Mathematics versus cancer

“In the fight against cancer, all tricks are allowed”, states Dr. Engineer Mariusz Bodzioch from the Department of Multimedia and Computer Graphics at the Faculty of Mathematics and Computer Science of the University of Warmia and Mazury. This young mathematician and computer scientist is trying to fight cancer using differential equations.

It is no mistake: Doctor of Mathematics Mariusz Bodzioch is searching for a method to cure cancer, not in modern biochemistry, radiotherapy or surgery, but in applied mathematics. Why is a mathematician trying to do so, and not a physician?

Cancer is responsible for the death of over 8 million people each year, making it one of the most common causes of death. Despite the significant advances in science, it still defeats even the most sophisticated treatment methods. That is why it has been studied by scientists from various fields of science, including mathematicians.

Oncologists are cautious about mathematics

Dr. Bodzioch is the manager of the project entitled “Mathematical modelling of drug resistance in cancer therapies” financed by the NCN (National Science Centre). He manages a team consisting of Professor Dr. Habilitated Urszula Foryś, the Head of the Department of Biomathematics and Game Theory of the University of Warsaw, and Piotr Bajger, M.Sc. – a doctoral student of the Inter-faculty Interdisciplinary Doctoral Studies of the University of Warsaw in the field of mathematics and natural sciences.

“Progress in the field of measurement techniques greatly facilitated the acquisition of experimental data on cancer processes. Experimental oncologists are now flooded with information. It seems that the next step in the development of oncology should be the use of mathematical, quantitative and qualitative methods. Meanwhile, the authors of articles in medical journals rarely refer to the possibilities offered to them by mathematics. Mathematical models allow us to study the course of diseases at the cellular level, which is of particular importance in research on the etiology of neoplastic diseases.” This is how Doctor Bodzioch presents the idea of his work.

Simulation will show it sooner

The main assumption of his project is the development of mathematical models that could be used in clinical settings. Such models usually take the form of systems of differential equations. Differential equations bind a certain physical value with its changes over time, Dr. Bodzioch explains. The finished model mathematically describes the biochemical and physical processes taking place inside tumours, such as cell division, mutations and diffusion, and the effects of the administered medications. Currently, cancer therapists prescribe cancer therapy based on their knowledge, professional experience and intuition. An oncologist armed with an appropriate mathematical model could carry out computer simulations of phenomena occurring at the cellular level, in both healthy and cancerous cells. Thanks to such simulations, he/she could predict tumour behaviour (e.g. growth rate, response to treatment or probability of metastasis) and plan the treatment accordingly.

Cells mutated with chemo and hunger

There are many types of cancer and many processes that are responsible for them. To prepare a mathematical model, researchers need to determine which processes are essential and to take them into account.

“In our project, we focus on building mathematical models that describe two key phenomena for planning anticancer therapy: acquired drug resistance and complicated relationships occurring in combination therapies (e.g. anti-angiogenic therapy combined with chemotherapy)”, Dr. Bodzioch explains.

Acquired drug resistance is a process by which malignant cells develop resistance to chemotherapy as treatment progresses. This process is usually explained by the rapid rate of multiplication and mutation typical for cancer cells. However, the preliminary results of our research show that the competition between cancer cells also significantly influences the acquired drug resistance. Cancer cells, like all organisms, exist in an environment with limited resources (e.g. oxygen and glucose). Hence, they have to compete with others for survival. Under normal conditions, i.e. before chemotherapy, cells sensitive to therapy win the competition for food resources with resistant cells. However, when physicians use chemotherapy, resistant cells win the competition with healthy ones because they are better adapted to exist in new, hostile conditions. Mathematical modelling will help physicians to contribute to a better understanding of the cellular processes responsible for drug resistance and to develop therapeutic schemes that minimize this process.

The formula for a dose of life

The second phenomenon refers to a very promising therapeutic strategy, i.e. combined therapy. Cancer cells have the ability to form blood vessels, with which they are then fed. Physicians know how to block their formation, which deprives the tumour of oxygen and nutrients. The “starved” tumour is dying, i.e. cancer is receding. Such therapy is called anti-angiogenic therapy. Doctors have high hopes for it, but unfortunately its effectiveness in prolonging the lives of patients has turned out to be insignificant.

Anti-angiogenic therapy combined with chemotherapy gives much better results. Limiting the number of blood vessels, however, means that a smaller amount of drugs enters the tumour. Therefore, the effectiveness of chemotherapy decreases.

“However, there is a hypothesis which assumes that although anti-angiogenic therapy reduces the number of blood vessels, it improves the quality of those that will survive it. This type of therapy may, therefore, lead to an improvement in the quality of the cardiovascular system of the tumour, and thus, at the same time, increase the effectiveness of chemotherapy. However, to make full use of this, it is necessary to dose both types of drugs very precisely. And it is the mathematical model that will calculate the optimal dose.”

Promising results

“Thanks to the model, we will know when to administer anti-angiogenic agents that limit the growth of blood vessels, when to administer chemo and in what amount”, Dr. Mariusz Bodzioch explains. In this case the saying “More is not always better” is very accurate: an excessive dose of medications can do more harm than good.

The grant managed by Dr Mariusz Bodzioch finishes in September this year. Can we talk about its results?

“The grant finishes in September, but further studies on the optimization of therapeutic protocols will take several more years. Without going into the details, I can say that the results are promising”, Dr. Bodzioch assures us. With each success or response, new questions arise and next hypotheses are made that need to be verified.

Dr. Mariusz Bodzioch was nominated for the Polish Intellectual Development Award 2019 in the category “Scientist of the future” for this project. It is awarded by Centrum Inteligentnego Rozwoju (Centre for Intelligent Development) - a Silesian company that promotes Polish science. Two years ago, the current Prime Minister Mateusz Morawiecki received such an award in the category of “Man of the Year”. Since 2018, the President of the Patent Office of the Republic of Poland has been the patron of the award.

Lech Kryszałowicz

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