- Email: [email protected]
Mathematical modelling of the response of tumour cells to radiotherapy
Nuclear Instruments and Methods in Physics Research B 188 (2002) 210–215 www.elsevier.com/locate/nimb
Mathematical modelling of the response of tumour cells to radiotherapy N.F. Kirkby b
a,* ,
N.G. Burnet b, D.B.F. Faraday
a
a Department of Chemical and Process Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK Oncology Centre, Addenbrooke’s Hospital, University of Cambridge, Department of Oncology, Hills Road, Cambridge CB2 2QQ, UK
Abstract Using a model of cell cycle (CelCyMUS) in which the cell population is described in terms of the cell age and position within the cell cycle typical radiotherapy strategies are simulated, and the effects compared to clinical results, showing reasonable qualitative agreement. Alternative strategies are tested on a slow and a fast-growing tumour and the strategy most likely to completely destroy the tumour is found to depend on the growth rate. This work may have important applications for the development of novel radiotherapy fractionation, for example, schedules to take advantage of hypersensitivity to small doses of radiation exhibited by some tumours, which might improve results compared to conventional dosing strategies. 2002 Published by Elsevier Science B.V. Keywords: Cancer; Radiotherapy; Mathematical model; Cell cycle
1. Introduction After surgery, radiotherapy, is the most effective curative treatment for cancer [1]. The pattern of fractionation of radiation treatment has an important effect on both tumour cell kill and the damage in the surrounding healthy normal tissue. Mathematical models of cell cycles have proved to be extremely effective in predicting the behaviour of in vitro cell cultures [2–4] and are now used in the design and control of such reactors. More recently, cell cycle models have begun to be adapted to encompass in vivo systems such as tumours [5,6]. In both types of system each phase of the cell cycle is modelled separately, to achieve age
*
Corresponding author. E-mail address: [email protected] (N.F. Kirkby).
distributed population balances. This paper describes a mathematical model of the cell cycle of tumour cells in an exponentially growing population. Superimposed on this model is a description of the effect that radiation has on each phase of the cell cycle, based on the experimental work of Sinclair and Morton [7]. The objectives of this work are: • to devise a specific model of a typical mammalian tumour cell; • to incorporate models of the damage caused by the radiation to each phase of the cell cycle; • to test the model against measured cell survival curves on populations normally distributed around the cell cycle; • to optimise existing patterns of radiotherapy and predict the effects of novel patterns not yet used in normal clinical use.
0168-583X/02/$ - see front matter 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 1 0 9 6 - 5
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
It is possible that this approach will facilitate improved clinical efficiency in the treatment of cancer. Before going on to discuss the results generated by the model, it is pertinent to describe, in a little more detail, the cell cycle, the generic model and the specific model used in this study, together with the assumptions and experimental data used to develop the specific model used in this study.
2. Theory The cell cycle is divided into four main phases (Fig. 1). In M phase cell division occurs via a two step process; the division of the cell’s nucleus (mitosis) followed by the division of the cell into daughter cells, each containing its own nucleus (cytokinesis). The period between one M phase and the next is called the interphase, during which the cells grow. The interphase is divided into the three remaining phases of the cell cycle; G1 , S and G2 . The G1 ðG ¼ gapÞ phase is the interval between the completion of the M phase and the start of the S phase, when the cell replicates its DNA, prior to the next mitosis. The G2 phase is the interval between DNA synthesis (S phase) and cell division (M phase).
Fig. 1. Cell cycle.
211
The generic model Cell Cycle Model University of Surrey (CelCyMUS) developed by Kirkby and Faraday [2,3], describes the cell population in terms of the cell age and position within the cell cycle. The system dynamics of the cell population can therefore be determined at any time during the simulation. The population balance equations used to describe this behaviour are transformed into ordinary differential equations (ODE) by applying the method of characteristics; the resultant ODE are solved using Euler integration. This model has some features in common with that employed by Smith and Martin [8]; their approach, however, did not account for cell death or the interaction of the cell population with the growth medium. Specific models employing CelCyMUS have been highly successful in predicting the growth of mouse–mouse 321 (MM-321) hybridoma cell lines for use in antibiotic production [2,3,9]. The specific model used to describe avascular mammalian tumour growth (where diffusion of nutrients does not inhibit cell growth) uses as its starting point the growth characteristics of an MM-321 hybridoma cell. The cell cycle modelled is the same as that described above, except that the G1 phase is further sub-divided into two parts, G1a and G1b . The G1a phase has a fixed duration of 2.5 h, while the time spent in the G1b phase can vary from 0 to 10 h depending on prevailing conditions. Where cancerous cells clump together to form a large tumour, nutrients are supplied and toxins removed by a dedicated blood supply. Such a tumour is described as a vascular tumour. The tumour modelled here is a small avascular tumour, in which it is assumed that the rate of arrival of nutrients and removal of toxins is not diffusion limited and therefore the tumour can be approximated to a well mixed system. In order to model the effects of radiation, the cell cycle is duplicated in its entirety to form a cell cycle within a cell cycle (Fig. 2). The outer cycle is termed normal and represents the normal tumour cells. Upon irradiation, cells from the outer normal cycle can jump into the inner cycle termed the doomed cycle. The duration of the phases and the parameter values used in the doomed cycle are identical to those in the normal cycle with one
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N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
Fig. 2. Cell cycle for cells undergoing radiation treatment.
important exception. During the M phase, the cell normally divides into two daughter cells, but in the doomed cycle all cells die while trying to divide. Thus, the proliferation factor in the doomed cycle is set to 0. If every normal cell gave rise to two viable daughters the proliferation factor for the normal cycle would be set to 2. However, this may give rise to an unrealistically small tumour doubling time. One method to slow the growth of the tumour is to assume that not all mitotic cells give rise to two viable daughters. This method has the advantage of simplicity and maybe the main limitation on the growth of small tumours before all the complications of nutrient diffusion, the effects of angiogenesis and even space restrictions become significant. There is a dearth of published data relating to how radiation affects the survival fraction for the different phases within the cell cycle. One of the few papers, by Sinclair and Morton [7], studies Chinese hamster ovary (CHO) cells throughout all four stages. Other data, from Steel [10], compare the total cell survival fraction for different human tumour cell lines and their response to radiation dose. Combining the data of Sinclair and Morton, to give a total cell survival fraction and comparing this with the data from Steel [10] it is found that the data for the CHO cells lie within the data range for the human tumour cell lines and have a similar dispersion. Thus the data from Sinclair and
Morton [7] appear to offer a valid approximation to human tumour cell growth and as more data becomes available the model can be developed accordingly. The model can represent any desired pattern of fractionation of the radiation treatment. When a dose of radiation is delivered, a fraction of normal cells survive in the normal cycle. The remainder are mortally damaged and are assumed to be transferred to the doomed cycle. Hence, cells are not killed immediately by a dose of radiation but are assumed to die at their next attempt at division. The relationship between survival fraction and radiation dose received can be determined by one of several commonly used models and each phase can use its own model and parameters. Thus it is possible to allow radiation to have very different effects depending on which part of the cell cycle a cell is in when it receives the radiation. In this work, the cell survival models allowed are the simple multi-target and single event model and the linear quadratic model [10]. However, the choice of model is not restricted to these, in general, and for instance the induced repair model of Joiner et al. [11] could equally well be used.
3. Results Fig. 3 shows the effect of proliferation factor on the tumour doubling time. The model predicts a doubling time of about 20 h when the proliferation
Fig. 3. Effect of proliferation factor on doubling time.
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
factor approaches 2. The tumour ceases to grow as the proliferation factor is reduced to approximately 1.2. It should be noted that a small number of cells die at the end of G1b as described in more detail in [9]. In the rest of this paper, a slow growing tumour is modelled by a proliferation factor of 1.22 which represents a tumour doubling time of approximately four weeks and a fast tumour doubles in one week and is modelled with a proliferation factor of 1.3. When the cell survival curves were fitted to the data of Sinclair and Morton [7], it was found that the linear quadratic equation gave the best fit to the data for the cells in the S phase. In contrast, the cells in the remainder of the cycle were best fitted by the multiple target single event model, as illustrated in Fig. 4. From Fig. 4, it is apparent that for the purposes of this paper the response of the cells in the G1 , G2 and M phases may be considered the same. Having fitted the response of the individual phases of the cell cycle it is now possible to reconstruct the average response of a population of these cells, during exponential growth (see Fig. 5). In Fig. 5, the overall cell survival fraction is shown with respect to radiation
213
Fig. 5. Overall survival curve for simulated typical tumour cell.
dose. It should be noted from this diagram that these cells are quite radioresistant compared to many of the human cell lines discussed by Steel [12]. The constituent parts of the model are now complete, and can be combined in simulations of complete courses of radiotherapy. A conventional strategy involves the delivery of 60 Gy in total delivered in 2 Gy fractions at intervals of one day, excluding weekends, i.e. six weeks in total. The effects of this strategy are
Fig. 4. Experimental and fitted cell survival curves.
214
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
Fig. 6. Slow tumour: total cells versus time for conventional radio therapy and CHART.
Fig. 7. Fast growing tumour: total cells versus time for conventional and CHART.
illustrated in Fig. 6 for the slow tumour and in Fig. 7 for the fast tumour. In each diagram, the results may be contrasted with the continuous hyperfractionated accelerated radiotherapy (CHART) strategy. In this treatment schedule 54 Gy is delivered in total, in three fractions of 1.5 Gy/day, every day including the weekend for 12 days. The treatment times are 8 a.m., 2 p.m. and 8 p.m. every day. It can be seen that the conventional strategy leaves about three times fewer cells alive than the CHART treatment.
have been ignored or simplified such as the delayed transition between cell phases while radiation damage is repaired. The experimental data of Sinclair and Morton has been fitted and different cell survival curves have been found to fit best to different phases. During the S phase that DNA replication occurs and any damage caused to the cell by incoming radiation is more likely to be repaired. The resulting survival curve for an exponentially growing population is shown in Fig. 5 and it is noted that the simulated tumours are relatively radioresistant compared to published results for human tumour cell lines. Thus it can be argued that the subsequently shown effects of different radiotherapy strategies are of little direct interest since neither the fast or slow tumour are obvious candidates for treatment by radiotherapy. However, several features of the results in Figs. 6 and 7 show strong similarities with establish clinical experience. In Fig. 6, the conventional pattern of treatment reduces the cell number to the lower level and therefore represents the best chance of a ‘cure’ for the slow growing tumour. In the simulated cases, neither strategy kills all the cells and the tumour regrows. Since the regrowth starts earlier with the CHART treatment, the eventual outcome is that the cell number in the CHART-treated tumour is higher. However, for the faster growing tumour, the better chance of a cure is afforded by the CHART treatment which leaves many fewer cell alive at the end of the treatment period. Fig. 7 also
4. Discussion Fig. 3 shows that reducing the proliferation factor to significantly less than 2, i.e. 1.22 for slow tumour growth and 1.3 for fast tumour growth provides an effective means of slowing the tumour doubling rate to realistic levels. The reduction of the effective proliferation factor is plausible, biologically, if some daughter cells have fatal abnormalities, some are attacked by the immune system and some are swept away from the tumour in the lymph. This method is thought to be valid for small tumours only where the diffusion of nutrients and toxins is not limiting growth. In this simple model, many other features of the cell biology, such as apoptosis, have been ignored. Fig. 4 shows the results of simulating the experiments necessary to establish a cell survival curve. Many established features of radiobiology
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
shows that although the treatment more likely to effect a cure has reversed, the final trends, if any cells are left viable, is the same. In both cases, if not cured, the tumour treated by CHART is bigger after 1300 h (8 weeks) from the commencement of treatment. It is to be hoped that the model described has sufficient flexibility and generality that it may be developed into an effective tool for guiding the clinical decisions affecting the treatment of tumours. Despite the very many simplifying assumptions in the present work, the qualitative behaviour of this model agrees well with clinical experience.
5. Conclusions and future work An age-distributed population balance model of a typical tumour cell cycle has been formulated. The response to radiation has been accounted for separately for each phase of the cell cycle. Cells permanently damaged by radiation are assumed to fall into a parallel cell cycle, finally dying at the next mitosis. This model has been adapted to represent the growth rates of real tumours by reducing the proliferation factor of the undamaged tumour cells. Although plausible and effective, this assumption is not supported by direct experimental evidence. The radiosensitivity of each phase has been derived from the results of Sinclair and Morton, and results in a simulated tumour that is rather more radioresistant that human tumours considered by clinicians to be good candidates for radiotherapy. However, the resulting model does illustrate the potential difficulties of selecting an effective treatment. A conventional treatment schedule is compared with CHART on one fast and one slow growing tumour. The slow growing tumour is more likely to be completely destroyed by the conventional therapy whereas the CHART treatment is more likely to complete destruction of the fast-growing tumour. Thus the model presented shows some encouraging similarities to established clinical experience and with further development may eventually provide clinicians with a useful tool in planning treatment.
215
The model described may be easily modified to include the hyperadiosensitivity described by Short et al. [13]. Similarly, effects of dose on phase durations and other phenomena such as apoptosis which may be simulated as memory effects may be included at the expense of very little extra computational effort. In the longer term, it is intended to develop this model to include the structure of vascular tumours and the diffusion of oxygen and nutrients within the structure. It is intended that such a detailed model will also be flexible enough to account for hadron-based therapy. Acknowledgements The authors would like to thank Mr. Anthony Skadorwa and Mr. Femi Oluboyede for their contributions in final year research projects and Dr. Karen Kirkby for all her help and encouragement. References [1] G.G. Steel, Phys. Med. Biol. 41 (1996) 205. [2] D.B.F. Faraday, The Mathematical Modelling of the Cell Cycle of a Hybridoma Cell line, Ph.D. dissertation, University of Surrey, 1994. [3] D.B.F. Faraday, N.F. Kirkby, Trans I Chem. E 70 (Part A) (2) (1992) 174. [4] S. Araujo, Modelling and Control of Cell Cultures, Ph.D. dissertation, University of Surrey, 1998. [5] B.D. Sleeman, Solid tumour growth: a case study in mathermatical biology, in: P.J. Aston (Ed.), Nonlinear Mathematics and its Applications, first ed., Cambridge University Press, London, 1996, p. 237. [6] M.A.J. Chaplain, B.D. Sleeman, J. Math. Biol. 31 (1993) 431. [7] W.K. Sinclair, R.A. Morton, Radiat. Res. 29 (1966) 450. [8] J.A. Smith, L.N. Martin, Proc. Nat. Acad. Sci. 70 (4) (1973) 1263. [9] D.B.F. Faraday, P. Hayter, N.F. Kirkby, Biochem. Eng. J 7 (2001) 49. [10] G.G. Steel, Basic Clinical Radiobiology, second ed., Arnold, Hodder Headline Group, Oxford University Press, London, 1997. [11] M.C. Joiner, B. Marples, H. Johns, Recent Results Cancer Res. 130 (1993) 27. [12] J. Deacon, M.J. Peckham, G.G. Steel, Radiother. Oncol. 2 (1984) 317. [13] S.C. Short, S.A. Mitchell, P. Boulton, M. Woodcock, M.C. Joiner, Int. J. Radiat. Biol. 75 (11) (1999) 1341.
Mathematical modelling of the response of tumour cells to radiotherapy N.F. Kirkby b
a,* ,
N.G. Burnet b, D.B.F. Faraday
a
a Department of Chemical and Process Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK Oncology Centre, Addenbrooke’s Hospital, University of Cambridge, Department of Oncology, Hills Road, Cambridge CB2 2QQ, UK
Abstract Using a model of cell cycle (CelCyMUS) in which the cell population is described in terms of the cell age and position within the cell cycle typical radiotherapy strategies are simulated, and the effects compared to clinical results, showing reasonable qualitative agreement. Alternative strategies are tested on a slow and a fast-growing tumour and the strategy most likely to completely destroy the tumour is found to depend on the growth rate. This work may have important applications for the development of novel radiotherapy fractionation, for example, schedules to take advantage of hypersensitivity to small doses of radiation exhibited by some tumours, which might improve results compared to conventional dosing strategies. 2002 Published by Elsevier Science B.V. Keywords: Cancer; Radiotherapy; Mathematical model; Cell cycle
1. Introduction After surgery, radiotherapy, is the most effective curative treatment for cancer [1]. The pattern of fractionation of radiation treatment has an important effect on both tumour cell kill and the damage in the surrounding healthy normal tissue. Mathematical models of cell cycles have proved to be extremely effective in predicting the behaviour of in vitro cell cultures [2–4] and are now used in the design and control of such reactors. More recently, cell cycle models have begun to be adapted to encompass in vivo systems such as tumours [5,6]. In both types of system each phase of the cell cycle is modelled separately, to achieve age
*
Corresponding author. E-mail address: [email protected] (N.F. Kirkby).
distributed population balances. This paper describes a mathematical model of the cell cycle of tumour cells in an exponentially growing population. Superimposed on this model is a description of the effect that radiation has on each phase of the cell cycle, based on the experimental work of Sinclair and Morton [7]. The objectives of this work are: • to devise a specific model of a typical mammalian tumour cell; • to incorporate models of the damage caused by the radiation to each phase of the cell cycle; • to test the model against measured cell survival curves on populations normally distributed around the cell cycle; • to optimise existing patterns of radiotherapy and predict the effects of novel patterns not yet used in normal clinical use.
0168-583X/02/$ - see front matter 2002 Published by Elsevier Science B.V. PII: S 0 1 6 8 - 5 8 3 X ( 0 1 ) 0 1 0 9 6 - 5
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
It is possible that this approach will facilitate improved clinical efficiency in the treatment of cancer. Before going on to discuss the results generated by the model, it is pertinent to describe, in a little more detail, the cell cycle, the generic model and the specific model used in this study, together with the assumptions and experimental data used to develop the specific model used in this study.
2. Theory The cell cycle is divided into four main phases (Fig. 1). In M phase cell division occurs via a two step process; the division of the cell’s nucleus (mitosis) followed by the division of the cell into daughter cells, each containing its own nucleus (cytokinesis). The period between one M phase and the next is called the interphase, during which the cells grow. The interphase is divided into the three remaining phases of the cell cycle; G1 , S and G2 . The G1 ðG ¼ gapÞ phase is the interval between the completion of the M phase and the start of the S phase, when the cell replicates its DNA, prior to the next mitosis. The G2 phase is the interval between DNA synthesis (S phase) and cell division (M phase).
Fig. 1. Cell cycle.
211
The generic model Cell Cycle Model University of Surrey (CelCyMUS) developed by Kirkby and Faraday [2,3], describes the cell population in terms of the cell age and position within the cell cycle. The system dynamics of the cell population can therefore be determined at any time during the simulation. The population balance equations used to describe this behaviour are transformed into ordinary differential equations (ODE) by applying the method of characteristics; the resultant ODE are solved using Euler integration. This model has some features in common with that employed by Smith and Martin [8]; their approach, however, did not account for cell death or the interaction of the cell population with the growth medium. Specific models employing CelCyMUS have been highly successful in predicting the growth of mouse–mouse 321 (MM-321) hybridoma cell lines for use in antibiotic production [2,3,9]. The specific model used to describe avascular mammalian tumour growth (where diffusion of nutrients does not inhibit cell growth) uses as its starting point the growth characteristics of an MM-321 hybridoma cell. The cell cycle modelled is the same as that described above, except that the G1 phase is further sub-divided into two parts, G1a and G1b . The G1a phase has a fixed duration of 2.5 h, while the time spent in the G1b phase can vary from 0 to 10 h depending on prevailing conditions. Where cancerous cells clump together to form a large tumour, nutrients are supplied and toxins removed by a dedicated blood supply. Such a tumour is described as a vascular tumour. The tumour modelled here is a small avascular tumour, in which it is assumed that the rate of arrival of nutrients and removal of toxins is not diffusion limited and therefore the tumour can be approximated to a well mixed system. In order to model the effects of radiation, the cell cycle is duplicated in its entirety to form a cell cycle within a cell cycle (Fig. 2). The outer cycle is termed normal and represents the normal tumour cells. Upon irradiation, cells from the outer normal cycle can jump into the inner cycle termed the doomed cycle. The duration of the phases and the parameter values used in the doomed cycle are identical to those in the normal cycle with one
212
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
Fig. 2. Cell cycle for cells undergoing radiation treatment.
important exception. During the M phase, the cell normally divides into two daughter cells, but in the doomed cycle all cells die while trying to divide. Thus, the proliferation factor in the doomed cycle is set to 0. If every normal cell gave rise to two viable daughters the proliferation factor for the normal cycle would be set to 2. However, this may give rise to an unrealistically small tumour doubling time. One method to slow the growth of the tumour is to assume that not all mitotic cells give rise to two viable daughters. This method has the advantage of simplicity and maybe the main limitation on the growth of small tumours before all the complications of nutrient diffusion, the effects of angiogenesis and even space restrictions become significant. There is a dearth of published data relating to how radiation affects the survival fraction for the different phases within the cell cycle. One of the few papers, by Sinclair and Morton [7], studies Chinese hamster ovary (CHO) cells throughout all four stages. Other data, from Steel [10], compare the total cell survival fraction for different human tumour cell lines and their response to radiation dose. Combining the data of Sinclair and Morton, to give a total cell survival fraction and comparing this with the data from Steel [10] it is found that the data for the CHO cells lie within the data range for the human tumour cell lines and have a similar dispersion. Thus the data from Sinclair and
Morton [7] appear to offer a valid approximation to human tumour cell growth and as more data becomes available the model can be developed accordingly. The model can represent any desired pattern of fractionation of the radiation treatment. When a dose of radiation is delivered, a fraction of normal cells survive in the normal cycle. The remainder are mortally damaged and are assumed to be transferred to the doomed cycle. Hence, cells are not killed immediately by a dose of radiation but are assumed to die at their next attempt at division. The relationship between survival fraction and radiation dose received can be determined by one of several commonly used models and each phase can use its own model and parameters. Thus it is possible to allow radiation to have very different effects depending on which part of the cell cycle a cell is in when it receives the radiation. In this work, the cell survival models allowed are the simple multi-target and single event model and the linear quadratic model [10]. However, the choice of model is not restricted to these, in general, and for instance the induced repair model of Joiner et al. [11] could equally well be used.
3. Results Fig. 3 shows the effect of proliferation factor on the tumour doubling time. The model predicts a doubling time of about 20 h when the proliferation
Fig. 3. Effect of proliferation factor on doubling time.
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
factor approaches 2. The tumour ceases to grow as the proliferation factor is reduced to approximately 1.2. It should be noted that a small number of cells die at the end of G1b as described in more detail in [9]. In the rest of this paper, a slow growing tumour is modelled by a proliferation factor of 1.22 which represents a tumour doubling time of approximately four weeks and a fast tumour doubles in one week and is modelled with a proliferation factor of 1.3. When the cell survival curves were fitted to the data of Sinclair and Morton [7], it was found that the linear quadratic equation gave the best fit to the data for the cells in the S phase. In contrast, the cells in the remainder of the cycle were best fitted by the multiple target single event model, as illustrated in Fig. 4. From Fig. 4, it is apparent that for the purposes of this paper the response of the cells in the G1 , G2 and M phases may be considered the same. Having fitted the response of the individual phases of the cell cycle it is now possible to reconstruct the average response of a population of these cells, during exponential growth (see Fig. 5). In Fig. 5, the overall cell survival fraction is shown with respect to radiation
213
Fig. 5. Overall survival curve for simulated typical tumour cell.
dose. It should be noted from this diagram that these cells are quite radioresistant compared to many of the human cell lines discussed by Steel [12]. The constituent parts of the model are now complete, and can be combined in simulations of complete courses of radiotherapy. A conventional strategy involves the delivery of 60 Gy in total delivered in 2 Gy fractions at intervals of one day, excluding weekends, i.e. six weeks in total. The effects of this strategy are
Fig. 4. Experimental and fitted cell survival curves.
214
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
Fig. 6. Slow tumour: total cells versus time for conventional radio therapy and CHART.
Fig. 7. Fast growing tumour: total cells versus time for conventional and CHART.
illustrated in Fig. 6 for the slow tumour and in Fig. 7 for the fast tumour. In each diagram, the results may be contrasted with the continuous hyperfractionated accelerated radiotherapy (CHART) strategy. In this treatment schedule 54 Gy is delivered in total, in three fractions of 1.5 Gy/day, every day including the weekend for 12 days. The treatment times are 8 a.m., 2 p.m. and 8 p.m. every day. It can be seen that the conventional strategy leaves about three times fewer cells alive than the CHART treatment.
have been ignored or simplified such as the delayed transition between cell phases while radiation damage is repaired. The experimental data of Sinclair and Morton has been fitted and different cell survival curves have been found to fit best to different phases. During the S phase that DNA replication occurs and any damage caused to the cell by incoming radiation is more likely to be repaired. The resulting survival curve for an exponentially growing population is shown in Fig. 5 and it is noted that the simulated tumours are relatively radioresistant compared to published results for human tumour cell lines. Thus it can be argued that the subsequently shown effects of different radiotherapy strategies are of little direct interest since neither the fast or slow tumour are obvious candidates for treatment by radiotherapy. However, several features of the results in Figs. 6 and 7 show strong similarities with establish clinical experience. In Fig. 6, the conventional pattern of treatment reduces the cell number to the lower level and therefore represents the best chance of a ‘cure’ for the slow growing tumour. In the simulated cases, neither strategy kills all the cells and the tumour regrows. Since the regrowth starts earlier with the CHART treatment, the eventual outcome is that the cell number in the CHART-treated tumour is higher. However, for the faster growing tumour, the better chance of a cure is afforded by the CHART treatment which leaves many fewer cell alive at the end of the treatment period. Fig. 7 also
4. Discussion Fig. 3 shows that reducing the proliferation factor to significantly less than 2, i.e. 1.22 for slow tumour growth and 1.3 for fast tumour growth provides an effective means of slowing the tumour doubling rate to realistic levels. The reduction of the effective proliferation factor is plausible, biologically, if some daughter cells have fatal abnormalities, some are attacked by the immune system and some are swept away from the tumour in the lymph. This method is thought to be valid for small tumours only where the diffusion of nutrients and toxins is not limiting growth. In this simple model, many other features of the cell biology, such as apoptosis, have been ignored. Fig. 4 shows the results of simulating the experiments necessary to establish a cell survival curve. Many established features of radiobiology
N.F. Kirkby et al. / Nucl. Instr. and Meth. in Phys. Res. B 188 (2002) 210–215
shows that although the treatment more likely to effect a cure has reversed, the final trends, if any cells are left viable, is the same. In both cases, if not cured, the tumour treated by CHART is bigger after 1300 h (8 weeks) from the commencement of treatment. It is to be hoped that the model described has sufficient flexibility and generality that it may be developed into an effective tool for guiding the clinical decisions affecting the treatment of tumours. Despite the very many simplifying assumptions in the present work, the qualitative behaviour of this model agrees well with clinical experience.
5. Conclusions and future work An age-distributed population balance model of a typical tumour cell cycle has been formulated. The response to radiation has been accounted for separately for each phase of the cell cycle. Cells permanently damaged by radiation are assumed to fall into a parallel cell cycle, finally dying at the next mitosis. This model has been adapted to represent the growth rates of real tumours by reducing the proliferation factor of the undamaged tumour cells. Although plausible and effective, this assumption is not supported by direct experimental evidence. The radiosensitivity of each phase has been derived from the results of Sinclair and Morton, and results in a simulated tumour that is rather more radioresistant that human tumours considered by clinicians to be good candidates for radiotherapy. However, the resulting model does illustrate the potential difficulties of selecting an effective treatment. A conventional treatment schedule is compared with CHART on one fast and one slow growing tumour. The slow growing tumour is more likely to be completely destroyed by the conventional therapy whereas the CHART treatment is more likely to complete destruction of the fast-growing tumour. Thus the model presented shows some encouraging similarities to established clinical experience and with further development may eventually provide clinicians with a useful tool in planning treatment.
215
The model described may be easily modified to include the hyperadiosensitivity described by Short et al. [13]. Similarly, effects of dose on phase durations and other phenomena such as apoptosis which may be simulated as memory effects may be included at the expense of very little extra computational effort. In the longer term, it is intended to develop this model to include the structure of vascular tumours and the diffusion of oxygen and nutrients within the structure. It is intended that such a detailed model will also be flexible enough to account for hadron-based therapy. Acknowledgements The authors would like to thank Mr. Anthony Skadorwa and Mr. Femi Oluboyede for their contributions in final year research projects and Dr. Karen Kirkby for all her help and encouragement. References [1] G.G. Steel, Phys. Med. Biol. 41 (1996) 205. [2] D.B.F. Faraday, The Mathematical Modelling of the Cell Cycle of a Hybridoma Cell line, Ph.D. dissertation, University of Surrey, 1994. [3] D.B.F. Faraday, N.F. Kirkby, Trans I Chem. E 70 (Part A) (2) (1992) 174. [4] S. Araujo, Modelling and Control of Cell Cultures, Ph.D. dissertation, University of Surrey, 1998. [5] B.D. Sleeman, Solid tumour growth: a case study in mathermatical biology, in: P.J. Aston (Ed.), Nonlinear Mathematics and its Applications, first ed., Cambridge University Press, London, 1996, p. 237. [6] M.A.J. Chaplain, B.D. Sleeman, J. Math. Biol. 31 (1993) 431. [7] W.K. Sinclair, R.A. Morton, Radiat. Res. 29 (1966) 450. [8] J.A. Smith, L.N. Martin, Proc. Nat. Acad. Sci. 70 (4) (1973) 1263. [9] D.B.F. Faraday, P. Hayter, N.F. Kirkby, Biochem. Eng. J 7 (2001) 49. [10] G.G. Steel, Basic Clinical Radiobiology, second ed., Arnold, Hodder Headline Group, Oxford University Press, London, 1997. [11] M.C. Joiner, B. Marples, H. Johns, Recent Results Cancer Res. 130 (1993) 27. [12] J. Deacon, M.J. Peckham, G.G. Steel, Radiother. Oncol. 2 (1984) 317. [13] S.C. Short, S.A. Mitchell, P. Boulton, M. Woodcock, M.C. Joiner, Int. J. Radiat. Biol. 75 (11) (1999) 1341.