Some optimal control problems in cancer chemotherapy with a toxicity limit

We investigate some models of a cancer chemotherapy problem where the normal cell population must be maintained above a lower limit and a measure of the total drug use is bounded as a limit of toxicity. Three sets of objective functions are analyzed, and some numerical solutions are provided for the last case. In all cases the structure of the optimal solution is similar, including a bolus application of drug followed later by continuous infusion.     

Function-valued adaptive dynamics and optimal control theory

In this article we further develop the theory of adaptive dynamics of function-valued traits. Previous work has concentrated on models for which invasion fitness can be written as an integral in which the integrand for each argument value is a function of the strategy value at that argument value only. For this type of models of direct effect, singular strategies can be found using the calculus of variations, with singular strategies needing to satisfy Euler’s equation with environmental feedback. In a broader, more mechanistically oriented class of models, the function-valued strategy affects a process described by differential equations, and fitness can be expressed as an integral in which the integrand for each argument value depends both on the strategy and on process variables at that argument value. In general, the calculus of variations cannot help analyzing this much broader class of models. Here we explain how to find singular strategies in this class of process-mediated models using optimal control theory. In particular, we show that singular strategies need to satisfy Pontryagin’s maximum principle with environmental feedback. We demonstrate the utility of this approach by studying the evolution of strategies determining seasonal flowering schedules.     

The effect of heterogeneity on optimal regimens in cancer chemotherapy

Optimal drug regimens for cancer chemotherapy are determined when knowledge is only available on the behaviour of the tumour and the drugs used, over a population of patients. The case of two drugs is investigated where they are equivalent on average. Our calculations indicate that the optimal regimen has both drugs given initially but then sequences the two drugs. Our calculations also indicate that as tumour heterogeneity increases, the benefit to be gained from the optimal regimen can decrease in comparison to reasonable regimens. This has the effect of complicating the calculation of optimal regimens in a clinical setting, and may explain why results in experimental oncology fail to carry over to clinical oncology.     

Cell cycle control and cancer chemotherapy

As detailed information accumulates about how cell cycle events are regulated, we can expect new opportunities for application to cancer therapy. The altered expression of oncogenes and tumor suppressor genes that commonly occurs in human cancers may impair the ability of the cells to respond to metabolic perturbations or stress. Impaired cell cycle regulation would make cells vulnerable to pharmacologic intervention by drug regimens tailored to the defects existing in particular tumors. Recent findings that may become applicable to therapy are reviewed, and the possible form of new therapeutic stratagems is considered.     

Sensitivity enhancement in NMR of macromolecules by application of optimal control theory

NMR of macromolecules is limited by large transverse relaxation rates. In practice, this results in low efficiency of coherence transfer steps in multidimensional NMR experiments, leading to poor sensitivity and long acquisition times. The efficiency of coherence transfer can be maximized by design of relaxation optimized pulse sequences using tools from optimal control theory. In this paper, we demonstrate that this approach can be adopted for studies of large biological systems, such as the 800 kDa chaperone GroEL. For this system, the ¹H–¹⁵N coherence transfer module presented here yields an average sensitivity enhancement of 20–25% for cross-correlated relaxation induced polarization transfer (CRIPT) experiments.Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1007/s10858-005-3592-0     

Optimal control problems arising in cell-cycle-specific cancer chemotherapy

We explore mathematical properties of models of cancer chemotherapy including cell-cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, (a) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or (b) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell-cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell-cycle dependent drug action.     

Combining optimal control theory and molecular dynamics for protein folding

A new method to develop low-energy folding routes for proteins is presented. The novel aspect of the proposed approach is the synergistic use of optimal control theory with Molecular Dynamics (MD). In the first step of the method, optimal control theory is employed to compute the force field and the optimal folding trajectory for the Cα atoms of a Coarse-Grained (CG) protein model. The solution of this CG optimization provides an harmonic approximation of the true potential energy surface around the native state. In the next step CG optimization guides the MD simulation by specifying the optimal target positions for the Cα atoms. In turn, MD simulation provides an all-atom conformation whose Cα positions match closely the reference target positions determined by CG optimization. This is accomplished by Targeted Molecular Dynamics (TMD) which uses a bias potential or harmonic restraint in addition to the usual MD potential. Folding is a dynamical process and as such residues make different contacts during the course of folding. Therefore CG optimization has to be reinitialized and repeated over time to accomodate these important changes. At each sampled folding time, the active contacts among the residues are recalculated based on the all-atom conformation obtained from MD. Using the new set of contacts, the CG potential is updated and the CG optimal trajectory for the Cα atoms is recomputed. This is followed by MD. Implementation of this repetitive CG optimization-MD simulation cycle generates the folding trajectory. Simulations on a model protein Villin demonstrate the utility of the method. Since the method is founded on the general tools of optimal control theory and MD without any restrictions, it is widely applicable to other systems. It can be easily implemented with available MD software packages.     

A mathematical model of cancer chemotherapy with an optimal selection of parameters

An optimal parameter selection model of cancer chemotherapy is presented which describes the treatment of a tumor over a fixed period of time by the repeated administration of a single drug. The drug is delivered at evenly spaced intervals over the treatment period at doses to be selected by the model. The model constructs a regimen that both minimizes the tumor population at the end of the treatment and satisfies constraints on the drug toxicity and intermediate tumor size. Numerical solutions show that an optimal regimen withholds the bulk of the doses until the end of the treatment period. When a drug used is of either moderate or low effectiveness, an optimal regimen is superior to a schedule that delivers all of the drug at the beginning of the treatment. This study questions whether the current method for the administration of chemotherapy is optimal and suggests that alternative regimens should be considered.     

Design of optimal cancer chemotherapy using a continuous-time state model of cell kinetics

A continuous bilinear model in state space is used to describe the cell kinetics of a tumor-cell population under the effects of chemotherapy. Firstly, the time-course behavior of a Chinese-hamster-ovary (CHO) cell population is simulated to demonstrate the utility of the model. Then, an optimal strategy for cancer treatment is derived, based on the need to balance the effects on both cancerous and normal tissues. The performance index minimized is the sum of the weighted tumor population and the weighted total drug dosage. The optimization problem has resulted in a two-point boundary-value problem (TPBVP) with a bang-bang control policy, which is solved by the switching-time variation method (STVM). Computer simulation of CHO cells is also carried out as a numerical example of determining optimal cancer chemotherapy.     

Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

Role of chemotherapy in the treatment of anaplastic thyroid cancer --personal observations]

Results of chemotherapy applied in 23 patients with anaplastic thyroid cancer are presented. Chemotherapy combined with local treatment is a suggested therapy for treatment the locally advanced anaplastic thyroid cancer.     

Early distant relapse after optimal local control in locally advanced rectal cancer

We present a case of locally advanced rectal cancer with initial optimal local control after neoadjuvant concurrent chemoradiotherapy followed by surgery; early liver recurrence then occurred and was treated again with curative intent with neoadjuvant combination chemotherapy followed by liver surgery. We reflect on this difficult problem and discuss relevant topics to this case report.     

The optimal dose of CAF regimen in adjuvant chemotherapy for breast cancer patients at stage IIB

This paper deals with the idea of balancing drug effects on tumor and normal cell populations based on a variety of criteria, which is evaluated by the oncologist for breast cancer patients at stage IIB. In this paper, the optimal controller represents the optimal drug dosage of CAF (Cyclophosphamide, Adriamycin and Fluorouracil) regimen in adjuvant chemotherapy after surgery for these patients. We determined the doses of CAF regimen by minimizing a cost function with some constraints. The cost function includes the cancer cell and the normal cell growth dynamics with prescribed weighting coefficients for each patient. The physician determines these weighting coefficients based on some individual parameters. The optimal treatment schedules are computed based on a trade-off between the cancer cell reduction and the normal cell preserving. Numerical simulations are given to illustrate the accuracy of the optimal controller.     

On control and optimal control in biodynamic systems

In any control system for which the number of independent controls is smaller than the number of degrees of freedom to be controlled, our choice of control in any state is restricted to a submanifold of smaller dimension than the tangent space. This simple fact has a number of important consequences for questions of biological import; we consider its implications for adaptation, for senescent phenomena and for the determination of tertiary structures of polypeptides through control of certain average properties. We also formulate the Pontryagin Maximum Principle of Optimal control theory in such a way as to inquire whether specific biodynamic systems can be regarded as optimal with respect to rate of accumulation of particular quantities of the system. We find that if this is possible, the quantity in question must play the role of a clock.     

A network control theory approach to modeling and optimal control of zoonoses: case study of brucellosis transmission in sub-Saharan Africa

Background

Developing control policies for zoonotic diseases is challenging, both because of the complex spread dynamics exhibited by these diseases, and because of the need for implementing complex multi-species surveillance and control efforts using limited resources. Mathematical models, and in particular network models, of disease spread are promising as tools for control-policy design, because they can provide comprehensive quantitative representations of disease transmission.

Methodology/Principal Findings

A layered dynamical network model for the transmission and control of zoonotic diseases is introduced as a tool for analyzing disease spread and designing cost-effective surveillance and control. The model development is achieved using brucellosis transmission among wildlife, cattle herds, and human sub-populations in an agricultural system as a case study. Precisely, a model that tracks infection counts in interacting animal herds of multiple species (e.g., cattle herds and groups of wildlife for brucellosis) and in human subpopulations is introduced. The model is then abstracted to a form that permits comprehensive targeted design of multiple control capabilities as well as model identification from data. Next, techniques are developed for such quantitative design of control policies (that are directed to both the animal and human populations), and for model identification from snapshot and time-course data, by drawing on recent results in the network control community.

Conclusions/Significance

The modeling approach is shown to provide quantitative insight into comprehensive control policies for zoonotic diseases, and in turn to permit policy design for mitigation of these diseases. For the brucellosis-transmission example in particular, numerous insights are obtained regarding the optimal distribution of resources among available control capabilities (e.g., vaccination, surveillance and culling, pasteurization of milk) and points in the spread network (e.g., transhumance vs. sedentary herds). In addition, a preliminary identification of the network model for brucellosis is achieved using historical data, and the robustness of the obtained model is demonstrated. As a whole, our results indicate that network modeling can aid in designing control policies for zoonotic diseases.     

Sustainable ecosystem management using optimal control theory: part 2 (stochastic systems)

Sustainable development of ecosystems through external ecosystem management is assuming importance for the environmentalists. To that effect, previous work by the authors looked at the option of manipulating population dynamics of the species in an ecosystem to achieve sustainability. Fisher information is used as the quantifying measure of sustainability and optimal control theory is used to derive the control profiles. However, that work considered only deterministic systems. Uncertainty being prevalent in all systems, particularly in natural systems, this paper extends that work to analyse uncertain systems. Predator-prey models are used to model the species populations and different control philosophies are compared. Ito mean reverting process is used to model the stochastic process, and stochastic maximum principle is used to derive the control profiles. The results for the objective of FI variance minimization qualitatively agree with those for the deterministic system, while the results for the FI maximization objective differ. It is observed that the instability associated with the FI maximization objective for deterministic systems is absorbed by the noise introduced by the uncertainty. Quantitatively, it is observed that the degree of uncertainty, along with its presence, is also important to identify the most appropriate management strategy.     

Sustainable ecosystem management using optimal control theory: part 1 (deterministic systems)

The concept of sustainability, an abstract one by its nature, has been given a mathematical representation through the use of Fisher information as a measure. It is used to propose the sustainability hypotheses for dynamical systems, which has paved the way to achieve sustainable development through externally enforced control schemes. For natural systems, this refers to the task of ecosystem management, which is complicated due the lack of clear objectives. This work attempts to incorporate the idea of sustainability in ecosystem management. The natural regulation of ecosystems suggests two possible control options, top-down control and bottom-up control. A comparison of these two control philosophies is made on generic food chain models using the objectives derived from the sustainability hypotheses. Optimal control theory is used to derive the control profiles to handle the complex nature of the models and the objectives. The results indicate a strong relationship between the hypotheses and the dynamic behavior of the models, supporting the use of Fisher information as a measure. As regards to ecosystem management, it has been observed that top-down control is more aggressive but can result in instability, while bottom-up control is guaranteed to give a stable and improved dynamic response. The results also indicate that bottom-up control is a better option to affect shifts in the dynamic regimes of a system, which may be required to recover the system from a natural disaster like the hurricane Katrina.     

Integrating optimal foraging and optimal oviposition theory in plant–insect research

The current approach for studying host selection by phytophagous insects is mainly based on optimal oviposition theory, i.e. the preference–performance hypothesis. Almost no attention has been given to optimal foraging theory. However, recent papers and additional evidence given in this work illustrate that also optimal foraging may shape host preference patterns of phytophagous insects. Therefore and because optimal foraging and optimal oviposition may oppose conflicting needs to phytophagous insects, we plea for an integration of optimal foraging and optimal oviposition in plant–insect research. We argue how this may improve our understanding of plant–insect interactions.