Optimal dynamic control approach in a multi-objective therapeutic scenario: Application to drug delivery in the treatment of prostate cancer

Patient-Specific, Mechanistic Models of Tumor Growth Incorporating Artificial Intelligence and Big Data

Despite the remarkable advances in cancer diagnosis, treatment, and management over the past decade, malignant tumors remain a major public health problem. Further progress in combating cancer may be enabled by personalizing the delivery of therapies according to the predicted response for each individual patient. The design of personalized therapies requires the integration of patient-specific information with an appropriate mathematical model of tumor response. A fundamental barrier to realizing this paradigm is the current lack of a rigorous yet practical mathematical theory of tumor initiation, development, invasion, and response to therapy. We begin this review with an overview of different approaches to modeling tumor growth and treatment, including mechanistic as well as data-driven models based on big data and artificial intelligence. We then present illustrative examples of mathematical models manifesting their utility and discuss the limitations of stand-alone mechanistic and data-driven models. We then discuss the potential of mechanistic models for not only predicting but also optimizing response to therapy on a patient-specific basis. We describe current efforts and future possibilities to integrate mechanistic and data-driven models. We conclude by proposing five fundamental challenges that must be addressed to fully realize personalized care for cancer patients driven by computational models.

Antifragile Control Systems: The Case of an Anti-Symmetric Network Model of the Tumor-Immune-Drug Interactions

A therapy’s outcome is determined by a tumor’s response to treatment which, in turn, depends on multiple factors such as the severity of the disease and the strength of the patient’s immune response. Gold standard cancer therapies are in most cases fragile when sought to break the ties to either tumor kill ratio or patient toxicity. Lately, research has shown that cancer therapy can be at its most robust when handling adaptive drug resistance and immune escape patterns developed by evolving tumors. This is due to the stochastic and volatile nature of the interactions, at the tumor environment level, tissue vasculature, and immune landscape, induced by drugs. Herein, we explore the path toward antifragile therapy control, that generates treatment schemes that are not fragile but go beyond robustness. More precisely, we describe the first instantiation of a control-theoretic method to make therapy schemes cope with the systemic variability in the tumor-immune-drug interactions and gain more tumor kills with less patient toxicity. Considering the anti-symmetric interactions within a model of the tumor-immune-drug network, we introduce the antifragile control framework that demonstrates promising results in simulation. We evaluate our control strategy against state-of-the-art therapy schemes in various experiments and discuss the insights we gained on the potential that antifragile control could have in treatment design in clinical settings.

Optimal control in pharmacokinetic drug administration

We consider a two-box model for the administration of a therapeutic substance and discuss two scenarios: First, the substance should have an optimal therapeutic concentration in the central compartment (typically blood) and be degraded in an organ, the peripheral compartment (e.g., the liver). In the other scenario, the concentration in the peripheral compartment should be optimized, with the blood serving only as a means of transport. In either case the corresponding optimal control problem is to determine a dosing schedule, i.e., how to administer the substance as a function u of time to the central compartment so that the concentration of the drug in the central or in the peripheral compartment remains as closely as possible at its optimal therapeutic level. We solve the optimal control problem for the central compartment explicitly by using the calculus of variations and the Laplace transform. We briefly discuss the effect of the approximation of the Dirac delta distribution by a bolus. The optimal control function u for the central compartment satisfies automatically the condition u≥0. But for the peripheral compartment one has to solve an optimal control problem with the non-linear constraint u≥0. This problem does not seem to be widely studied in the current literature in the context of pharmacokinetics. We discuss this question and propose two approximate solutions which are easy to compute. Finally we use Pontryagin's Minimum Principle to deduce the exact solution for the peripheral compartment.

A Model-Based Framework to Identify Optimal Administration Protocols for Immunotherapies in Castration-Resistance Prostate Cancer

Prostate cancer (PCa) is one of the most frequent cancer in male population. Androgen deprivation therapy is the first-line strategy for the metastatic stage of the disease, but, inevitably, PCa develops resistance to castration (CRPC), becoming incurable. In recent years, clinical trials are testing the efficacy of anti-CTLA4 on CRPC. However, this tumor seems to be resistant to immunotherapies that are very effective in other types of cancers, and, so far, only the dendritic cell vaccine sipuleucel-T has been approved. In this work, we employ a mathematical model of CRPC to determine the optimal administration protocol of ipilimumab, a particular anti-CTLA4, as single treatment or in combination with the sipuleucel-T, by considering both the effect on tumor population and the drug toxicity. To this end, we first introduce a dose-depending function of toxicity, estimated from experimental data, then we define two different optimization problems. We show the results obtained by imposing different constraints, and how these change by varying drug efficacy. Our results suggest administration of high-doses for a brief period, which is predicted to be more efficient than solutions with prolonged low-doses. The model also highlights a synergy between ipilimumab and sipuleucel-T, which leads to a better tumor control with lower doses of ipilimumab. Finally, tumor eradication is also conceivable, but it depends on patient-specific parameters.

OptiDose: Computing the Individualized Optimal Drug Dosing Regimen Using Optimal Control

Providing the optimal dosing strategy of a drug for an individual patient is an important task in pharmaceutical sciences and daily clinical application. We developed and validated an optimal dosing algorithm (OptiDose) that computes the optimal individualized dosing regimen for pharmacokinetic-pharmacodynamic models in substantially different scenarios with various routes of administration by solving an optimal control problem. The aim is to compute a control that brings the underlying system as closely as possible to a desired reference function by minimizing a cost functional. In pharmacokinetic-pharmacodynamic modeling, the controls are the administered doses and the reference function can be the disease progression. Drug administration at certain time points provides a finite number of discrete controls, the drug doses, determining the drug concentration and its effect on the disease progression. Consequently, rewriting the cost functional gives a finite-dimensional optimal control problem depending only on the doses. Adjoint techniques allow to compute the gradient of the cost functional efficiently. This admits to solve the optimal control problem with robust algorithms such as quasi-Newton methods from finite-dimensional optimization. OptiDose is applied to three relevant but substantially different pharmacokinetic-pharmacodynamic examples.

Control Theory and Cancer Chemotherapy: How They Interact

Control theory arises in most modern real-life applications, not least in biological and medical applications. In particular, in biological and medical contexts, the role of control theory began to take shape in the early 1980s when the first works appeared on the application of control theory in models of pharmacokinetics and pharmacodynamics for antitumor therapies. Forty years after those first works, the theory of control continues to be considered a mathematical analysis tool of extreme importance and usefulness, but the challenges it must overcome in order to manage the complexity of biological processes are in fact not yet overcome. In this article, we introduce the reader to the basic ideas of control theory, its aims and its mathematical formalization, and we review its use in cell phase-specific models for cancer chemotherapy. We discuss strengths and limitations of the control theory approach to the analysis pharmacokinetics and pharmacodynamics models, and we will see that most of them are strongly related to data availability and mathematical form of the model. We propose some future research directions that could prove useful in overcoming the these limitations and we indicate the crucial steps preliminary to a useful and informative application of control theory to cancer chemotherapy modeling.

Optimal Control Theory for Personalized Therapeutic Regimens in Oncology: Background, History, Challenges, and Opportunities

Optimal control theory is branch of mathematics that aims to optimize a solution to a dynamical system. While the concept of using optimal control theory to improve treatment regimens in oncology is not novel, many of the early applications of this mathematical technique were not designed to work with routinely available data or produce results that can eventually be translated to the clinical setting. The purpose of this review is to discuss clinically relevant considerations for formulating and solving optimal control problems for treating cancer patients. Our review focuses on two of the most widely used cancer treatments, radiation therapy and systemic therapy, as they naturally lend themselves to optimal control theory as a means to personalize therapeutic plans in a rigorous fashion. To provide context for optimal control theory to address either of these two modalities, we first discuss the major limitations and difficulties oncologists face when considering alternate regimens for their patients. We then provide a brief introduction to optimal control theory before formulating the optimal control problem in the context of radiation and systemic therapy. We also summarize examples from the literature that illustrate these concepts. Finally, we present both challenges and opportunities for dramatically improving patient outcomes via the integration of clinically relevant, patient-specific, mathematical models and optimal control theory.

Optimization of Cancer Treatment in the Frequency Domain

Thorough exploration of alternative dosing frequencies is often not performed in conventional pharmacometrics approaches. Quantitative systems pharmacology (QSP) can provide novel insights into optimal dosing regimen and drug behaviors which could add a new dimension to the design of novel treatments. However, methods for such an approach are currently lacking. Recently, we illustrated the utility of frequency-domain response analysis (FdRA), an analytical method used in control engineering, using several generic pharmacokinetic-pharmacodynamic case studies. While FdRA is not applicable to models harboring ever increasing variables such as those describing tumor growth, studying such models in the frequency domain provides valuable insight into optimal dosing frequencies. Through the analysis of three distinct tumor growth models (cell cycle-specific, metronomic, and acquired resistance), we demonstrate the application of a simulation-based analysis in the frequency domain to optimize cancer treatments. We study the response of tumor growth to dosing frequencies while simultaneously examining treatment safety, and found for all three models that above a certain dosing frequency, tumor size is insensitive to an increase in dosing frequency, e.g., for the cell cycle-specific model, one dose per 3 days, and an hourly dose yield the same reduction of tumor size to 3% of the initial size after 1 year of treatment. Additionally, we explore the effect of drug elimination rate changes on the tumor growth response. In summary, we show that the frequency-domain view of three models of tumor growth dynamics can help in optimizing drug dosing regimen to improve treatment success.