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Comparison of a Single Optimized Coil and a Helmholtz Pair for Magnetic Nanoparticle Hyperthermia Michael D. Nieskoski and B. Stuart Trembly∗

Abstract—Magnetic nanoparticles in a tumor can induce therapeutic heating when energized by an alternating magnetic field from a current-carrying coil outside the body. We analyzed a singleturn, air-core coil carrying a filamentary current to quantify the power absorbed by: a) magnetic nanoparticles at depth in tissue and b) superficial tissue in response to induced eddy currents; we defined this quotient as power ratio (PR). Given some limit on the eddy current heating tolerated by an alert patient, maximizing the PR maximizes the power absorbed in the tumor; all else being equal, this increases the thermal dose delivered to the tumor. The mean eddy current heating rate tolerated in four clinical studies we reviewed equaled 12.5 kW/m3 . We differentiated our analytical expression for PR with respect to the radius of the coil to find the value of radius that maximizes PR. Under reasonable simplifying assumptions, the optimal value of coil radius equaled 1.187 times the depth of the nanoparticle target below the body surface. We also derived the PR of two coils surrounding the body configured as a Helmholtz pair. We computed PR for combinations of nanoparticle depths below the surface and axial locations with respect to the coils. At depths less than 4.6 cm, the optimized single coil had a higher PR than that of the Helmholtz pair and furthermore produced less total ohmic heating within the coil. These results were independent of driving frequency, nanoparticle concentration, tissue electrical conductivity, and magnetic nanoparticle heating rate, provided the latter is assumed to be proportional to the product of frequency and the square of the local magnetic field. This paper supports the clinical application of current-carrying coils to deliver efficacious hyperthermia therapy to tumors injected with magnetic nanoparticles. Index Terms—Helmholtz pair, magnetic nanoparticle heating, optimization, single coil.

I. INTRODUCTION AGNETIC nanoparticles (mNPs) have many biomedical applications, including drug delivery, magnetic actuation for the control of cellular function, contrast enhancement of magnetic resonance imaging, and hyperthermia therapy [1]. In mNP hyperthermia therapy, mNPs are localized in a tumor region by means of direct injection or systemic delivery. A

M

Manuscript received September 11, 2013; revised November 13, 2013; accepted December 14, 2013. Date of publication March 27, 2014; date of current version May 15, 2014. This work was supported by a gift from R. Crump and M. Crump to the Thayer School of Engineering. Asterisk indicates corresponding author. M. D. Nieskoski is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA (e-mail: Michael.D.Nieskoski. [email protected]). ∗ B. S. Trembly is with the Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TBME.2013.2296231

¯ at current-carrying coil applies an alternating magnetic field, H, a frequency, f , to the tumor region, thereby causing the mNPs to generate heat and elevate the tumor temperature. Heating mammalian cells to 44.5 ◦ C for 30 min results in a cell survival fraction of 0.6% [2]. In practice, hyperthermia is used to potentiate radiation therapy and chemotherapy. Faraday’s Law states that a time-varying magnetic field passing through a conductive surface induces electric currents (eddy currents) around the perimeter of the surface, which generate local ohmic heating. These eddy currents heat healthy tissue ¯ overlaying a tumor and thereby limit the maximum value of H and f that can be safely applied to a patient. Atkinson et al. [3] reported maximum tolerable values of the product Hf while treating cancer patients with ferromagnetic thermal seeds; others reported the same from clinical mNP hyperthermia studies [4]–[6]. The Hf-limitation originates on the periphery of the surface penetrated by magnetic fields, where eddy current density is highest; other hot spots may arise at skin folds [4]. Eddy currents generate heating power in tissue according to [7] ARD =

1 2 2 2 σμ r π (Hf )2 2 0

(1)

where σ is the conductivity of tissue, r is the radius of the circular surface exposed to the magnetic field, μ0 is the permeability of a vacuum, and absorption rate density (ARD) is the power deposited per unit volume of tissue. An optimal single coil for mNP hyperthermia would minimize the eddy current ARD at the surface of the patient and maximize the ARD of mNPs in a tumor below the surface. Researchers have developed many coil geometries for induction of hyperthermia by eddy currents, localized hyperthermia with millimeter ferromagnetic implants, and mNP and magnetic fluid hyperthermia. These coil geometries either induce superficial eddy current heating [8] or heat an implanted magnetic target at depth; none have been designed with consideration of both effects. In eddy current hyperthermia, superficial tumors are treated with coils that surround the patient [9] or external coils placed parallel to the surface of the patient [10], [11]. To achieve significant eddy current heating in tumors at depth, Oleson et al. [12] designed a coaxial pair of coils, one on either side of the patient, which achieved a modest improvement in heating, compared to a single current loop. These coils induced  eddy current heating, but were not designed to maximize H at a magnetic target at depth. External coils, such as a Helmholtz pair, can deliver local hyperthermia by producing a magnetic field of sufficient strength

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NIESKOSKI AND TREMBLY: COMPARISON OF A SINGLE OPTIMIZED COIL AND A HELMHOLTZ PAIR

in ferromagnetic targets implanted in a deep-seated tumor. For ferromagnetic implants, ARD is proportional to the square of the ¯ [13]. A Helmholtz pair consists of two identical, coaxlocal H ial, circular coils that are separated by a distance equal to the coil radius; it generates a nearly uniform magnetic field along the centerline between the coils. Mier et al. [14] constructed a Helmholtz pair to heat ferromagnetic implants, while Huang et al. [15] used what they described as a Helmholtz pair to heat an array of ferrite needles implanted in a mouse liver. Solenoids are also used to heat ferromagnetic implants. Park et al. [16] used a nine-turn solenoid to induce heat by eddy current losses in stainless-steel implants. Kimura and Katsuki [17] used a solenoid joined to a conical solenoid for treating oral cancer with metallic implants. Stauffer et al. [18] designed solenoids, including a double-layer reverse-wound solenoid, a two-layer solenoid, in which the coils in each layer are wound in the opposite direction. This design aimed to reduce the electric field generated between close windings of the solenoid, [11], which can contribute to superficial heating. This electric field effect is significant in solenoids, but is not relevant to single-turn external coils and Helmholtz pairs. Surface coils, such as the spiral pancake coil [18], [19], deliver heat energy to implants at small tumor depths [18]. To increase treatment depth, Stauffer et al. [18] designed two multilayered solenoids, each containing three layers of four turns, placed on either side of the patient to form a coaxial pair, as well as a conformal surface coil, which is an oval spiral coil that is bent slightly at each end to form a saddle shape. These authors analyzed heating ferromagnetic implants at depth, but they did not consider eddy current heating at the surface of the patient. In mNP hyperthermia, the alternating magnetic field deposits heat energy in nanometer-sized magnetic particles suspended in a fluid that has been injected directly or delivered systemically to the tumor. Once administered to the tumor, mNPs are endocytosed by the tumor cells [20]. To couple energy into mNPs, Salloum et al. [21] designed two circular coils of equal diameter with an axial separation smaller than the coil radius, i.e., not a Helmholtz pair. Di Barba et al. [22] modified the radii and axial separation of two coils, a main and a correcting coil, to minimize the magnetic field variation across a finite target region. Bordelon et al. [23] modified a solenoid coil with magnetic flux-concentrating material to increase field uniformity within the solenoid. Stigliano et al. [24] used a single turn, sheet-like coil with a magnetic core. Jordan et al. [25] developed a magnetic fluid hyperthermia (MFH) therapy system consisting of a C-shaped core of high permeability, around which two solenoidal coils were wrapped. The patient was placed in the air gap of the core. All of these coils are intended to produce a therapeutic value of magnetic field in mNPs injected at depth, but the tradeoff with superficial eddy current heating was not analyzed explicitly. In this study, we explicitly optimized the magnetic field produced at depth by a single air-core coil with respect to the superficial eddy current heating it produces. We defined a power ratio (PR) as the ratio of the heating power generated by the mNPs to the heating power induced on the surface of the patient by eddy currents. Our analysis defined the value of coil radius

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Fig. 1. I(t) denotes (a) single filamentary current and (b) Helmholtz pair of filamentary currents heating magnetic nanoparticles at depth, d, within a patient. The offset between filamentary current and patient is z0 .

a1 that maximizes PR for the single coil. We also quantified the effect on PR of an offset z0 between the coil and surface of the patient [see Fig. 1(a)]. This offset is necessary to accommodate the finite diameter of the coil conductor and its cooling system. We compared the values of PR of the optimized single coil to the values of PR of the Helmholtz pair at combinations of mNP target depth, d, and axial location, z  [see Fig. 1(b)]. We did not include a solenoid surrounding a patient, because its closelyspaced coils produce an additional electric field component that contributes further to superficial heating. To our knowledge, this study is the first to optimize deep heating of a magnetic target with respect to superficial eddy current heating. II. THEORY Table I defines the variables used in this analysis. In this analysis, we assumed that each coil carried a sinusoidal filamentary current with a frequency between 50 and 200 kHz. In this frequency range, the retarded potential is negligible, resulting in a quasistatic magnetic field. The attenuation in tissue is negligible at 100 kHz, with penetration depths δ equal to 3.48 m (skin

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TABLE I NOMENCLATURE

electric potential Φ [11] and [27] − →  ∂  ∂B → − − → =− ∇× A ∇× E =− ∂t ∂t − → ∂A → − E = −∇Φ − ∂t

(2)

where ∇Φ represents the electric field from the electrical poten→ − tial around the coil conductor and ∂ A represents induced eddy ∂t

currents. We treated the skin surface as a ground plane for our alternating field, due to the high permittivity of skin at 100 kHz. Guy [11] used this technique, which applies the solution of Weber [28] for the potential, Φ, of a ring of charge over a ground plane. In our case, this approximation is still more justified, because our frequency is much lower than that of Guy, who used 27.12 MHz. This makes the electrical length of our coil smaller and gives a much higher permittivity of skin. Furthermore, our single coil produces no intercoil electric field, such as Guy considered. With these assumptions, the electric field ∇Φ is normal to the skin surface, but vanishes within, thus contributing no power deposition in the skin.  In cylindrical coordinates with a θ-directed current, the vec→ − → −  tors E and A are θ-directed. We express the coil current as a complex sinusoid (I(t) = I0 ej ω t ) and the complex exponential is suppressed in the manipulations below. Due to the large wavelength at 100 kHz, we approximate a single turn coil with a circular filamentary current that produces a quasistatic magnetic → − → − vector potential A and magnetic field H . Smythe [29] derived Aθ for a single coil carrying a filamentary current as  

μ0 · I0 a1 1 2 Aθ = (3) 1 − k K(k) − E(k) π·k r 2 where K(k) and E(k) are complete elliptic integrals of the first and second kind, a1 is the radius of the single coil, r is the radial position from the center of the coil, and k 2 is the modulus of the elliptic integral for this problem k2 = and muscle) and 8 m (fat), much larger than the diameter of a patient. Penetration depth is given by 

δ= ω

μ0 εr ε0 2

1



1+

σ2 ω 2 (ε r ε 0 ) 2

−1



where the properties of wet skin and muscle are approximated as εr = 28000, σ = 0.33 S/m, and those of fat are approximated as εr = 70, σ = 0.04 S/m at 100 kHz [26].

A. Optimization of a Single Coil 1) Induced Eddy Current Heating for a Single Coil: Using Faraday’s Law, the electric field generated by a single coil can → − be expressed in terms of the magnetic vector potential A and

4 · a1 · r (a1 + r)2 + z 2

where z equals the coil offset, z0 [see Fig. 1(a)]. The power density produced by eddy currents from a single coil is [9] ARDEC =

1 · σ · Eθ2 . 2

(4)

The maximum Eθ occurs very close to r = a1 [see Fig. 1(a)], as will be discussed below. 2) Power Generated by mNPs on the Axis of a Single Coil: Rosensweig [30] expressed volumetric power deposition as ARDm NP = μ0 πχ f H 2 where χ is the imaginary component of the ferrofluid susceptibility. Kallumadil [31] showed χ above is approximately constant under the conditions discussed below, which yield ARDm NP = ρ · SAR = ILP · ρ · f · ((Hr )2 + (Hz )2 )

(5)

NIESKOSKI AND TREMBLY: COMPARISON OF A SINGLE OPTIMIZED COIL AND A HELMHOLTZ PAIR

where ρ is the iron concentration in the tumor, Hr and Hz are the radial and axial components of the magnetic field, respectively, and specific absorption rate (SAR) is the power absorbed per unit mass of iron. Intrinsic loss power (ILP) is a property associated with the magnetic material. It can be treated as a constant under the simplifying assumption that SAR is proportional to the product of frequency and the square of the magnitude of the magnetic field. The concept of ILP as a constant also requires the following conditions: low frequencies (between 105 and 106 Hz), polydispersity index (uniformity of mNP diameter) greater than 0.1, and an applied magnetic field that is much smaller than the mNP saturation magnetic field [31]. Measurements may  show that the ARD of a given type of mNP varies → − with  H  raised to a power slightly different from 2. In this case, an average ILP could be computed from the data, or the measured data could be used to compute ARD directly. Smythe derived the axial and radial components of the magnetic field from the curl of Aθ in cylindrical coordinates [29]  1  → − − → ∇× A H = μ0 I0  Hz = 2π

 a21 − r2 − z 2 E(k) K(k) + (a1 − r)2 + z 2 (a1 + r)2 + z 2 1

(6)

 2 2 2 z I0 a1 + r + z  E(k) −K(k)+ Hr = 2π (a1 − r)2 + z 2 r (a1 +r)2 + z 2 (7) where z = z0 + d, where d is target depth [see Fig. 1(a)]. In this paper, we assumed that the radius of the circular coil is much larger than that of the tumor containing the mNPs, and that the tumor lies on the axis of the coil; thus, the magnetic field remains nearly constant across the tumor. With r = 0, the radial component of the magnetic field Hr vanishes in (5) and the axial component of the magnetic field (6) reduces to the well-known, on-axis expression [27], using K(0) = E(0) = π/2 [32]

 1 I0 a21 − (z0 + d)2 → − E(0) K(0) + 2 Hz = 2π a21 + (z0 + d)2 a1 + (z0 + d)2 I(t) · a21 =  . 3 2 [a21 + (z0 + d)2 ]

(8)

3) Optimal Radius for a Single Coil: We defined the PR as the quotient of the ARD due to mNPs and the ARD due to eddy currents at the skin surface PR =

ILP · ρ · f · Hz2 ARDm NP = 1 2 ARDEC 2 · σ · Eθ

ILP · ρ · f ·

PR = 1 2

·σ·



ω ·μ 0 ·I 0 π ·k

a1 r



2

I 0 ·a 21 3

[a 21 +(z 0 +d) 2 ]   2 . (9) 1 − 12 k 2 K(k) − E(k) 2

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TABLE II EDDY CURRENT HEATING TOLERATED IN THE CLINIC

Differentiation with respect to a1 determines the maximum value of PR for given values of mNP target depth d and coil offset z0 dPR 4 · ILP · ρ · f · Hz = da1 σ · Eθ3  dHz dEθ × Eθ · − Hz · = 0. da1 da1

(10)

We evaluated the derivatives of Eθ and HZ [32] in (10) under the simplifying assumption that the maximum Eθ occurs at the coil radius r = a1 3 · a31 · I0 −  (11) 3 5 [a21 + (z0 + d)2 ] 2 [a21 + (z0 + d)2 ]  k dEθ ω · μ0 · I0 1 = − · K(k) da1 π a1 a1 · k

 2 · a1 · k 4 · a1 6 · a1 + − · E(k) . (12) + z 2 · k3 z2 · k z2

dHz = da1

a1 · I0

For this assumption, the error in PR is less than 1% for values of d ≥ 1 cm. In the results section, we explain why values of d < 1 cm are not relevant to our optimization process. As seen in (10), the optimal value of a1 is independent of parameters ILP, f , ρ, and σ. We substituted (11) and (12) into (10) and solved numerically for the value of a1 (resolution of 10−6 m) that made dPR/da1 vanish, and we confirmed that PR was a maximum, not a minimum. We checked the accuracy of our optimization by computing the values of Hz2 and Eθ2 for the optimized single coil using finiteelement analysis (COMSOL Multiphysics Version 4.3 (Classkit License) [33], 3-D model: ac/dc module: magnetic fields physics solver: frequency-domain study, 1 754 058 tetrahedral mesh elements). 4) Determining Coil Current and Ohmic Power in a Single Coil: The maximum tolerable eddy current heating may constrain a hyperthermia treatment. Table II lists values of Hf that

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were tolerated by alert hyperthermia patients heated continuously by coils in four clinical studies. These studies did not state values of patient radius; instead, we estimated these values based on the treatment anatomy. We converted Hf to a maximum value of ARDEC using σ = 0.33 S/m and the relevant radius value in (1). Through (2)–(4), we related the mean maximum tolerated ARDEC (Table II) to I02 , z0 , and a1 , then solved for I02 as a function of a1 for a fixed value of z0 . The quantity I02 in turn expressed the resistive power losses in the coil through POhm ic = R · I02 /2, where R is the resistance of the coil, accounting for the skin effect. In practice, a cooling system must have the capacity to transport this waste heat from the coil if eddy current heating must be set to the value in Table II to attain a therapeutic temperature in the tumor. B. Helmholtz Pair Surrounding a Patient

III. RESULTS

1) Induced Eddy Current from Helmholtz Pair: Fig. 1(b) illustrates a Helmholtz pair that surrounds a patient. We assumed that a filamentary sinusoidal current flows through each coil in series and that the two currents are in phase. Each coil [subscript 1 for left coil, subscript 2 for right coil in Fig. 1(b)] induces an electric field, which contributes to ARDEC ARDEC =

1 σ · (Eθ ,1 + Eθ ,2 )2 . 2

(13)

Electric field is computed using (2) and (3) with these moduli in the elliptic integrals k12 =

4 · a2 · rpt (a2 + rpt )2

k22 =

4 · a2 · rpt . (a2 + rpt )2 + a22

The maximum electric field occurs at the surface of the patient, r = rpt , and approximately at the axial position of each coil; we calculated the electric field only at the location of coil #1, due to symmetry. 2) Power Generated by mNPs in a Helmholtz Pair: The ARD generated by mNPs heated by a Helmholtz pair is ARDm NP = ILP · ρ · f · ((Hr,1 + Hr,2 )2 + (Hz ,1 + Hz ,2 )2 ). (14) The axial and radial components of the magnetic fields are computed from (6) and (7) with these moduli in the elliptic integrals k12 =

4a2 (rpt − d) (a2 + (rpt − d))2 + (z  )2

k22 =

4a2 (rpt − d) (a2 + (rpt − d))2 + (a2 − z  )2

Fig. 2. Optimal single coil radius, a 1 , plotted against mNP target depth, d, at a fixed coil offset, z0 .

where z  is the axial position of mNPs within the central volume of the Helmholtz pair; the radial position of the mNPs is r = rpt − d [see Fig. 1(b)]. We computed PR (9) for the Helmholtz pair using (14) for ARDm NP and (13) for ARDEC .

To compute the PR, we used σ = 0.33 S/m for skin tissue [26], f = 100 kHz, and the measured SAR function, SAR = 0.36 · H 1.6 for the MFL A nanoparticles at a concentration of 112 mgFe /mL of Gneveckow et al. [34]. These are superparamagnetic magnetite nanoparticles with an average core diameter of 15 nm and an aminosilane coating [4]. In this SAR function, the exponent is somewhat less than 2, implying ILP is not a constant. We set ILP equal to the mean value of 1.534 nHm2 /kg (SD = 0.35) computed over the range of 3 to 17 kA/m. We calculated a typical postinjection value of mNP concentration in tissue as follows. According to Wust el al. [4], a SAR of 50 W/kg was achieved if 1 mL of MFL 082AS magnetic fluid (112 mgFe /mL) was injected into 10 mL of tumor tissue, with a field strength of 5 kA/m and frequency of 100 kHz, or SAR = 10 · (vm NP /VTissue ) · (H/5 kA/m)2 · (f /100 kHz)2 · 50 W/kg. Their SAR function varied with the square of the frequency, indicating that ILP was not a constant for these mNPs. However, the frequency dependence was not relevant, because we computed with their value of 100 kHz. Using their median SAR value of 130 W/kg in twenty-two patients and their mean magnetic field strength of 4.5 kA/m, we computed that 0.32 mL of magnetic fluid existed in 1 mL of tumor tissue postinjection. We multiplied this ratio by the syringe concentration, 112 mgFe /mL, to get a typical, clinical tissue concentration of 35.84 mgFe /mL. Our numerical solution of (10) showed that the optimal radius a1 is proportional to d, with a slope of 1.187 and R2 of 0.9998 for z0 < d (see Fig. 2). The optimal value of a1 is independent of ILP, f , and ρ. It is not physically meaningful to maximize PR for d < 1 cm, as will be explained below. Fig. 3 shows how PR varies with d and a1 , for a fixed z0 ; the optimal value of a1 falls on the same straight-line locus defined in Fig. 2. Also plotted are lines of constant I02 ; these vary with a1 , but are independent of d, as shown by (2)–(4). PR is not plotted for d < 1 cm, because PR goes to infinity at a rate of 1/a21 , and I02 goes to infinity at a rate of 1/a61 . For such small values of d, a therapeutic tumor temperature could be attained with a PR far less than the maximum attainable value. This would entail smaller values of coil current and a value of

NIESKOSKI AND TREMBLY: COMPARISON OF A SINGLE OPTIMIZED COIL AND A HELMHOLTZ PAIR

Fig. 3. Contour plot of PR for a single coil against coil radius, a 1 , and mNP target depth, d. Coil offset, z0 , is fixed at 1 cm. The open dots indicate the optimal coil radius for each mNP target depth, from Fig. 2. The dashed lines show loci of constant I02 , in (kA)2 (there is no distinction between kA and kA·turns for a single turn coil).

Fig. 4. Contour plot of PR for the single coil against mNP target depth, d, and coil offset, z0 . At each target depth, the coil radius a 1 has been set to the optimal value from Fig. 2.

ARDEC less than the maximum tolerable value, both practical advantages. Therefore, there is no practical need to maximize PR for d < 1 cm. Fig. 4 shows the effect of z0 and d on PR and I02 , using the optimal value of a1 defined in the straight-line locus in Fig. 2. Fig. 5 shows the PR of a Helmholtz pair in a central crosssection of the patient’s body. The normalized axial position is the axial distance z  from coil #1, divided by coil separation a2 . For example, a normalized axial position of 0.5 indicates that the mNPs are midway between the two coils. Fig. 5 has the same orientation as the Helmholtz pair in Fig. 1(b). Fig. 6 shows the effect of increasing the Helmholtz pair radius a2 with a fixed patient radius, effectively increasing the offset z0 . The normalized axial position z  /a2 was set equal to the value in Fig. 5 with the highest PR for each value of d (shown as circles). Fig. 7 compares the PR of the optimized single coil to that of the Helmholtz pair, with the normalized axial position for the Helmholtz set equal to the value with the largest PR in Fig. 5. To confirm the theoretical expression for PR (9), we computed the values for Hz2 and Eθ2 for the optimized single coil radius a1 = 3.6 cm at target depth d = 3 cm numerically using

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Fig. 5. Contour plot of PR for the Helmholtz pair against normalized axial position, z  /a 2 , and target depth, d. The Helmholtz coil radius a 2 = 16 cm, and coil offset z0 = 1 cm. PR equals 3.87 at the array center (d = 15 cm and z  /a 2 = 0.5). The quantity I02 is not plotted, because it equals 0.44 (kA)2 for all values of d and z  . There is no distinction between kA and kA·turns because I02 flows through each coil in series. This plot has the same orientation as Fig. 1(b).

Fig. 6. Contour plot of PR for the Helmholtz pair against target depth, d, and coil radius, a 2 . The radius of the patient is fixed at 15 cm. For each target depth, the target is assumed to be at the optimal axial location, as shown in Fig. 5.

COMSOL Multiphysics [33]. The maximum error for Hz2 and Eθ2 , when comparing the numerical results to theory, was 0.47% and 0.35%, respectively. The results for ARDEC (4) are shown in Fig. 8. The coil current, I0 , was set equal to 1.57 kA to produce the mean tolerated ARDEC of 12.5 kW/m3 (see Table II). In the numerical results, the maximum value of ARDEC occurs at a1 = r, as assumed in our theory. IV. DISCUSSION The goal of this study was to compare the performance of a Helmholtz pair and an optimized single coil in mNP hyperthermia. Our analysis demonstrates that optimizing the single coil maximizes PR for only a single value of d, as shown in Fig. 2. Fig. 3 quantifies how departing from the optimal a1 reduces PR. To treat multiple depths d with a single coil, it would be best to optimize that single coil for the largest expected value of d. For example, a1 = 4.76 cm is optimal for treating d = 4 cm, resulting in a PR equal to 8.96. If d were to decrease to 2 cm with coil radius fixed at 4.76 cm, the PR increases to 30.7 (although it is not optimal), indicating that a1 = 4.76 cm treats 2 cm ≤ d ≤ 4 cm effectively. We note that for shallow depths, it may be

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TABLE III COMPARISON OF SINGLE OPTIMIZED COIL AND HELMHOLTZ PAIR

Fig. 7. Plot of PR of the optimized single coil and the Helmholtz pair against mNP target depth, d. For both types of coil, z0 = 1 cm. For each target depth, the single coil has the optimal radius from Fig. 2, and the Helmholtz pair has the target at the optimal normalized axial location z  /a 2 from Fig. 5.

Fig. 8. Plot of ARDE C against the radial position from the center of the single coil, r, for z = z0 [see Fig. 1(a)]. ARDE C was computed using (4), (2), (3), and using COMSOL Multiphysics [33]; every tenth data point computed by COMSOL was plotted above. The single coil radius a 1 = 3.6 cm and coil current I0 = 1.57 kA from Table III were used to set the maximum ARDE C equal to the mean tolerable value in Table II.

desirable to supplement heating of mNPs with intentional eddy current heating, but we have not analyzed that strategy here. Fig. 4 shows that the PR of the optimized single coil increases with an increasing z0 . However, the quantity I02 increases at a faster rate. For example, at d = 2 cm, an increase in z0 from 1cm to 3 cm results in a 14.3 fold increase in I02 , but only a 1.56 fold increase in PR. A larger I02 generates more heat in the coil and thus requires a more powerful cooling system. In practice, a limit on cooling power likely will require using the smallest value of z0 that gives a safe clearance between the coil and patient. Fig. 5 illustrates the well-known uniformity of the magnetic field along the central axis of a Helmholtz pair [35]. Fig. 5 also shows that the maximum PR is not fixed midway between the two coils for the case of shallow depths. Specifically, as d decreases, the maximum PR shifts toward a normalized axial position z  /a2 of 0 or 1 (location of coil #1 or coil #2). A future study will quantify PR as a function of mNP target location with coil separation closer than the Helmholtz distance. For a fixed patient radius, the PR and I02 of the Helmholtz pair both rise with increasing coil radius a2 , but not in proportion. For example, in Fig. 6, at d = 12 cm, as a2 increases from 17

to 18 cm, PR increases by 1.2 fold, while I02 increases by 1.27 fold. Actual ohmic heating in the coils would increase by a still larger factor, due to the larger circumference of the coil. Fig. 6 shows the effect is far more pronounced for d ≤ 6 cm. As with z0 for the single optimized coil, limits on cooling power likely require use of the smallest value of a2 for the Helmholtz pair that gives safe clearance to the patient. Fig. 7 directly compares the PR of the Helmholtz pair to the optimized single coil. The Helmholtz pair yields a greater PR for 4.6 cm < d ≤ 15 cm. However, for d < 4.6 cm, the optimized single coil gives a greater PR and less ohmic power, as shown in Table III. At a target depth of 3 cm, the single optimized coil has a 2.1 times larger PR than the Helmholtz pair. The optimized single coil has a value of I02 5.5 times larger than the Helmholtz pair, but its circumference is nine times smaller than the sum of the circumferences of the two Helmholtz coils. Therefore, the ohmic power generated by the optimized single coil is 0.62 times the power generated by the Helmholtz pair, assuming constant coil conductivity per unit length. By rearranging (9), the magnitude of the magnetic field at a target depth, d, can be computed from  ARDEC · PR . |H| = ILP · ρ · f Using ILP, f , and ρ as defined at the beginning of the results section, the mean ARDEC (Table II), and a PR of 3.87 for a target depth of 15 cm (see Fig. 7), the magnetic field equals 2.97 kA/m or 37.3 Oersteds. Although not shown in our results, the addition of a second optimized coil with an equal radius on the opposite side of the patient, such as described in [12], increased the PR four-fold at d = 15 cm. However, the two coils achieved a less significant increase of 1.05 at shallow depths, such as d < 5 cm. The point of intersection in Fig. 7 did not significantly change with the addition of this second coil, and we did not consider this case further in this work. A single filamentary current can be used to model sheetlike coils designed for hyperthermia, because alternating

NIESKOSKI AND TREMBLY: COMPARISON OF A SINGLE OPTIMIZED COIL AND A HELMHOLTZ PAIR

current concentrates towards the inside radius of a circular sheet geometry. Papagiannopoulos et al. [36] showed such concentration through numerical analysis of pairs of circular sheets. There are two physical causes for this effect. Not only is the inner radius the shortest path for current flow, but the sinusoidal current generates a magnetic field in the coil, inducing eddy currents which oppose current flow in the region between the inner and outer radii while augmenting current flow at the inner and outer radii. Therefore, the circular sheet coil can be modeled approximately as two filamentary currents, one at the inner radius and one at the outer radius of the coil. However, the magnitude of the current flowing at the outer radius is a small fraction of that at the inner radius current and could be ignored in a first-order model. The first-order model also would apply to the singleturn sheet-like coil designed for mNP hyperthermia described by Stigliano et al. [24], and consequently the results of our optimization would apply to it as well. The steady-state temperature in the superficial tissue heated by eddy currents can be computed by the method of Hand et al. [37]. However, the value of this temperature is of less interest, because we have included the maximum tolerable eddy current heating rate known from clinical studies in our definition of PR. The values of the Hf product that produced the greatest superficial heating tolerated by patients were quantified during these clinical studies, and these values of Hf directly express eddy current ARD, according to (1). Once it is known that a certain heating rate built into PR is barely tolerable, the associated temperature at that anatomical site has less significance. The value of PR does not directly imply the values of steadystate temperatures in the tumor heated by mNPs. To find tumor temperature, one can apply the Green’s function derived by Giordano et al. [38] for the bioheat equation expressed in onedimensional, spherical coordinates. A solution for steady-state temperature would follow from integrating the Green’s function numerically over time, to represent steady thermal power deposition from the mNPs, and by integrating over the radial positions defined by the tumor, to represent the thermal power contributed by each radial shell of tumor tissue containing mNPs. Of great significance to this calculation of steady-state temperature is the value of tumor blood perfusion, which may not be well known in the clinical setting; similarly, the blood perfusion in adjacent normal tissue must be known as a function of local temperature [38]. It is nonetheless advantageous to maximize PR, regardless of the particular values of tumor and normal tissue blood perfusion in a given patient. For example, Table III shows that the optimized single coil has more than twice the PR of the Helmholtz pair for a mNP target 3 cm below the skin surface. All else being equal, the mNP density, ρ, in tissue heated by the optimized single coil could be half the value required by the Helmholtz pair, according to (9). This fact extends the number of clinical cases in which therapeutic heating could be attained through use of the optimized single coil. V. CONCLUSION This study compared an optimized single coil to a Helmholtz pair for use in mNP hyperthermia. We defined the PR as the

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ratio of induced heat in mNPs to surface eddy current heating, and we used this metric to compare coil geometries. The mean eddy current heating rate tolerated in four clinical studies we reviewed equaled 12.5 kW/m3 . Analysis demonstrated that the optimal coil geometry depends on the mNP target depth. For a patient radius of 15 cm, the optimized single coil achieves a greater PR and less ohmic heating than the Helmholtz pair at mNP target depths d < 4.6 cm. However, at greater depths, the PR of the Helmholtz pair exceeds the PR of the optimized single coil. Increasing the PR decreases the mNP concentration in tissue required for a therapeutic temperature, all else being equal. In future papers, further improvements to PR will be investigated by modifying the Helmholtz pair presented here and by optimizing the radius and offset of a pair of coils external to the patient.

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Michael Nieskoski received the B.S. degree in physics from Fairfield University, Fairfield, CT, USA, in 2011. He is currently working toward the Ph.D. degree in engineering science at the Thayer School of Engineering at Dartmouth College, Hanover, NH, USA. His research interests include coil optimization and treatment planning for magnetic nanoparticle hyperthermia.

B. Stuart Trembly received the B.S. degree from Yale University, New Haven, CT, USA, in 1975, and the Ph.D. degree from Dartmouth College, Hanover, NH, USA, in 1983. Since 1983, he is currently with the faculty of the Thayer School of Engineering at Dartmouth College. His research interest is the application of electrical engineering to medical problems. Prof. Trembly was a recipient of the National Science Foundation Presidential Young Investigator Award in 1984.