- Email: [email protected]
Modelling and simulation of radiotherapy
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 255 (2007) 13–17 www.elsevier.com/locate/nimb
Modelling and simulation of radiotherapy Norman F. Kirkby
*
Fluids and Systems Research Centre, School of Engineering, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom Available online 26 December 2006
Abstract In this paper, models are described which have been developed to model both the way in which a population of cells respond to radiation and the way in which a population of patients respond to radiotherapy to assist the conduct of clinical trials in silico. Population balance techniques have been used to simulate the age distribution of tumour cells in the cell cycle. Sensitivity to radiation is not constant round the cell cycle and a single fraction of radiation changes the age distribution. Careful timing of further fractions of radiation can be used to maximize the damage delivered to the tumour while minimizing damage to normal tissue. However, tumour modelling does not necessarily predict patient outcome. A separate model has been established to predict the course of a brain cancer called glioblastoma multiforme (GBM). The model considers the growth of the tumour and its effect on the normal brain. A simple representation is included of the health status of the patient and hence the type of treatment offered. It is concluded that although these and similar models have a long way yet to be developed, they are beginning to have an impact on the development of clinical practice. Ó 2006 Elsevier B.V. All rights reserved. PACS: 87.18.Bb Keywords: Mathematical modelling; Brain cancer; Radiotherapy; Glioblastoma; Patient survival; Tumour growth
1. Introduction The diseases, collectively known as cancer, are the second highest cause of adult mortality in the Western World [1]. Clinicians have three treatment modalities available: chemotherapy, radiotherapy and surgery. It is sometimes forgotten that surgery cures more patients and effectively destroys more cancer cells than either of the other modalities. It is almost always forgotten that radiotherapy is the second most effective treatment for cancer [2]. The purpose of this paper is to discuss the role of mathematical modelling and simulation in the planning, delivery and development of radiotherapy. This is an area which is developing extremely rapidly at present; so rapidly that it is very difficult to give a comprehensive and completely inclusive review of all the work currently in progress. Therefore, this paper will refer mainly to work on brain cancer in pro-
*
Tel.: +44 1483 686577; fax: +44 1483 686581. E-mail address: [email protected]
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.005
gress in the author’s group as illustration of what it is being attempted, what some of the challenges are and what it is hoped may be achieved. This paper will briefly review some aspects of the radiobiology, medical physics and current clinical practice of radiotherapy before describing the models and presenting some of the results.
2. Background The central dogma of radiobiology is that cancer cells can be destroyed when radiation (usually photons or protons) creates double strand breaks in DNA which are subsequently either not repaired or misrepaired by the cells involved. Cancer cells evolve or originate from normal cells by abnormal processes of DNA replication. Cancer is a process by which errors in the DNA are replicated and amplified and which results in abnormal growth and division of cells. Cancer cells can often be extremely similar
14
N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
in many other respects to the surrounding healthy tissue [3]. In the case of many cancers, a small solid tumour forms which grows to a size at which the central cells become starved of nutrients, especially oxygen. These small, avascular tumours issue a variety of normal and abnormal distress signals which can cause the surrounding health tissue to develop a blood supply for the tumour. This process of angiogenesis is the target of intensive research activity, not least in the drug discovery industries. Some tumours can grow very rapidly partly because they can invade the normal tissue and annex the existing structures and processes of nutrient supply. This process of invasive growth is sometimes associated with cancers derived from normal cells which are designed to be motile within normal tissue. An example here is an extremely aggressive and dangerous form of brain cancer called glioblastoma (GBM) which is derived from the glial cells of the brain. The glial cells are ‘helper cells’ some of which are designed to be able to move to sites of damage within the brain [4]. As illustrated in Fig. 1, mature gliomas generally have a dead, or necrotic, core outside which is a nutrient deprived shell of glioma cells which are in turn surrounded by rapidly growing cells. Further out from the centre of the tumour more normal, invaded tissue can be found. This region usually represents the margin of the tumour on a CT scan, but it should not be forgotten that beyond here there will be glioma cells moving out into the normal tissue by a process often modelled as diffusion [5]. This diffusion is known to be especially rapid down the cabling of the brain (the white matter) and somewhat slower in the bulk of the brain (the grey matter). GBM patients can present with a very wide range of sizes of tumour some of which are smaller than the current resolution of CT scanners through to tumours the size of a small orange. Some of these tumours are known to have doubled in volume in as little as 4 days. GBM is not a particularly common form of cancer; it represents about 1% of all cases of cancer. However it is very dangerous; less than 3% of all GBM patients are long term survivors of the condition. GBM is responsible for
Anisotropic diffusion into normal tissue Necrotic (dead) Core Quiescent Tumour Cells ‘Healthy’ Tumour Cells Invaded Tissue: Tumour & Normal Cells Fig. 1. The structure of a solid tumour.
more years of life lost per patient, on average, than any other common form of cancer [6]. The primary tumour in GBM is called a glioma and is usually responsible for mortality. Gliomas grow inside the blood-brain barrier and this means that they are relatively protected from attack by the immune system. Secondary growth or metastasis is rare in GBM. 50% of all GBM patients are dead within 10 months of presentation at hospital [7]. 3. Challenges for radiotherapy Briefly, clinicians require delivery of radiation to cancer cells only. Dose to normal tissue causes acute toxicity and long term side effects. Currently, acute toxicity in normal tissue controls the dose per day and total dose delivered to patients. This causes clinicians to seek ever better ways of targeting the tumour with radiation to reduce normal tissue exposure [8]. There is currently increasing interest in particle therapy for two main reasons [9]: (1) particle beams have narrow penumbra so that higher doses can be delivered closer to critical normal structures, and (2) particle beams have a Bragg peak with almost no dose beyond, so there is no ‘exit beam’ passing through normal tissue.
4. Challenges for radiobiology Understanding DNA damage and repair is one of the current ‘great questions’. For radiotherapy, an increase in the number of double strand breaks in cancer cells and a reduction in normal cells is required. Similarly a reduction in DNA repair in cancer cells is required and an increase in repair is required in normal cells. For instance, it is known that certain types of cells are abnormally sensitive to damage by low-doses of radiation, where the same cell lines become very resistant to higher doses. This is called low-dose hypersensitivity and why it exists is not understood. It may have very serious consequences for environmental exposure as well as being potentially exploitable in radiotherapy. Understanding what other processes are caused by radiation damage in cells is also an exciting area. It is known that some cells, when hit by radiation, can cause surrounding cells, which have not been hit, to die. This is called a non-targetted or bystander effect and its mechanisms are very poorly understood. Cancer cells progress from one cell division to the next through a sequence of phases known as the cell cycle. The cell cycle is illustrated in Fig. 2. For instance, cells replicate their DNA during the synthesis phase (S phase). This phase typically lasts several hours during which time the cells are often relatively insensitive to radiation because DNA replication requires that error detection and repair systems are at their most active.
N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
Mitosis x2
M Second Gap Phase
Cell division
G2 Check Points G1 First Gap Phase
S DNA Synthesis
Fig. 2. The cell cycle.
In other parts of the cell cycle, DNA damage may go undetected or may be misrepaired. Thus it is to be expected that radiation delivered to a population of cells distributed around the cell cycle may produce a partial synchronization because the probability of survival depends on the position in the cell cycle [10]. The cell cycle also has check-points where cells have systems that permit or delay progress to subsequent phases. These check-points partly regulate growth rate by not permitting progress unless the nutritional status of the cell is sufficient, but progress is also delayed if the cell can detect that DNA damage is still being repaired. Cells can be directed at check-points into quiescent states if, for instance, oxygen is in short supply. This is the basis of hypoxic radio-resistance because, in the absence of oxygen, broken double strand breaks stand more chance of repair and less chance of the damage being ‘fixed’ to an unrepairable state by oxidative processes [10]. 5. Challenges for modelling and simulation There are at least seven length scales potentially involved in the modelling of cancer, as illustrated in
15
Fig. 3. Each length scale presents unique challenges for mathematical modelling and simulation and combining these scales presents further difficulties. Modelling activity has already served many useful purposes including drawing attention to how little is known in each of these areas. Modelling at the multi-cellular level is a particularly active field. These models represent the growth rate of cells, their consumption of nutrients and their responses to radiation in a variety of ways. In the author’s group and elsewhere, the established methods are to write PDEs for the population balances on the cells. The independent variables of these formulations may include time, physical position, phase of the cell cycle, age within the cell cycle, cell type, cell size and DNA damage status. Many difficulties exist, for example: (1) the efficient numerical solution of these equations requires specialized techniques which are not readily available in standard software libraries, and (2) it is effectively impossible to represent the development or emergence of new structures such as capillaries resulting from angiogenesis. Agent-based computing models, underpinned by strongly object-oriented programming techniques, have completely changed the approach to modelling and simulation. Biological cells are represented as agents with associated data and rules for interaction with neighbours. Diffusion of nutrients, signals and products can also be computed using cellular automaton or agent-based techniques. In Fig. 4, results are presented for the population balance approach [11,12]. In this model, a spatially well-mixed population of cells, originally developed for the growth of animal cells in a bioreactor is developed [13]. Data for cell survival to typical doses of radiation are available for the main phases of the cell cycle. The changes in age distribution due to the delivery of a single fraction of radiotherapy are calculated, and from this model it is possible to suggest
Molecular
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Fig. 3. Length scales in modelling cancer.
Fig. 4. Population radiotherapy.
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N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
how the timing between fractions can be optimized to maximize tumour destruction. This model includes observed data for low-dose hypersensitivity in parts of the cell cycle [14–16]. Currently it is not practicable to determine which patients may have tumours which are low-dose sensitive, therefore it would not be ethical to deliver treatments which could enhance the treatment of some patients but be of very less benefit than existing treatments to others. However the modelling does suggest that it might be possible to deliver ‘asymmetric’ hyper-fractionated treatment where three fractions are delivered per day, two small doses are given to tune the age distribution of the cells in the tumour if they are low-dose sensitive, and then one large fraction is given at the end of each day. This large fraction ensures that this form of treatment is as effective as traditional fractionation patterns in those patients who do not show low-dose sensitivity. However, in patients who do have low-dose sensitivity, this larger fraction can be timed to maximize the destruction of tumour given that the two smaller fractions have prepared the age distribution of the cells in the cell cycle. The problem with models of tumours is that they do not predict directly the main features of the disease process that the patient and the clinician are most concerned with such as patient survival. In the final part of this paper, a brief description is given of a new model [17,18] which currently addresses only the top two scales represented in Fig. 3. 6. GBM patient model The objective of this model is to see if it is possible to extract biological information from clinical data. The model consists of three components: (1) a patient model, (2) a treatment model, and (3) a clinical trial or population model. The patient model assumes first order growth for a tumour in the brain and a bimolecular model for the destruction of normal brain as a result of the presence of the tumour. It is assumed that patients die when the amount of normal brain falls below a critical level. The model for treatment currently assumes that radiotherapy is fractionated but that the fractions are delivered instantaneously after a wait for diagnosis, treatment planning and waiting for the availability of radiotherapy equipment. Early in the development of this model it became clear that some representation of the clinical decision making process would be required. The model contains several simple approximations to these processes. Firstly, when a patient presents at hospital an assessment is made about what type of treatment to offer. Radical treatment, i.e. with curative intent, is offered only if the patient is not ‘too poorly’ as assessed by WHO guidelines. Patients not offered radical treatment may be offered palliative treat-
ment where a very different plan of fractionation and dose is given to extend life. In the model, it is possible to calculate how long the patient would survive without treatment and this is used as the basis for modelling the ‘too poorly at presentation’ criterion. After an indeterminate wait for treatment, patients have to be reassessed when treatment is due to start because treatment is itself arduous and it is unfortunately true that some patients have deteriorated so badly by the time treatment is due to commence that it would not be in the patients best interests to continue. A cohort of patients is constructed and Monte Carlo simulation is based on the assumption that a number of parameters are distributed according to probability distributions. The following five parameters are assumed to be distributed normally in the population: (1) number of normal brain cells at presentation, (2) number of normal brain cells at death, (3) interaction parameter between the normal brain cells and cancer cells, (4) tumour doubling time, and (5) delay to start radiotherapy treatment. A sixth parameter, the number of cancer cells that survive a single fraction of radiation, is assumed to follow a skewed distribution constrained between 0 and 1. This model can be fitted to clinical data using optimization based on established simulated annealing and folding polygon techniques [19]. An example is shown in Fig. 5 where this model has been fitted to data from palliative patients at the Addenbrooke’s Hospital. The construction of this model has asked some important and often overlooked questions of the clinicians and scientists involved. One example, for instance, is that very little work has ever been done on the neurological basis of death in GBM patients. More interestingly, these models can be used to ask some very interesting questions such as: (1) What might be the basis of the conflicting evidence in the literature about the effects of waiting times? 1
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Fig. 5. The BJJK model fitted to Addenbrooke’s patients treated with palliative intent.
N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
and protocols for patient assessment and selection at the largest scale.
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Acknowledgements
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The author wishes to thank the Life Sciences Interface of the Engineering and Physical Sciences Research Council for the Discipline Hopping Grant, the Master and Fellows of Fitzwilliam College, Cambridge, for the Visiting Fellowship and the Marie Curie Research Training Network ‘‘Cellion’’ for assistance to attend COSIRES 2006.
Tumour age (days)
Fig. 6. Prediction of tumour age distribution at presentation.
(2) Assuming this could be done safely, how would increasing dose relate to patient survival? (3) What is the age distribution of tumours at presentation? These are interesting questions and a model such as that described can be a guide in designing clinical trials or experiments to find answers. Some of these questions will be extremely difficult and expensive to answer by other means. Fig. 6 shows predictions for the age distribution of tumours at presentation, these data would be very hard to verify clinically because screening the population for such a rare condition could not be justified by a cost – benefit analysis. 7. Conclusions and future work The use of advanced mathematical models in clinical oncology and radiotherapy is at an early stage. The most advanced models tend to concentrate on the formation and growth of solid tumours. These models do not always include the effects that radiotherapy has on either the tumour or the patient. Advanced models rarely include factors such as the mechanisms by which tumour growth causes death or the effect that patient selection for different treatment strategies has on outcome. There is much work to be done in this area and multiscale models will play an important role in combining and integrating knowledge of the molecular biology of cells at the smallest scale, up to models of hospital operations
References [1] J.R. Toms (Ed.), Cancer Stats Monograph, Cancer Research UK, London, 2004. [2] L. Werko et al., Acta Oncol. 35 (6) (1996) 9. [3] R.J.B. King, Cancer Biology, second ed., Pearson Education Ltd., Harlow, 2000. [4] L.E. Abrey, W.P. Mason, Fast Facts-Brain Tumours, Health Press Ltd., Oxford, 2003. [5] K.R. Swanson, C. Bridge, J.D. Murray, E.C. Alvord Jr., J. Neurol. Sci. 216 (1) (2003) 1. [6] N.G. Burnet, S.J. Jefferies, R.J. Benson, D.P. Hunt, F.P. Treasure, Br. J. Cancer 92 (2) (2005) 241. [7] N.M. Bleehen, S.P. Stenning, Br. J. Cancer 64 (4) (1991) 769. [8] R. Jena, S.J. Price, C. Baker, S.J. Jefferies, J.D. Pickard, J.H. Gillard, N.G. Burnet, Clin. Oncol. 17 (8) (2005) 581. [9] N.G. Burnet, K.J. Kirkby, N.F. Kirkby, Clin. Oncol. 16 (5) (2004) 381. [10] G.G. Steel, Basic Clinical Radiobiology, third ed., Arnold, London, 2002. [11] N.F. Kirkby, D.B.F. Faraday, S.C. Short, M. Woodcock, M.C. Joiner, N.G. Burnet, Radiother. Oncol. 64 (Suppl. 1) (2002) S57. [12] N.F. Kirkby, N.G. Burnet, D.B.F. Faraday, Nucl. Instr. and Meth. B 188 (2002) 210. [13] D.B.F. Faraday, P. Hayter, N.F. Kirkby, Biochem. Eng. J. 7 (2001) 49. [14] S.C. Short, M. Woodcock, B. Marples, M.C. Joiner, Int. J. Radiat. Biol. 79 (2) (2003) 99. [15] M.C. Joiner, B. Marples, P. Lambin, S.C. Short, I. Turesson, Int. J. Radiat. Oncol. Biol. Phys. 49 (2) (2001) 379. [16] S.C. Short, S.A. Mitchell, P. Boulton, M. Woodcock, M.C. Joiner, Int. J. Radiat. Biol. 75 (11) (1999) 1341. [17] N.G. Burnet, R. Jena, S.J. Jefferies, S.P. Stenning, N.F. Kirkby, Clin. Oncol. 18 (2) (2006) 93. [18] N.F. Kirkby, R. Jena, S.P. Stenning, S.J. Jefferies, N.G. Burnet, Radiother. Oncol. 73 (2004) S367. [19] W.H. Press, S.A. Tenkolsky, W.T. Vetterling, B.P. Flannery, Fortran Numerical Recipes in Fortran 90, Cambridge University Press, Cambridge, 1996.
Nuclear Instruments and Methods in Physics Research B 255 (2007) 13–17 www.elsevier.com/locate/nimb
Modelling and simulation of radiotherapy Norman F. Kirkby
*
Fluids and Systems Research Centre, School of Engineering, University of Surrey, Guildford, Surrey GU2 7XH, United Kingdom Available online 26 December 2006
Abstract In this paper, models are described which have been developed to model both the way in which a population of cells respond to radiation and the way in which a population of patients respond to radiotherapy to assist the conduct of clinical trials in silico. Population balance techniques have been used to simulate the age distribution of tumour cells in the cell cycle. Sensitivity to radiation is not constant round the cell cycle and a single fraction of radiation changes the age distribution. Careful timing of further fractions of radiation can be used to maximize the damage delivered to the tumour while minimizing damage to normal tissue. However, tumour modelling does not necessarily predict patient outcome. A separate model has been established to predict the course of a brain cancer called glioblastoma multiforme (GBM). The model considers the growth of the tumour and its effect on the normal brain. A simple representation is included of the health status of the patient and hence the type of treatment offered. It is concluded that although these and similar models have a long way yet to be developed, they are beginning to have an impact on the development of clinical practice. Ó 2006 Elsevier B.V. All rights reserved. PACS: 87.18.Bb Keywords: Mathematical modelling; Brain cancer; Radiotherapy; Glioblastoma; Patient survival; Tumour growth
1. Introduction The diseases, collectively known as cancer, are the second highest cause of adult mortality in the Western World [1]. Clinicians have three treatment modalities available: chemotherapy, radiotherapy and surgery. It is sometimes forgotten that surgery cures more patients and effectively destroys more cancer cells than either of the other modalities. It is almost always forgotten that radiotherapy is the second most effective treatment for cancer [2]. The purpose of this paper is to discuss the role of mathematical modelling and simulation in the planning, delivery and development of radiotherapy. This is an area which is developing extremely rapidly at present; so rapidly that it is very difficult to give a comprehensive and completely inclusive review of all the work currently in progress. Therefore, this paper will refer mainly to work on brain cancer in pro-
*
Tel.: +44 1483 686577; fax: +44 1483 686581. E-mail address: [email protected]
0168-583X/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2006.11.005
gress in the author’s group as illustration of what it is being attempted, what some of the challenges are and what it is hoped may be achieved. This paper will briefly review some aspects of the radiobiology, medical physics and current clinical practice of radiotherapy before describing the models and presenting some of the results.
2. Background The central dogma of radiobiology is that cancer cells can be destroyed when radiation (usually photons or protons) creates double strand breaks in DNA which are subsequently either not repaired or misrepaired by the cells involved. Cancer cells evolve or originate from normal cells by abnormal processes of DNA replication. Cancer is a process by which errors in the DNA are replicated and amplified and which results in abnormal growth and division of cells. Cancer cells can often be extremely similar
14
N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
in many other respects to the surrounding healthy tissue [3]. In the case of many cancers, a small solid tumour forms which grows to a size at which the central cells become starved of nutrients, especially oxygen. These small, avascular tumours issue a variety of normal and abnormal distress signals which can cause the surrounding health tissue to develop a blood supply for the tumour. This process of angiogenesis is the target of intensive research activity, not least in the drug discovery industries. Some tumours can grow very rapidly partly because they can invade the normal tissue and annex the existing structures and processes of nutrient supply. This process of invasive growth is sometimes associated with cancers derived from normal cells which are designed to be motile within normal tissue. An example here is an extremely aggressive and dangerous form of brain cancer called glioblastoma (GBM) which is derived from the glial cells of the brain. The glial cells are ‘helper cells’ some of which are designed to be able to move to sites of damage within the brain [4]. As illustrated in Fig. 1, mature gliomas generally have a dead, or necrotic, core outside which is a nutrient deprived shell of glioma cells which are in turn surrounded by rapidly growing cells. Further out from the centre of the tumour more normal, invaded tissue can be found. This region usually represents the margin of the tumour on a CT scan, but it should not be forgotten that beyond here there will be glioma cells moving out into the normal tissue by a process often modelled as diffusion [5]. This diffusion is known to be especially rapid down the cabling of the brain (the white matter) and somewhat slower in the bulk of the brain (the grey matter). GBM patients can present with a very wide range of sizes of tumour some of which are smaller than the current resolution of CT scanners through to tumours the size of a small orange. Some of these tumours are known to have doubled in volume in as little as 4 days. GBM is not a particularly common form of cancer; it represents about 1% of all cases of cancer. However it is very dangerous; less than 3% of all GBM patients are long term survivors of the condition. GBM is responsible for
Anisotropic diffusion into normal tissue Necrotic (dead) Core Quiescent Tumour Cells ‘Healthy’ Tumour Cells Invaded Tissue: Tumour & Normal Cells Fig. 1. The structure of a solid tumour.
more years of life lost per patient, on average, than any other common form of cancer [6]. The primary tumour in GBM is called a glioma and is usually responsible for mortality. Gliomas grow inside the blood-brain barrier and this means that they are relatively protected from attack by the immune system. Secondary growth or metastasis is rare in GBM. 50% of all GBM patients are dead within 10 months of presentation at hospital [7]. 3. Challenges for radiotherapy Briefly, clinicians require delivery of radiation to cancer cells only. Dose to normal tissue causes acute toxicity and long term side effects. Currently, acute toxicity in normal tissue controls the dose per day and total dose delivered to patients. This causes clinicians to seek ever better ways of targeting the tumour with radiation to reduce normal tissue exposure [8]. There is currently increasing interest in particle therapy for two main reasons [9]: (1) particle beams have narrow penumbra so that higher doses can be delivered closer to critical normal structures, and (2) particle beams have a Bragg peak with almost no dose beyond, so there is no ‘exit beam’ passing through normal tissue.
4. Challenges for radiobiology Understanding DNA damage and repair is one of the current ‘great questions’. For radiotherapy, an increase in the number of double strand breaks in cancer cells and a reduction in normal cells is required. Similarly a reduction in DNA repair in cancer cells is required and an increase in repair is required in normal cells. For instance, it is known that certain types of cells are abnormally sensitive to damage by low-doses of radiation, where the same cell lines become very resistant to higher doses. This is called low-dose hypersensitivity and why it exists is not understood. It may have very serious consequences for environmental exposure as well as being potentially exploitable in radiotherapy. Understanding what other processes are caused by radiation damage in cells is also an exciting area. It is known that some cells, when hit by radiation, can cause surrounding cells, which have not been hit, to die. This is called a non-targetted or bystander effect and its mechanisms are very poorly understood. Cancer cells progress from one cell division to the next through a sequence of phases known as the cell cycle. The cell cycle is illustrated in Fig. 2. For instance, cells replicate their DNA during the synthesis phase (S phase). This phase typically lasts several hours during which time the cells are often relatively insensitive to radiation because DNA replication requires that error detection and repair systems are at their most active.
N.F. Kirkby / Nucl. Instr. and Meth. in Phys. Res. B 255 (2007) 13–17
Mitosis x2
M Second Gap Phase
Cell division
G2 Check Points G1 First Gap Phase
S DNA Synthesis
Fig. 2. The cell cycle.
In other parts of the cell cycle, DNA damage may go undetected or may be misrepaired. Thus it is to be expected that radiation delivered to a population of cells distributed around the cell cycle may produce a partial synchronization because the probability of survival depends on the position in the cell cycle [10]. The cell cycle also has check-points where cells have systems that permit or delay progress to subsequent phases. These check-points partly regulate growth rate by not permitting progress unless the nutritional status of the cell is sufficient, but progress is also delayed if the cell can detect that DNA damage is still being repaired. Cells can be directed at check-points into quiescent states if, for instance, oxygen is in short supply. This is the basis of hypoxic radio-resistance because, in the absence of oxygen, broken double strand breaks stand more chance of repair and less chance of the damage being ‘fixed’ to an unrepairable state by oxidative processes [10]. 5. Challenges for modelling and simulation There are at least seven length scales potentially involved in the modelling of cancer, as illustrated in
15
Fig. 3. Each length scale presents unique challenges for mathematical modelling and simulation and combining these scales presents further difficulties. Modelling activity has already served many useful purposes including drawing attention to how little is known in each of these areas. Modelling at the multi-cellular level is a particularly active field. These models represent the growth rate of cells, their consumption of nutrients and their responses to radiation in a variety of ways. In the author’s group and elsewhere, the established methods are to write PDEs for the population balances on the cells. The independent variables of these formulations may include time, physical position, phase of the cell cycle, age within the cell cycle, cell type, cell size and DNA damage status. Many difficulties exist, for example: (1) the efficient numerical solution of these equations requires specialized techniques which are not readily available in standard software libraries, and (2) it is effectively impossible to represent the development or emergence of new structures such as capillaries resulting from angiogenesis. Agent-based computing models, underpinned by strongly object-oriented programming techniques, have completely changed the approach to modelling and simulation. Biological cells are represented as agents with associated data and rules for interaction with neighbours. Diffusion of nutrients, signals and products can also be computed using cellular automaton or agent-based techniques. In Fig. 4, results are presented for the population balance approach [11,12]. In this model, a spatially well-mixed population of cells, originally developed for the growth of animal cells in a bioreactor is developed [13]. Data for cell survival to typical doses of radiation are available for the main phases of the cell cycle. The changes in age distribution due to the delivery of a single fraction of radiotherapy are calculated, and from this model it is possible to suggest
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Fig. 3. Length scales in modelling cancer.
Fig. 4. Population radiotherapy.
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how the timing between fractions can be optimized to maximize tumour destruction. This model includes observed data for low-dose hypersensitivity in parts of the cell cycle [14–16]. Currently it is not practicable to determine which patients may have tumours which are low-dose sensitive, therefore it would not be ethical to deliver treatments which could enhance the treatment of some patients but be of very less benefit than existing treatments to others. However the modelling does suggest that it might be possible to deliver ‘asymmetric’ hyper-fractionated treatment where three fractions are delivered per day, two small doses are given to tune the age distribution of the cells in the tumour if they are low-dose sensitive, and then one large fraction is given at the end of each day. This large fraction ensures that this form of treatment is as effective as traditional fractionation patterns in those patients who do not show low-dose sensitivity. However, in patients who do have low-dose sensitivity, this larger fraction can be timed to maximize the destruction of tumour given that the two smaller fractions have prepared the age distribution of the cells in the cell cycle. The problem with models of tumours is that they do not predict directly the main features of the disease process that the patient and the clinician are most concerned with such as patient survival. In the final part of this paper, a brief description is given of a new model [17,18] which currently addresses only the top two scales represented in Fig. 3. 6. GBM patient model The objective of this model is to see if it is possible to extract biological information from clinical data. The model consists of three components: (1) a patient model, (2) a treatment model, and (3) a clinical trial or population model. The patient model assumes first order growth for a tumour in the brain and a bimolecular model for the destruction of normal brain as a result of the presence of the tumour. It is assumed that patients die when the amount of normal brain falls below a critical level. The model for treatment currently assumes that radiotherapy is fractionated but that the fractions are delivered instantaneously after a wait for diagnosis, treatment planning and waiting for the availability of radiotherapy equipment. Early in the development of this model it became clear that some representation of the clinical decision making process would be required. The model contains several simple approximations to these processes. Firstly, when a patient presents at hospital an assessment is made about what type of treatment to offer. Radical treatment, i.e. with curative intent, is offered only if the patient is not ‘too poorly’ as assessed by WHO guidelines. Patients not offered radical treatment may be offered palliative treat-
ment where a very different plan of fractionation and dose is given to extend life. In the model, it is possible to calculate how long the patient would survive without treatment and this is used as the basis for modelling the ‘too poorly at presentation’ criterion. After an indeterminate wait for treatment, patients have to be reassessed when treatment is due to start because treatment is itself arduous and it is unfortunately true that some patients have deteriorated so badly by the time treatment is due to commence that it would not be in the patients best interests to continue. A cohort of patients is constructed and Monte Carlo simulation is based on the assumption that a number of parameters are distributed according to probability distributions. The following five parameters are assumed to be distributed normally in the population: (1) number of normal brain cells at presentation, (2) number of normal brain cells at death, (3) interaction parameter between the normal brain cells and cancer cells, (4) tumour doubling time, and (5) delay to start radiotherapy treatment. A sixth parameter, the number of cancer cells that survive a single fraction of radiation, is assumed to follow a skewed distribution constrained between 0 and 1. This model can be fitted to clinical data using optimization based on established simulated annealing and folding polygon techniques [19]. An example is shown in Fig. 5 where this model has been fitted to data from palliative patients at the Addenbrooke’s Hospital. The construction of this model has asked some important and often overlooked questions of the clinicians and scientists involved. One example, for instance, is that very little work has ever been done on the neurological basis of death in GBM patients. More interestingly, these models can be used to ask some very interesting questions such as: (1) What might be the basis of the conflicting evidence in the literature about the effects of waiting times? 1
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and protocols for patient assessment and selection at the largest scale.
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The author wishes to thank the Life Sciences Interface of the Engineering and Physical Sciences Research Council for the Discipline Hopping Grant, the Master and Fellows of Fitzwilliam College, Cambridge, for the Visiting Fellowship and the Marie Curie Research Training Network ‘‘Cellion’’ for assistance to attend COSIRES 2006.
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Fig. 6. Prediction of tumour age distribution at presentation.
(2) Assuming this could be done safely, how would increasing dose relate to patient survival? (3) What is the age distribution of tumours at presentation? These are interesting questions and a model such as that described can be a guide in designing clinical trials or experiments to find answers. Some of these questions will be extremely difficult and expensive to answer by other means. Fig. 6 shows predictions for the age distribution of tumours at presentation, these data would be very hard to verify clinically because screening the population for such a rare condition could not be justified by a cost – benefit analysis. 7. Conclusions and future work The use of advanced mathematical models in clinical oncology and radiotherapy is at an early stage. The most advanced models tend to concentrate on the formation and growth of solid tumours. These models do not always include the effects that radiotherapy has on either the tumour or the patient. Advanced models rarely include factors such as the mechanisms by which tumour growth causes death or the effect that patient selection for different treatment strategies has on outcome. There is much work to be done in this area and multiscale models will play an important role in combining and integrating knowledge of the molecular biology of cells at the smallest scale, up to models of hospital operations
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