Mathematical modeling of PDGF-driven glioma reveals the dynamics of immune cells infiltrating into tumors

Abstract

Background: Tumor-infiltrated immune cells compose a significant component of many cancers. They have been observed to have contradictory impacts on tumors. Although the primary reasons for these observations remain elusive, it is important to understand how immune cells infiltrating into tumors is regulated. Recently our group conducted a series of experimental studies, which showed that muIDH1 gliomas have a significant global reduction of immune cells and suggested that the longer survival time of mice with CIMP gliomas may be due to the IDH mutation and its effect on reducing of the tumor-infiltrated immune cells. However, to comprehend how IDH1 mutants regulate infiltration of immune cells into gliomas and how they affect the aggressiveness of gliomas, it is necessary to integrate our experimental data into a dynamical system to acquire a much deeper understanding of subtle regulation of immune cell infiltration.

Methods: The method is integration of mathematical modeling and experiments. According to mass conservation laws and assumption that immune cells migrate into the tumor site along a chemotactic gradient field, a mathematical model is formulated. Parameters are estimated from our experiments. Numerical methods are developed to solve the problem. Numerical predictions are compared with experimental results.

Results: Our analysis shows that the net rate of increase of immune cells infiltrated into the tumor is approximately proportional to the 4/5 power of the chemoattractant production rate, and it is an increasing function of time while the percentage of immune cells infiltrated into the tumor is a decreasing function of time. Our model predicts that wtIDH1 mice will survive longer if the immune cells are blocked by reducing chemotactic coefficient. For more aggressive gliomas, our model shows that there is little difference in their survivals between wtIDH1 and muIDH1 tumors, and the percentage of immune cells infiltrated into the tumor is much lower. These predictions are verified by our experimental results. In addition, wtIDH1 and muIDH1 can be quantitatively distinguished by their chemoattractant production rates, and the chemotactic coefficient determines possibilities of immune cells migration along chemoattractant gradient fields.

Conclusions: The chemoattractant gradient field produced by tumor cells may facilitate immune cells migration to the tumor cite. The chemoattractant production rate may be utilized to classify wtIDH1 and muIDH1 tumors. The dynamics of immune cells infiltrating into tumors is largely determined by tumor cell chemoattractant production rate and chemotactic coefficient.

Keywords: Chemotactic gradient field; Immune cell; Infiltration dynamics; muIDH1; wtIDH1.

Fig. 1

(A) Numerical fitting: if $m_1=5.5$ and $m_2=1.5$ are assigned to wtIDH1 and muIDH1 tumors respectively, our mathematical model shows wtIDH1 and muIDH1 tumor mice die at 42.3 days and 54.2 days respectively, when their tumor grow to the volume of radius 5 millimeters. (B) Experimental data: we have 12 wtIDH1 tumor mice and their median survival is 42 days, 20 muIDH1 tumor mice and their median survival is 56 days. (C) For chosen m values, percentage profiles of infiltrated immune cells produced by our mathematical model. (D) Comparison of immune cell percentages when glioma-bearing mice die between experimental data and numerical results produced by our mathematical model, noticing we lumped macrophages, monocytes, PNMs, etc. together.

Fig. 2

Infiltrating dynamics of immune cells into the tumor. (A) Profiles of each cell population in the first 10 days of both types of tumors. (B) Profiles of each cell population within both types of tumors after initial growth. (C) Numerical results about the net rate of increase of immune cell infiltrating into gliomas. (D) Numerical results about the net rate of increase of glioma cells.

Fig. 3

Immune cell blockage hypothesis in wtIDH1 glioma: immune cells are blocked by reducing the chemotactic coefficient to one fourth of its estimated value in wtIDH1 tumors. (A) Numerical results about wtIDH1 tumor mice with and without immune blocked, ones with immune blocked have two-week longer survival. (B) Experimental data: 15 wtIDH1 tumor mice with Ly6G blocking antibody have a median survival of 55 days, while18 wtIDH1 tumor mice have a median survival of 42 days. (C) Numerical results about the net rate of increase of immune cell infiltrating into gliomas. D: Percentage profiles of immune cells: the infiltrated immune cells decreases 25% when tumor mice die.

Fig. 4

Aggressive tumor growth: tumors with doubling the proliferation rate λ, increasing tumor cell lysis rate by one third, and reducing the chemotactic coefficient to be one thirtieth of its estimated value are considered to be aggressive gliomas. (A) Numerical results show that there is only one day difference in survivals for wtIDH1 and muIDH1 aggressive tumors. (B) Experimental data for tumor mice in Ink4a/Arf^−/− background: 42 wtIDH1 tumor mice have a median survival of 36 days while 65 muIDH1 tumor mice have a median survival of 35 days. (C) Mathematical model prediction for profiles of infiltrated immune cells. (D) Truncations of infiltrated immune cell percentages, 0.5% and 0.1% for wtIDH1 tumor and muIDH1 tumor, respectively, when mice die.

Fig. 5

Tumor growth with two different chemotactic coefficients, one is estimated value, and the other is half of the estimated value. (A) Comparison of growth curves for muIDH1 tumors correspond to two chemotactic coefficients. (B) Comparison of growth curves for wtIDH1 tumors correspond two chemotactic coefficients. (C) Profiles of infiltrated immune cells for muIDH1 tumors correspond two chemotactic coefficients. (D) Profiles of infiltrated immune cells for wtIDH1 tumors correspond two chemotactic coefficients.

Fig. 6

(A) Tumor radius growth for different chemoattractant production rates. (B) Profiles of immune cell percentages for different chemoattractant production rates.