Download Dose uniformity of ferromagnetic seed implants in tissue with

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
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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
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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
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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
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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)).
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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
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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.
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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
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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.
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