INSTITUTE OF PHYSICS PUBLISHING
PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 46 (2001) 2503–2514
PII: S0031-9155(01)20830-6
A fast algorithm to find optimal controls of multiantenna applicators in regional hyperthermia Torsten K¨ohler1 , Peter Maass1 , Peter Wust2 and Martin Seebass3 1
Universit¨at Bremen, Zentrum f¨ur Technomathematik, PF 33 04 40, 28334 Bremen, Germany Humboldt-Universit¨at Berlin, Charit´e, Campus Virchow-Klinikum, Augustenburger Platz 1, 13353 Berlin, Germany 3 Konrad-Zuse-Zentrum f¨ ur Informationstechnik, Takustr. 7, 14195 Berlin, Germany
2
E-mail:
[email protected]
Received 9 January 2001, in final form 11 July 2001 Published 22 August 2001 Online at stacks.iop.org/PMB/46/2503 Abstract The goal of regional hyperthermia is to heat up deeply located tumours to temperatures above 42 ◦ C while keeping the temperatures in normal tissues below tissue-dependent critical values. The aim of this paper is to describe and analyse functions which can be used for computing hyperthermia treatment plans in line with these criteria. All the functionals considered here can be optimized by efficient numerical methods. We started with the working hypothesis that maximizing the quotient of integral absorbed power inside the tumour and a weighted energy norm outside the tumour leads to clinically useful power distributions which also yield favourable temperature distributions. The presented methods have been implemented and tested with real patient data from the Charit´e Berlin, Campus Virchow-Klinikum. The results obtained by these fast routines are comparable with those obtained by relatively expensive global optimization techniques. Thus the described methods are very promising for online optimization in a hybrid system for regional hyperthermia where a fast response to MR-based information is important.
1. Introduction The basic idea of regional hyperthermia (RHT) is to heat up a deeply located tumour region to the required temperature as uniformly as possible while keeping the temperature in the surrounding healthy tissue G\ below specified critical values. Satisfactory heat treatment may enhance the efficacy of simultaneous or subsequent chemo- and/or radio-therapy. The energy deposition inducing the temperature increase is achieved using a set of N radiowave antennas surrounding the patient. The latest generation of hyperthermia equipment uses up to N = 24 antennas, which are positioned on an elliptic cylinder in three rings and operate at the same frequency in a defined phase relation. The free parameters are the phase 0031-9155/01/092503+12$30.00 © 2001 IOP Publishing Ltd
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delays ϕj with respect to a reference phase and the amplitudes aj (j = 1, . . . , N) of the emitted radiowaves. A basic hyperthermia treatment plan (for N antennas) therefore consists of N complex numbers pj = aj exp(−iϕj ) which determine the pattern of power deposition and the resulting temperature distribution in the body. More precisely the forward problem of computing the steady state temperature distribution inside the body G for a given set of parameters p = (p1 , . . . , pN )t ∈ CN can be split into two steps: • Computation of the resulting electric field distribution E(x) inside the inhomogeneous body by solving Maxwell’s equations. • Computation of the temperature distribution T (x) by solving the bioheat transfer equation, assuming perfusion in every tissue point x of G. Now we can state the inverse or optimization problem of hyperthermia treatment planning in more detail. Let us assume that critical temperatures Tc (x) have been assigned on medical grounds for every point x ∈ G\ in healthy tissues. Then we are looking for max T90 subject to T (x) Tc (x) ∀x ∈ G\
p∈CN
(1)
where T90 is the temperature achieved by 90% of x ∈ . This is a high-dimensional nonlinear optimization problem in a Banach space formulation which requires substantial computational effort. Most of this paper deals with adequate simplifications of this functional which can be solved efficiently. Recently a second generation of hyperthermia systems has been installed in some oncological centres. The above-mentioned triple-ring applicator with 12 antenna pairs, i.e. 24 antennas, is now commercially available as the so-called SIGMA-Eye applicator (BSD Medical Corp., UT, USA). Simulation studies employing a global optimization routine have formerly demonstrated that this 24-antenna applicator can significantly improve the control of SAR patterns (SAR: specific absorption rate, see section 2.1) in comparison to the clinically used SIGMA-60 applicator (consisting of one ring of four antenna pairs, i.e. eight antennas) (Wust et al 1996). Modern hyperthermia systems are built using a hybrid technique, i.e. the RHT applicators operate in the field of view of a magnetic resonance tomograph (MRT). Online registration of temperature distributions (non-invasive thermography) (see e.g. Wlodarczyk et al 1998, 1999, Liauh and Roemer 1993), can elucidate regions exceeding critical temperatures (typically >43.5 ◦ C, so-called hot spots), and in conjunction with clinical information (e.g. localized discomfort or other intolerances) enables reformulation of the optimization problem in (1). The latter is performed by modifying perfusions and adapting weighting factors as described later in this paper. However, a fast and reliable solution of the inverse problem (1) will be required in real time (preferably in seconds) during heat treatments under MR monitoring by matching the adjustments to the clinical characteristics of each patient. This paper deals with the development of efficient and flexible optimization strategies for regional radiofrequency hyperthermia which are useful for interactive online control. A global optimization method, optimizing suitable functionals (objective functions) by efficient search algorithms (as used in Wust et al (1996) for example), is used as a reference method. The first basic ideas for an elegant formulation of hyperthermia optimization problems as an eigenvalue problem were described in B¨ohm et al (1993) and Bardati et al (1995). A considerable generalization of this approach with respect to three-dimensional models, with inclusion of temperature distribution, weight functions and time dependences, is outlined in the following.
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2. Methods In section 2.1 we outline the basic equations for E-field, power and temperature distribution. In the following we define functionals which are suitable for optimizing the power deposition pattern (section 2.2) or the temperature distribution directly (section 2.3). The formulation of the optimization problem is performed in a way that requires the solution of a generalized eigenvalue problem. Efficient routines are available for this task. 2.1. Basic mathematical models We assume N antennas operating in a coherent mode at frequency ω. The free parameters consist of N complex amplitudes pj = aj exp(−iϕj ), which are combined in the vector p. Each antenna generates an electric field pj Ej (x), where Ej describes the normalized electric field of antenna j alone, assuming ai = 0 and a 50 load for the other antennas i = j . The relation between homogeneous and inhomogeneous media, given by Maxwell’s equations, is included in the Ej s. Integral equation methods (Wust et al 1999b), finite difference methods (Nadobny et al 1998) as well as finite element methods (Beck et al 1999) are employed to calculate Ej (x). Various problems and controversies concerning the E-field calculation are discussed elswhere (Wust et al 1999b, Gellermann et al 1999). The total electric field is then generated by a superposition of the basic electric fields Ej : E(p, x) =
N
pj Ej (x).
(2)
j =1
A relevant quantity for hyperthermia treatment is not the electric field itself but rather the absorbed power, which also depends on the electric conductivity σ (in S m−1 ): ARD(p, x) = σ (x)|E(p, x)|2 . Here ARD describes the absorbed power per volume (in W m−3 ). This quantity is directly related to the absorbed power per mass of tissue, which is described by SAR = ARDρ −1 (in W kg−1 ), by consideration of the tissue density ρ. The first mathematical approaches to hyperthermia treatment planning focused on optimizing the ARD distribution (see e.g. B¨ohm et al 1993). We will describe this approach in more detail in section 2.2. The ultimate aim of hyperthermia treatment planning is to optimize the resulting temperature distribution. So far no complete and efficient model exists which incorporates all aspects of the heat transfer (diffusion, perfusion, large vessels, whole vessel free temperature regulation, etc). However, a standard simplified model, the stationary bioheat transferequation, governs major aspects and produces results which agree sufficiently with reality. More precisely its solution determines the increase in temperature, Thyp (x), achieved by a given ARD distribution: div(κgradThyp ) − cW Thyp + ARD = 0.
(3)
Here c combines several quantities describing the material properties of the blood flow, W (x) models the local perfusion and κ(x) denotes thermal conductivity. Thus the computation of a temperature distribution essentially consists of two steps. Given a control vector p one has to compute the Ej by solving Maxwell’s equations (see above) and to compute T by solving (3), exploiting the fact that a slightly more complicated superposition principle analogous to (2) holds for Thyp . Equation (3) is solved very efficiently using an adaptive finite-element code (Bornemann et al 1993, Deuflhard et al 1989). The inverse or optimization problem is formulated in (1).
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2.2. Functionals for hyperthermia treatment planning As already stated, optimizing the functional in (1) requires solution of a high-dimensional, nonlinear optimization problem. This approach is not suitable for computer-aided optimizations in real time. One specific optimization criterion was introduced in B¨ohm et al (1993). This approach starts with a functional which compares the absorbed energy in the tumour region with the absorbed energy in the healthy tissues G\: σ (x)|E(x)|2 dx
E L2 (,σ ) . (4) = max max 2 N N p∈C , p =1 E L2 (G\,σ ) p∈C , p =1 G\ σ (x)|E(x)| dx This functional differs substantially from (1). However, from a mathematical point of view, the advantage of this new functional lies in its Hilbert space structure, which allows an efficient computation. Using (2) this functional is equivalent to p, Ap CN max (5) p∈CN , p =1 p, Bp CN with N × N matrices A, B: Aij = σ (x)Ei (x)Ej (x) dx Bij = σ (x)Ei (x)Ej (x) dx.
(6)
G\
This problem is solved by the normalized eigenvector corresponding to the largest generalized eigenvalue of (7)
Ap = λBp.
The normalized largest eigenvector only gives the relationship between the different antenna parameters pj . Hence, the final treatment plan is achieved by multiplying p with an additional amplitude factor a, such that all restrictions in the healthy tissue are met. In its original form this elegant approach was realized in a 2D setting. A generalization to higher dimensions is easily performed, but has several additional severe drawbacks: (a) Averaging over G\ gives little weight to small overheated areas in the healthy tissue, i.e. the resulting hyperthermia treatment plan can produce pronounced hot spots. (b) The approach is limited to optimize the ARD distribution, i.e. the influence of the heat transport with respect to the temperature distribution is neglected. The main purpose of this paper is to find a functional which has the same structure as (4) but which better models clinical demands, i.e. we want to maintain the efficient way of maximizing the resulting functional using a generalized eigenvalue problem. Of course we would like to compare the results of our optimization procedure with the global optimum obtained by (1). However, this functional is time-consuming to maximize. Hence, we use as reference the results obtained by a method described in Lang et al (1997). The authors solve min f3 (x) dx (8) f2 (x) dx + v3 f1 (x) dx + v2 p∈CN
G\
G\
with f1 (x) =
(43 − T (x))2 0
for T (x) < 43 ◦ C otherwise
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f2 (x) =
(T (x) − 42)2 0
f3 (x) =
(T (x) − Tc (x) − ε)2 0
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for T (x) > 42 ◦ C otherwise for T (x) > Tc (x) − ε otherwise
(with weight factors v1 and v2 for the corresponding terms and ε < 0.1 ◦ C) by an iterative method (damped Gauss–Newton method). This approach is a compromise between numerical efficiency and medical/clinical demands. We would like to stress that approximating the global minimum of this functional can be done with reasonable numerical effort, but even a single step of the Gauss–Newton iteration requires substantially more computation time than maximizing the functional (4) by solving (7) as described. 2.3. Efficient functionals for optimizing temperature distribution Again we start with the basic functional (4). We will discuss various improvements of this functional leading to a clinically relevant and practical methodology for computing hyperthermia treatment plans. All of these functionals have the same structure as (4), hence they can be maximized by the simple eigenvalue computation. More specifically, we will follow two ideas: • Temperature optimization: we introduce a functional which allows us to incorporate the temperature solution of the bioheat equation in a suitable Hilbert space functional. • Generation of weight functions: theory offers the possibility of introducing suitable weight functions which can be used to consider particular attributes of the patient. In section 2.2 we described the method used by B¨ohm et al (1993) for optimizing the ARD distribution N ARD(x) = σ (x)|E(x)|2 = pi pj σ (x)Ei (x)E j (x). (9) i,j =1
In a first attempt we have experimented with functionals like (4), where the L2 norm has been replaced by more general Sobolev norms of order s > 0. This change was done in order to give more importance to localized structures with large derivatives like hot spots. The success of this approach was limited because the electric field E also exhibits discontinuities at every boundary between organs, bones, muscles, etc, thus not only hot spots are weighted by using those norms. Moreover optimizing ARD instead of T is often not a reflection of clinical conditions, in particular if we are not considering any kind of heat transport. So we turn to optimizing the temperature distribution itself. The main problem is to find adequate function spaces and functionals which also allow optimization using a generalized eigenvalue problem. The temperature T (x) which is finally reached by the hyperthermia treatment plan p has two components: the basal temperature Tbas , which describes the temperature prior to the treatment, and Thyp , the temperature increase due to p and the related ARD distribution. Thus for each x ∈ G we have T (x) = Tbas (x) + Thyp (x). The quantity Thyp is computed by solving the bioheat transfer equation (3). Equation (9) states that the resulting ARD(p) can be considered as a sum of N 2 terms pi p j ARD(ij ) with ARD(ij ) := σ Ei E j . Moreover, (3) describes a linear dependency between ARD and Thyp , i.e. instead of solving this differential equation for each ARD(p) one can use an elegant concept to determine Thyp , which is of particular use for our approach and also reduces the numerical effort: if we solve (3) first for each separate ARD(ij ) (the respective solution shall
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be denoted with Thyp ), than for any input ARD(p) the resulting Thyp arises from a summation analogous to (9): Thyp (p, x) =
N
(ij )
pi pj Thyp (x) = p, M(x)p
i,j =1 (ij ) Thyp (x).
with M(x)ij := Hence we restrict ourselves to optimizing the increase in temperature 1 Thyp in an L setting. This is reasonable because the basal temperature is rather homogeneous in the relevant interior of the body. We define Thyp (x) dx max . (10) N p∈C , p =1 G\ Thyp (x) dx The functional (10) again is maximized by solving a generalized eigenvalue problem as stated in (7), where A, B now are N × N matrices with Aij = Mij (x) dx. Mij (x) dx, Bij =
G\
For more details see K¨ohler (1999). Significant improvements could be achieved by introducing suitable weight functions w. For this purpose we modify (10) to Thyp (x)w(x) dx max (11) p∈CN , p =1 G\ Thyp (x)w(x) dx
which leads to matrices A and B where the integrands of the elements have changed to Mij (x)w(x). The main advantage of this approach in connection with efficient eigenvalue optimization is its adaptivity: starting with a preliminary weight function one obtains a temperature distribution which might exhibit hot spots. This information can be used to formulate an adapted weight function for a second optimization step. 3. Results 3.1. The steady state case
In the following we outline some results obtained with these different functionals. Calculations have been done for two patients of the Charit´e Berlin, Campus Virchow-Klinikum: first a female patient with a recurrence of a cervical carcinoma (patient 1), and secondly a male patient with a rectal carcinoma (patient 2). Initially, patient data were obtained on finite element grids to calculate field and temperature. For an efficient implementation of our techniques all these data have been interpolated to regular grids. As the result of the optimization process a phase vector p includes information on the relation of the antenna’s amplitudes only, but not on the absolute amplitude value, i.e. the amplitude is merely determined up to a constant factor. In all examples this factor has been chosen in the following way: outside the tumour in a sensitive tissue (e.g. kidney, lung) a temperature of 42 ◦ C or in an insensitive tissue (e.g. muscle, bone) of 44 ◦ C is reached in at least one point, but not exceeded. This scaling strategy leads to comparable results for the different functionals. Table 1 displays three different quality measures for each of the hyperthermia treatment plans. The index temperature T90 (see legend of table 1) is accepted among clinicians as the
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Table 1. Results for optimizing the temperature distribution with different functionals. T90 is the temperature achieved or exceeded by 90% of the target (near the minimum temperature), Tmean is the averaged temperature and V42 the percentage of the target at 42 ◦ C or more. Patient 1
Patient 2
Antenna control
T90 (◦ C)
Tmean (◦ C)
V42 (%)
T90 (◦ C)
Tmean (◦ C)
V42 (%)
‘sync’ ‘ARD − L2 ’ ‘Thyp ’ ‘Thyp wsr ’
39.09 39.15 39.43 39.52
40.33 41.08 41.39 41.62
3.48 30.13 35.51 41.71
38.48 38.48 38.28 40.23
39.89 39.71 39.43 41.99
2.96 0.00 0.00 51.07
39.78
41.43
34.94
40.51
42.50
64.42
adapt
‘Thyp wsr
’
most powerful predictor of the quality of the heat treatment (and is derived from various clinical data, see e.g. Rau et al (2000)). The treatment plan ‘sync’ refers to a standard adjustment typically used if no optimization is available. Here constant p with pi = pj for i, j = 1, . . . , N was used. ‘ARD − L2 ’ is obtained with the method of B¨ohm et al (1993) as described in section 2.2. The result ‘Thyp ’ is obtained with the (unweighted) functional (10). The values for ‘Thyp wsr ’ are caused by a control where an a priori weight function which gives higher weights to sensitive regions was used. The resulting temperature distributions better fit the scaling rules as described above. adapt Finally ‘Thyp wsr ’ stands for a control based on the adaptive strategy as described at the end of section 2. Here the information about the localization of hot spots, taken from a preliminary computed temperature distribution, is used to create a new weight function which is very well adapted to the patient’s geometry and the control of a particular antenna. This strategy yields the best results for the steady state case. Figures 1 and 2 display temperature distribution in a representative cross section and longitudinal section of patients 1 and 2 respectively, where for each section a yellow line marks the tumour region. Images are given for different optimization strategies which obviously lead to distinct temperature distribution characteristics. The particular comparison with a synchronous control (i.e. equal phases and amplitudes at every antenna) shows the benefit of the different optimization strategies. It should be stressed that the scaling of the temperature increase by the choice of an appropriate total power as described above strictly respects the critical temperatures of the different tissues. This leads to temperature distributions where in the true sense of the word no hot spots will occur (i.e. temperatures above 42 ◦ C and 44 ◦ C respectively). Following this scaling strategy one can only choose the total power in such a way that the index temperature T90 is high if the temperature distribution outside the tumour is fairly uniform. Thus not only figures 1 and 2 but also the given numbers are suitable for evaluating the occurrence of hot spots. 3.2. Modelling time dependency So far we have only attempted to optimize a single set of phases and power amplitudes p in the steady state. However, this approach neglects the dynamic process of how the temperature increases from its original temperature Tbas (x) to the steady state temperature T (x) = Tbas (x) + Thyp (x). Modelling the time dependency reveals new opportunities for optimizing hyperthermia treatment plans. By following the time evolution of T (x) and switching over the power deposition pattern before reaching a critical temperature value we
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Figure 1. Temperature distributions on a cross section of patient 1 for the functionals ‘sync’ (left) and ‘ARD − L2 ’ to illustrate the small benefit obtained following the approach of B¨ohm et al (1993). The bar plot on the right-hand side shows the associated temperatures (in ◦ C). For reasons of layout and for a better combination with sagittal longitudinal sections below all cross sections have been set up on edge.
Figure 2. Temperature distributions on a cross section and longitudinal section of patient 2 for difadapt ferent functionals: ‘sync’ (left), ‘Thyp wsr ’ (centre) and the result obtained by global optimization according to Lang et al (1997) (right). Here the temperature scaling is the same as in figure 1.
can start out by selecting adjustments which exhibit hot spots in their steady states. Let us assume a set of hyperthermia treatment plans which in their steady states exhibit hot spots in
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Figure 3. Heating/cooling profile in a certain point x for a combination of two hyperthermia treatment plans.
different locations. This allows us to adjust higher power amplitudes for these plans if we control the heating process in time and switch between these plans accordingly. Of course a heuristic approach of this sort is already used in clinical practice. A typical hyperthermia treatment session lasts approximately 75–90 min and the medical supervisor modifies the original p according to the patient’s reactions. Our aim is to describe and implement some basic ideas for a strict mathematical optimization which can be used in a systematic way during a heat treatment. In order to put this into a mathematical framework we need a model as well as optimality criteria for the switching points. Therefore, we assume a reasonable temporal transition (1) (2) between two steady state temperature distributions from Thyp to Thyp according to Newton’s cooling law, which leads to (1) (2) (1) (x) + (Thyp (x) − Thyp (x))[1 − exp(−ξ(x)t)] T (x, t) = Tbas (x) + Thyp
(for further details see K¨ohler (1999)). The exponent ξ(x) depends on perfusion and thermal conductivity (via the thermal gradient) and consequently varies with space and time, especially depending on the tissue. However, in order to evaluate roughly the idea of this approach, a good approximation is to choose a medium and constant value ξ(x) = 0.5 which is derived from clinical practice. For a detailed presentation of those issues see Roemer et al (1985). Figure 3 shows the temperature–time curve at a fixed point x for a combination of antenna controls p (1) and p (2) . The diagram on the left shows a combination of treatment plans where p (2) achieves a higher Thyp than p (1) ; the diagram on the right shows the opposite, which leads to cooling in position x after switching over. In the following we describe the results obtained by combining two power deposition patterns p(1) and p (2) . For this we have to specify the duration %t1 and %t2 of each treatment plan. Due to our simple but realistic heating model, this low-dimensional optimization problem can be solved by a direct method; for details and implementational issues see K¨ohler (1999). The weight function w used in the computation of p(2) has been defined adaptively depending on the temperature distribution achieved with p (1) by choosing rather high weight factors in normal tissues, where p(1) achieved high temperatures in the steady state. Here one important optimality strategy is to make the controls differ sufficiently with respect to the location of any hot spots which arise. For this reason the T90 values of these controls differ substantially from the best steady state solutions. So for patient 1 the combination of such two controls with T90 = 39.45 ◦ C and T90 = 39.52 ◦ C respectively lead to a combination result of T90 = 39.81 ◦ C. In case of patient 2 the resulting value T90 = 40.57 ◦ C was achieved by
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steady state controls of T90 = 40.25 ◦ C and T90 = 40.19 ◦ C. Thus the described method of a time-dependent combination of different antenna controls leads to a remarkable suppression of hot spots. In table 2 for clarity we repeat some results starting from the method introduced in B¨ohm adapt et al (1993), continued by the best steady state solution (which was ‘Thyp wsr ’) for comparison with the descriptors of an average temperature distribution achieved by a combination of two suitable controls as described above (‘comb’). Finally for further comparison we have listed the descriptors achieved using the global optimization of the functional (8). Table 2. Some more optimization results, combined with two examples from table 1 for comparison. Patient 1
Patient 2
Antenna control
T90 (◦ C)
Tmean (◦ C)
V42 (%)
T90 (◦ C)
Tmean (◦ C)
‘ARD − L2 ’
39.15
41.08
30.13
38.48
39.71
0.00
39.78 39.81 39.83
41.43 41.46 41.53
34.94 35.80 37.85
40.51 40.57 40.66
42.50 42.63 42.74
64.42 65.85 69.79
adapt
‘Thyp wsr ‘comb’ ‘global’
’
V42 (%)
Tables 1 and 2 demonstrate that the temperatures in the target (tumour) can be increased remarkably by stepwise utilization of our optimization methods. By combining just two antenna controls employing the efficient procedure sketched above we achieved the best improvement for both examples. We would like to state that within the accuracy of the used mathematical models the latter method gives results which are comparable with those obtained using the comparatively expensive global optimization. 4. Discussion The next generation of RF applicators (Sigma-Eye applicator) will require hyperthermia treatment planning because the number of adjustable parameters (11 relative phases, 11 relative amplitudes) is much too high to find the ‘best adjustment’ by intuition or trial and error (see Wust et al (1996)). Treatment planning systems are now at the stage of clinical practicability, and a preliminary verification has also been performed (Gellermann et al 1999). In particular, antenna models are available employing the FDTD method or FE method which quite accurately describe the E-field in the water bolus for the Sigma-60 applicator (Wust et al 1999a) and for the Sigma-Eye applicator (data to be published). Even though quite a pronounced sensitivity of the patient model to the E-field (SAR pattern) in the interior of the body is known (details of the segmentation, grid, numerical method) (Gellermann et al 1999, Wust et al 1999b), accumulation of further experimental and clinical data will probably enable a valid patient-specific prediction of the power deposition pattern. Therefore the forward problem up to the level of E(x) appears manageable. The numerical calculation of the temperature distribution stands on a weaker basis (from the bioheat transfer equation), mostly because of the variable tissue/tumour perfusion that has considerable influence on the temperature. The basic problem is that the perfusion under hyperthermia conditions is not well known, and furthermore can vary during heat treatment (e.g. because of local and systemic thermoregulatory process). Solutions of the inverse problem depend even more sensitively on numerous parameters including the objective function itself, and especially the assumed perfusions and other details of the input. Clearly, all uncertainties of the prior planning process (patient model, E-field calculation, temperature calculation),
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and in addition individual variations such as positioning errors, shifts of organs and tissues, thermoregulation, systemic influences, intolerances, etc are accumulating. Therefore, in clinical use a continuous online adaptation of the optimization procedure according to the individual situation, i.e. during the heat treatment, appears obligatory. A hybrid system is designed to deliver online information in order to adjust the optimization process: • Registration of temperature distribution during heating-up after switching on the power (i.e. delivering information about the gradient of temperature increase) can characterize the SAR distribution in vivo. • Registration of temperature distribution during the steady state can characterize perfusion, and furthermore localize hot spots (which may or may not be predicted by the SAR distribution previously determined). In addition, other methods to measure perfusion patterns are available using MRI, e.g. utilizing a contrast media bolus or other dedicated sequences. • Individual intolerances might be found and there may be reduced perfusion postsurgically or temperature-dependent sensations of any kind. The optimization method described in this paper, especially the strategy to maximize the functional in (11), is particularly suitable for use with the data from the hybrid system. The following CPU times are given for a Silicon Graphics Indigo 2 with 250 MHz MIPS and 192 MB RAM: (a) For a reclassified perfusion case (estimated form MRI) a computation of M(x)ij to determine Thyp (x) according to (10) needs 6.5 s for four channels (i, j = 1, . . . , 4), and consequently 58.8 s for 12 channels (i, j = 1, . . . , 12). (b) An adaptation of weighting factors needs 2 s for four channels and 18 s for 12 channels to recompute the matrices as in (11) (multiplying the integrand with w(x)) for solution of a generalized eigenvalue problem. (c) The solution of a generalized eigenvalue problem itself is performed very rapidly, i.e. in 0.1 s for four and less than 1 s for 12 channels respectively. The result of such an optimization process is an improved set of control parameters pj (j = 1, . . . , 4 or 1, . . . , 12) of phases and amplitudes. Other functionals, as in (4), are also useful, e.g. if a change of conductivity σ (x) is registered and considered ex posteriori. The computation time for a completely updated optimization of the Sigma-Eye applicator (12 channels) is clinically feasible (≪80 s), and further reduction is expected with evolving computer technology (at least a factor 10). Considerable flexibility is available to customize the control parameters to changes in electrical conductivity, perfusions and individual intolerances of the patient. Therefore, the developed eigenvalue approach is especially useful for online optimization in a hybrid RHT system. Strictly speaking changes in electrical conductivity σ would enforce a complete recalculation of the electric field. However, a first-order correction is achieved for ARD(x) = σ (x)|E(x)|2 by assuming the same E(x) and varying σ alone. Of course, further refinement of these procedures will be possible with increasing practical experience. Acknowledgments This work has been supported in part by the Deutsche Forschungsgemeinschaft (DFG) (SFB 273, GraKo 331) and BMBF (grant MA7POT-6). Furthermore we are grateful to the referees of this paper for their thorough and careful analysis and very helpful comments.
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References Bardati F, Borrani A, Gerardino A and Lovisolo G A 1995 SAR optimization in a phased array radiofrequency hyperthermia system IEEE Trans. Biomed. Eng. 42 1201–7 Beck R, Deuflhard P, Hege H C, Seebass M and Stalling D 1997 Numerical algorithms and visualization in medical treatment planning Visualization and Mathematics ed H C Hege and K Poltier (Berlin: Springer) pp 303–25 Beck R, Deuflhard P, Hiptmair R, Hoppe R and Wohlmuth B 1999 Adaptive multilevel methods for edge element discretizations of Maxwell’s equations Surv. Math. Indust. 8 271–312 B¨ohm M, Kremer J and Louis A K 1993 Efficient algorithm for computing optimal control of antennas in hyperthermia Surv. Math. Indust. 3 233–51 Bornemann F, Erdmann B and Kornhuber R 1993 Adaptive multilevel methods in three space dimensions Int. J. Numer. Methods Eng. 36 3187–202 Deuflhard P, Leinen P and Yserentant H 1989 Concepts of an adaptive hierarchical finite element code IMPACT Comp. Sci. Eng. 1 3–35 Gellermann J, Wust P, Stalling D, Seebass M, Nadobny J, Beck R, Beier J, Hege H C, Deuflhard P and Felix R 1999 Clinical evaluation and verification of the hyperthermia treatment planning system HyperPlan Int. J. Radiat. Oncol. Biol. Phys. 47 1145–56 K¨ohler T 1999 Effiziente Algorithmen f¨ur die Optimierung der Therapie-Planung zur regionalen Hyperthermie PhD Thesis Universit¨at Potsdam Kremer J and Louis A K 1990 On the mathematical foundation of hyperthermia treatment Math. Methods Appl. Sci. 13 467–79 Lang L, Erdmann B and Seebass M 1997 Impact of nonlinear heat transfer on temperature control in regional hyperthermia Preprint SC 97-73 (Berlin: Konrad-Zuse-Zentrum f¨ur Informationstechnik) Liauh C and Roemer R B 1993 Multiple minima in inverse hyperthermia temperature estimation problem ASME J. Biomech. Eng. 115 239–46 Nadobny J, Sullivan D, Wust P, Seebass M, Deuflhard P and Felix R 1998 A high-resolution interpolation at arbitrary interfaces for the FDTD method IEEE Trans. Microwave Theory Tech. 46 1759–66 Nikita K S, Maratos N G and Uzunoglu N K 1993 Optimal steady-state temperature distribution for a phased array hyperthermia system IEEE Trans. Biomed. Eng. 40 1299–1306 Rau B, Wust P, Tilly W, Gellermann J, Harder C, Riess H, Budach V, Felix R and Schlag M D 2000 Preoperative radiochemotherapy in locally rectal cancer: regional radiofrequency hyperthermia correlates with clinical parameters Int. J. Radiat. Oncol. Biol. Phys. 48 381–91 Roemer R B, Fletcher A M and Cetas T C 1985 Obtaining local SAR and blood perfusion data from temperature measurements: steady state and transient techniques compared Int. J. Radiat. Oncol. Biol. Phys. 11 1539–50 Wlodarczyk W, Boroschewski R, Hentschel M, Wust P, M¨onich G and Felix R 1998 Three-dimensional monitoring of small temperature changes for therapeutic hyperthermia using MR J. Magn. Reson. Imaging 8 165–74 Wlodarczyk W, Hentschel M, Wust P, Noeske R, Hosten N, Rinneberg H and Felix R 1999 Comparison of four magnetic resonance methods for mapping small temperature changes Phys. Med. Biol. 44 607–24 Wust P, Berger J, F¨ahling H, Nadobny J, Gellermann J, Tilly W, Rau B, Petermann K and Felix R 1999a Scanning E-field sensor device for on-line control of annular-phased-array systems Int. J. Radiat. Oncol. Biol. Phys. 143 927–37 Wust P, Nadobny J, Seebass M, Stalling D, Gellermann J, Hege H C, Deuflhard P and Felix R 1999b Influence of patient models and numerical methods on predicted power deposition patterns Int. J. Hyperth. 15 519–40 Wust P, Seebass M, Nadobny J, Deuflhard P, M¨onich G and Felix R 1996 Simulation studies promote technological development of radiofrequency phased array hyperthermia Int. J. Hyperth. 12 477–94
