< DS The degree of vascular stasis, a measure of chronic thermal necrosis [28], can be expressed in terms of the survival fraction α: DS( r, t) = 1- α( r, t ). (5) The baseline perfusion rate in normal human liver tissue is ω 0 = s -1 [46]. The perfusion profile, as described in Eq. 4, encapsulates both the cessation of perfusion in thermally damaged regions Table 2: Experimental SLP values in function of the applied magnetic field amplitude (peak-to-peak) (ka/m) for sinusoidal waveform at 150 khz [5]. Applied magnetic field amplitude (peak-topeak) (ka/m) Experimental SLP (W/g Fe) Fitted SLP (W/g Fe) ΔSLP (W/g Fe) The approximated SLP values are calculated using sixth-order polynomial curve fitting. Finally, the difference between experimental and fitted SLP values is shown.

5 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 495 between fitted and experimental SLP), errors are introduced in the P MNP -term of Eq. 1. Being the source of heat in the tumor, the spatial distribution of MNPs within tissue significantly dominates the spatial temperature profile of the tissue in MNH. We assume a 3D normal distribution [34], centered in the center of the tumor and with decay following a standard deviation σ in every direction: f MNP ( 1 x² + y² + z² x, y, z ) = exp -. 3/2 σ³( 2π) 2 σ² We neglect diffusion of the nanoparticles during the hyperthermia treatment (i.e. f MNP is time-independent). The total heating power distribution P MNP (x, y, z, t) is thus the product of the injected iron mass, the SLP, and the nanoparticle distribution: P ( x, y, z, t ) = m SLP ( f, H ( t )) f ( x, y, z ). (8) MNP Fe To simulate different forms of amplitude modulated MNH, we calculate the temperature distribution using four different magnetic field amplitude profiles H(t): constant (i.e. no modulation), rectangular, sinusoidal, and triangular modulations. These profiles are modulations superimposed on a 150-kHz sine waveform. As described in Eq. 8, the SLP is the only time-dependent component of P MNP. To incorporate the fact that all modulations with their specific shape parameters have a different specific loss energy (SLE) deposited in the tissue, we use the time integral of the SLP value, t i.e. SLE= SLP( f, H( τ))dτ (unit J/g Fe), instead of time as a measure. When varying the m Fe 0, the following measure can be used as well: LE = SLE m Fe (unit J). When depositing a certain amount of loss energy, or LE, thermal damage will be inflicted to the tissues and can be represented by 1- α( t ) Thermal damage per LE =. (9) LE ( t ) The larger the value of Eq. 9, the more damage is done to the tissue for a certain amount of applied energy. Note that this value is a linearization and merely represents an approximation of the rate of change of inflicted damage vs. the LE. Numerical methods We consider the Pennes bioheat equation (Eq. 1) in the sphere V and apply a Dirichlet boundary condition on the outer edge of the sphere at the constant body temperature T V = 37 C (also see Figure 1). The weak formulations of Eqs. 1 and 2 are used to solve this problem numerically, with test functions ϕ T and ϕ Ω related to the temperature and Arrhenius damage index, respectively: T ρc ϕ dv+ k T ϕ d V+ ω( Ω ) ρ c [ TT - ] ϕ dv- P ϕ dv = 0 t T T b b b T MNP T V V V V MNP V Ω E ϕ V A V t a d - exp - ϕ d 0 Ω = Ω RT V (7) (10) (11) Consequently, we solve a coupled set of partial differential equations: solutions of the temperature problem depend on the Arrhenius damage index and vice versa. Note that in the case of a symmetrical spatial distribution f MNP (x, y, z) of MNPs (spread in the three directions is the same) and a spherically symmetrical geometry, the calculations can be performed in one dimension (1D). The partial differential equation (Eq. 10) then becomes T 2k T T ϕt ρc ϕ d- r ϕ dr+ k dr T T t r r r r L L L e + ω( Ω ) ρ c [ T-T ] ϕ dr- m SLP d 0, b b b T ϕ r= 3 3/ 2 Fe T L L σ ( 2π) (12) with r 2 = x 2 +y 2 +z 2 and L being the 1D line considered that varies from 0 to the outer radius of the domain. Note that in the second term of Eq. 12, a 1/r term arises, which can generate singularities near the center r = 0. We therefore multiplied every term in the integrands with r. The meshing procedure for the 3D and 1D case is performed using Gmsh [23] and the numerical calculations using the finite element solver GetDP [19]. The time-stepping scheme is Crank-Nicholson with fixed time step. Gmsh is used as a graphical front-end through the ONELAB [38] interface. Post-processing of the results is performed using MATLAB (R2013a; Mathworks, Natick, MA, USA). The proposed methodology can be solved three-dimensionally, as we wish to develop a generic tool. It enables to assess the spatiotemporal temperature evolution and the thermal damage (in terms of the survival fraction) for cases where the spread σ in f MNP differs for the three directions or the applied field is inhomogeneous, where ellipsoidal or other shapes of tumor and healthy tissue, multiple tumor regions, inhomogeneous material properties having different thermal conductivities, heat capacities, perfusion rates, etc. are considered. Moreover, the inhomogeneous spatial distribution of power dissipation due to eddy currents can be included in the 3D model. So to proceed toward actual a priori treatment planning, the model needs extensive validation and the effect of material, thermal damage, geometrical, AMF field distribution parameters needs to be clarified. The 1D simulations can be utilized for validating the 3D implementation and enables swift analysis of various parameters such as iron mass m Fe, the spread σ of the MNP distribution, and the AMF profile H(t). The parameters considered in this article and that would be of importance when applying MNH are the AMF amplitude, frequency, MNP type, concentration, and spread. So to quantitatively measure the efficiency of the heating and thermal damage inflicted to the tissues, we post-process the temperature T and survival fraction α at various spatial points, as well as the average temperature in the tumor < T > t and healthy tissue < T > h (with the healthy tissue considered here as a shell of approximately 1 cm around the tumor) and the average damage in the tumor < α > t and healthy tissue < α > h. Finally, we calculate the thermal damage per loss energy, as described in Eq. 9. Results and discussion The clinical realities associated with MNH place demanding requirements on the nanoparticle material properties to realize significant heat production using low-amplitude AMFs. Although many aspects of this nanoparticle-amf and tissue heating interaction have been studied, the 2 -r 2 2 σ

6 496 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH effects of fixed frequency amplitude modulation on nanoparticle heating and the thermoregulatory response measured by perfusion have not been studied. We undertook a computational effort to elucidate features of this complex interaction, hypothesizing that modulating AMF amplitude can enhance heating to generate more tumor tissue damage. The Pennes bioheat equation was modified to include thermal damage-dependent perfusion. Temperatures were calculated using the finite element method and the tissue thermal damage by means of the survival fraction of cells based on an Arrhenius rate function. The accuracy of the temperature and survival fraction depends on the spatial finite element size and time-step discretization in the numerical finite elements. The validity and accuracy of both spatial and time-step discretization was examined by quantifying the convergence of the temperature and survival fraction for increasing levels of discretization. The compromise between achieving results at an acceptable computational time with acceptable accuracy (i.e. convergence for increasing levels of discretization) led to the use of approximately elements (corresponding with vertices) when performing 3D simulations. A fixed time-stepping method based on Crank-Nicholson was used. The ideal time step was identified to be 0.1 s for constant magnetic field amplitudes (no modulation) and rectangular modulations, whereas for sinusoidal and triangular modulations, 0.05 s was preferred as time step. Comparison between the 3D and 1D simulations was made to validate the results obtained by the 3D solver. Figure 2 shows the time variation of the average temperature in the tumor < T > t for the 1D and 3D case. The 1D case has a mesh consisting of about 200 elements and was simulated with a time step of 0.05 s. The dependence of the accuracy of the 3D simulations on the number of elements and time step is illustrated. When comparing 1D simulations (with time step 0.05 s) to 3D simulations having 5500 elements and time step of 0.1 s, the temperature deviated 5.1%. In case of 9000 elements, the discrepancy with the 1D simulations was 4% when using a time step of 0.1 s and 0.6% in case of a time step of 0.05 s. Further results are generated using the 3D model unless otherwise mentioned. As the MNPs are the heat source in MNH, the characteristics of the MNP distribution will have an important influence on the resulting temperature distribution. Therefore, we will analyze the effect of the total MNP concentration or m Fe and the MNP tissue distribution, characterized by the spread σ, on the temperature and survival fraction. Listed in Table 3 are the average temperatures and survival fractions calculated for both tumor and healthy tissue using a constant magnetic field amplitude (no modulation) H ampl = 20 ka/m (1DFEM) and varied particle concentrations and distributions. Also listed in Table 3 are calculated temperature (T b ) and survival fraction (α b ) at the boundary of the tumor. Note that the results are described from the perspective of Figure 2: Comparison between 3D and 1D simulations with the 1D simulations performed for 200 finite elements and a time step of 0.05 s. The accuracy of the 3D simulations can be assessed through comparison with the 1D solutions by varying the number of elements (5500 and 9000 elements) and the time step (0.05 and 0.1 s). These simulations were performed for a constant magnetic field (no modulation) with amplitude 30 ka/m.

7 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 497 Table 3: Average temperature and survival fraction in the tumor and healthy tissue together with the temperature and survival fraction on the boundary of the tumor for various MNP concentrations (given by the total injected iron mass m Fe ) and distributions (given by spread σ). m Fe (mg) σ (m) < T > t ( C) < α > t < T> h ( C) < α > h T b ( C) α b LE (J) Average thermal damage per LE in tumor (10-4 J -1 ) The results are put in perspective to the LE. The average thermal damage per LE in the tumor represents the average rate of change of the thermal damage inflicted to the tumor vs. the deposited loss energy. loss energy (LE). If a certain LE is enforced, the rate of decay of the survival fraction in the tumor < α > t is listed in the last column as being the average thermal damage per LE. The larger this value, the faster the decay of the survival fraction in the tumor. As can be anticipated, the smaller the spread, the more damage is inflicted to the tumor, whereas the larger the spread, the more damage is inflicted to the healthy tissue when compared to the tumor. When injecting a relatively large amount of MNPs (e.g. 5 mg), it is possible that the tumor is not affected because the spread of the MNP distribution is too large. For instance, m Fe = 5 mg and σ = m result in smaller average thermal damage when compared to lower m Fe with e.g. σ = m. These extensive computations were obtained using the 1D FEM. In case we presume that 95% of all particles are situated in the 1-cm 3 tumor (diameter d = cm), we have to assign σ x = σ y = σ z = σ = m as spread parameters for the MNP distribution. In case we increase the spread parameters to σ = m, corresponding with approximately 7.5% of the particles being in the tumor, a significant increase in survival fraction and a decrease in temperature elevation in tumor are evidently observed. The spread of nanoparticles has an important clinical relevance. If nanoparticles diffuse into regions of healthy tissue, adequate tumor heating cannot occur without risk of significant damage to normal tissues. Heat damage to adjacent normal tissues can arise, depending upon the nanoparticle distribution (i.e. concentrated near the tumor-tissue boundary), even if nanoparticles are confined to the tumor boundary. A safe and reproducible application of nanoparticles to tumors has often been cited as one of the most important challenges of MNH [20]. The difference in survival fraction α, as defined in Eq. 3, between σ = m and σ = m is illustrated in Figure 3 at two discrete points (center of the tumor and 1.92 mm from the center) when applying a constant magnetic field amplitude (i.e. no modulation) H ampl = 20 ka/m (peak-topeak value is 2H ampl ). Even though the spatial integration of Eq. 7 equals 1 for both distributions, the spatial spread changes the distribution of power deposition in the tumor, and thus the temperature gradients that arise due to thermoregulatory response (perfusion) in the tumor and adjacent healthy tissue. The spatial distribution of P MNP in Eq. 1 indeed significantly affects the temperature changes. As a comparison, we only have a maximum volumetric heating power of W/m 3 in the center of the tumor in the case of σ = m, whereas the maximum power dissipation is W/m 3 for σ = m. By comparing the time evolution of the survival fraction between a thermal damage-dependent perfusion model ω(ds) and a constant perfusion rate ω 0, we can assess the impact of the perfusion term. Figure 4 shows the survival fraction at two discrete points (in center and in a point at 1.92 mm from center) when using ω(ds) or ω 0 in Eq. 1, while applying a constant amplitude H ampl = 30 ka/m. Initially, we observe a bigger decrease of survival fraction in the ω 0 case compared to ω(ds), due to the increased blood perfusion when the thermal damage is limited in the ω(ds) case (Eq. 4). However, once the thermal damage exceeds 8% (Eq. 4), the survival fraction starts decreasing dramatically when using the ω(ds) perfusion profile. The constant perfusion case ω 0 thus has a much slower thermal damage compared to the ω(ds)

8 498 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH Figure 3: Influence of the fraction of particles situated in the tumor (95%, σ = m, and 7.5%, σ = m) on the survival fraction α in time at two discrete points (center and 1.92 mm from center). Figure 4: Survival fraction α as function of time at two discrete points for a thermal damage-dependent perfusion term (blue lines) and a constant perfusion ω 0 (red lines). A constant magnetic field amplitude (no modulation) of H ampl = 30 ka/m is applied to a MNP distribution having σ = m. model. The shapes of the survival fractions are fundamentally different when using the thermal damage-dependent perfusion rate compared to the constant perfusion rate. In the former case, the rate of decay of the survival fraction increases in time (concave shape), whereas the latter case is characterized by a long tail with a decreasing rate of decay (convex shape). Figure 4 therefore demonstrates the dramatic influence of the perfusion profile in Eq. 1 and highlights the importance of an accurate perfusion modeling. The survival fraction, when using ω 0 or ω(ds), depends on the applied magnetic field amplitude. Figure 5 depicts the effect of various constant magnetic field amplitudes H ampl (i.e. no modulation) on the survival fraction in the center of the tumor. We observe a non-linear thermal damage response related to the effect of the amplitude. For instance, at 200 s, we have α 0 for H ampl = 40 ka/m and α 0.4 for H ampl = 30 ka/m, whereas the survival fractions are α 0.95 and α 0.99 for H ampl = 20 ka/m and H ampl = 10 ka/m, respectively. These

9 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 499 Figure 5: Influence of constant magnetic field amplitudes (no modulation) on the survival fraction α in time in the center of the tumor for a MNP distribution with σ = m. large differences are related to the dependence of P MNP on the SLP (see Eq. 8). Until now, we displayed the effect of key components in the bioheat equation (Eq. 1), being the perfusion rate ω and parameters included in P MNP (MNP concentration/ injected iron mass, m Fe, and MNP distribution, σ). Besides constant magnetic field amplitudes (no modulation), we applied amplitude modulations as well. For instance, we consider a rectangular modulation varying between constant H min during time t d and during time t u. Figure 6 shows the time evolution of the survival fraction for a rectangular modulation with H min = 10 ka/m, = 40 ka/m, Figure 6: Survival fraction α as function of time at several radial distances from the center for a thermal damage-dependent perfusion term (solid lines) and a constant perfusion ω 0 (dashed lines). A rectangular modulation is applied with parameters H min = 10 ka/m, = 40 ka/m, t d = 1 s, and t u = 1 s. The MNP distribution has a spread σ = m.

10 500 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH Figure 7: Calculated temperatures for various radial distances from the center at different time intervals ( s) using a thermal damage-dependent perfusion term based on ω(ds). A rectangular modulation is applied with parameters H min = 10 ka/m, = 40 ka/m, t d = 1 s, and t u = 1 s. The MNP distribution has a spread σ = m. t d = 1 s, and t u = 1 s. The impact of the perfusion term is again illustrated: the dashed lines show the survival fraction when we apply a constant perfusion rate ω 0 in contrast to solid lines, representing the implemented perfusion model ω(ds). The tumor region follows a different trajectory in terms of survival fraction when presuming a thermal damage-dependent perfusion term. Similar trends as in Figure 4 can be observed for this magnetic field profile. The survival fraction follows a slower decrease in Figure 6 than in Figure 4 because the total amount of power deposited is lower. The temperature evolution is shown in Figure 7 on a radial line starting from the center, for the applied rectangular modulation. At the tumor boundary (i.e. at r = 6.2 mm), we achieve a therapeutic temperature of around 42 C. The different shape parameters of the magnetic field profile (modulation) have an effect on the thermal damage as well. Figure 8 shows the impact of the t d and t u parameters of the rectangular modulation on the survival fraction in the center of the tumor. This figure illustrates the sensitivity of the thermal damage on the magnetic field profile s parameters. A change in magnetic field amplitude indeed changes the deposited heating power. These curves need to be related to the total deposited power because the total deposited power of t u = 1 s and t d = 2 s at a certain time instant is lower than in case of t u = 1 s and t d = 1 s. The amount of time during which we deposit heat is relatively larger for the latter and even more for the case of t u = 2 s and t d = 1 s. Therefore, we compare the thermal damage for a certain amount of deposited loss energy (LE). A similar approach of comparing tumor growth rate in terms of the deposited energy has been done for power-modulated animal experiments [10]. The thermal damage per LE is evaluated for different profile modulations having different shape parameters and is listed in Table 4 using the 1DFEM. We consider an MNP concentration where the total amount of injected iron mass is m Fe = 3 mg and an MNP distribution with spread σ = m. The rate of decay of the survival fraction clearly depends on the magnetic field amplitude modulation and its shape parameters. It is important to notice that the average inflicted damage to the tumor needs to be put in perspective to the inflicted damage at the boundary of the tumor. The influence of the AMF profile on the tumor is demonstrated here. When increasing the amplitude of the magnetic field in a non-modulated profile, the average thermal damage per LE in the tumor increases. However, this does not necessarily increase the average thermal damage per LE at the boundary of the tumor. When applying a rectangular modulation of the magnetic field amplitude with relatively long switching constants t d = 100 s and t u = 100 s between H min = 20 ka/m and = 30 ka/m, the average thermal damage per LE at the boundary is increased compared to the non-modulated profile with H ampl = 30 ka/m. When a sinusoidal modulation with frequency of 1 or 2 Hz is applied, the average thermal damage per LE in the

11 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 501 Figure 8: Survival fraction α in the center of the tumor for different t d and t u parameters of the rectangular modulation with H min = 10 ka/m and = 40 ka/m. Table 4: Average thermal damage in the tumor and at the boundary of the tumor per LE for different magnetic amplitude profiles ( modulations), average maximum temperature in the tumor < T > t, and temperature at the boundary T b. Profile type < T > t ( C) < α > t T b ( C) α b LE (J) Average thermal damage per LE in tumor (10-4 J -1 ) Average thermal damage per LE at boundary (10-4 J -1 ) Constant, H ampl = 20 ka/m Constant, H ampl = 30 ka/m Constant, H ampl = 40 ka/m Rectangular, H min = 10 ka/m, = 40 ka/m, t d = 1 s, t u = 1 s Rectangular, H min = 10 ka/m, = 40 ka/m, t u = 20 s, t u = 20 s Rectangular, H min = 20 ka/m, = 30 ka/m, t u = 100 s, t u = 100 s Sine, H min = 10 ka/m, = 40 ka/m, 1 Hz Sine, H min = 10 ka/m, = 40 ka/m, 2 Hz Sine, H min = 20 ka/m, = 30 ka/m, 1 Hz Triangular, H min = 10 ka/m, = 40 ka/m, t d = 1 s, t u = 1 s Triangular, H min = 10 ka/m, = 40 ka/m, t d = 20 s, t u = 10 s The maximum temperature is recorded because the end temperature may be lower than the maximum temperature due to the modulation of the magnetic field amplitude. The results are put in perspective to the LE. The average thermal damage per LE in the tumor represents the average rate of change of the thermal damage inflicted to the tumor vs. the deposited loss energy. Calculations are performed using 1DFEM for m Fe = 3 mg and σ = m.

12 502 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH Figure 9: Survival fraction α at various radial distances from center when applying a rectangular modulation and no modulation of the magnetic field amplitude having a loss energy of LE = 238 J. This LE corresponds with applying a non-modulated magnetic field amplitude of H ampl = ka/m for a total time of 200 s, a rectangular modulation with H min = 20 ka/m, = 40 ka/m, t u = 100 s, and t d = 100 s for 200 s and a rectangular modulation with H min = 20 ka/m, = 40 ka/m, t u = 50 s, and t d = 50 s for 200 s. The MNP distribution has a spread σ = m and m Fe = 3 mg. The inset figures show the variation of the survival fraction in time and radial distance r for the non-modulated magnetic field amplitude H ampl = ka/m and the rectangular modulation with H min = 20 ka/m, = 40 ka/m, t u = 100 s, and t d = 100 s. tumor and at the boundary does not change. The triangular waveform illustrates the variability in the average thermal damage per LE with respect to the applied magnetic field modulation. Note that the results depicted in this table vary depending on the spread of the MNP distribution as illustrated in Table 3. In case a certain spread is given, the thermal damage inflicted to the tumor can be assessed in this way. Figure 9 depicts the survival fraction at various radial distances from center when applying no modulation and a rectangular modulation of the magnetic field amplitude for the same MNP distribution and concentration as in Table 4. Calculations were performed using the 1DFEM. The total deposited LE is equal for all considered modulations: LE = 238 J. A magnetic field amplitude of ka/m corresponds with SLP = W/g Fe (see Eq. 6 and Table 2 with peak-to-peak value of ka/m) and thus LE = W/g Fe 3 mg Fe 200 s = J. The spatial variation of the survival fraction clearly depends on the applied magnetic field modulation and the spread of the MNPs, when depositing an equal LE within a given time. For instance, at 2 mm from the center, the survival fraction is approximately α 0.3 in the case of no modulation. However, the survival fraction for both rectangular modulations is α 0. At 3 mm from the center, the differences are even more pronounced. When we apply no modulation, the survival fraction α 0.73, indicating that tumor cells are likely to survive the thermal damage. In the case of the rectangular modulation with H min = 20 ka/m, = 40 ka/m, t d = 50 s, and t u = 50 s, the survival fraction α Meanwhile, the survival fraction is only α 0.22 for the rectangular modulation with H min = 20 ka/m, = 40 ka/m, t d = 100 s, and t u = 100 s, effectively damaging the majority of the tumor cells at 3 mm from the center. Despite the fact that an equal LE is deposited in all cases, large differences can thus be denoted among all modulations. The time evolution of the spatial variation of the survival fraction is depicted in the inset of Figure 9 and illustrates the difference in evolution between no modulation and the rectangular modulation. The survival fraction in the tumor was the lowest when applying the rectangular modulation with H min = 20 ka/m, = 40 ka/m, and relatively long switching constants t d = 100 s and t u = 100 s compared to the non-modulated magnetic field amplitude when applying 238 J in 200 s. The rectangular modulation with t d = 50 s and t u = 50 s was situated in between those profiles. These and previous results exhibit the strong variability of the decay in survival fraction due to the applied amplitude modulation. The variability is the result of the effect of the heat source term in the Pennes equation (P MNP ) that has a non-linear dependence on the applied amplitude modulation (Eq. 6). This has profound effects on the spatiotemporal temperature

13 F. Soetaert et al.: Computational evaluation of amplitude modulation for MNH 503 distribution, which influences the thermal damage (survival fraction). The thermal damage then acts as a positive feedback due to the thermal damage-dependent perfusion model. Note that the use of amplitude modulations for enhanced MNH does enable to stay within the therapeutic temperature limits, such that safe operation is warranted. Acknowledgments: F.S. is a PhD fellow of the Research Foundation Flanders (FWO) and a Fulbright Belgium grantee of the Commission for Educational Exchange between the United States, Belgium, and Luxembourg. G.C. acknowledges the support from the Research Foundation Flanders (FWO). Conclusion This computational study focuses on the application of amplitude modulation in MNH. A 3D FEM was implemented to solve the Pennes bioheat equation that describes macroscopic heat transfer in biological tissues. Experimental non-linear SLP data of BNF-Starch particles was used to calculate the volumetric heating power of the MNPs. Starting from the resulting temperature distribution, the thermal damage was estimated as the established survival fraction that describes cell thermal damage by an Arrhenius rate function. First, we assessed the influence of the MNP distribution, characterized by the spread. Furthermore, we modified the Pennes equation to include a realistic thermal damage-dependent perfusion profile. Comparison with the classic constant perfusion rate demonstrated the importance of accurate perfusion modeling as the survival fractions had fundamentally different shapes. In addition to 3D simulations, the symmetry of our model allowed us to perform 1D simulations as well. The 1D simulations validate the 3D implementation and enable efficient analyses of various parameters including the MNP distribution and the applied amplitude modulation. The influence of amplitude modulation was tested by applying different modulations of a 150-kHz sinusoidal waveform. More concretely, we considered constant (i.e. no modulation), sinusoidal, rectangular, and triangular modulations. We observed significant differences among the different modulations with varying shape parameters in terms of temperature, survival fraction, and thermal damage per loss energy. The presented evaluation and methods are possibly useful components of a priori treatment planning because it enables calculations for an inhomogeneous spread of MNPs and inhomogeneous magnetic fields, realistic non-symmetrical geometries, multiple tumor regions, and inhomogeneous material properties. To be clinically relevant, experimental and clinical investigations need to be performed as a next step for validating the model. Additionally, the effect of eddy currents as a volumetric power dissipation term needs examination. 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