Model Represents Metastatic Tumor Growth to Explore Treatments
By Jillian Kunze
According to the World Health Organization, cancer accounts for nearly one in six deaths worldwide. Modeling tumor growth and the ways in which it is impacted by different treatment options is an essential effort, and the expansion of efficient numerical methods that can solve said models is similarly important. During a minisymposium presentation at the 2023 SIAM Conference on Control and Its Applications, which took place this week in Philadelphia, Pa., Iulia Martina Bulai of the University of Sassari described the development of a metastatic tumor growth model with the ultimate goal of incorporating different types of treatments. She worked with Maria Carmela De Bonis, Concetta Laurita, and Valeria Sagaria (all of the University of Basilicata) to develop the model and associated numerical methods.
In the basic biological framework of this problem, the primary tumor will evolve and grow over time and may also metastasize, with a single cancer cell or cluster of multiple cells splitting off to form a secondary tumor at another location — which may then undergo further metastasis itself. Bulai specifically focused on lung and breast cancer, as information about their growth laws is available based on data from lab experiments in mice.
The modeling framework uses partial differential equations (PDEs) to describe tumor growth and metastatic spreading. It assumes that there are no metastases at the starting time, the primary and secondary tumors both initially appear with multiple cells, the primary and secondary tumors can metastasize at different rates, and each metastasis does indeed become a new tumor that grows at a rate depending on the total tumor volume. Finally, the model includes five different laws that can potentially describe the tumor growth: exponential, power law, Gompertz, generalized logistic, and von BertalanffyWest. “We have focused on these growth laws because we have data for them,” Bulai said. “We will see a comparison between these five different growth laws for two case studies.”
Some medical and mathematical challenges arise in this work. Predicting the number or mass of metastatic tumors at a certain time based on the primary tumor’s dimensions is a difficult problem. And current clinical methods are unable to detect metastases under a certain size, so it is important to be able to mathematically predict when they will occur. Furthermore, the model should be able to show how different treatments exert different effects on the disease without any changes to its methods, which requires being able to numerically solve the general version of the model under different growth laws and emission rates. “It’s also important to have an efficient method from a computational point of view,” Bulai said.
Bulai and her collaborators assumed that for the growth rate of the primary and secondary tumors, the tumor volume would be the solution of an ordinary differential equation. A transport equation then describes the metastatic growth. “We also need to model the metastatic emission,” Bulai said. “It is modeled by a nonlinear boundary condition, and we also have an initial condition for the density,” which states that no metastatic tumor initially exists.
Figure 1. Metastatic mass and cumulative number of metastases over time for a numerical simulation in the lung tumor case. Figure courtesy of Iulia Martina Bulai.
Several aspects of this work are novel as compared to previous modeling efforts, such as the particular investigation of the five different tumor growth laws for lung and breast tumors. The model’s assumption that the metastases that are emitted at the initial timestamp can comprise a cluster of several cells, rather than a single cell, is also new. “And we have assumed that the colonization rate can be different depending on if the metastasis is generated by the primary tumor or by the metastases themselves,” Bulai said.
While the unknown solution to the model is the metastatic density, Bulai stated that the particular quantities of interest are the biological observables: the total metastatic mass at a particular time, and the cumulative number of metastases with a volume above a certain size. Both of these observables are representable as weighted integrals of metastatic density, so Bulai and her collaborators wanted to find the values of those integrals. To do so, they reformulated the PDE model as a Volterra integral equation (VIE) of the second type and applied a Nyströmtype method to approximate its solution. “We have shown the sufficient conditions of the method that ensure existence and uniqueness of the solution, wellconditioning of the linear system, and convergence,” Bulai said.
To perform numerical simulations, the collaborators created the VIE Toolbox in MATLAB. This code allowed them to compute the total metastatic mass and cumulative number of metastases for case studies with the aforementioned tumor growth laws. They set the primary tumor’s initial value based on experiments in mice, and also fixed the secondary tumor’s initial value to a much smaller size.
Figure 2. Metastatic mass and cumulative number of metastases over time for a numerical simulation in the breast tumor case. Figure courtesy of Iulia Martina Bulai.
Figure 1 displays the results for both the metastatic mass and the cumulative number of metastases in the case study of a lung tumor. The different tumor growth laws give rise to markedly different outputs, though several of the simulations exhibit similar behavior for the cumulative number of metastases across approximately the first 20 days. Similarly, Figure 2 displays the breast tumor case; for the cumulative number of metastases, only the simulation with the von BertalanffyWest law behaves quite differently.
“We have proposed an efficient numerical method for the resolution of PDE models, in this case describing the metastatic tumor growth,” Bulai said. “This model was reformulated in terms of VIEs of the second kind.” She and her colleagues are currently in the process of extending their method to account for a combined chemotherapy/antiangiogenic therapy. The implementation is working well so far, and shows interesting changes in model outcomes for different treatments. “This is only an initial result,” Bulai said. “Once we know it’s working well, we aim to build our own models for the treatments.”

Jillian Kunze is the associate editor of SIAM News. 