On optimal delivery of combination therapy for tumors

Next-generation chemotherapy treatments based on black hole algorithms: From cancer remission to chronic disease management

PROBLEM

Therapeutic planning strategies have been developed to enhance the effectiveness of cancer drugs. Nevertheless, their performance is highly limited by the inefficient biological representativeness of predictive tumor growth models, which hinders their translation to clinical practice.

OBJECTIVE

This study proposes a disruptive approach to oncology based on nature-inspired control using realistic Black Hole physical laws, in which tumor masses are trapped to experience attraction dynamics on their path to complete remission or to become a chronic disease. This control method is designed to operate independently of individual patient idiosyncrasies, including high tumor heterogeneities and highly uncertain tumor dynamics, making it a promising avenue for advancing beyond the limitations of the traditional survival probabilistic paradigm.

DESIGN

Here, we provide a multifaceted study of chemotherapy therapeutic planning that includes: (1) the design of a pioneering controller algorithm based on physical laws found in the Black Holes; (2) investigation of the ability of this controller algorithm to ensure stable equilibrium treatments; and (3) simulation tests concerning tumor volume dynamics using drugs with significantly different pharmacokinetics (Cyclophosphamide and Atezolizumab), tumor volumes (200 mm3 and 12 732 mm3) and modeling characterizations (Gompertzian and Logistic tumor growth models).

RESULTS

Our results highlight the ability of this new astrophysical-inspired control algorithm to perform effective chemotherapy treatments for multiple tumor-treatment scenarios, including tumor resistance to chemotherapy, clinical scenarios modelled by time-dependent parameters, and highly uncertain tumor dynamics.

CONCLUSIONS

Our findings provide strong evidence that cancer therapy inspired by phenomena found in black holes can emerge as a disruptive paradigm. This opens new high-impacting research directions, exploring synergies between astrophysical-inspired control algorithms and Artificial Intelligence applied to advanced personalized cancer therapeutics.

Combination Chemotherapy Optimization with Discrete Dosing

Chemotherapy drug administration is a complex problem that often requires expensive clinical trials to evaluate potential regimens; one way to alleviate this burden and better inform future trials is to build reliable models for drug administration. This paper presents a mixed-integer program for combination chemotherapy (utilization of multiple drugs) optimization that incorporates various important operational constraints and, besides dose and concentration limits, controls treatment toxicity based on its effect on the count of white blood cells. To address the uncertainty of tumor heterogeneity, we also propose chance constraints that guarantee reaching an operable tumor size with a high probability in a neoadjuvant setting. We present analytical results pertinent to the accuracy of the model in representing biological processes of chemotherapy and establish its potential for clinical applications through a numerical study of breast cancer.

Tumour growth control: analysis of alternative approaches

In this work we address the problem of tumour growth control by properly exploiting a low-dimensional model that grounds on the Chemical Reaction Network (CRN) formalism. Originally conceived to work both in deterministic and stochastic frameworks, it is shown that, except for the case of very low number of tumour cells, the deterministic approach is appropriate to characterize the system behaviour, especially for control planning purposes. Two alternative control approaches are here investigated. One trivially assumes a constant infusion of external drug administration, the other is designed according to a state-feedback control scheme, with complete or partial knowledge of the state. Pros and cons of both control laws are investigated, showing that the tumour size at the beginning of the therapy plays a role of paramount importance for fixed infusion therapies, whilst only state-feedback laws can eradicate arbitrarily large tumours.

Design and implementation of an adaptive fuzzy sliding mode controller for drug delivery in treatment of vascular cancer tumours and its optimisation using genetic algorithm tool

In this paper, the side effects of drug therapy in the process of cancer treatment are reduced by designing two optimal non-linear controllers. The related gains of the designed controllers are optimised using genetic algorithm and simultaneously are adapted by employing the Fuzzy scheduling method. The cancer dynamic model is extracted with five differential equations, including normal cells, endothelial cells, cancer cells, and the amount of two chemotherapy and anti-angiogenic drugs left in the body as the engaged state variables, while double drug injection is considered as the corresponding controlling signals of the mentioned state space. This treatment aims to reduce the tumour cells by providing a timely schedule for drug dosage. In chemotherapy, not only the cancer cells are killed but also other healthy cells will be destroyed, so the rate of drug injection is highly significant. It is shown that the simultaneous application of chemotherapy and anti-angiogenic therapy is more efficient than single chemotherapy. Two different non-linear controllers are employed and their performances are compared. Simulation results and comparison studies show that not only adding the anti-angiogenic reduce the side effects of chemotherapy but also the proposed robust controller of sliding mode provides a faster and stronger treatment in the presence of patient parametric uncertainties in an optimal way. As a result of the proposed closed-loop drug treatment, the tumour cells rapidly decrease to zero, while the normal cells remain healthy simultaneously. Also, the injection rate of the chemotherapy drug is very low after a short time and converges to zero.

Numerical Investigations of the Fractional-Order Mathematical Model Underlying Immune-Chemotherapeutic Treatment for Breast Cancer Using the Neural Networks

The aim of this work is to design a stochastic framework to solve the fractional-order differential model based on the breast cancer progression during the immune-chemotherapeutic treatment phase, including certain control parameters such as anti-cancer medications, ketogenic diet and immune boosters. The developed model considers tumor density progression throughout chemotherapy treatment, as well as an immune response during normal cell–tumor cell interaction. This study’s subject seems to be to demonstrate the implications and significance of the fractional-order breast cancer mathematical model. The goal of these studies is to improve accuracy in the breast cancer model by employing fractional derivatives. This study also includes an integer, nonlinear mathematical system with immune-chemotherapeutic treatment impacts. The mathematical system divides the fractional-order breast cancer mathematical model among four manifestations: normal cell population (N), tumor cells (T), immune response class (I), and estrogen compartment (E), i.e., (NTIE). The fractional-order NTIE mathematical system is still not published previously, nor has it ever been addressed employing the stochastic solvers’ strength. To solve a fractional-order NTIE mathematical system, stochastic solvers based on the Levenberg–Marquardt backpropagation scheme (LMBS) and neural networks (NNs), namely, LMBNNs, are been constructed. To solve the fractional-order NTIE mathematical model, three cases with varying values for this same fractional order have been supplied. The statistics used to offer the numerical solutions of the fractional-order NTIE mathematical model are divided as follows: 75% in training, 15% in testing, and 10% in the authorization. The acquired numerical findings were compared using the reference solutions to determine the accuracy of the LMBNNs using Adams–Bashforth–Moulton. The numerical performances employing error histograms (EHs), state transitions (STs), regression, correlation, including mean square error (MSE) have been further supplied to authenticate overall capability, competence, validity, consistency, as well as exactness of such LMBNNs.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

Optimization of additive chemotherapy combinations for an in vitro cell cycle model with constant drug exposures

Proliferation of an in vitro population of cancer cells is described by a linear cell cycle model with n states, subject to provocation with m chemotherapeutic compounds. Minimization of a linear combination of constant drug exposures is considered, with stability of the system used as a constraint to ensure a stable or shrinking cell population. The main result concerns the identification of redundant compounds, and an explicit solution formula for the case where all exposures are nonzero. The orthogonal case, where each drug acts on a single and different stage of the cell cycle, leads to a version of the classic inequality between the arithmetic and geometric means. Moreover, it is shown how the general case can be solved by converting it to the orthogonal case using a linear invertible transformation. The results are illustrated with two examples corresponding to combination treatment with two and three compounds, respectively.

Analysis of the effectiveness of the treatment of solid tumors in two cases of drug administration

A complete stability analysis of the equilibrium solutions of a system modeling tumor chemotherapy is performed in two cases of administration of the treatment, by continuous infusion and by periodic infusion. Several numerical simulations illustrate and complement the theory.

Mathematical modeling of the immune-chemotherapeutic treatment of breast cancer under some control parameters

We consider a system of fractional-order differential equations to analyze breast cancer growth in the immune-chemotherapeutic treatment process under some control parameters: ketogenic diet, immune booster, and anti-cancer drugs. The established model assumes the growth of the tumor density under chemotherapy treatment and the immune response during the interaction between the normal cells and tumor cells. For the local stability of the critical points (tumor-free critical point, dead critical point, and co-existing critical point), we used the Routh-Hurwitz criteria to show the necessary effect of the immune booster; moreover, we addressed the ketogenic rate in the treatment process. Our theoretical and numerical studies pointed out that on early detection of the tumor density (with weak Allee effect) the treatment should be supported by ketogenic nutrition. Several examples are shown to present our theoretical findings.

Optimizing adaptive cancer therapy: dynamic programming and evolutionary game theory

Recent clinical trials have shown that adaptive drug therapies can be more efficient than a standard cancer treatment based on a continuous use of maximum tolerated doses (MTD). The adaptive therapy paradigm is not based on a preset schedule; instead, the doses are administered based on the current state of tumour. But the adaptive treatment policies examined so far have been largely ad hoc. We propose a method for systematically optimizing adaptive policies based on an evolutionary game theory model of cancer dynamics. Given a set of treatment objectives, we use the framework of dynamic programming to find the optimal treatment strategies. In particular, we optimize the total drug usage and time to recovery by solving a Hamilton-Jacobi-Bellman equation. We compare MTD-based treatment strategy with optimal adaptive treatment policies and show that the latter can significantly decrease the total amount of drugs prescribed while also increasing the fraction of initial tumour states from which the recovery is possible. We conclude that the use of optimal control theory to improve adaptive policies is a promising concept in cancer treatment and should be integrated into clinical trial design.

Designing combination therapies using multiple optimal controls

Strategic management of populations of interacting biological species routinely requires interventions combining multiple treatments or therapies. This is important in key research areas such as ecology, epidemiology, wound healing and oncology. Despite the well developed theory and techniques for determining single optimal controls, there is limited practical guidance supporting implementation of combination therapies. In this work we use optimal control theory to calculate optimal strategies for applying combination therapies to a model of acute myeloid leukaemia. We present a versatile framework to systematically explore the trade-offs that arise in designing combination therapy protocols using optimal control. We consider various combinations of continuous and bang-bang (discrete) controls, and we investigate how the control dynamics interact and respond to changes in the weighting and form of the pay-off characterising optimality. We demonstrate that the optimal controls respond non-linearly to treatment strength and control parameters, due to the interactions between species. We discuss challenges in appropriately characterising optimality in a multiple control setting and provide practical guidance for applying multiple optimal controls. Code used in this work to implement multiple optimal controls is available on GitHub.

On the Role of the Objective in the Optimization of Compartmental Models for Biomedical Therapies

We review and discuss results obtained through an application of tools of nonlinear optimal control to biomedical problems. We discuss various aspects of the modeling of the dynamics (such as growth and interaction terms), modeling of treatment (including pharmacometrics of the drugs), and give special attention to the choice of the objective functional to be minimized. Indeed, many properties of optimal solutions are predestined by this choice which often is only made casually using some simple ad hoc heuristics. We discuss means to improve this choice by taking into account the underlying biology of the problem.

An Agent-based Model for Investigating the Effect of Myeloid-Derived Suppressor Cells and its Depletion on Tumor Immune Surveillance

BACKGROUND

To predict the behavior of biological systems, mathematical models of biological systems have been shown to be useful. In particular, mathematical models of tumor-immune system interactions have demonstrated promising results in prediction of different behaviors of tumor against the immune system.

METHODS

This study aimed at the introduction of a new model of tumor-immune system interaction, which includes tumor and immune cells as well as myeloid-derived suppressor cells (MDSCs). MDSCs are immune suppressor cells that help the tumor cells to escape the immune system. The structure of this model is agent-based which makes possible to investigate each component as a separate agent. Moreover, in this model, the effect of low dose 5-fluorouracil (5-FU) on MDSCs depletion was considered.

RESULTS

Based on the findings of this study, MDSCs had suppressive effect on increment of immune cell number which consequently result in tumor cells escape the immune cells. It has also been demonstrated that low-dose 5-FU could help immune system eliminate the tumor cells through MDSCs depletion.

CONCLUSION

Using this new agent-based model, multiple injection of low-dose 5-FU could eliminate MDSCs and therefore might have the potential to be considered in treatment of cancers.

Maximization of viability time in a mathematical model of cancer therapy

In this paper, we study a dynamic optimization problem for a general nonlinear mathematical model for therapy of a lethal form of cancer. The model describes how the populations of cancer and normal cells evolve under the influence of the concentrations of nutrients (oxygen, glucose, etc.) and the applied therapeutic agent (drug). Regulated intensity of the therapy is interpreted as a time-dependent control strategy. The therapy (control) goal is to maximize the viability time, i. e., the duration of staying in a so-called safety region (which specifies safe living conditions of a patient in terms of constraints on the amounts of cancer and normal cells), subject to limited resources of the therapeutic agent. In a specific benchmark case, a novel optimality principle for admissible therapy strategies is established. It states that the optimal trajectories should finally reach a certain corner of the safety region or at least the upper constraint on the quantity of cancer cells. The results of numerical simulations appear to be in good agreement with the proposed principle. Typical qualitative structures of optimal treatment strategies are also obtained. Furthermore, important characteristics of the model are the competition coefficient (describing the negative influence of cancer cells on normal cells), the upper bound in the drug integral constraint, and the ratio between the therapy and damage coefficients (i. e., the ratio between the positive primary and negative side effects of the therapy).

Quantifying the effects of antiangiogenic and chemotherapy drug combinations on drug delivery and treatment efficacy

Tumor-induced angiogenesis leads to the development of leaky tumor vessels devoid of structural and morphological integrity. Due to angiogenesis, elevated interstitial fluid pressure (IFP) and low blood perfusion emerge as common properties of the tumor microenvironment that act as barriers for drug delivery. In order to overcome these barriers, normalization of vasculature is considered to be a viable option. However, insight is needed into the phenomenon of normalization and in which conditions it can realize its promise. In order to explore the effect of microenvironmental conditions and drug scheduling on normalization benefit, we build a mathematical model that incorporates tumor growth, angiogenesis and IFP. We administer various theoretical combinations of antiangiogenic agents and cytotoxic nanoparticles through heterogeneous vasculature that displays a similar morphology to tumor vasculature. We observe differences in drug extravasation that depend on the scheduling of combined therapy; for concurrent therapy, total drug extravasation is increased but in adjuvant therapy, drugs can penetrate into deeper regions of tumor.

Dose Titration Algorithm Tuning (DTAT) should supersede 'the' Maximum Tolerated Dose (MTD) in oncology dose-finding trials

Background. Absent adaptive, individualized dose-finding in early-phase oncology trials, subsequent 'confirmatory' Phase III trials risk suboptimal dosing, with resulting loss of statistical power and reduced probability of technical success for the investigational therapy. While progress has been made toward explicitly adaptive dose-finding and quantitative modeling of dose-response relationships, most such work continues to be organized around a concept of 'the' maximum tolerated dose (MTD). The purpose of this paper is to demonstrate concretely how the aim of early-phase trials might be conceived, not as 'dose-finding', but as dose titration algorithm (DTA)-finding. Methods. A Phase I dosing study is simulated, for a notional cytotoxic chemotherapy drug, with neutropenia constituting the critical dose-limiting toxicity. The drug's population pharmacokinetics and myelosuppression dynamics are simulated using published parameter estimates for docetaxel. The amenability of this model to linearization is explored empirically. The properties of a simple DTA targeting neutrophil nadir of 500 cells/mm ³ using a Newton-Raphson heuristic are explored through simulation in 25 simulated study subjects. Results. Individual-level myelosuppression dynamics in the simulation model approximately linearize under simple transformations of neutrophil concentration and drug dose. The simulated dose titration exhibits largely satisfactory convergence, with great variance in individualized optimal dosing. Some titration courses exhibit overshooting. Conclusions. The large inter-individual variability in simulated optimal dosing underscores the need to replace 'the' MTD with an individualized concept of MTD _i . To illustrate this principle, the simplest possible DTA capable of realizing such a concept is demonstrated. Qualitative phenomena observed in this demonstration support discussion of the notion of tuning such algorithms. Although here illustrated specifically in relation to cytotoxic chemotherapy, the DTAT principle appears similarly applicable to Phase I studies of cancer immunotherapy and molecularly targeted agents.

Dose Titration Algorithm Tuning (DTAT) should supersede 'the' Maximum Tolerated Dose (MTD) in oncology dose-finding trials

Background. Absent adaptive, individualized dose-finding in early-phase oncology trials, subsequent 'confirmatory' Phase III trials risk suboptimal dosing, with resulting loss of statistical power and reduced probability of technical success for the investigational therapy. While progress has been made toward explicitly adaptive dose-finding and quantitative modeling of dose-response relationships, most such work continues to be organized around a concept of 'the' maximum tolerated dose (MTD). The purpose of this paper is to demonstrate concretely how the aim of early-phase trials might be conceived, not as 'dose-finding', but as dose titration algorithm (DTA)-finding. Methods. A Phase I dosing study is simulated, for a notional cytotoxic chemotherapy drug, with neutropenia constituting the critical dose-limiting toxicity. The drug's population pharmacokinetics and myelosuppression dynamics are simulated using published parameter estimates for docetaxel. The amenability of this model to linearization is explored empirically. The properties of a simple DTA targeting neutrophil nadir of 500 cells/mm ³ using a Newton-Raphson heuristic are explored through simulation in 25 simulated study subjects. Results. Individual-level myelosuppression dynamics in the simulation model approximately linearize under simple transformations of neutrophil concentration and drug dose. The simulated dose titration exhibits largely satisfactory convergence, with great variance in individualized optimal dosing. Some titration courses exhibit overshooting. Conclusions. The large inter-individual variability in simulated optimal dosing underscores the need to replace 'the' MTD with an individualized concept of MTD _i . To illustrate this principle, the simplest possible DTA capable of realizing such a concept is demonstrated. Qualitative phenomena observed in this demonstration support discussion of the notion of tuning such algorithms. Although here illustrated specifically in relation to cytotoxic chemotherapy, the DTAT principle appears similarly applicable to Phase I studies of cancer immunotherapy and molecularly targeted agents.

Fractal dimension and universality in avascular tumor growth

For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation-the Bertalanffy-Richards model-even if they do not necessarily share the same biological properties.

Novel computational method for predicting polytherapy switching strategies to overcome tumor heterogeneity and evolution

The success of targeted cancer therapy is limited by drug resistance that can result from tumor genetic heterogeneity. The current approach to address resistance typically involves initiating a new treatment after clinical/radiographic disease progression, ultimately resulting in futility in most patients. Towards a potential alternative solution, we developed a novel computational framework that uses human cancer profiling data to systematically identify dynamic, pre-emptive, and sometimes non-intuitive treatment strategies that can better control tumors in real-time. By studying lung adenocarcinoma clinical specimens and preclinical models, our computational analyses revealed that the best anti-cancer strategies addressed existing resistant subpopulations as they emerged dynamically during treatment. In some cases, the best computed treatment strategy used unconventional therapy switching while the bulk tumor was responding, a prediction we confirmed in vitro. The new framework presented here could guide the principled implementation of dynamic molecular monitoring and treatment strategies to improve cancer control.

Optimizing the delivery of combination therapy for tumors: A mathematical model

We present in this paper a new mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent for a tumor. Our model takes into account the process of angiogenesis and the quality of the vasculature discriminating between stable blood vessels and unstable blood vessels. We characterize theoretically the optimal controls on drug distribution to minimize the number of cancer cells at the end of the treatment in a free horizon time problem with restrictions on the total amount of drug doses. Finally, we solve the optimal control problem by using numerical simulations, obtaining as a result that, despite the number of the tumor cells decrease with anti-angiogenic treatment, the best results are reached at the end of the chemotherapy treatment.

Local controllability and optimal control for a model of combined anticancer therapy with control delays

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.g., normalization of the vascular network. The optimized protocols of the combined therapy for the model, considered as solutions to an optimal control problem with delays in control, are found using necessary conditions of optimality and numerical computations. Our numerical approach uses dicretization and nonlinear programming methods as well as the direct optimization of switching times. The structural sensitivity of the considered control properties and optimal solutions is also discussed.

Optimization of an in vitro chemotherapy to avoid resistant tumours

Chemotherapy use against solid tumours often results in the resistance of the cancer cells to the molecule used. In this paper, we will set up and analyse an ODE model for heterogeneous in vitro tumours, consisting of cells that are sensitive or resistant to a certain drug. We will then use this model to develop different protocols, that aim at reducing the tumour volume while preserving its heterogeneity. These drug administration schedules are determined through analysis of the system dynamics, and optimal control theory.

Determination of an optimal control strategy for drug administration in tumor treatment using multi-objective optimization differential evolution

The mathematical modeling of physical and biologic systems represents an interesting alternative to study the behavior of these phenomena. In this context, the development of mathematical models to simulate the dynamic behavior of tumors is configured as an important theme in the current days. Among the advantages resulting from using these models is their application to optimization and inverse problem approaches. Traditionally, the formulated Optimal Control Problem (OCP) has the objective of minimizing the size of tumor cells by the end of the treatment. In this case an important aspect is not considered, namely, the optimal concentrations of drugs may affect the patients' health significantly. In this sense, the present work has the objective of obtaining an optimal protocol for drug administration to patients with cancer, through the minimization of both the cancerous cells concentration and the prescribed drug concentration. The resolution of this multi-objective problem is obtained through the Multi-objective Optimization Differential Evolution (MODE) algorithm. The Pareto's Curve obtained supplies a set of optimal protocols from which an optimal strategy for drug administration can be chosen, according to a given criterion.

Deterministic and Stochastic Study for a Microscopic Angiogenesis Model: Applications to the Lewis Lung Carcinoma

Angiogenesis modelling is an important tool to understand the underlying mechanisms yielding tumour growth. Nevertheless, there is usually a gap between models and experimental data. We propose a model based on the intrinsic microscopic reactions defining the angiogenesis process to link experimental data with previous macroscopic models. The microscopic characterisation can describe the macroscopic behaviour of the tumour, which stability analysis reveals a set of predicted tumour states involving different morphologies. Additionally, the microscopic description also gives a framework to study the intrinsic stochasticity of the reactive system through the resulting Langevin equation. To follow the goal of the paper, we use available experimental information on the Lewis lung carcinoma to infer meaningful parameters for the model that are able to describe the different stages of the tumour growth. Finally we explore the predictive capabilities of the fitted model by showing that fluctuations are determinant for the survival of the tumour during the first week and that available treatments can give raise to new stable tumour dormant states with a reduced vascular network.

Modeling of Optimal Targeted Therapies Using Drug-Loaded Magnetic Nanoparticles for Liver Cancer

To enhance locoregional therapies for liver cancer treatment, we propose in this study a mathematical model to optimize the transcatheter arterial delivery of therapeutical agents. To maximize the effect of the treatment and minimize adverse effects on the patient, different mathematical models of the tumor growth are considered in this study to find the optimal number of the therapeutic drug-loaded magnetic nanoparticles to be administered. Three types of therapy models are considered, e.g., angiogenesis inhibition therapy, chemotherapy and radiotherapy. We use state-dependent Riccati equations (SDRE) as an optimal control methodology framework to the Hahnfeldt's tumor growth formulation. Based on this, design optimal rules are derived for each therapy to reduce the growth of a tumor through the administration of appropriate dose of antiangiogenic, radio- and chemo-therapeutic agents. Simulation results demonstrate the validity of the proposed optimal delivery approach, leading to reduced intervention time, low drug administration rates and optimal targeted delivery.

Optimizing Chemotherapeutic Anti-cancer Treatment and the Tumor Microenvironment: An Analysis of Mathematical Models

We review results about the structure of administration of chemotherapeutic anti-cancer treatment that we have obtained from an analysis of minimally parameterized mathematical models using methods of optimal control. This is a branch of continuous-time optimization that studies the minimization of a performance criterion imposed on an underlying dynamical system subject to constraints. The scheduling of anti-cancer treatments has all the features of such a problem: treatments are administered in time and the interactions of the drugs with the tumor and its microenvironment determine the efficacy of therapy. At the same time, constraints on the toxicity of the treatments need to be taken into account. The models we consider are low-dimensional and do not include more refined details, but they capture the essence of the underlying biology and our results give robust and rather conclusive qualitative information about the administration of optimal treatment protocols that strongly correlate with approaches taken in medical practice. We describe the changes that arise in optimal administration schedules as the mathematical models are increasingly refined to progress from models that only consider the cancerous cells to models that include the major components of the tumor microenvironment, namely the tumor vasculature and tumor-immune system interactions.

On probabilistic certification of combined cancer therapies using strongly uncertain models

This paper proposes a general framework for probabilistic certification of cancer therapies. The certification is defined in terms of two key issues which are the tumor contraction and the lower admissible bound on the circulating lymphocytes which is viewed as indicator of the patient health. The certification is viewed as the ability to guarantee with a predefined high probability the success of the therapy over a finite horizon despite of the unavoidable high uncertainties affecting the dynamic model that is used to compute the optimal scheduling of drugs injection. The certification paradigm can be viewed as a tool for tuning the treatment parameters and protocols as well as for getting a rational use of limited or expensive drugs. The proposed framework is illustrated using the specific problem of combined immunotherapy/chemotherapy of cancer.

Analyzing the quality robustness of chemotherapy plans with respect to model uncertainties

Mathematical models of chemotherapy planning problems contain various biomedical parameters, whose values are difficult to quantify and thus subject to some uncertainty. This uncertainty propagates into the therapy plans computed on these models, which poses the question of robustness to the expected therapy quality. This work introduces a combined approach for analyzing the quality robustness of plans in terms of dosing levels with respect to model uncertainties in chemotherapy planning. It uses concepts from multi-criteria decision making for studying parameters related to the balancing between the different therapy goals, and concepts from sensitivity analysis for the examination of parameters describing the underlying biomedical processes and their interplay. This approach allows for a profound assessment of a therapy plan, how stable its quality is with respect to parametric changes in the used mathematical model.

Effect of treatment on the global dynamics of delayed pathological angiogenesis models

For three different types of angiogenesis models with variable delays, we consider either continuous or impulse therapy that eradicates tumor cells and suppresses angiogenesis. For the cancer-free solution, explicit conditions of global stability for the continuous and impulsive systems are obtained, together with delay-dependent estimates for the rates of decay for the tumor volume and pathological angiogenesis.

Model-based angiogenic inhibition of tumor growth using modern robust control method

Cancer is one of the most destructive and lethal illnesses of the modern civilization. In the last decades, clinical cancer research shifted toward molecular targeted therapies which have limited side effects in comparison to conventional chemotherapy and radiation therapy. Antiangiogenic therapy is one of the most promising cancer treatment methods. The dynamical model for tumor growth under angiogenic stimulator/inhibitor control was posed by Hahnfeldt et al. in 1999; it was investigated and partly modified many times. In this paper, a modified version of the originally published model is used to describe a continuous infusion therapy. In order to generalize individualized therapies a robust control method is proposed using H(∞) methodology. Uncertainty weighting functions are determined based on the real pathophysiological case and simulations are performed on different tumor volumes to demonstrate the robustness of the proposed method.

Characterizing cancer subtypes as attractors of Hopfield networks

MOTIVATION

Cancer is a heterogeneous progressive disease caused by perturbations of the underlying gene regulatory network that can be described by dynamic models. These dynamics are commonly modeled as Boolean networks or as ordinary differential equations. Their inference from data is computationally challenging, and at least partial knowledge of the regulatory network and its kinetic parameters is usually required to construct predictive models.

RESULTS

Here, we construct Hopfield networks from static gene-expression data and demonstrate that cancer subtypes can be characterized by different attractors of the Hopfield network. We evaluate the clustering performance of the network and find that it is comparable with traditional methods but offers additional advantages including a dynamic model of the energy landscape and a unification of clustering, feature selection and network inference. We visualize the Hopfield attractor landscape and propose a pruning method to generate sparse networks for feature selection and improved understanding of feature relationships.

Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers

Although optimal control theory has been used for the theoretical study of anti-cancerous drugs scheduling optimization, with the aim of reducing the primary tumor volume, the effect on metastases is often ignored. Here, we use a previously published model for metastatic development to define an optimal control problem at the scale of the entire organism of the patient. In silico study of the impact of different scheduling strategies for anti-angiogenic and cytotoxic agents (either in monotherapy or in combination) is performed to compare a low-dose, continuous, metronomic administration scheme with a more classical maximum tolerated dose schedule. Simulation results reveal differences between primary tumor reduction and control of metastases but overall suggest use of the metronomic protocol.

Gompertz model with delays and treatment: mathematical analysis

In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.

T model of growth and its application in systems of tumor-immune dynamics

In this paper we introduce a new growth model called T growth model. This model is capable of representing sigmoidal growth as well as biphasic growth. This dual capability is achieved without introducing additional parameters. The T model is useful in modeling cellular proliferation or regression of cancer cells, stem cells, bacterial growth and drug dose-response relationships. We recommend usage of the T growth model for the growth of tumors as part of any system of differential equations. Use of this model within a system will allow more flexibility in representing the natural rate of tumor growth. For illustration, we examine some systems of tumor-immune interaction in which the T growth rate is applied. We also apply the model to a set of tumor growth data.

Understanding cancer mechanisms through network dynamics

Cancer is a complex, multifaceted disease. Cellular systems are perturbed both during the onset and development of cancer, and the behavioural change of tumour cells usually involves a broad range of dynamic variations. To an extent, the difficulty of monitoring the systemic change has been alleviated by recent developments in the high-throughput technologies. At both the genomic as well as proteomic levels, the technological advances in microarray and mass spectrometry, in conjunction with computational simulations and the construction of human interactome maps have facilitated the progress of identifying disease-associated genes. On a systems level, computational approaches developed for network analysis are becoming especially useful for providing insights into the mechanism behind tumour development and metastasis. This review emphasizes network approaches that have been developed to study cancer and provides an overview of our current knowledge of protein-protein interaction networks, and how their systemic perturbation can be analysed by two popular network simulation methods: Boolean network and ordinary differential equations.

Passing to the limit 2D-1D in a model for metastatic growth

We prove the convergence of a family of solutions to a two-dimensional transport equation with a non-local boundary condition modelling the evolution of a population of metastases. We show that when the data of the repartition along the boundary tend to a Dirac mass, then the solution of the associated problem converges and we derive a simple expression for the limit in terms of the solution of a 1D equation. This result permits us to improve the computational time needed to simulate the model.

Data collapse, scaling functions, and analytical solutions of generalized growth models

We consider a nontrivial one-species population dynamics model with finite and infinite carrying capacities. Time-dependent intrinsic and extrinsic growth rates are considered in these models. Through the model per capita growth rate we obtain a heuristic general procedure to generate scaling functions to collapse data into a simple linear behavior even if an extrinsic growth rate is included. With this data collapse, all the models studied become independent from the parameters and initial condition. Analytical solutions are found when time-dependent coefficients are considered. These solutions allow us to perceive nontrivial transitions between species extinction and survival and to calculate the transition's critical exponents. Considering an extrinsic growth rate as a cancer treatment, we show that the relevant quantity depends not only on the intensity of the treatment, but also on when the cancerous cell growth is maximum.