Mathematical Modeling Turns Cancer Discoveries into Effective Treatments

By Trachette L. Jackson

According to Time magazine, “The hero scientist who defeats cancer will likely never exist” [5].

Unfortunately, the reasoning behind this prediction is sound because “cancer” is the designation for hundreds of genetically different diseases. Furthermore, a “team-based, cross-disciplinary approach to cancer research is upending tradition and delivering results faster,” meaning that many heroes may emerge rather than just one [5]. Mathematicians can bring essential expertise to medical teams that are searching for more effective methods to fight cancer, and mathematical oncologists play an important role in transforming new biological discoveries into effective treatment strategies through mathematical and computational modeling.

Mathematical modeling is especially valuable when the hallmark of medical treatments—such as cytotoxic chemotherapy—fails for a particular type of cancer. In these types of situations, targeted molecular cancer therapeutics may provide an answer. These drugs are designed to interfere with specific cells, enzymes, proteins, or other molecules that are necessary for tumor growth and progression.

The cancer stem cell (CSC) is a special type of cell and potential target that is relevant to a wide variety of tumor types (see Figure 1). These cells function as key drivers of tumor initiation, metastasis, and therapeutic evasion. When they are present, CSCs are typically (i) the minority, as they only make up a small fraction of the total tumor volume; (ii) self-renewing, as they can create an unlimited number of themselves; (iii) tumorigenic, as they can differentiate into other cell types and are responsible for tumor progression; and (iv) difficult to treat, as they are often resistant to drugs, radiation, and cell stress. All of these features make CSCs appealing conceptual targets for cancer therapy, especially because they play a disproportionately important role in determining tumor growth and treatment outcomes (despite comprising only a small portion of the total tumor burden). Unfortunately, not many CSC-directed cancer therapies exist; the plot thickens due to evidence that cisplatin—the mainstay chemotherapeutic option—induces an increase in CSC self-renewal in head and neck squamous cell carcinomas (HNSCC).

Figure 1. Conventional versus cancer stem cell (CSC)-targeted therapy. CSCs (in red) self-renew and differentiate, thereby creating transient amplifying cells (in green), fully differentiated tumor cells (in blue), and more CSCs. Conventional therapies primarily impact nontumorigenic cancer cells and can result in tumor reduction, but they cannot eliminate CSCs. Targeting CSCs can lead to long-term tumor regression. Figure courtesy of [6].

How Can We Target CSCs?

Finding effective ways to target CSCs requires additional knowledge of their preferred niches. In many tumors, CSCs reside near the invasive edge and close to blood vessels. Furthermore, endothelial cells that line blood vessel walls secrete interleukin 6 (IL-6) — a vital protein that enhances CSC survival, self-renewal, and tumor initiation potential [2]. Notably, treatment with tocilizumab (TCZ)—an antibody against the IL-6 receptor that inhibits the IL-6 pathway and is often prescribed for arthritis—decreases the CSC fraction.

Modeling Motivation and Goals

Experiments clearly illustrate that the standard-of-care (i.e., cisplatin chemotherapy) and CSC-targeted therapy (i.e., TCZ) affect the CSC pool in opposing ways. It is difficult to determine the order and timing of dual drug delivery that will achieve maximum tumor and CSC reduction with experiments alone. However, one can use a validated, multiscale mathematical model of IL-6’s impact on tumor growth and CSC fraction to propose a rational schedule for combination drug administration.

Multiscale Modeling

My group’s general modeling strategy links receptor-ligand binding on the cell surface to intracellular signaling pathways that are critical for cell proliferation and apoptosis. These pathways then mediate population-level tumor growth dynamics and response to treatments that target them. We believe that integrating these tiers of information will provide precisely the necessary level of detail to uncover possible hidden mechanisms that mediate expected and potentially counterintuitive therapeutic effects of novel targeted drugs, both alone and in combination with traditional chemotherapy.

We employ a strategy of isolation followed by integration. We first isolate critical biological subsystems, then integrate these components into the larger, modular framework. Our endothelial cell-tumor cell crosstalk modules are the result of more than a decade of modeling-experimental collaboration to isolate the parameters and calibrate the subsystems. We also developed, analyzed, and calibrated the CSC, IL-6, TCZ, and cisplatin treatment optimization subsystems as separate modules. To develop these modules, we begin at the cellular and intracellular scales and write equations for protein \((A)\) that directly binds to cell surface receptors \((R)\), thereby forming a signal-initiating complex \((C)\):

\[\frac{dA}{dt}=-k_fAR+k_rC-\lambda_A A+\textrm{production}\] \[\frac{dR}{dt}=-k_fAR+k_rC+k_pC+\textrm{production}-\textrm{loss}\] \[\frac{dC}{dt}=k_fAR-k_rC-k_pC-\textrm{loss}.\]

In order to link receptor binding and intracellular signaling with cell growth and survival, we assume that all cells have the same total number of receptors \((R_T)\) and that the formation of the signaling complex \((C)\) elicits two responses in cancer cells \((N)\): (i) a direct increase in tumor cell proliferation and (ii) up-regulation of intracellular, pro-survival proteins \((B)\). In this way, fractional occupancy of signal-initiating complexes—\(\phi_N=\frac{c}{R_TN}\)—connects the different scales of the mathematical model:

\[\frac{dN}{dt}=f(\phi_N)N-\frac{\delta N}{1+\gamma B^2}\] \[\frac{dB}{dt}=\chi+\eta\phi_N-\lambda_B B.\]

We then expand this general modeling framework to represent CSC-driven tumor growth and targeted treatment against IL-6. The resulting mathematical model of the full system contains approximately 20 equations and more than 40 carefully calibrated parameters.

Figure 2. Schematic representation of the model’s different scales. 2a. Molecular- and cellular-level details of IL-6’s impact on cancer stem cell (CSC)-driven tumor growth. 2b. Molecular- and cellular-level details of endothelial cell (EC)–tumor cell (TC) crosstalk: TC-secreted vascular endothelial growth factor stimulates EC proliferation and survival. EC- secreted IL-6 enhances proliferation and survival of tumor cells. Figure 2a courtesy of [4], 2b courtesy of [3].

Data-driven Parameter Estimation

We compiled data across many biological scales to calibrate each model subsystem (see Figure 2). At the intracellular and cellular levels, we utilized Bcl-2 mRNA expression data and vascular endothelial growth factor secretion rates. At the tissue level, we incorporated data that demonstrated the effect of Bcl-2 overexpression on both endothelial cell and tumor cell proliferation and survival. We then used data from primary human HNSCC CSCs that were implanted without human endothelial cells to estimate the baseline pre-treatment parameter values [2]. My team was also fortunate to have access to time-course data from two separate treatment experiments that were explicitly designed for this modeling study. These experiments assigned mice to four groups: (i) treated with 5 mg/kg cisplatin combined with 5 mg/kg TCZ; (ii) treated with 5 mg/kg cisplatin; (iii) treated with 5 mg/kg TCZ; and (iv) control. Figure 3 depicts experimental data and the corresponding parameter fitting for both the endothelial cell-tumor cell crosstalk and the CSC modules.

Figure 3. A small sampling of experimental data and the corresponding parameter fitting for both the endothelial cell-tumor cell crosstalk module and the cancer stem cell (CSC)-IL-6 module. Figure courtesy of [1] and [3].

Major Findings

After experimentally validating the model by directly comparing its predictions to the TCZ therapy data without any additional parameter fitting, we employed it to find better dose scheduling options for stem-cell-targeted and traditional chemotherapy. We specifically wanted to minimize the amount of targeted therapy that is required for a fixed amount of chemotherapy while simultaneously maximizing CSC and tumor reduction. To accomplish this objective, we varied the time and frequency at which the two drugs were administered with respect to each other.

Computational analysis showed that these two drugs could not achieve optimal synergistic activity with the conventional weekly pre-, post-, or co-treatment scheduling regimens. Because traditional dosing schedules were antagonistic and TCZ has long-lasting influences on tumor growth compared to cisplatin’s short cytotoxic time interval, we investigated the impact of administering the same amount of cisplatin on longer time intervals. For instance, we questioned the outcome of administering cisplatin once every two weeks instead of every week. Our results demonstrated that a biweekly administration of cisplatin in concert with a weekly administration of TCZ reduces the frequency of chemotherapy while significantly increasing the synergism between the two drugs.

Overall, our multiscale mathematical modeling platform provides a framework for the preclinical exploration of cisplatin and TCZ dose and frequency optimization. Medical oncologists can test the model predictions in future clinical studies. Our mathematical modeling framework operates at the intracellular, cellular, and tissue levels and contains enough detail to capture the mechanism of action for targeted cancer therapeutic strategies. It can investigate cellular crosstalk’s impact on tumor growth and reduction due to therapy and predict potential synergism/antagonism of combination therapies. Though our approach is based on tumors of the head and neck, one can easily adapt it for the study of any receptor-targeted therapy in any tumor type — both alone and in combination with chemotherapy.

References
[1] Jain, H., & Jackson, T. (2017). Mathematical modeling of cellular cross-talk between endothelial and tumor cells highlights counterintuitive effects of VEGF-targeted therapies. Bull. Math. Biol., 80, 971-1016.
[2] Krishnamurthy, S., Warner, K.A., Dong, Z., Imai, A., Nör, C., Ward, B.B., … Nör, J.E. (2014). Endothelial Interleukin-6 defines the tumorigenic potential of primary human cancer stem cells. Stem Cells, 32(11), 2845-2857.
[3] Nazari, F., Oklejas, A.E., Nör, J.E., Pearson, A.T., & Jackson, T.L. (2020). In silico models accurately predict in vivo response for IL6 blockade in head and neck cancer. Cancer Res., 80(7), 1451-1460. 
[4] Nazari, F., Pearson, A.T., Nör, J.E., & Jackson, T.L. (2018). A mathematical model for IL-6-mediated, stem cell driven tumor growth and targeted treatment. PLoS Comput. Biol., 14(1), e100592
[5] Saporito, B. (2013, April 1). The conspiracy to end cancer. TIME Magazine. Retrieved from https://healthland.time.com/2013/04/01/the-conspiracy-to-end-cancer.
[6] Yen, A., Zhang, K., Daneshgaran, G., Kim, H.-J., & Ho, D. (2016). A chemopreventive nanodiamond platform for oral cancer treatment. J. Calif. Dent. Assoc., 44(2), 121-127.

Trachette L. Jackson is a professor of mathematics at the University of Michigan, where she specializes in mathematical oncology. She is an award-winning educator and scholar who has received numerous honors for her accomplishments in both areas, including as the first African American SIAM Fellow.