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Phys. Med. Biol. 43 (1998) 121–138. Printed in the UK PII: S0031-9155(98)82478-0 Dose uniformity of ferromagnetic seed implants in tissue with discrete vasculature: a numerical study on the impact of seed characteristics and implantation techniques N van Wieringen†‡, A N T J Kotte†, G M J van Leeuwen†, J J W Lagendijk†, J D P van Dijk§ and G J Nieuwenhuysk † Department of Radiotherapy, University Hospital Utrecht, Utrecht, The Netherlands § Department of Radiotherapy, Academic Medical Centre, University of Amsterdam, Amsterdam, The Netherlands k Kamerlingh Onnes Laboratory, Leiden State University, Leiden, The Netherlands Received 7 March 1997 Abstract. The results from simulations with a new three-dimensional treatment planning system for interstitial hyperthermia with ferromagnetic seeds are presented in this study. The thermal model incorporates discrete vessel structures as well as a heat sink and enhanced thermal conductivity. Both the discrete vessels and the ferroseeds are described parametrically in separate calculation spaces. This parametric description has the advantage of an arbitrary orientation of the structures within the tissue grid, easy manipulation of the structures and independence from the resolution of the tissue voxels (tissue calculation space). The power absorption of the selfregulating seeds is according to empirical data. The thermal effects of an unlimited number of thin layers surrounding the seed (coatings, catheters) can be modelled. The initial calculations have been performed for an array of 12 identical ferromagnetic seeds in a tissue volume with a computer generated artificial vessel network spanning four vessel generations in both the arterial and venous tree. The heterogeneously distributed large isolated vessels impair the temperature distribution significantly, indicating the limited accuracy of continuum models. Simulations with different types of ferromagnetic seeds have confirmed that the efforts of previous studies to optimize the self-regulating temperature control and the implantation techniques of the ferroseeds will improve the homogeneity of the temperature distribution in the target volume. Multifilament seeds implanted in brachytherapy needles and tubular seeds appear to be the most favourable configurations. The division of long seeds into shorter segments with the appropriate Curie temperature will further improve the homogeneity of the temperature distribution without increasing the average temperature in the volume of interest. Given the proper thermal tissue data, the model presented in this study will prove to be a useful tool in making choices for the implant geometry, seed spacing and Curie temperature. 1. Introduction Whereas several clinical trials have demonstrated that reasonable results are achieved with hyperthermia even if the intended minimum temperature is not reached (Overgaard et al 1995, Gonzalez Gonzalez et al 1995), other studies have shown that there is a direct relation ‡ Address for correspondence: N van Wieringen, University Hospital Utrecht, Department of Radiotherapy, Heidelberglaan 100, 3584 CX, Utrecht, The Netherlands. E-mail address: [email protected] c 1998 IOP Publishing Ltd 0031-9155/98/010121+18$19.50 121 122 N van Wieringen et al between the thermal parameters and the treatment response (Dewhirst et al 1984, Cox and Kapp 1992, Sneed et al 1992, Overgaard et al 1996). The latter results have stimulated efforts to optimize the temperature control of hyperthermia techniques aiming at a more uniform temperature distribution throughout the target volume. Interstitial hyperthermia using ferromagnetic seeds has been investigated in recent years by several groups. These studies have resulted in better reproducible seeds with improved self-regulating properties and better implantation techniques (Brezovich et al 1984, Kobayashi et al 1986, Brezovich and Liu 1987, Chen et al 1988, Haider et al 1991, Ferguson et al 1993, Meijer et al 1995, Paulus et al 1996, Van Wieringen et al 1996, 1997a, Cetas et al 1997). The objective of the present paper is to investigate whether these improvements will result in a more uniform temperature distribution in the target volume. For this purpose we will use a threedimensional treatment planning system which has been developed at the University Hospital in Utrecht (Kotte et al 1998). A large number of thermal models for ferromagnetic seed heating have been developed previously (Brezovich et al 1984, Matloubieh et al 1984, Chen et al 1991, 1992, Chin and Stauffer 1991, Haider et al 1993, Tompkins et al 1994, Indik and Indik 1994, Indik et al 1994). In all numerical studies performed with these models the blood flow has been taken homogeneously (in some cases the perfusion rate in the tumour core is reduced) and is represented by a heat sink, which describes the behaviour of blood vessels in a collective way. Consequently, the applied models are unable to simulate the local effects of large blood vessels, which have to be treated individually (Crezee et al 1991, Moros et al 1993, Rawnsley et al 1994, Lagendijk et al 1994, Kolios et al 1995). If much detail on the temperature distribution is required, for instance for clinical treatment planning, a discrete vessel description will be necessary (Mooibroek and Lagendijk 1991, Huang et al 1996, Kotte et al 1996). In the three-dimensional treatment planning system used in this study discrete vessel structures as well as a heat sink and enhanced thermal conductivity are incorporated. The discrete vessels are described parametrically in the ‘vessel calculation space’ which is separate from the voxel based tissue space (Kotte et al 1996). This parametric description has the advantage of (i) easy manipulation of the vessel tree (creation of vessels, shifting, rotation), (ii) compatibility with vessel data reconstructed from magnetic resonance angiography (MRA) images, (iii) the ability to model a wide range of vessel structures (strong curvatures, changing diameter) and (iv) independence from the resolution of the tissue voxels (tissue calculation space). Similarly, the ferromagnetic seeds are described in the ‘seed calculation space’ (Kotte et al 1998). This allows an arbitrary orientation of the seeds within the tissue grid, modelling of an unlimited number of thin layers that surround the seed (coatings, catheters) and thermal conduction of heat along the longitudinal axis of the seed. The heat production of the seeds is described according to experimental data. In this paper the initial calculations with this new three-dimensional treatment planning model are presented. Various types of seeds have been modelled to investigate whether past efforts to optimize the self-regulating properties of the ferromagnetic seeds and the applied implantation techniques have improved the homogeneity of the temperature distribution in the target volume. ‘Ideal seeds’ have been modelled to study whether future efforts to further optimize the seeds will be beneficial. These simulations have been performed in a tissue volume with an inhomogeneous discrete vessel network to emphasize possible hot and cold spots and to obtain insight in the local temperature control of the various seeds. Finally, some additional simulations have been performed, in which the blood flow is represented by a more traditional continuum model, that is a heat sink either with or without an enhanced thermal conductivity. Dose uniformity of ferromagnetic seed implants 123 2. Method 2.1. General set-up of the three-dimensional treatment planning system The tissue volume which is heated is discretized three-dimensionally on a regular grid of voxels. In the tissue volume, the so-called tissue calculation space, heat transport takes place by conduction (a heat sink, enhanced thermal conductivity and an additional power deposition term can be included). A finite difference method is applied to model the time evolution of the tissue temperature. In the present study we will not consider the transient effects, but we will concentrate on the steady-state temperature distribution. The descriptions of the vessel tree and the array of seeds have been separated from the tissue volume. This enhances the flexibility of the system and reduces the need for a highresolution tissue model. Both vessels and seeds are described as three-dimensional curves with associated diameters. This parametric description as well as the interaction of seeds and vessels with the tissue are described extensively in papers by Kotte et al (1996, 1998). The approach used enables modelling of an unlimited number of catheters or coatings surrounding the seed. Thermal conduction of heat along the seed is also incorporated. Convective heat transport is included in the vessel calculation space. Figure 1. The modelled tissue volume: a hexagonal array of 12 identical ferromagnetic seeds with interseed spacing equal to 13 mm is placed in a tissue volume (70 mm × 70 mm × 80 mm) with discrete blood vessels. The modelled vasculature consists of an arterial (shaded dark) and venous tree which are roughly counter-current. 2.2. Implant geometry and seed properties Simulations are performed for a hexagonal array of 12 identical ferromagnetic seeds (depicted schematically in figure 1). Each seed is 5 cm long and is discretized in 25 one-dimensional segments, the so-called buckets (Kotte et al 1996). The spacing between 124 N van Wieringen et al the seeds is 13 mm. The different kinds of seeds that have been modelled represent the results of past and present efforts to optimize the self-regulating properties (Burton et al 1971, Chen et al 1988, Ferguson et al 1993, Meijer et al 1995), the heat production (Haider et al 1991, Van Wieringen et al 1996) and the implantation technique of ferromagnetic seeds (Brezovich and Liu 1987, Van Wieringen et al 1996, 1997a). The seeds are characterized by their radius RS , thermal conductivity kS and heat production per unit volume PV (T ). All seeds have equal density ρ = 11.27 × 103 kg m−3 and heat capacity c = 305 J kg−1 K−1 (Kobayashi et al 1986). Figure 2. The heat production per unit volume as a function of temperature of a multifilament PdNi seed (11 filaments of cross-sectional diameter 0.307 mm, measurement repeated three times N, × and +). The amplitude of the applied magnetic field strength is 2.3 kA m−1 . The fit to the data is used as input for model calculations. The heat production of self-regulating thermoseeds is dependent on the temperature of the seed, as is shown in figure 2 for a multifilament palladium–nickel, PdNi, seed (11 filaments of 0.307 mm diameter). In the current study this relation is approximated by a stepwise linear function (also depicted in figure 2), which is a fit to the experimental data PV (T ) = PV 0 PV (T ) = PV 0 PV (T ) = 0 TC − T TC − Tlow for T < Tlow for Tlow 6 T 6 TC for T > TC . (1) Here PV 0 is the heat production per unit volume of the seed for temperatures below Tlow . The temperature TC is the Curie temperature of a seed. In an experimental study Van Wieringen et al (1996) showed that the heat production of PdNi seeds is limited by magnetic saturation. The maximum heat production is approached asymptotically with increasing magnetic field strength. The asymptote represents the situation of optimal heat production and temperature control for PdNi seeds. It is equal for seeds which have the same amount of PdNi per unit length regardless of their geometry (either solid, multifilament or tubular). For temperatures close to the Curie temperature the asymptote is approached at relatively low magnetic field strengths (between 1 and 2 kA m−1 ). Consequently, in this temperature range the heat Dose uniformity of ferromagnetic seed implants 125 production of solid and multifilament seeds which are composed of the same amount of PdNi is identical. At lower temperatures a higher field intensity is needed to reach the asymptote, especially for solid seeds or filaments with a large radius. Therefore, the heat production of different seeds with an equal amount of PdNi may differ for temperatures below Tlow . An increase in the amount of PdNi per unit length of a seed (e.g. by using more filaments of the same diameter) will result in increased heat production and quality of temperature control for all type of seeds. The implants which have been modelled in this study have been divided in two groups. The various seeds in the first group of implants have different heat production and/or thermoregulating properties. They are not covered by a catheter or coating, but are placed directly into the tissue. In the first group the following seeds have been modelled: (i) Constant power seed: the heat production of this type of seed is independent of its temperature. Temperature regulation can only be achieved by the adaptation of the magnetic field strength. It is not possible to compensate for local variations in tissue cooling, for instance by inhomogeneous blood flow. Three simulations have been performed corresponding to different intensities of the applied field strength: constant power seed 1: low field strength/heat production to prevent hot spots within the target volume; constant power seed 2: average field strength/heat production; constant power seed 3: high field strength/heat production to prevent cold spots. (ii) Solid PdNi seed: the heat production of this seed is dependent on its temperature. The magnetic transition of PdNi has been optimized, aiming at a homogeneous temperature distribution (Meijer et al 1995). The heat production of a solid seed was considered to be insufficient to raise the temperature of a tumour with relatively high perfusion to a therapeutic level. (iii) Multifilament PdNi seed (1): the heat production of the self-regulating PdNi seeds was improved by application of a number of filaments with a small diameter. As it is not possible to model the various filaments within a multifilament seed, this seed is treated as a solid seed with improved heat production. (iv) Tubular PdNi seed: tubular ferromagnetic seeds have been developed for placement on the outside of catheters or needles. The application of tubular seeds has the following advantages: (1) the heat transfer to the surrounding tissue has improved compared with seeds placed inside catheters; (2) it enables the placement of radioactive sources in the lumen of the catheter for simultaneous treatment of hyperthermia and interstitial radiotherapy; (3) the heat production and quality of temperature control have improved compared with solid or multifilament seeds due to an increase in the amount of PdNi. As the thermal model does not support hollow seeds, these seeds have been modelled as solid rods. However, the heat production inside a bucket as well as the conduction of heat between adjacent buckets is proportional to the total cross section of the bucket. To correct for the hollow centre of tubular seeds, the heat production and thermal conductivity are multiplied by a factor (rS2 − rI2 )/rS2 where rS (1.0 mm) and rI (0.75 mm) are the outer and inner radius of the tube respectively. (v) Tubular PdNi seed composed of short segments with different Curie temperatures: to compensate for local hot and cold spots in the target volume the tubular seeds have been replaced by shorter segments with different Curie temperatures (equal to the Curie temperature of the long tubular seeds or 2.5 ◦ C higher or lower). Note that for all segments Tlow changes by the same amount as TC . The choice for the Curie temperature of each segment has been based on the results found in the previous calculation, rather than using an extensive optimization routine to find the best implant. The length of the segments 126 N van Wieringen et al varied from 10 to 35 mm. It is assumed that the heat production of the short segments is not reduced significantly by a demagnetizing field. Van Wieringen et al (1997b) have shown this is true for temperatures close to the Curie temperature. (vi) Ideal seed: the most desirable thermoseed would reach a predetermined temperature independent of the tissue type and blood perfusion. The heating curve of such an ideal thermoseed would have to be a step function, that is, the heat production would have to remain high up to the desired temperature and drop to zero at that temperature. In this study ideal seeds have been modelled to study the possible benefits of future efforts to further optimize PdNi seeds. The outer diameter of the ideal seeds has been taken as 1.0 and 2.0 mm. The Curie temperature of these seeds has been set equal to the average temperature of the seeds in an array with bare multifilament seeds. In this way the total amount of heat dissipated by these two implants will be approximately equal. In clinical practice ferromagnetic seeds are usually afterloaded in brachytherapy catheters (Stea et al 1992, Mack et al 1993, Gannet et al 1995). The catheters or coatings which are needed to implant the seeds or to guarantee biocompatibility generally impair the selfregulating properties of the ferromagnetic seeds (Van Wieringen et al 1997a). In the second group of implants modelled in this study various implantation methods have been simulated. The seeds have equal heat production and thermoregulating properties. (vii) Multifilament PdNi seed (2) inserted in a plastic catheter filled up with air: catheters have a low thermal conductivity and are expected to reduce the self-regulating temperature control of the seeds. The intermediate layer between the seed and the catheter is filled with air. (viii) Multifilament PdNi seed (3) inserted in a plastic catheter filled up with water: this implant is similar to the previous one, but the catheter is filled up with water to reduce the influence of the intermediate air layer. (ix) Multifilament PdNi seed (4) inserted in a metallic needle: the catheter as used for the previous implants has been replaced by a metallic brachytherapy needle to improve the heat transfer to tissue. As has been shown by Van Wieringen et al (1997a) these needles do not impair the heat production of the PdNi seed. In correspondence with clinical practice the outer diameter of a needle is smaller than the outer diameter of a catheter. The intermediate layer between the seed and the needle is filled with water. (x) Multifilament PdNi seed (5) inserted in a PDR (pulsed dose rate) brachytherapy needle: instead of using a standard brachytherapy needle as was done for the previous implant the multifilament seed is inserted in a PDR brachytherapy needle. This PDR needle has a larger inner diameter and consequently the seed can be composed of more filaments. This will result in a higher heat production and Curie gradient. As a seed composed of nine filaments of 0.32 mm diameter appears to fit snugly into the needle the thin layer of water is not modelled. The characteristics of all seeds are summarized in table 1. The heat production is given per unit length for ease of comparison. The quality of temperature control of the seeds is characterized by the Curie gradient which is defined as the gradient of the heat production per unit length of the implant (Van Wieringen et al 1996). The properties of the PdNi seeds are in accordance with experimental data measured at a magnetic field strength of clinical relevance (Van Wieringen et al 1996, 1997a). The outer radius and the thermal conductivity of layers surrounding a seed are indicated by rS1 , rS2 and kS1 , kS2 respectively (two layers maximum). The various modelled layers can be distinguished by their thermal conductivity: for air 0.024, for plastic 0.22, for water 0.6 and for metal 75 W m−1 K−1 . layer 1 layer 2 Seed type R (mm) Tlow (◦ C) TC (◦ C) PL0 (mW cm−1 ) Curie gradient (mW cm−1 ◦ C−1 ) r1 (mm) k1 (W m−1 K−1 ) r2 (mm) k2 (W m−1 K−1 ) T50 (◦ C) T90 (◦ C) T10 –T90 (◦ C) HC (◦ C) Const. pow. 1 Const. pow. 2 Const. pow. 3 Solid M-filament 1 M-filament 2 M-filament 3 M-filament 4 M-filament 5 Tubular Tubular TC∗ Ideal 1 Ideal 2 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.6 1.0 1.0 0.5 1.0 — — — 47.6 44.9 44.9 44.9 44.9 44.9 44.9 44.9∗ — — — — — 52.8 52.8 52.8 52.8 52.8 52.8 52.8 52.8∗ 48.5 48.5 62 113 178 135 206 206 206 206 297 361 361 ∞ ∞ 0 0 0 26.1 26.1 26.1 26.1 26.1 37.6 45.7 45.7 ∞ ∞ — — — — — 1.1 1.1 1.1 1.5 — — — — — — — — — 0.024 0.6 0.6 75 — — — — — — — — — 1.7 1.7 1.5 — — — — — — — — — — 0.22 0.22 75 — — — — — 41.59 45.46 50.47 44.49 44.52 41.87 43.50 44.85 45.76 46.48 45.57 44.29 45.50 38.52 39.77 41.40 39.98 40.10 38.79 39.59 40.31 40.86 41.42 41.66 40.42 41.48 5.05 9.33 14.91 6.96 6.85 4.99 6.23 7.00 7.44 7.64 5.89 5.81 5.83 3.33 3.37 3.39 2.33 2.21 2.79 2.41 2.12 1.93 1.73 1.26 1.70 1.30 Dose uniformity of ferromagnetic seed implants Table 1. Properties of the modelled thermoseeds: the thermoseed radius, RS , the heat production of the seed per unit length, PL0 , for temperatures below Tlow , the Curie temperature, TC , and the Curie gradient. The outer radius and thermal conductivity of the possible layers surrounding a seed are indicated by r1 and r2 and k1 and k2 respectively. Simulations have been performed for a hexagonal array of 12 seeds in tissue with a discrete vessel network. The vessels in the applied network are distributed heterogeneously over the tissue volume to simulate inhomogeneous tissue perfusion. Parameters for evaluation of the resulting temperature distributions are given in the last four columns. The parameter Tn quantifies the temperature which is exceeded in n per cent of the volume of interest. const. pow = constant power, m-filament = multifilament. TC∗ means tubular seeds with Curie temperature as given or ±2.5 ◦ C (see section 2.2 for a description of all seeds). HC is the heterogeneity coefficient (equation (4)). 127 128 N van Wieringen et al 2.3. Tissue volume and vessel tree The tissue volume which has been modelled has dimensions of 70 mm×70 mm×80 mm (the longer axis is parallel to the ferromagnetic seeds). This allows a margin of at least 15 mm from each seed to the boundary of the volume, which is kept at a fixed baseline temperature of 37 ◦ C. The volume is discretized on a regular grid of voxels of size 1 mm×1 mm×1 mm. The tissue has an intrinsic thermal conductivity of 0.6 W m−1 K−1 , density of 1000 kg m−3 and a heat capacity equal to 4000 J kg−1 K−1 . The vascular tree which has been applied, consists of an irregular branching network of curved vessel segments as is depicted in figure 1. The main part of the vessel tree is situated in the upper half of the target volume in order to realize an inhomogeneous cooling field. This will emphasize the presence of hot and cold spots, and consequently a better insight into the local temperature control of the various seeds can be obtained. The vascular tree has been generated using a vessel network generation algorithm designed by Van Leeuwen et al (1997a). The network consists of complex counter-current arterial and venous vessel trees. The incoming vessel of the arterial tree has a diameter of 2.6 mm and enters the modelled tissue volume at the top of the implant. A bifurcation at the centre of the hexagonal array of seeds creates a Y-shaped vessel structure whose legs cross the target volume and leave the tissue volume at the opposite side approximately half-way down the array. This Y-shaped vessel is used as the starting point for the generation of an extensive tree with 70 branches of the fourth and also highest generation. The number of termination points in the lower half of the volume was set at one-quarter of the number in the upper half, making the perfusion one-quarter of that in the upper half. The average diameter of the fourth-generation branches is 0.47 mm. It is unlikely that the blood leaving these terminal arterial branches will have reached thermal equilibrium with the surrounding tissue. However, the branches of a higher generation are too numerous to model discretely due to the limited computer power. Therefore, it is modelled that the blood leaving the terminal branches perfuses a subvolume in the adjacent tissue and it is assumed to reach thermal equilibrium with this tissue instantaneously, much like a heat sink. Note, however, that in contrast to a Pennes heat sink (Pennes 1948), the actual temperature of the ‘preheated’ blood is taken into account. The geometry of the subvolume in which the ‘heat sink like’ energy transfer takes place can be chosen freely. In this study a sphere with its centre located at the end point of the terminal arterial branch and a radius of 6 mm has been used. This radius is of the same order as the thermal equilibrium length of the fifth-generation arterial branches which are not being modelled discretely. A venous tree, roughly counter-current to the arterial network, has been generated according to the same procedure. Like the arterial tree, it spans four vessel generations with a total of 66 terminating vessel branches with an average diameter of 0.44 mm. The inflow temperature of the venous blood is taken as equal to the average temperature of the tissue nodes adjacent to the terminal venal branch. The entire tree contains 188 vessel segments with a total length of 2.79 m. The interaction of the vessel segments with the surrounding tissue has been the subject of an earlier study (Kotte et al 1996). The accuracy of the description for non-symmetrical situations (i.e. for counter-current vessel pairs or at curvatures or branching points in the network) has been described by Van Leeuwen et al (1997b, c). The Nusselt number used for calculating the heat flow rate between the vessels and the surrounding tissue is taken constant for the entire vessel tree. The applied value, N u = 3.66, is valid for a Newtonian fluid in a thin walled tube with constant wall temperature and a thermally developed laminar flow (Kays and Crawford 1980, Crezee and Lagendijk 1992). Higher Nusselt values apply for temperature profiles which are not fully developed (entrance Dose uniformity of ferromagnetic seed implants 129 length phenomena). Huang et al (1996) have shown that changes in the Nusselt number of about 50% affect the temperature distribution only insignificantly. The inflow of blood in the main artery is equal to 1.06 × 10−6 m3 s−1 . Approximately 60% of the inflowing blood leaves the tissue volume by the two large arterial branches. Taking this into account, the effective perfusion rates in the upper and lower half of the volume become 10 and 2.5 ml/100 g/min respectively. Iterations are continued until a steady-state situation is achieved (at 25 to 30 min after the heating started). Run on an SGI-R4400 machine, it takes one to two days to reach the stationary situation. The thermal model has not yet been optimized for speed. 2.4. Alternative blood flow models Due to the limited capacity of our workstations, vessels with a diameter smaller than 0.47 mm have not been modelled discretely. The thermally unequilibrated blood leaving the terminal arterial branches was assumed to perfuse a subvolume in the adjacent tissue and to reach thermal equilibrium with this tissue instantaneously. However, even if thermal equilibrium has been reached the numerous small vessels which connect the fourthgeneration arteries to the fourth-generation veins may be of significant importance for the energy transfer within the tissue volume. To include some of the effects of these smaller vessels, an additional perfusion term was added to the discrete vasculature. Several investigators (Charny et al 1990, Crezee et al 1994, Zhu et al 1995) have indicated that the best way to account for these small vessels using a continuum description is by increasing the thermal conductivity of the tissue. The enhanced effective thermal conductivity, keff , is assumed to be isotropic and is calculated from the volumetric perfusion rate, wb , of the tissue keff = kt (1 + αwb ) (2) with kt the thermal conductivity of tissue and α a constant depending on vessel sizes and density. From experiments with isolated perfused bovine kidney at different blood flow rates Crezee and Lagendijk (1990, 1992) found α = 0.72 m3 s kg−1 . The same value has been used for the calculations in this study. This will slightly overestimate the real keff as part of keff is already accounted for by the discrete vasculature. The volumetric perfusion rate was set at 10 and 2.5 ml/100 g/min in the upper respectively lower half of the tissue volume in accordance with the effective perfusion rate of the discrete vessel tree. The emphasis in this study is put on the modelling of tissue volumes with discrete vasculature. In addition tissues with only a heat sink either with or without an enhanced thermal conductivity are modelled for comparison of the different thermal models. In these cases the blood perfusion rate was taken 20 ml/100 g/min in the upper half of the volume and 5 ml/100 g/min in the bottom half. The ratio of 4:1 is in accordance with the division of the terminating branches in the inhomogeneous discrete vessel network. The enhanced effective thermal conductivity has been calculated using equation (2). These additional simulations are limited to the most promising PdNi implant, i.e. the one which realizes the most uniform temperature distribution in the simulations with the discrete vessel network. 2.5. The target volume and evaluation of the results For the evaluation of the resulting temperature distributions one has to select a volume in the modelled tissue which is clinically relevant. Of the many plausible alternatives the convex 130 N van Wieringen et al hull of the implant is chosen as the tissue volume of interest or target volume in this study. A cumulative temperature–volume histogram (CTVH) has been constructed for the volume of interest for all simulations. To facilitate the comparison of different temperature distributions the dimensionless index HC (heterogeneity coefficient) is introduced as a relative measure for the uniformity of the temperature distribution. In correspondence with Van der Koijk et al (1997) HC is defined as: T10 − T90 (3) HC = T90 − Tcore with Tcore equal to the initial temperature of the tissue volume, i.e. 37 ◦ C. The parameters Tn quantify the temperatures which are exceeded in n per cent of the volume of interest. Additionally, T50 , T90 and T10 − T90 are determined. Figure 3. The 42 ◦ C (left) and 46 ◦ C (right) isotemperature surface plots in a tissue volume with discrete vasculature implanted with an array of tubular PdNi seeds. 3. Results The impact of large vessels on the uniformity of the temperature distribution which is realized during interstitial hyperthermia using ferromagnetic seeds can be seen in the figures 3 and 4. Figure 3 shows the 42 ◦ C and 46 ◦ C isotemperature surfaces in a tissue volume which is heated by an array of tubular PdNi seeds. The arteries and veins are shaded dark and light respectively. Figure 4 shows the isotemperature contours in two planes which are perpendicular to the seeds. Obviously, the highest tissue temperatures in the volume are found close to the ferromagnetic seeds. Relatively low temperatures are found further away from the seeds and in tissue regions that are close to large vessels. The isotemperature contour plot of the plane which is located 10 mm below the top of the seeds (left plot in figure 4) clearly shows a number of cold spots which are caused by the large vessels that cross this plane. No major blood vessels cross the plane located 35 mm below the top of the seeds. The cumulative temperature–volume histograms of the simulations of implants with uncoated thermoseeds are given in figure 5. Within an implant all seeds are identical. For each simulation a different type of seed has been used (see table 1 and section 2.2). The impact of the applied implantation technique on the temperature distribution in the volume Dose uniformity of ferromagnetic seed implants 131 Figure 4. Isotemperature contour plots of a tissue volume with discrete vasculature implanted with an array of tubular PdNi seeds. The two planes are perpendicular to the seeds and are located 10 (left) and 35 mm (right) below the top of the seeds. The contour levels are 2 ◦ C apart. The edges of the contour plots are at 37 ◦ C. Figure 5. Cumulative temperature–volume histograms of the volume of interest in an implant of thermoseeds. The tissue perfusion is modelled by a discrete vessel network. Different types of uncoated thermoseeds have been modelled as indicated in the legend (see section 2.2 and table 1 for specifications). of interest can be seen in the CTVHs in figure 6 (the CTVH of a bare multifilament PdNi seed is added for comparison). The statistical parameters which have been deduced from the histograms, e.g. T50 , are presented in table 1. The tails of the CTVHs at higher temperatures are a result of the strictly conductive nature of the applied hyperthermia technique. In a small volume close to the seeds the temperatures will be relatively high. Tissue temperatures further away from the seeds are much lower as the temperature gradients close to the seeds are relatively steep. 132 N van Wieringen et al Figure 6. Cumulative temperature–volume histograms of the volume of interest in an implant of ferromagnetic seeds. The tissue perfusion is modelled by a discrete vessel network. Different implantation techniques have been modelled as indicated in the legend (see section 2.2 and table 1 for specifications). Except for the implant with tubular seeds, multifilament PdNi seeds have been modelled inside the catheter/needle. As the lumen of the PDR needle is larger than the lumen of the other catheters and needles it allows for the insertion of a larger number of filaments. A comparison of the results of simulations with various seeds shows that the past efforts to improve the properties and implantation techniques for thermoseeds have indeed resulted in improved temperature distributions. For constant power seeds the risk of hot spots is substantial (table 1: constant power seed 3: T10 = 56.3 ◦ C). Temperature regulation for this type of seed can only be achieved by changing the intensity of the magnetic field. However, this will change the heat production of all seeds in an array equally and will not improve the temperature uniformity. The application of self-regulating seeds instead of constant power seeds reduces the risk of hot spots and improves the temperature uniformity. An increase in the heat production of PdNi seeds, realized by the application of multifilament seeds, results only in a limited improvement of the temperature distribution. However, it should be noted that using multifilament seeds instead of solid seeds only increases the heat production for temperatures where the asymptote for the heat production has not yet been reached. The Curie gradient of both seeds is therefore equal. Apparently, the heat production of the solid seed is high enough to reach a temperature in the range of the ferromagnetic transition in the tissue volume modelled in this study. An increase in the amount of PdNi per unit length of the seed improves both its heat production and its Curie gradient. These improvements result in a better homogeneity of the temperature distribution, as is demonstrated by simulations with tubular seeds and multifilament seeds in a PDR brachytherapy needle. An additional advantage of the application of seeds with a larger outer diameter is the fact that the radial temperature gradient just outside the seed will become more gradual. For conductive heating systems the radial temperature profile around a source is to first order proportional to the natural logarithm of the radial distance from the centre of the source (this approximation is based on the temperature profile around a source implanted in a cylindrically symmetric tissue volume without a heat sink). The radial temperature gradient in the tissue will be proportional to the reciprocal of the radial distance and consequently with increasing seed radius the gradient in the adjacent tissue Dose uniformity of ferromagnetic seed implants 133 will decrease. The influence of the seed diameter on the uniformity of the temperature distribution is also demonstrated by the results of simulations of implants with ideal seeds with an outer diameter of 1.0 and 2.0 mm respectively. The application of seeds composed of segments with the appropriate Curie temperature (see table 1, not presented graphically) leads to a significant improvement of the temperature uniformity. The most obvious difference between the CTVHs of the various implantation methods is the average temperature in the volume of interest (figure 6). In practice this difference can be compensated for by the application of seeds with an adjusted Curie temperature. However, one cannot compensate for the decrease in the temperature homogeneity if plastic afterloading catheters are applied. An air layer between the thermoseed and catheter further impairs the self-regulating temperature control of the seed. By the application of metallic needles the radial temperature gradient just outside the seeds will become more gradual as was also the case for seeds with a larger outer diameter (e.g. tubular seeds). Table 2. Various types of tissue perfusion used for comparison of different thermal models and the resulting temperature profile. Simulations have been performed for a hexagonal array of 12 tubular PdNi seeds. Parameters for evaluation of the resulting temperature distributions are given in the last four columns of this table. The parameter Tn quantifies the temperature which is exceeded in n per cent of the volume of interest. HC is the heterogeneity coefficient (equation (4)). Tissue perfusion Discrete vessels Heat sink (ml/100 g/min) keff (W m−1 K−1 ) T50 (◦ C) T90 (◦ C) T10 –T90 (◦ C) HC (◦ C) Yes Yes No No 0 0 20/5 20/5 0.6 1.32/0.78 0.6 2.04/0.96 46.48 46.30 45.58 45.67 41.42 41.63 42.78 43.27 7.64 7.12 4.88 4.17 1.73 1.54 0.84 0.67 Some additional simulations have been performed in which the blood flow is modelled by some alternative perfusion models. In all cases the tissue volumes are heated by an array of tubular PdNi seeds (the most promising PdNi seed based on the heterogeneity coefficient HC). A calculation of the temperature profile in tissue with both a discrete vessel network and an enhanced effective thermal conductivity showed that the addition of the latter term improves the temperature uniformity (table 2). The average temperature in the volume of interest has decreased a little due to the higher removal of heat. The CTVHs of this simulation and the simulation with only a discrete vessel tree are shown in figure 7. In this figure we also added the results of simulations in which the tissue perfusion is modelled by a more traditional continuum model only. For identical implants the temperature distribution is more homogeneous if a heat sink is applied to describe the tissue perfusion instead of a discrete vessel tree. The addition of an enhanced thermal conductivity to the heat sink term further increases the uniformity of the temperature distribution. The statistical parameters which are derived from the histograms are presented in table 2. 4. Discussion In the past decade the heat production and self-regulating properties of thermoseeds have been improved, aiming at a more uniform temperature distribution in the target volume. According to the simulations presented in this study, the change from constant power 134 N van Wieringen et al Figure 7. Cumulative temperature–volume histograms of the volume of interest in an implant of tubular PdNi seeds. Different types of blood flow models have been applied: tree = discrete vessel network, tree + keff = discrete vessel network + an enhanced effective thermal conductivity, hs = heat sink and hs + keff = heat sink + an enhanced effective thermal conductivity. In all cases the blood perfusion rate in the modelled volume is taken inhomogeneous (the perfusion rate in one-half of the volume 4× the rate in the other half). seeds to self-regulating seeds has had the biggest impact of all the adaptations made. The improvement of the Curie gradient towards an ideal step-like transition would further increase the temperature control of thermoseeds significantly. However, the Curie gradient has been optimized in earlier studies (Chen et al 1988, Meijer et al 1995, Van Wieringen et al 1996). It was also found to be limited by the asymptote for the heat production which is a result of magnetic saturation (Van Wieringen et al 1996). Instead of further optimizing the temperature regulating properties of the ferromagnetic seeds, it seems to be more efficient to apply the currently available seeds (in optimal configuration) and to optimize the Curie temperature of the seeds and the geometry of the array (a higher density of seeds close to large vessels). The improvement in the quality of temperature control of thermoseeds has resulted in a smaller variation of the seed temperatures in the array. For instance, for constant power seeds with a heat production of 113 mW cm−1 the seed temperatures vary as much as 11.4 ◦ C. For an array of bare multifilament PdNi seeds this difference has been reduced to 4.5 ◦ C and of course for ideal seeds it is 0 ◦ C. However, the improvement of the uniformity of the seed temperatures does not result in an equivalent improvement of the uniformity of the tissue temperatures. This is a consequence of the indirect response of thermoseeds to cold spots in the tissue. The steering ability at a distance of hot sources in general is limited, and consequently a rather dense implant array will be required. However, the heterogeneity of the temperature distribution for the simulations presented in section 3 may be exaggerated as vessels with a smaller diameter than 0.47 mm have not been modelled. The addition of smaller vessels might improve this response as it enhances the heat transfer in tissue. Several investigators (Charny et al 1990, Crezee et al 1994, Zhu et al 1995) indicated that vessels with a diameter smaller than 500 µm should be modelled by an enhanced thermal conductivity. Such a term reduces the equilibrium length of large discrete vessels and it Dose uniformity of ferromagnetic seed implants 135 will increase the vessel wall temperature (Crezee and Lagendijk 1990). Calculations of the temperature profile in tissue with both discrete vessels and an enhanced thermal conductivity showed that the addition of the latter term indeed improves the uniformity of the temperature distribution. The average temperature in the volume of interest decreased due to the higher removal of heat. Comparing the temperature distributions realized in tissue with discrete vasculature to those realized in tissue with blood flow modelled using continuum description it appears that the large vessels significantly impair the temperature uniformity. The large vessels cool the adjacent tissue and cause cold tracks through the tissue volume. It is therefore important that in treatment planning systems large vessels are accounted for individually as has been suggested by several authors (Crezee et al 1991, Moros et al 1993, Lagendijk et al 1994, Rawnsley et al 1994, Kolios et al 1995, Van der Koijk et al 1997). The vascular tree which has been used in this study has not been generated with the ambition of creating a completely realistic vessel network. However, the network which has been described in section 2.4 is useful for the investigation of trends and for comparison of the quality of different types of thermoseeds. The dimensionless heterogeneity coefficient HC has been introduced by Van der Koijk et al (1997) for a comparison of the temperature distributions realized with various interstitial hyperthermia techniques. They showed that systems which provide longitudinal temperature control (hot water tubes (Kapp et al 1993), multi-electrode current source applicators (Van der Koijk et al 1997) and ferromagnetic seeds (Van Wieringen et al 1997b)) are favourable compared with systems with only two-dimensional power steering (microwave antennas (Seegenschmiedt 1993), RF systems (Prionas et al 1994, Ryan et al 1994)). However, even with this relative index HC, a direct comparison of the various techniques is hard to make as the geometry and perfusion of the simulated and the treated tissue volumes are not identical. Furthermore, the CTVH and HC calculated from clinical measurements are constructed from temperature data of only a limited number of locations in the target volume. For numerical simulations the temperature in every volume element is known. The CTVHs from simulations will take into account all extremes in the temperature distribution resulting in a less favourable HC. 5. Conclusions The initial results from simulations with a new three- dimensional treatment planning system for interstitial hyperthermia with ferromagnetic seeds have been presented in this study. Calculations using a computer generated artificial vessel network have shown that large isolated vessels significantly impair the temperature distribution. For hot source techniques in particular, which respond to local cold spots indirectly (by thermal conduction), the use of a continuum model (with homogeneous tissue perfusion) will not be adequate. However, due to limited computer power it was not possible to model a fully developed discrete vessel network. The vessels which are too small and numerous to describe individually have to be modelled by an enhanced thermal conductivity and/or a heat sink term. The application of such hybrid models seems promising (Charny et al 1990, Crezee et al 1994, Van Leeuwen et al 1994, Zhu et al 1995), but further investigations are needed to find the most appropriate continuum model and the range of vessels which have to be modelled discretely. Different types of ferromagnetic seeds have been modelled to investigate whether past efforts to optimize the self-regulating properties and implantation technique of the seeds will improve the homogeneity of the temperature distribution in the target volume. In 136 N van Wieringen et al general, these computations confirmed the results of earlier studies and can be summarized as follows: (i) the use of self-regulating thermoseeds instead of constant power seeds improves the homogeneity of the temperature distribution significantly; (ii) coatings with a thermal conductivity below/above the intrinsic conductivity of tissue decrease/increase the average temperature in the tissue volume as well as the temperature uniformity; (iii) an increase in the heat production or Curie gradient results in an improvement of the homogeneity of the temperature distribution and average tissue temperature. The multifilament seeds which are implanted in PDR brachytherapy needles and the tubular seeds appear to be the most favourable configurations. For interstitial hyperthermia using ferromagnetic seeds the choice of the Curie temperature, seed spacing and implant geometry will have to be based on patient-specific data concerning the perfusion level and the presence of large isolated vessels in the volume of interest. With the proper thermal data, the model used in this study will prove to be a useful tool in making these choices. For instance, a simulation of seeds composed of segments with different Curie temperatures showed that a proper choice of Curie temperature will lead to a more uniform distribution without increasing the average temperature in the volume of interest. To fully exploit this feature, the planning system should be expanded with a routine which optimizes the Curie temperatures of the segments or the entire seeds. References Brezovich I A, Atkinson W J and Chakraborty D P 1984 Temperature distributions in tumor models heated by self-regulating nickel-copper alloy thermoseeds Med. Phys. 11 145–52 Brezovich I A and Liu I S 1987 Effect of catheters on the performance of self-regulating thermoseeds Proc. Ninth Ann. Conf. of the Engineering in Medicine and Biology Society (New York: IEEE) pp 1629–30 Burton C, Hill M and Walker A E 1971 The RF thermoseed—a thermally self-regulating implant for the production of brain lesions IEEE Trans. Biomed. Eng. 18 104–9 Cetas T C, Gross E J and Contractor Y 1997 A ferrite core/metallic sheath thermoseed for interstitial thermal therapies IEEE Trans. Biomed. Eng. accepted Charny C K, Weinbaum S and Levin R L 1990 An evaluation of the Weinbaum–Jiji bioheat equation for normal and hyperthermic conditions ASME J. Biomech. Eng. 112 80–7 Chen J S, Poirier D R, Damento M A, Demer L J, Biancaniello F and Cetas T C 1988 Development of Ni-4 wt.% Si thermoseeds for hyperthermia cancer treatment J. Biomed. Mater. Res. 22 303–19 Chen Z P, Roemer R B and Cetas T C 1991 Errors in the two-dimensional simulation of ferromagnetic implant hyperthermia Int. J. Hyperth. 7 735–9 ——1992 Three-dimensional simulations of ferromagnetic implant hyperthermia Med. Phys. 19 989–97 Chin R B and Stauffer P R 1991 Treatment planning for ferromagnetic seed heating Int. J. Radiat. Oncol. Biol. Phys. 21 431–39 Crezee J and Lagendijk J J W 1990 Experimental verification of bioheat transfer theories: measurement of temperature around large artificial vessels in perfused tissue Phys. Med. Biol. 35 905–23 ——1992 Temperature uniformity during hyperthermia: the impact of large vessels Phys. Med. Biol. 37 1321–37 Crezee J, Mooibroek J, Bos C K and Lagendijk J J W 1991 Interstitial heating: experiments in artificially perfused bovine tongues Phys. Med. Biol. 36 823–33 Crezee J, Mooibroek J, Lagendijk J J W and Van Leeuwen G M J 1994 The theoretical and experimental evaluation of the heat balance in perfused tissue Phys. Med. Biol. 39 813–32 Cox R S and Kapp D S 1992 Correlation of thermal parameters with outcome in combined radiation therapyhyperthermia trials Int. J. Hyperth. 8 719–32 Dewhirst M W, Sim D A, Sapareto S and Connor W G 1984, Importance of minimum tumor temperature in determining early and long-term responses of spontaneous canine and feline tumors to heat and radiation Cancer Res. 44 43–50 Ferguson S D, Paulus J A, Tucker R D, Loening S A and Park J B 1993 Effect of thermal treatment on heating characteristics of Ni-Cu alloy for hyperthermia: preliminary studies J. Appl. Biomater. 4 55–60 Gannet D E, Stea B, Shimm D S, Sneed P K, Fu K K, Steeves R and Vora N L 1995 Interstitial thermoradiotherapy for locally advanced and recurrent neoplasms of the head and neck Endocurie/Hyperth. Oncol. 11 143–53 Dose uniformity of ferromagnetic seed implants 137 Gonzalez Gonzalez D, Van Dijk J D P and Blank L E C M 1995 Radiotherapy and hyperthermia Eur. J. Cancer 31A 1351–5 Haider S A, Cetas T C and Roemer R B 1993 Temperature distribution in tissues from a regular array of hot source implants: an analytical approximation IEEE Trans. Biomed. Eng. 40 408–17 Haider S A, Chen J S, Wait J R and Cetas T C 1991 Power absorption in ferromagnetic implants from radio frequency magnetic fields and the problem of optimization IEEE Trans. Microwave Theory Tech. 39 1817–27 Huang H W, Chen Z P and Roemer R B 1996 A counter current vascular network model of heat transfer in tissues ASME J. Biomech. Eng. 118 120–9 Indik R A and Indik J H 1994 A new computer method to quickly and accurately compute steady-state temperatures from ferromagnetic seed heating Med. Phys. 21 1135–44 Indik R A, Indik J H and Cetas T C 1994 Fast and efficient computer modeling of ferromagnetic seed arrays of arbitrary orientation for hyperthermia treatment planning Int. J. Radiat. Oncol. Biol. Phys. 30 653–62 Kapp K S, Kapp D S, Stuecklschweiger G, Berger A and Geyer E 1993 Interstitial hyperthermia and high dose rate brachytherapy in the treatment of anal cancer: a phase I/II study Int. J. Radiat. Oncol. Biol. Phys. 28 189–99 Kays W M and Crawford M E 1980 Convective Heat and Mass Transfer (New York: McGraw-Hill) pp 88–132 Kobayashi T, Kida Y, Tanaka T, Kageyama N and Kobayashi H 1986 Magnetic induction hyperthermia for brain tumors using ferromagnetic implants with low Curie temperature J. Neuro-Oncol. 4 175–81 Kolios M C, Sherar M D and Hunt J W 1995 Large blood vessel cooling in heated tissues: a numerical study Phys. Med. Biol. 40 477–94 Kotte A N T J, Van Leeuwen G M J, de Bree J, Van der Koijk J, Crezee J and Lagendijk J J W 1996 A description of discrete vessel segments in thermal modelling of tissues Phys. Med. Biol. 41 865–84 Kotte A N T J, Van Wieringen N and Lagendijk J J W 1998 Modelling tissue heating with ferromagnetic seeds Phys. Med. Biol. 43 105–20 Lagendijk J J W, Crezee J and Hand J W 1994 Dose uniformity in scanned focused ultrasound hyperthermia Int. J. Hyperth. 10 775–84 Mack C F, Stea B, Kittelson J M, Shimm D S, Sneed P K, Phillips T L, Swift P S, Luk K, Stauffer P R, Chan K W, Steeves R, Cassady J R and Cetas T C 1993 Interstitial thermoradiotherapy with ferromagnetic implants for locally advanced and recurrent neoplasms Int. J. Radiat. Oncol. Biol. Phys. 27 109–15 Matloubieh A Y, Roemer R B and Cetas T C 1984 Numerical simulation of magnetic induction heating of tumors with ferromagnetic seed implants IEEE Trans. Biomed. Eng. 31 227–34 Meijer J G, Van Wieringen N, Koedooder C, Nieuwenhuys G J and Van Dijk J D P 1995 The development of PdNi thermoseeds for interstitial hyperthermia Med. Phys. 22 101–4 Mooibroek J and Lagendijk J J W 1991 A fast and simple algorithm for the calculation of convective heat transfer by large vessels in three-dimensional inhomogeneous tissues IEEE Trans. Biomed. Eng. 38 490–501 Moros E G, Dutton A W, Roemer R B, Burton M and Hynynen K 1993 Experimental evaluation of two simple thermal models using hyperthermia in muscle in vivo Int. J. Hyperth. 9 581–98 Overgaard J, Gonzalez Gonzalez D, Hulshof M C C M, Arcangeli G, Dahl O, Mella O and Bentzen S M 1995 Randomized trial of hyperthermia as adjuvant to radiotherapy for recurrent or metastatic malignant melanoma Lancet 345 540–3 ——1996 Hyperthermia as an adjuvant to radiation therapy of recurrent metastatic malignant melanoma. A multicentre randomized trial by the European Society for Hyperthermic oncology Int. J. Hyperth. 12 3–20 Paulus J A, Richardson J S, Tucker R D and Park J B 1996 Evaluation of inductively heated ferromagnetic alloy implants for therapeutic interstitial hyperthermia IEEE Trans. Biomed. Eng. 43 406–13 Pennes H H 1948 Analysis of tissue and arterial temperatures in the resting forearm J. Appl. Phys. 1 93–122 Prionas S D, Kapp D S, Goffinet D R, Ben-Yosef R, Fessenden P and Bagshaw M A 1994 Thermometry of interstitial hyperthermia given as an adjuvant to brachytherapy for the treatment of carcinoma of the prostate Int. J. Radiat. Oncol. Biol. Phys. 28 151–62 Rawnsley R J, Roemer R B and Dutton A W 1994 The simulation of discrete vessel effects in experimental hyperthermia ASME J. Biomech. Eng. 116 256–62 Ryan T P, Trembly B S, Roberts D W, Strohbehn J W, Coughlin C T and Hoopes P J 1994 Brain hyperthermia: I. Interstitial microwave antenna array techniques—the Darthmouth experience Int. J. Radiat. Oncol. Biol. Phys. 29 1065–78 Seegenschmiedt M H 1993 Clinical experience of interstitial thermoradiotherapy using microwave techniques Interstitial and Intracavitary Thermoradiotherapy ed M H Seegenschmiedt and R Sauer (Berlin: Springer) pp 49–53 Sneed P K, Gutin P H, Stauffer P R, Phillips T L, Prados M D, Weaver K A, Suen S, Lamb S A, Ham B, Ahn D K, Lamborn K, Larson D A and Wara W M 1992 Thermoradiotherapy of recurrent malignant brain tumors Int. J. Radiat. Oncol. Biol. Phys. 23 853–61 138 N van Wieringen et al Stea B, Kittelson J, Cassady J R, Hamilton A, Guthkelch N, Lulu B, Obbens E, Rossman K, Shapiro W, Shetter A and Cetas T 1992 Treatment of malignant gliomas with interstitial irradiation and hyperthermia Int. J. Radiat. Oncol. Biol. Phys. 24 657–67 Tompkins D T, Vanderby R, Klein S A, Beckman W A, Steeves R A and Paliwal B R 1994 Effect of interseed spacing, tissue perfusion, thermoseed temperatures and catheters in ferromagnetic hyperthermia: results from simulations using finite element models of thermoseeds and catheters IEEE Trans. Biomed. Eng. 41 975–85 Van der Koijk J F, Lagendijk J J W, Crezee J, de Bree J, Kotte A N T J, Van Leeuwen G M J and Batterman J J 1997 The influence of vasculature on temperature distributions in MECS interstitial hyperthermia: importance of longitudinal control Int. J. Hyperth. 13 365–85 Van Leeuwen G M J, Crezee J, Mooibroek J and Lagendijk J J W 1994 Alternative representation of small vessels in a discrete vessel thermal model Advances in Heat and Mass Transfer in Biological Systems (HTD Vol 288) (New York: ASME) pp 103–6 Van Leeuwen G M J, Kotte A N T J, de Bree J, Van der Koijk J F, Crezee J and Lagendijk J J W 1997b Accuracy of geometrical modelling of heat transfer from tissue to blood vessels Phys. Med. Biol. 42 1451–60 Van Leeuwen G M J, Kotte A N T J, Crezee J and Lagendijk J J W 1997c Tests of the geometrical description of blood vessels in a thermal model using counter current geometries Phys. Med. Biol. 42 1515–32 Van Leeuwen G M J, Van der Koijk J F, Kotte A N T J and de Bree J 1997a A flexible algorithm for construction of vessel networks for use in thermal modelling. Submitted to IEEE Trans. Biomed. Eng. accepted Van Wieringen N, Van Dijk J D P, Nieuwenhuys G J, Snel C E and Cetas T C 1996 Power absorption and temperature control of multi-filament palladium-nickel thermoseeds for interstitial hyperthermia Phys. Med. Biol. 41 2367–80 Van Wieringen N, Van Dijk J D P, Van Veldhuizen J and Nieuwenhuys G J 1997a The effect of catheters and coatings on the performance of palladium-nickel thermoseeds: design and evaluation of implantation techniques Int. J. Hyperth. 13 187–204 ——1997b Three-dimensional temperature control of palladium–nickel thermoseeds: a computer aided and experimental evaluation Int. J. Hyperth. 13 269–86 Zhu L, Lemons D E and Weinbaum S 1995 A new approach for predicting the enhancement in the effective conductivity of perfused muscle tissue due to hyperthermia Ann. Biomed. Eng. 23 1–12