- Email: [email protected]
Optimal control of drug administration in cancer chemotherapy
1066
Book
effectively sacrifice linearity for accuracy in the sense that they remain single-step methods, but the formula for integration is not z-transformable. The accuracy is gained via a multiple function evaluation. The analysis is more difficult; however, some techniques that were useful when analysing the LMM case can be applied here as well. In this case the ‘gain’ is a polynomial function of AT, and a method for constructing the stability regions is also presented. Chapter 7 deals with stiff systems. Such systems are characterized by the fact that the simulation timestep required for stability is much smaller than that required for accuracy in the important dynamics. An analysis using stability region gives good insight into the problems encountered when simulating stiff systems. Various methods for successfully simulating stiff system are presented, including stability region placement, matrix stability region placement and an enlargement of the RK stability region at the price of increasing the local truncation error. Chapter 8 is devoted to nonlinear systems. No specific integration method is presented here, but the possible behaviours of nonlinear systems are demonstrated for continuous systems of orders 1, 2 and 3, the last of these being capable of chaotic behaviour. The purpose of the chapter is to provide some background information on the possible dynamics of nonlinear systems. The approach is more intuitive than mathematical, but it provides a nice introduction into the exciting area of nonlinear differential equations. Chapter 9 presents method for performing multiple integration. It starts with the problem of double integration resulting from the solution of important equations of classical physics (equations of motion without friction). The problem is solved by applying a linear multistep method. The equations for the coefficients of the integration formulae for the pre-specified local truncation error are derived and the
‘views stability regions discussed. Further, it is shown that the approach can easily be generalized to high-order integration. Chapter 10 contains concluding discussions. The first topic treated is the trade-off among accuracy, stability and computation time. As an example, it is shown that the use of a more precise method does not necessarily imply higher precision of the solution for some particular case. The second topic discussed concerns hints for the choice of a method and a timestep. In conclusion, I found the book very interesting, and it gives me sufficient information in an area that is only ancillary to my work. This book is not a cookbook. Although you can find many methods in the appendix, they are not commented upon, Also, those who seek a recommendation for a specific method will not be completely satisfied. The book tries to provide the reader with genera1 insight to the problems of digital simulation. It gives an excellent introduction to the problems of digital simulation, and a nice analysis of typical methods. The method of presentation is clear and suitable for students and any beginner in the area, especially those who are familiar with control theory. About
rhe reviewer
Josef Bohm is currently working in the Institute of Information Theory and Automation at the Czech Academy of Sciences in the group dealing with adaptive control systems. His initial interests were in the field of identification and the approximation of systems. They then switched to problems of adaptive control: control algorithms, methods for the numerical solution of LQ optimization, possibilities of incorporating some important practical limitations into the optimization process, and the design of effective algorithms for on-line LQ optimization. Recently, he has concentrated on problems of mismodeling and its influence on the robustness of LQ design.
Optimal Control of Drug Administration
in Cancer
Chemotherapy* R. Martin and K. L. Teo Reviewer: MIKE CHAPPELL Department of Engineering, Coventry CV4 7AL, U.K.
University
of
Warwick,
The aim of this book is to present new mathematical techniques developed in order to optimize the scheduling of drugs in the treatment of cancer. The book is a research monograph, not only containing work published in research papers by the authors but also including new theoretical material developed in order to solve the control problems considered. A comprehensive survey of published articles related to the general problem of cancer treatment and the associated methods of drug administration is provided. Very simple mathematical models are used to describe the treatment and its aim. Each of these models includes the growth of tumour cells, the corresponding concentration of anti-cancer drug present, the manner in which the drug is scheduled, its effect upon the tumour and also any toxic side-effects on surrounding tissues and organs. The actual dynamical system describing both the tumour growth and the drug concentration includes just these two state variables, and is a combined first-order nonlinear differential/algebraic model. Three different possible forms of nonlinearity for tumour growth are considered, namely exponential, logistic *Optimal
Control
of
Drug
Administration
in
Cancer
by R. Martin and K. L. Teo. World Scientific (1994). ISBN 9810214286. Chemotherapy
and Gompertz (logarithmic) type growth. The anti-cancer drug input to the system is described by either a continuous function or a set of discrete doses, and choice of these inputs gives rise to optimal control or optimal parameter selection models respectively. Various system constraints are incorporated in the models considered in order to take account of the maximum permissable drug concentration, the cumulative toxicity of the drug and the maximum permissable tumour burden or number of cells present during or at the end of the treatment. Various drug administration scenarios are considered, and in the majority of cases the resulting optima1 control problem is solved numerically since general techniques are not available for solving such problems analytically. Even numerical solutions prove difficult to obtain in some cases where a large number of inequality constraints are included. For these cases the constraint transcription method is applied in order to reduce all these constraints to just one equivalent constraint. This single equivalent constraint is described as a ‘multiple characteristic time constraint’, and new theory is introduced in order to deal with this. This theory, it seems, could be applied to more general optimal control or optimal parameter selection problems where a similar reduction of constraints is possible. An apparent feature of this new theory is that the corresponding costate function related to the multiple characteristic time constraint has jump discontinuities. This may be an undesirable characteristic for other more general optimal control problems.
Book Reviews For all of the models considered the optimal control problem effectively falls into two distinct categories: one where the treatment time is fixed, the other where this time is optimized subject to certain criteria. Initially the treatment is considered to last for a fixed period of time (in practice covering a number of months) where the drug is administered at discrete intermediate times (every few weeks). The problem is then to optimise the dose in order to keep the tumour size and growth below a certain level and simultaneously keep the toxic effects of the drug within feasible bounds. The drawback of this approach is that the actual length of time for the treatment is arbitrary, and it is unknown what effect this has on the optimal drug regimen. The periods of treatment and drug scheduling used correspond to those currently employed in practice. The most interesting results obtained from the analysis indicate that the optimal drug regimen would involve the application of small doses over the initial phase of the treatment, followed by substantially larger doses over the later phase. This is contrary to current practice, where substantially larger doses are administered over the initial phase. However, some evidence from clinical trials concurring with the theoretically obtained optimal schedule is offered in the text. An inherent problem in chemotherapy is that some tumour cells may be resistant to the drug(s) administered. Further models are developed to include this factor, and the optimal control problems are developed for situations where the time of the treatment is optimized so that either the tumour is destroyed or an overgrowth of tumour cells resistant to the drug arises. Further cases are considered where the time over which the mass of a drug resistant tumour lies below a certain level is maximized-the survival time. For a drug-resistant tumour the application of two cross-resistant drugs is also considered. An interesting feature of the results obtained is that low-intensity therapy generally appears more effective than high-intensity therapy (where the greatest proportion of the tumour is killed as quickly as possible)-particularly where Gompertz growth is considered. Such therapy is contrary to current practice. Several suggestions for further research in this field are offered by the authors. The underlying model used to characterise tumour growth and drug concentration is a very simple one and raises a number of issues. A number of factors could also be incorporated in the model to make it somewhat more realistic. A cell-loss parameter is included in the system, but this is assumed to be a constant parameter, which is certainly not the case for a number of anti-cancer drugs, particularly at high dose levels. Perhaps a time varying cell-loss parameter might be more appropriate? The simplicity of the model also does not allow for the whole-body distribution of the drug administered to be assessed. Chemotherapeutic drugs may bind within the body to proteins or other agents, thus creating greater and more prolonged background toxicity. Knowledge of the wholebody distribution is therefore a key factor in terms of
assessment of the amount of the toxicity to which bodily tissues and organs are subjected. Another factor not included is cell repopulation or relative effectiveness of the anti-cancer drug. An assumption made is that all cells killed by the drug were killed instantaneously and killed outright. This may certainly not be the case in practice. Inclusion of these, and other, factors would certainly make the model more complex, maybe computationally intractable, but would perhaps enhance its realism. The results obtained without these factors may consequently be somewhat over-optimistic. However, the simplicity of the models considered does have the advantage that it is tractable, and thus results can be generated that highlight the important trends in the optimal control of anti-cancer drug administration. The optimal control problems formulated are standard in format and correspond closely to optimal control problems incorporating inequality constraints in other application areas. The introduction of new theory related to multiple characteristic time constraints is notable, and it would appear that the techniques presented could be applied to many other similarly formulated optimal control or optimal parameter selection problems that have a large number of constraints or parameters. Overall, the book seems very readable and is clearly written, albeit somewhat repetitive in content at times. The book is aimed at postgraduate scientific level and beyond. As it is essentially a research monograph, the book would not be particularly useful as a teaching aid. It does, however, contain some very interesting results and applications for those with interests in optimal control. The results concerning the administration of anti-cancer drugs in chemotherapy would naturally be of primary interest to medics working in this field. However, the medical fraternity may be somewhat disinclined to read the book when encountering the mathematical terminology, format and subtleties employed.
About
the reviewer
Mike Chappell has BSc, MSc and PhD degrees in Mathematics, all obtained from the University of Warwick. He is currently a Lecturer in the Department of Engineering at the same University, and is also Deputy Director of the Control and Instrument Systems Centre based within that Department. His research interests lie in the fields of mathematical modelling of physical processes (and in particular biomedical and pharmacokinetic systems), compartmental modelling, control and parameterised system identifiability and indistinguishability. In particular, over a recent number of years he has been closely involved with members of the Department of Immunology at the Medical School of the University of Birmingham in the formulation and development of a mathematical model characterising tumour targeting by monoclonal antibodies.
Book
effectively sacrifice linearity for accuracy in the sense that they remain single-step methods, but the formula for integration is not z-transformable. The accuracy is gained via a multiple function evaluation. The analysis is more difficult; however, some techniques that were useful when analysing the LMM case can be applied here as well. In this case the ‘gain’ is a polynomial function of AT, and a method for constructing the stability regions is also presented. Chapter 7 deals with stiff systems. Such systems are characterized by the fact that the simulation timestep required for stability is much smaller than that required for accuracy in the important dynamics. An analysis using stability region gives good insight into the problems encountered when simulating stiff systems. Various methods for successfully simulating stiff system are presented, including stability region placement, matrix stability region placement and an enlargement of the RK stability region at the price of increasing the local truncation error. Chapter 8 is devoted to nonlinear systems. No specific integration method is presented here, but the possible behaviours of nonlinear systems are demonstrated for continuous systems of orders 1, 2 and 3, the last of these being capable of chaotic behaviour. The purpose of the chapter is to provide some background information on the possible dynamics of nonlinear systems. The approach is more intuitive than mathematical, but it provides a nice introduction into the exciting area of nonlinear differential equations. Chapter 9 presents method for performing multiple integration. It starts with the problem of double integration resulting from the solution of important equations of classical physics (equations of motion without friction). The problem is solved by applying a linear multistep method. The equations for the coefficients of the integration formulae for the pre-specified local truncation error are derived and the
‘views stability regions discussed. Further, it is shown that the approach can easily be generalized to high-order integration. Chapter 10 contains concluding discussions. The first topic treated is the trade-off among accuracy, stability and computation time. As an example, it is shown that the use of a more precise method does not necessarily imply higher precision of the solution for some particular case. The second topic discussed concerns hints for the choice of a method and a timestep. In conclusion, I found the book very interesting, and it gives me sufficient information in an area that is only ancillary to my work. This book is not a cookbook. Although you can find many methods in the appendix, they are not commented upon, Also, those who seek a recommendation for a specific method will not be completely satisfied. The book tries to provide the reader with genera1 insight to the problems of digital simulation. It gives an excellent introduction to the problems of digital simulation, and a nice analysis of typical methods. The method of presentation is clear and suitable for students and any beginner in the area, especially those who are familiar with control theory. About
rhe reviewer
Josef Bohm is currently working in the Institute of Information Theory and Automation at the Czech Academy of Sciences in the group dealing with adaptive control systems. His initial interests were in the field of identification and the approximation of systems. They then switched to problems of adaptive control: control algorithms, methods for the numerical solution of LQ optimization, possibilities of incorporating some important practical limitations into the optimization process, and the design of effective algorithms for on-line LQ optimization. Recently, he has concentrated on problems of mismodeling and its influence on the robustness of LQ design.
Optimal Control of Drug Administration
in Cancer
Chemotherapy* R. Martin and K. L. Teo Reviewer: MIKE CHAPPELL Department of Engineering, Coventry CV4 7AL, U.K.
University
of
Warwick,
The aim of this book is to present new mathematical techniques developed in order to optimize the scheduling of drugs in the treatment of cancer. The book is a research monograph, not only containing work published in research papers by the authors but also including new theoretical material developed in order to solve the control problems considered. A comprehensive survey of published articles related to the general problem of cancer treatment and the associated methods of drug administration is provided. Very simple mathematical models are used to describe the treatment and its aim. Each of these models includes the growth of tumour cells, the corresponding concentration of anti-cancer drug present, the manner in which the drug is scheduled, its effect upon the tumour and also any toxic side-effects on surrounding tissues and organs. The actual dynamical system describing both the tumour growth and the drug concentration includes just these two state variables, and is a combined first-order nonlinear differential/algebraic model. Three different possible forms of nonlinearity for tumour growth are considered, namely exponential, logistic *Optimal
Control
of
Drug
Administration
in
Cancer
by R. Martin and K. L. Teo. World Scientific (1994). ISBN 9810214286. Chemotherapy
and Gompertz (logarithmic) type growth. The anti-cancer drug input to the system is described by either a continuous function or a set of discrete doses, and choice of these inputs gives rise to optimal control or optimal parameter selection models respectively. Various system constraints are incorporated in the models considered in order to take account of the maximum permissable drug concentration, the cumulative toxicity of the drug and the maximum permissable tumour burden or number of cells present during or at the end of the treatment. Various drug administration scenarios are considered, and in the majority of cases the resulting optima1 control problem is solved numerically since general techniques are not available for solving such problems analytically. Even numerical solutions prove difficult to obtain in some cases where a large number of inequality constraints are included. For these cases the constraint transcription method is applied in order to reduce all these constraints to just one equivalent constraint. This single equivalent constraint is described as a ‘multiple characteristic time constraint’, and new theory is introduced in order to deal with this. This theory, it seems, could be applied to more general optimal control or optimal parameter selection problems where a similar reduction of constraints is possible. An apparent feature of this new theory is that the corresponding costate function related to the multiple characteristic time constraint has jump discontinuities. This may be an undesirable characteristic for other more general optimal control problems.
Book Reviews For all of the models considered the optimal control problem effectively falls into two distinct categories: one where the treatment time is fixed, the other where this time is optimized subject to certain criteria. Initially the treatment is considered to last for a fixed period of time (in practice covering a number of months) where the drug is administered at discrete intermediate times (every few weeks). The problem is then to optimise the dose in order to keep the tumour size and growth below a certain level and simultaneously keep the toxic effects of the drug within feasible bounds. The drawback of this approach is that the actual length of time for the treatment is arbitrary, and it is unknown what effect this has on the optimal drug regimen. The periods of treatment and drug scheduling used correspond to those currently employed in practice. The most interesting results obtained from the analysis indicate that the optimal drug regimen would involve the application of small doses over the initial phase of the treatment, followed by substantially larger doses over the later phase. This is contrary to current practice, where substantially larger doses are administered over the initial phase. However, some evidence from clinical trials concurring with the theoretically obtained optimal schedule is offered in the text. An inherent problem in chemotherapy is that some tumour cells may be resistant to the drug(s) administered. Further models are developed to include this factor, and the optimal control problems are developed for situations where the time of the treatment is optimized so that either the tumour is destroyed or an overgrowth of tumour cells resistant to the drug arises. Further cases are considered where the time over which the mass of a drug resistant tumour lies below a certain level is maximized-the survival time. For a drug-resistant tumour the application of two cross-resistant drugs is also considered. An interesting feature of the results obtained is that low-intensity therapy generally appears more effective than high-intensity therapy (where the greatest proportion of the tumour is killed as quickly as possible)-particularly where Gompertz growth is considered. Such therapy is contrary to current practice. Several suggestions for further research in this field are offered by the authors. The underlying model used to characterise tumour growth and drug concentration is a very simple one and raises a number of issues. A number of factors could also be incorporated in the model to make it somewhat more realistic. A cell-loss parameter is included in the system, but this is assumed to be a constant parameter, which is certainly not the case for a number of anti-cancer drugs, particularly at high dose levels. Perhaps a time varying cell-loss parameter might be more appropriate? The simplicity of the model also does not allow for the whole-body distribution of the drug administered to be assessed. Chemotherapeutic drugs may bind within the body to proteins or other agents, thus creating greater and more prolonged background toxicity. Knowledge of the wholebody distribution is therefore a key factor in terms of
assessment of the amount of the toxicity to which bodily tissues and organs are subjected. Another factor not included is cell repopulation or relative effectiveness of the anti-cancer drug. An assumption made is that all cells killed by the drug were killed instantaneously and killed outright. This may certainly not be the case in practice. Inclusion of these, and other, factors would certainly make the model more complex, maybe computationally intractable, but would perhaps enhance its realism. The results obtained without these factors may consequently be somewhat over-optimistic. However, the simplicity of the models considered does have the advantage that it is tractable, and thus results can be generated that highlight the important trends in the optimal control of anti-cancer drug administration. The optimal control problems formulated are standard in format and correspond closely to optimal control problems incorporating inequality constraints in other application areas. The introduction of new theory related to multiple characteristic time constraints is notable, and it would appear that the techniques presented could be applied to many other similarly formulated optimal control or optimal parameter selection problems that have a large number of constraints or parameters. Overall, the book seems very readable and is clearly written, albeit somewhat repetitive in content at times. The book is aimed at postgraduate scientific level and beyond. As it is essentially a research monograph, the book would not be particularly useful as a teaching aid. It does, however, contain some very interesting results and applications for those with interests in optimal control. The results concerning the administration of anti-cancer drugs in chemotherapy would naturally be of primary interest to medics working in this field. However, the medical fraternity may be somewhat disinclined to read the book when encountering the mathematical terminology, format and subtleties employed.
About
the reviewer
Mike Chappell has BSc, MSc and PhD degrees in Mathematics, all obtained from the University of Warwick. He is currently a Lecturer in the Department of Engineering at the same University, and is also Deputy Director of the Control and Instrument Systems Centre based within that Department. His research interests lie in the fields of mathematical modelling of physical processes (and in particular biomedical and pharmacokinetic systems), compartmental modelling, control and parameterised system identifiability and indistinguishability. In particular, over a recent number of years he has been closely involved with members of the Department of Immunology at the Medical School of the University of Birmingham in the formulation and development of a mathematical model characterising tumour targeting by monoclonal antibodies.