Cluster Models in Cellular Level Electron Dose Calculations

ORIGINAL ARTICLE

Cluster Models in Cellular Level Electron Dose Calculations Juha S. Lampinen, Petteri J. Va¨lima¨ki, Antti A. Kuronen, Jiri Stepanek and Sauli E. Savolainen From the Department of Physics (J.S. Lampinen, P.J. Va¨lima¨ki), University of Helsinki, Department of Radiology (J.S. Lampinen), Helsinki University Central Hospital Helsinki, Finland, Laboratory of Computational Engineering, (A.A. Kuronen), Helsinki University of Technology, Espoo, the Accelerator Laboratory, (A.A. Kuronen), University of Helsinki, Finland, Institute for Medical Radiobiology (MRI) of the University of Zurich and the Paul Scherrer Institute (J. Stepanek), Villigen, Switzerland, and the Departments of Radiology and Laboratory Medicine, (S.E. Savolainen), Helsinki University Central Hospital, Helsinki, Finland Correspondence to: Dr Juha S. Lampinen, Department of Physics, P.O. Box 9, FIN-00014 University of Helsinki, Finland. Fax: + 358 9 191 8378. E-mail: [email protected]

Acta Oncologica Vol. 38, No. 3, pp. 367–372, 1999 A program for calculating absorbed dose was developed for radioimmunotherapy (RIT) purposes. It was used to determine the difference in the therapeutic effect of 111In electrons when using a close-packed cubic geometry and a cell cluster model developed in this project. Our cluster model piles the cells individually. The cells were modelled as spheres of diameters of 12 (tumour) and 30 (healthy) mm. Both models were used to generate clusters with spherical tumours inside healthy tissue. The program uses Monte Carlo-based dose kernels. The radiation spectra were calculated from the Auger and x-ray transition strengths and fluorescence yields of 111In. The results show the importance of the cluster model in cellular level dose calculations. Near the tumour/healthy tissue interface in particular, the doses differ because of geometrical differences. In the case of a small cluster with tumour and total diameters of 30 and 150 mm, the ratio of the therapeutic effects is 20. Recei6ed 6 April 1998 Accepted 6 October 1998

To optimize the efficacy of radioimmunotherapy (RIT), the ideal antibody-radioisotope combinations should be used to deliver the highest tumour and the lowest normal tissue doses (1) In selecting the right isotope to the RIT treatment many different factors have to be taken into consideration (2). These include the physical, chemical, and biological properties of the isotope and its carrier regarding the properties of the tissue. The selection of the isotope is part of the treatment planning, which in the case of systemic radiation therapy is still unimplemented (3). Strand et al. (3) have in their review concluded that for future development and evaluation of systemic radiation therapy, dosimetry should be as accurate as possible. Owing to the high localization of the monoclonal antibodies (MoAbs) in RIT, the decay radiation spectra used in dosimetric calculations have to be as complete as possible, including also the low-energy Auger electrons. The range of these Auger electrons is small, and therefore when calculating cross-doses from one cell to another, the geometry of the cell cluster model could have an effect on the results. © Scandinavian University Press 1999. ISSN 0284-186X

Goddu et al. (4) have presented a method to calculate the cellular self-absorbed dose. They also studied the selfdose versus cross-dose dependence in cell clusters (5). They used experimental range-energy relationships (6) in calculating the absorbed fractions, fi (rk rh ), which is the fraction of energy emitted from the source region rh that is absorbed in the target region rk for each radiation component (i ) (7). The cell cluster model used by Goddu et al. (4) consisted of spherical cells of one diameter, 10 mm, with a concentric spherical nucleus 8 mm in diameter (5). The cells were arranged in close-packed cubic geometry (8) such that 74% of the cluster volume is occupied by the cells and 26% by interstitial spaces. The cluster size ranged from 26 mm to 400 mm. Rassow et al. (9) have presented cell models for normal and tumour cells. In their model, the diameters of the spherical cells are 30 mm and 12 mm for normal and tumour cells, respectively. The relative amounts of intracellular, vascular and intercellular spaces are 73%, 11% and 16% for normal cells and 50%, 5% and 45% for tumour cells, respectively (9). Acta Oncologica

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The Auger electron spectra have been previously calculated by Monte Carlo (MC) (10) or by combined deterministic and MC methods (11, 12). The decay energy spectra are used to calculate point dose kernels, which describe the absorbed dose from a point source as a function of distance. The absorbed fractions, fi (rk rh ), can be calculated from the known dose distribution. Some point dose kernels for cellular lever dosimetry have been previously calculated by Kuronen et al. (13, 14). These kernels have been used to calculate therapeutic effects in some simple cases (15). In this work, two different cell cluster models constructed of normal and tumour cells were used to study the effect of the cell cluster model on electron dose calculations. The clusters used were based on the close-packed cubic geometry model and on a new cluster model where the cells are individually packed as close to each other as possible. The electron decay spectra and point dose kernels for 111In were generated using deterministic and Monte Carlo methods. The point dose kernels were used to calculate absorbed electron doses to the cells in each cell cluster. The absorbed dose distributions in different cluster models were then compared.

MATERIAL AND METHODS The decay radiation spectra of 111In were calculated using the program IMRDEC (11, 12), which calculates the radiation cascades deterministically or by employing the Monte Carlo method. The IMRDEC program takes into Fig. 2. Dose point kernels for 111In electrons as a function of distance, r, from the centre of the source cell. Kernels for two geometries are given; activity inside ( —) or on the surface (……) of a sphere of diameter 12 mm. A, B, and C correspond to the different parts of the decay spectrum shown in Fig. 1.

consideration all types of decay processes and the radiation resulting from them. Point dose kernel F(r) is defined as the expectation value of the fractional energy deposition per decay of a point source at a certain distance r from the source. In Monte Carlo simulations it is readily calculated as (16) F(r) =

Fig. 1. Decay radiation spectra of 111In electrons. Boxes A, B and C denote energy groups, for which the point dose kernels were produced. In the upper part of the figure a continuous slowing down approximation (CSDA) estimate of the electron mean range as a function of energy (19) is presented.

dE(r)/T0 dr

[1]

where dE(r) is the energy deposited into a spherical shell with radius r and thickness dr, T0 is the mean total electron (or photon) kinetic energy released in one decay. Point dose kernels were calculated by the Monte Carlo method using the EGS4 code system (17) and by applying the PRESTA algorithm (18). The cut-off energies used in the simulation were 1.0 keV. Particles with a lower kinetic energy than this value were assumed to deposit their kinetic energy on-site. Particles with energies higher than


the cut-off were divided into 3 categories, so three-point dose kernels were produced. The kernels were calculated for two different source geometries: activity inside or on the surface of a sphere of diameter 12 mm. All cell models were built from spheres of two diameters, 12 mm and 30 mm for tumour and normal cells, respectively (9) In the case of the tumour cell (12 mm in diameter), an additional distance of 0.5 mm from the surface was kept clear of other cells. Two different methods were used to pack the cells into clusters. The first method was based on a close-packed cubic geometry (8). A new method, hereafter referred to as CellPacker, was developed in order to pack the spheres representing cells in a less geometric way. For both methods, the cell cluster was assumed to be spherical. A spherical cluster of tumour cells was surrounded by a layer of normal cells. CellPacker sets the spheres representing cells into a three-dimensional coordinate system, as near the origin as possible. The first cell is laid in the origin and the others are piled around it in a systematic way. CellPacker uses spherical coordinates while seeking a location for the central point of the cell. First, it goes through all the capable values of the f-coordinate in one round with certain fixed values of the u- and r-coordinates. After that the value of the u-coordinate is changed and in this way the whole surface of the sphere with a certain radius is explored. Then the r-coordinate is increased and the process starts again. The calculation accuracy of the program is adjustable. The ratio of the cluster volume occupied by the cells packed by CellPacker was calculated by dividing the sum of the cell volumes by the volume calculated from the outer diameter of the whole cluster. This was done sepa-

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rately for clusters filled with only tumour or healthy cells. For the calculations of this work, cell clusters sized as 600 mm in outer diameter were used. In the middle of each cluster there was a tumour cluster of variable size, ranging from 30 mm to 300 mm. A computer program, hereafter referred to as CellDose, was developed to calculate absorbed doses in a cell cluster. Dose calculations are based on point dose kernels described earlier. CellDose either creates a close-packed cubic geometry cell cluster, the tumour size specified by the user, or it loads previously made coordinates of the cells (made by CellPacker). The absorbed dose D( (rk rh ) to a target cell rk from activity in source cell rh is given by the MIRD schema (7): D( (rk rh ) =A0 h % Difi (rk rh )mk

[2]

i

where A0 h is the cumulative activity of the source cell, Di is the mean energy emitted per nuclear transition, fi (rk rh ) is the fraction of the energy emitted from the source cell that is absorbed in the target cell for the ith radiation component and mk is the mass of the target cell. fi (rk rh ) values for those source-target combinations needed are calculated from the point dose kernels. For a close-packed cubic geometry model, CellDose calculates the radial distribution of the cells in the cluster, i.e. the number of cells, Ni, at each discrete distance, Ri, from the origin. Equation (2) is used to calculate absorbed doses from every source cell rh to one target cell rk at each discrete distance, Ri, where there is at least one cell. The mean absorbed dose inside a cell cluster, D( (R), as a function of cluster radius R, is calculated as %i Ni Di D( (R) =

[3] %i Ni

Fig. 3. Cell clusters based on a close-packed cubic geometry (left) and on the output of CellPacker (right). The radius of the inner (tumour) cell cluster is 30 mm, the outer radius is 75 mm. Both clusters have been sliced in half for a better view of the geometry.

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RESULTS The electron decay spectra of 111In are shown in Fig. 1. Point dose kernels for electrons from different energy ranges of 111In are presented in Fig. 2. Illustrations of cell clusters, one based on a closedpacked cubic geometry and another made by CellPacker, are presented in Fig. 3. In a cell cluster made by CellPacker, the relative amount of volume occupied by the cells was 48% and 64% for clusters comprising only tumour or healthy cells, respectively. The cumulative number of cells in both cluster models as a function of cluster radius, for clusters with different tumour sizes, is presented in Fig. 4. The therapeutic effects, i.e. the ratio of the mean absorbed dose in tumour cells (normalized to unity) to the dose of the healthy cells, of 111In radionuclide as a func-

Fig. 4. The number of cells inside a cluster of radius, r, for close-packed cubic geometry clusters and for clusters made with CellPacker. Parts (I), (II) and (III) present cases for different tumour sizes, 150 mm, 75 mm and 15 mm, respectively.

where Di is the total absorbed dose to one cell at distance i. For a cell cluster made by CellPacker, the absorbed dose (Equation (2)) is calculated from every source cell to every target cell. Energies of the low-energy electrons (EB1.0 keV) are deposited at site, full energy when the activity is inside the sphere or half energy when the activity is on the surface of the sphere. The therapeutic effect, T/N, i.e. the ratio of the mean absorbed dose in tumour cells to the dose of the healthy cells, in a cell cluster inside a radius R is calculated as T D( tumour (R)= N D( (R)

[4]

where D( tumour is the mean absorbed dose in the tumour cells, and D( (R) is the mean absorbed dose in the healthy cell inside radius R. In the comparisons of this work, D( tumour is normalized to unity.

Fig. 5. The therapeutic effect of 111In as a function of cluster radius, r, for different cell cluster models. Parts I, II and III present therapeutic effects caused by electrons for tumour sizes of 150 mm, 75 mm and 15 mm of radius. Activity was distributed inside the tumour cells. The graphs given present the ratios of mean absorbed dose in tumour cells (normalized to unity) to the mean absorbed dose in healthy cells inside radius, r.


Fig. 6. The therapeutic effect of 111In as a function of cluster radius, r, for two activity distributions, activity inside or on the surface of the tumour cells. The cluster model used is the same as in part III of Fig. 5. The graphs represent the ratios of mean absorbed dose in tumour cells (normalized to unity) to the mean absorbed dose in healthy cells inside radius, r.

tion of distance from the tumour/healthy tissue interface for different tumour sizes are plotted in Fig. 5. The effect of the activity distribution in the tumour cells (activity inside or on the surface) on the therapeutic effect is demonstrated in Fig. 6. DISCUSSION In this work we developed a cell cluster model and calculated absorbed doses from 111In electrons to cells in the new cluster model as well in a cluster based on a closepacked cubic geometry model. The dose calculations were based on the dose point kernels produced from the calculated electron decay spectrum of 111In. The calculation method used in this work is applicable to any isotope and to different cell models. Owing to the large variation in the electron decay spectra of 111In, the spectrum (see Fig. 1) was simulated in three parts. With this method, the dose point kernels (Fig. 2) provided more detailed information about the dose absorption, especially at very short distances from the source cell. The close-packed cubic geometry is the most efficient method to pack spheres with constant radii into a volume. The cell cluster illustrations (Fig. 3) reveal the limitation of the close-packed cubic geometry model when spheres with different radii are used; there is a wide gap between the outermost tumour cells and the innermost healthy cells. The gap is produced by the difficulty in matching two differently sized grids (the tumour and the healthy cell grid) at the interface. In the case of the cluster made by CellPacker, this gap is avoided since the cells are piled individually, i.e. when the outer radius of the tumour is reached the process continues immediately with healthy cells. The width of the gap in the close-packed cubic geometry cluster varies from zero to the diameter of a healthy cell,

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from 0 mm to 30 mm in the case of the cell models used in this study. As can be seen from Fig. 2, the maximum ranges of the electrons in groups A and B are about 7 mm and 16 mm, respectively. Therefore, in the tumour/healthy tissue interface, part of the radiation from these energy groups is lost, if the close-packed cubic geometry model is used. The efficacy of the two cluster models is compared in Fig. 4, where the cumulative number of cells as a function of cluster radius is shown. As the tumour size decreases, the effect caused by the gap between the tumour and the healthy tissue in the close-packed cubic geometry becomes more evident. The same phenomenon is present in the graphs showing the therapeutic effects (see Fig. 5). As the tumour size decreases, the doses to the healthy cells near the tumour/healthy tissue interface (absent in the closepacked cubic geometry) become dominant. For example, in a tumour with a radius of 15 mm, the close-packed cubic geometry models suggests that in a cluster with total radius of 75 mm, the mean absorbed dose in the tumour cells is about 4000 times higher than the mean dose in the healthy cells, while the value given by the CellPacker model is only 200. When the activity is distributed on the surfaces of the tumour cells, instead of being inside the cells, the therepautic effect (Fig. 6) changes as expected. When the activity is on the surface of the cells, the mean dose to healthy cells does not change, while the dose in the tumour cells decreases. Therefore, the therapeutic effect, if the activity is inside the cells, is about two times higher. CONCLUSIONS In this study we present a cell cluster model for cellular level dose calculations and compare the doses calculated in our model with those in a close-packed cubic geometry model. According to our results, the selection of the cluster model influences the calculated therapeutic effect. This effect is emphasized when small clusters are selected. ACKNOWLEDGEMENTS We gratefully acknowledge the funding of this study by the State Subsidy for University Hospitals, research grant of the Departments of Radiology and Neurology (Helsinki University Central Hospital) (JL) and the Finnish Society of Nuclear Medicine (JL).

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