Cancer is one of the five leading causes of death. And yet, despite decades of research, there is no standardised first-line treatment for most cancers. In addition, disappointing results from predominant second-line treatments like chemotherapy have established the need for alternative methods.

Mathematical modeling of cancer usually involves describing the evolution of tumours in terms of differential equations and stochastic or agent-based models, and testing the effectiveness of various treatments within the chosen mathematical framework. Tumour progression (or regression) is evaluated by studying the dynamics of tumour cells under different treatments, such as immune therapy, chemotherapy and drug therapeutics while optimising dosage, duration and frequencies.

In a paper published in the SIAM Journal on Applied Mathematics, 'Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis,' authors Andrea Minelli, Francesco Topputo, and Franco Bernelli-Zazzera propose a differential equation model to describe tumour–immune interactions. "We study the dynamics of the competition between the tumour and the immune system," Topputo explains.

The relationship between cancers and the immune system has been studied for many years, and immune therapy has been known to influence tumour regression. Clinically called immunotherapy, it involves using external factors to induce, enhance, or suppress a patient's immune response for treatment of disease. In this study, the therapy consists of injecting a type of immune cells called dendritic cells, which generate tumour-specific immunity by presenting tumour-associated antigens.

"In particular, cancer immunotherapy has the purpose of identifying and killing tumour cells," says Topputo. "Our research considers a model that describes the interaction between the neoplasia [or tumor], the immune system, and drug administration." Such modeling and simulation can be used to assess the impact of drugs and therapies before clinical application.

Using ordinary differential equations, the authors model the progress of different cell populations in the tumour environment as a continuous process. Within the dynamical system presented by the tumour environment, they apply the theory of optimal control—a mathematical optimisation method—to design ad-hoc therapies and find an optimal treatment.

The end goal of the control policy is to minimise tumour cells while maximising effectors, such as immune cells or immune-response chemicals. "The aim is to minimise the tumor concentration while keeping the amount of administered drug below certain thresholds, to avoid toxicity," says Topputo. "In common practice, one searches for effective therapies; in our approach, we look for efficiency and effectiveness."

Elaborating on a prior study where indirect methods used to solve the optimal control problem are not effective, the authors use direct methods that apply algorithms from aerospace engineering to solve the associated optimal control problem in this paper. Optimal protocols are analysed, and the duration of optimal therapy is determined.

The robustness of the optimal therapies is then assessed. In addition, their applicability toward personalised medicine is discussed, where treatment is customised to each individual based on various factors such as genetic information, family history, social circumstances, environment and lifestyle.

"We have shown that personalised therapy is robust even with uncertain patient conditions. This is relevant as the model coefficients are characterised by uncertainties," Topputo explains. "Further studies would include designing optimal protocols by considering personalised constraints based on individual patient conditions, such as maximum amount of drug, therapy duration, and so on."

Other future directions would be the use of more diverse models and studying the effectiveness of treatment combinations. "More detailed approaches like agent-based models that describe tumour-immune interactions and hybrid therapies that consist of combined chemotherapy-immunotherapy treatments should also be considered," says Topputo.

Source: Society for Industrial and Applied Mathematics