The Promise and Limitations of Cell-Cell Mathematical Models in Cancer Treatment

Mathematical Models are known for decades to be a powerful tool to study, understand, and predict natural processes. Moreover, as computational power becomes more accessible and affordable, mathematical models become more popular and useful over time. Indeed, one can find mathematical models to be in the heart of multiple products and ideas in our daily life without even noticing. A good example is the traffic lights. By using them, we promise optimal flow of movement in the road in the shortest time. Maybe you are waiting for the end of the news to know the weather for tomorrow. These are a few examples from an uncountable amount surrounding us.

It is not surprising that mathematics also entered the healthcare domain, which is one of the most important domains we have in our life according to the average person on the street (I did not formally check that, so it’s just a guess for now). Since the healthcare system is an extensive field with a large knowledge base, it will be hard to pinpoint the contribution of mathematical science in recent developments and will be out of the scope of this text. Moreover when we are referring to a large body of work that focuses on mathematical modeling of cancer development and treatment [1]. It is fair to say that these models contributed to the development of drugs and the improvement of treatment protocols. However, they are not without flaws. As such, in this blog post, I would like to present a quick overview of the promise and limitations of cell-cell mathematical models in cancer treatment.

Background

As a tool to make sense of the interactions among the many components of the tumor microenvironment, researchers have used mathematical models since these models can investigate interactions on different biological scales and also investigate the emergent properties of the system. These mathematical models are used to distill the essential components of the interactions, thus identifying the most plausible mechanisms that can lead to the observed outcomes, if done correctly.

To be exact, in recent years, evidence has accumulated indicating that the immune system can recognize and eliminate malignant tumors. This finding leads to the development of a powerful cancer treatment called immunotherapies. In order to use immunotherapies, one must first understand the mechanisms governing the dynamics of tumor growth and its interactions with the immune system.

To do so, early modeling attempts focused on the tumor growth dynamics while trying to understand how normal cells can mutate into cancer cells. More recently, biologists have come to the conclusion that the oversimplified concept of a single type of normal cell is inadequate. Hence, the models extended accordingly to include interactions among multiple cell types and chemicals which play fundamental roles in the initiation and progression of tumors. To accurately represent the bio-clinical dynamics, the tumor microenvironment was defined as a one that includes multiple cell types with unique rules and properties. For example, immune cells, fibroblasts, endothelial cells, signaling molecules, and tumor cells.

Due to a large number of components, the interactions between them, understanding how these interactions are sufficient to derive cancer immunotherapies and the overtime changing dynamics, has proven to be a very challenging task [2]. As a tool to tackle this challenge, researchers have used mathematical models, mainly based on the cell-cell interaction of cell populations as a function of their size. These models are usually described by ordinary differential equations (ODEs for short).

Examples

Kiselyov et al. [3] modeled and investigated intravesical Bacillus Calmette–Guerin (BCG) based treatment for non-muscle invasive bladder carcinoma (NMIBC) in order to prevent recurrence and progression of cancer. The authors used a cell-cell model with a pre-/post-BCG treatment classification of biological markers. To name a few, the authors used clinical pathologies such as tumor size, patient's age, tumor stage, and a number of tumor polyps. From a cellular point of view, the cellular proliferation Ki-67 parameter was used along with others.

Arciero et al. [4] presented a mathematical model that describes the growth, immune escape, and siRNA treatment of tumors. The proposed model consists of a system of nonlinear ODEs describing tumor cells and immune effectors, as well as the immune-stimulatory and suppressive cytokines IL-2 and TGF-beta. The authors developed a computer simulation that stimulates tumor growth by promoting angiogenesis, explaining the inclusion of an angiogenic switch mechanism for TGF-beta activity and ultimately predicted that by increasing the rate of TGF-beta production for reasonable values of tumor antigenicity will enhance the tumor’s growth and its ability to escape host detection.

Novak and Tyson [5] constructed a mathematical model for the regulation of the Restriction Point based on the Cyclin/CDK complexes, the Rb-E2F interaction and the signal-transduction pathway, Growth Factors–Early response genes–Delayed response genes. The authors assumed rapid message turnover for mRNA, which occurs in a longer time scale. One of the most important features of this model is the growth and division simulation of the cell. The rate of mass is determined by the level of General machinery, controlled by the concentration of Retinoblastoma Protein and the absence of growth factors.

The Promise

Cell-cell models exploit biological and clinical knowledge to capture complex dynamics in a relatively straightforward way.

Of note, as machine learning (ML) becomes more popular by the day, ODE-based models are providing a feasible alternative. ODE-based models (and others in the same category) are outperforming ML models on three fronts: 1) Low-data. Since the ODE-based model performs fitting rather than learning, they require just a small sample to converge into a useful model while ML-based models usually require at least hundreds to thousands of samples which are extremely hard to obtain especially in clinical settings. 2) Explainable. ODE-based models by definition are explainable. I will not argue that it is easy to understand or use them but with the right training, it is possible. On the other hand, ML-based models are often inexplicable by nature. It would be fair to say the explainable-ML is a hot topic nowadays and multiple companies and labs all over the world are trying to achieve results in this direction. Nonetheless, currently, we are far from both high-performance and explainable ML models. 3) Inexpensive computation. As training and even using ML models require a lot of computational power, ODE-based models are usually based on simple computation that is known to be optimized for some time already. Again, more computation-efficient ML models are promised to become available in the near future but it would be naive to assume they will be as computation-efficient as ODE-based models any time soon.

The Limitations

Cell-cell models usually assume the dynamics taking place in an isolated system where no other processes are taking place and affecting the results. As such, cross-body dynamics are usually neglected. This results in highly specific modeling of the body's systems in a narrow context. In the same manner, as the different processes are not taken into consideration, global and local changes over time as a result of versus activities.

In addition, due to the complexity of gathering clinical data and numerically solving large scale and complex models, most of the models were developed and evaluated for an average person. This practice is common in medical studies on the one hand. On the other hand, as time goes by, researchers better understand the shortcomings of these approaches. The main issue is that this hypothetical average person does not exist (And if it does, 99.99 percent chance it is not you). As such, almost all the cases will obtain a sub-optimal treatment using such a model. The novel model should go the extra mile and propose a personalization option of their results given the required data, even if it is not available at the time of the model development.

Moreover, most of the cell-cell models are not taking into consideration the spatial dynamics of cell-cell interaction and the influence of distances in chemical, biological, and physical interactions between cell populations. Recent studies show that these dynamics have a non-neglected effect on the biological dynamics and can greatly alter a model's results.

Summary

Cell-cell interaction models can fairly represent the biological and clinical processes in general and for oncological patients in particular. The current model is starting to take into consideration the spatial effect of cells and organs but much more research should be conducted so we fully understand and be able to model these dynamics.

Call for papers - Cells (3.2022-9.2022)

With the ever-increasing access to high-performance computing and data alongside advances in theoretical understanding of biological and clinical processes throughout mathematical models, new open questions in theoretical and applied biologically have recently emerged in the scientific community. These include the ability to personalize a clinical treatment as well as predict treatment outcomes without the need to widely test it, etc. Mathematical modeling is shown to be a useful tool in oncology, allowing to investigate both the disease and possible treatments. A particular group of models that are able to obtain accurate predictions of biological and clinical outcomes is the ones based on cell-cell interactions. In parallel, in recent years, immunotherapy treatments showing promising results with fewer side effects, long-term effects, and better success rates for multiple diseases and cancer in particular. However, many opportunities in cell-cell interaction-based models for cancer immunotherapy treatment are still available, enriching both our understanding of biomathematical models, clinical processes, and the interactions between. Specificities underlying each particular setting usually trigger interesting disclosures concerning models and methods of resolution. The purpose of this Special Issue is to give a stage for modern applied mathematical models tackling immunotherapy treatments in multiple forms of cancer in order to straighten the connection between theoretical and applied biology and present novel ideas for clinical investigation. We invite authors to submit research articles that fit this purpose.

References

1. R. Eftimie, J. L. Bramson. "Interactions Between the Immune System and Cancer: ABrief Review of Non-spatial Mathematical Models". Bullten Mathematical Biology. 2011; 73, 2-32.
2. T. Gajewski. "Failure at the effector phase: immune barriers at the level of melanoma tumor microenvironment". Clin. Cancer Res. 2011; 13:18, 5256-5261.
3. A. Kiselyov, S. Bunimovich-Mendrazitsky, V. Startsev. "Treatment of non-muscle invasive bladder cancer with Bacillus Calmette–Guerin (BCG): Biological markers and simulation studies". BBA Clinical. 2015; 4, 27-34.
4. J. Arciero, T. Jackson, D. Kirschner. "A mathematical model of tumor-immune evasion and siRNA treatment". Discrete Contin. Dyn. Syst., Ser. B. 2004; 4:1, 39-58.
5. B. Novak, J. J. Tyson. "A model for restriction point control of the mammalian cell cycle". J. Theor. Biol 2004; 230, 563-579.