Mathematical modeling of combinatorial antigen targeting with multiple CAR T-cell products for glioblastoma treatment

Abstract

Glioblastoma is a highly aggressive and difficult-to-treat brain cancer that resists conventional therapies. Recent advances in chimeric antigen receptor (CAR) T-cell therapy have shown promising potential for treating glioblastoma; however, achieving optimal efficacy remains challenging due to tumor antigen heterogeneity, the tumor microenvironment, and T-cell exhaustion. In this study, we developed a mathematical model of CAR T-cell therapy for glioblastoma to explore combinatorial antigen targeting with multiple CAR T-cell treatments that take into account the spatial heterogeneity of antigen expression. Our hybrid model, created using the multicellular modeling platform PhysiCell, couples partial differential equations that describe the tumor microenvironment with agent-based models for glioblastoma and CAR T-cells. The model captures cell-to-cell interactions between the glioblastoma cells and CAR T-cells throughout treatment, focusing on three target antigens–IL-13Rα2, HER2, and EGFR. We analyze tumor antigen expression heterogeneity informed by expression patterns identified from human tissues and investigate patient-specific combinatorial multiple CAR T-cell treatment strategies. Our model demonstrates that an early intervention is the most effective approach, especially in glioblastoma tumors characterized by mixed antigen expression. However, in tissues with clustered antigen patterns, we find that sequential administration with specific CAR T-cell types can achieve efficacy comparable to simultaneous administration. For instance, the percent tumor reduction is 7.1% for simultaneous administration versus 6.7% for sequential administration. In addition, spatially targeted delivery of CAR T-cells to specific tumor regions with matching antigen is an effective strategy as well, resulting in up to 19.6% greater tumor reduction with multi-location administration compared to baseline injection. Our model provides a valuable platform for developing patient-specific CAR T-cell treatment plans with the potential to optimize scheduling and locations of CAR T-cell injections based on individual antigen expression profiles.

Introduction

Chimeric Antigen Receptor (CAR) T-cell therapy is regarded as one of the most effective forms of adoptive cell-based immunotherapy for cancer treatment, with FDA approval occurring in 2017^1–3. While CAR T-cell therapy has proven highly effective in treating leukemias and lymphomas⁴, it is particularly well-suited for hematological cancers due to the ability of CAR T-cells to circulate through the bloodstream and lymphatic system. However, when applied to solid tumors, CAR T-cells face several challenges, including: the difficulty of delivering CAR T-cells to the tumor mass, a hostile tumor microenvironment that can inhibit T-cell activity, and tumor antigen heterogeneity^4–6. Additionally, the effectiveness of CAR T-cell therapy may be further compromised by various tumor-infiltrating cells, such as stromal cells, which support tumor growth⁵.

While glioblastoma is one of the most lethal brain tumors, for decades, the standard of care has consisted of maximal surgical resection followed by a combination of radiation and chemotherapy, with the more recent addition of an electric field stimulation^7,8. Despite these efforts, glioblastoma remains highly aggressive and difficult to treat, with a median overall survival time of 12–18 months^9,10 and a 5-year survival rate of 6.7%, which is the lowest among brain tumors^11,12. New treatment strategies for glioblastoma are actively being explored to improve clinical outcomes. Some recent advances include immunotherapy, focused ultrasound, and targeted treatments to overcome immunosuppressive tumor microenvironment¹³. Among these approaches, immunotherapy has emerged as a promising direction, aiming to reverse the immunosuppressive microenvironment and enhance the effectiveness of immune cells in recognizing and attacking glioblastoma cells. Such immunotherapies include immune checkpoint blockade, CAR T-cell therapies, oncolytic viruses and vaccines, gene therapy, and bispecific antibody therapy^14,15.

Mathematical models of immunotherapy for cancer have provided significant insights by predicting patient responses to treatment, identifying optimal treatment regimens, designing personalized treatment plans, and understanding the complex interactions within the tumor^16–20. One of the first mathematical models to explain the interaction between immune cells and cancer cells is the dynamical system model developed by Kuznetsov et al. (1994)²¹. This model not only captures the population dynamics during the treatment, but also describes the formation of tumor dormant states and immune system evasion. Subsequent work by Kirschner and Panetta (1998) introduced the cytokine interleukin-2 into the dynamics between cancer cells and immune effector cells²². This model was able to interpret short-term tumor oscillations and long-term tumor relapses. Following this, later studies added periodic treatment and time delay to the model, along with corresponding stability analysis, to explain persistent oscillations observed in immune systems^23,24. Models built afterward incorporated additional cell types to capture tumor immune escape and explain multiple equilibrium phases of coexisting immune cells and cancer cells^25,26.

While the aforementioned mathematical models have provided important insights into immunotherapy, the recent emergence of CAR T-cell therapy has prompted the development of models tailored to its distinct biological features and therapeutic challenges. Sahoo et al. develops a dynamical system model that investigates the correlation between the CAR T-cell dosage and the rates of proliferation and exhaustion²⁷. Kimmel et al. explore the role of stochasticity in the therapy effect, offering insight into optimizing the tumor-killing rate and CAR T-cell adaptability for improved therapy outcomes²⁸. In addition, Singh et al. (2019) and Laughlin et al. develop pharmacokinetic-pharmacodynamic (PK/PD) models for CAR-T-cells, demonstrating that PK/PD models not only aid in understanding the treatment, but can also guide CAR T-cell design^29,30. The following dynamical systems models incorporate additional biological processes and cell types, enabling further exploration of CAR T-cell dynamics. Cess and Finley model the signaling mechanisms of CAR T-cells to investigate intracellular modulations that could enhance cellular response³¹. Li et al. incorporate the dynamics of multiple CAR T-cells conjugates binding to cancer cells to better explain experimental results³². Kara et al. consider antigen-positive and antigen-negative cancer cells, as well as bystander CAR T-cells, to study tumor heterogeneity and bystander effects in treating solid tumors³³. Bodnar et al. integrate dual-target CAR T-cells and demonstrate their effectiveness, while providing insights into dosing regimens for dual-target CAR T-cell therapies³⁴. However, the aforementioned non-spatial models assume a well-mixed cell population and therefore cannot capture the spatial structure and heterogeneity of glioblastoma.

To investigate the impact of spatial structure, several works have modeled the spatial-temporal interaction of CAR T-cell treatment and solid tumors, showing that spatial models can significantly contribute to designing effective treatment strategies. Fischel et al.³⁵ utilize a 3D agent-based model to incorporate CAR T-cell therapy, showing that the percentage of antigen-presenting cancer cells is critical in treatment success and antigen non-presenting cells can form a shield over antigen-presenting cells. Prybutok et al.³⁶ develops a hybrid partial differential equation and agent-based model to study CAR T-cell therapy across various cancer cell lines. Luque et al.³⁷ develops an agent-based model to evaluate different strategies of CAR T-cell therapy for tumor-derived 3D organoids, where their findings suggest that a single dose of CAR T-cells may reduce tumor size but does not lead to complete elimination, and higher dosages can increase the number of free CAR T-cells and potential side effects. Camacho-Gomez et al.³⁸ presents a 3D agent-based model of CAR T-cells to compare the migratory dynamics of conventional T-cells and CAR T-cells. The authors reveal distinct migration patterns of CAR T-cells and show that a CXCL12 chemical gradient enhances the motility of CAR T-cells to be similar to conventional T-cells. In Santurio et al. ³⁹, the authors study resistance mechanisms of CAR T-cell immunotherapy using a system of integral-partial differential equations. By modeling continuous levels of antigen expression of cancer cells, their model enhances the understanding of relapses in CAR T-cell treatment. Owens et al. develops a system of partial differential equations of cancer and CAR T-cells to study local CAR T-cell administration, and predicts that locally administered CAR T-cells are most effective against slowly proliferating and highly diffusive tumors⁴⁰.

Although mathematical models have been increasingly applied to investigate CAR T-cell therapy, existing studies have primarily focused on single-antigen targeting. To our knowledge, no prior work has studied combinatorial antigen targeting with multiple CAR T-cell therapies for solid tumors using spatial mathematical modeling. Moreover, spatial clinical data–such as patient-derived tissue samples–have yet to be integrated into these models, limiting their relevance to real tumor heterogeneity. In this paper, we develop a 3D multiscale model to study the combination of IL-13Rα2, HER2, and EGFR targeting CAR T-cell treatment for glioblastoma. The model combines partial differential equation (PDE) and agent-based models (ABM), using PhysiCell⁴¹, an open-source multicellular model development framework. In the following section, we provide the clinical motivation for using a combination of IL-13Rα2, HER2, and EGFR targeting CAR T-cell derived from experimental data provided in ref. ⁶. Then, we investigate three treatment strategies: (1) sequential treatment, (2) multi-location spatially targeted treatment, (3) Dose-frequency-dependent treatment, and finally, propose a treatment plan that integrates our findings. The treatment strategies tested in this work are summarized in Fig. 1. Our work demonstrates that glioblastoma tissue with distinct patterns respond differently to each treatment strategies we tested, indicating the importance of personalized assessment and treatment decisions.

Fig. 1. Summary of the treatment strategies tested in this work.

The dosing schedules for the short-term treatments for clustered tissue are shown. The CAR T-cell injection locations targeting IL-13Rα2 and EGFR are denoted as yellow and orange, respectively.

Results

Selection of CAR T-cell target antigens for Glioblastoma

The study by Barish et al.⁶ investigates spatial heterogeneity of antigens relevant to CAR T-cell immunotherapy in glioblastoma. The antigens, IL-13Rα2, HER2, and EGFR, represent the primary targets of CAR T-cell therapy in glioblastoma across many clinical trials, due to their frequent over-expression in cancer cells^3,42–44. Barish et al.⁶ analyzed tumor samples from 43 patients at single-cell resolution, mapping the expression of the three antigens. The findings indicate that antigen expression is not randomly distributed, but is instead clustered into distinct regions within the tumor. For example, the expression of IL-13Rα2 and HER2 tends to be inversely related to that of EGFR, and specific patterns emerge around hypoxic areas near necrosis. This complex spatial organization may facilitate antigen escape, a phenomenon wherein the cancer cells evade CAR T-cell killing either by a lack of presentation of the target antigen or by reducing antigen expression. Antigen escape is one mechanism by which treatment with single-antigen targeting CAR T-cells becomes less effective. The study suggests that combinatorial antigen targeting could enhance therapeutic outcomes and prevent antigen escape.

In this context, our work focuses on developing a mathematical model of glioblastoma and CAR T-cell treatment that can study different combinatorial antigen targeting with multiple CAR T-cell treatments in silico. We use three tissue samples from Barish et al.⁶ (PBT025, PBT018, PBT030) that display a variety of antigen expression patterns. In Fig. 2, the expressions of IL-13Rα2, HER2, and EGFR from the patient tissue samples are colored in red, green, and blue, respectively. Darker colors represent cells that express two or three antigens simultaneously, while gray represents cells that do not express any of the three antigens. We remark that the triple-negative cells are treated as healthy cells in our simulation.

Fig. 2. Selected glioblastoma patient tissue samples– mixed (PBT025), partially-clustered (PBT018), clustered (PBT030)–from Barish et al.⁶ (top) and initial 3D tissues created by stacking 10 copies of the cross-sections (bottom).

Cells are color-coded according to the type of antigen that they express, with IL-13Rα2 (red), HER2 (green), and EGFR (blue). Darker-colored cells (teal, khaki, purple, and brown) represent double or triple-antigen-expressing cells, and the gray cells represent cells that do not express any of the three antigens and are considered to be non-cancerous cells in our simulation.

The three selected patient samples exhibit distinct antigen expression profiles, ranging from mixed to clustered. In particular, PBT030 primarily contains IL-13Rα2 and EGFR antigens, which form distinct clusters in the tissue. In contrast, PBT018 contains a complex spatial mixture of IL-13Rα2 and HER2 antigens, while also containing large regions of single antigen-positive cells. PBT025 predominantly consists of double and triple-positive antigen-expressing cells that are spatially mixed. Thus, we denote the three samples PBT030, PBT018, and PBT025 as clustered tissue, partially-clustered tissue, and mixed tissue. Together, these three samples from ref. ⁶ form a range of tumor micro-environments that could be encountered in patients. Other patient tissue samples from ref. ⁶ are analyzed in Supplementary note S1 to further justify the choice of patient samples used in our simulations. Moreover, in most tissue samples, single-antigen targeting CAR T-cell treatment will have limited efficacy, in some cases leaving up to 30% of the cancer tissue untreated when only the primarily expressed antigen is targeted.

Sequential treatment plan

We aim to test a sequential injection strategy motivated by the contact inhibition of locomotion observed in CAR T-cells within cancer tissue, especially when dealing with heterogeneous glioblastoma tissue with clustered patterns. For example, consider the partially-clustered tissue (See Fig. 2). Suppose a biopsy at a particular location indicates the presence of IL-13Rα2-expressing glioblastoma cells. In that case, we hypothesize that administering IL-13Rα2 CAR T-cells first, followed by HER2 CAR T-cells once the IL-13Rα2-expressing glioblastoma cells have been cleared, could enhance the mobility and effectiveness of HER2 CAR T-cells in targeting the remaining cancer tissue. However, this strategy may be less effective in tissues with mixed or dispersed antigen-expressing patterns compared to those with more clustered patterns. To assess the efficacy of sequential treatment plans, we simulate and compare the baseline simultaneous and sequential injections on the mixed and clustered tissue samples.

For the partially-clustered tissue sample, we implemented two treatment plans: a baseline simultaneous injection and a sequential injection. The baseline treatment administers IL-13Rα2 and HER2 CAR T-cells at the same time on day 0, while the sequential injection plan first administers IL-13Rα2 CAR T-cells on day 0, and then injects HER2 CAR T-cells after a designated break period. IL-13Rα2 CAR T-cells are injected first, as IL-13Rα2-positive cells are the predominant population at the center of the tissue. The injection locations are denoted as arrows in Fig. 4. We tested four different periods of break: a 5-day break, a 7-day break, a 9-day break, and an 11-day break. In both plans, we introduce 100 IL-13Rα2 CAR T-cells and 100 HER2 CAR T-cells.

Fig. 4. Time-course response of the partially-clustered glioblastoma tissue sample under combinatorial targeting CAR T-cell treatment, comparing the baseline (simultaneous) and sequential administration with a 7-day break.

The images show the center slice of the simulated 3D tissue. The tumor antigens expressed in this tissue are IL-13Rα2 (red), HER2 (green), and IL-13Rα2 {\&} HER2 (khaki). The baseline and sequential administrations show similar results considering tumor reduction (see Fig. 3) and CAR T-cell infiltration.

Figure 3 illustrates the dynamics of cancer cells during the treatment period, plotting the number of cells in different subpopulations based on antigen expression. While the total number of cancer cells decreases in both plans, the sequential injection does not show any advantages over the baseline injection plan; however, it demonstrates similar efficacy. The average percentage of tumor reduction after three weeks is 9.5% in the baseline administration and 9.9% in sequential administration using a 7-day break. The final outcome is summarized in Table 1, and similar results are observed in the clustered tissue. It is important to note that in the baseline simultaneous injection plan on the partially-clustered tissue, the HER2-expressing cancer cells do not begin to decay until around day 7. This corresponds to the time that the IL-13Rα2 CAR T-cells eliminate the local IL-13Rα2 cancer tissue, and the HER2 tissue is exposed. See Fig. 4 for the spatial pattern of the tissue slice. Furthermore, the number of HER2 cancer cells decays more rapidly in the sequential plan with an 11-day break, as HER2 CAR T-cells have more space to move around and find the target. Nevertheless, the baseline injection plan is more effective in eliminating the double-positive cancer cells expressing both IL-13Rα2 and HER2 at the injection location than the sequential plan.

Fig. 3. Comparison of the number of cancer cells between baseline (simultaneous) and sequential administration of multiple CAR T-cell treatments for the partially-clustered tissue.

In the sequential administration of IL-13Rα2 and HER2 CAR T-cell treatments, four different time breaks are considered: 5-day, 7-day, 9-day, and 11-day breaks. The figures show the reduction of cancer cells in all categories, expressing HER2, IL-13Rα2, and IL-13Rα2 {\&} HER2 antigens, including the total number of cancer cells. Considering the total number of cancer cells reduced, the sequential administration shows a similar or slightly less effective result compared to the baseline treatment.

Table 1.

Comparison of short-term treatment outcomes between the baseline (simultaneous) and sequential treatment plans across the three representative tissue samples, ranging from clustered to mixed antigen patterns

	Treatment	Initial	Final number	Percent tumor reduction
Mixed	Baseline	37110	23756 ([23290,24296])	36.0% ([34,5,37.2])
tissue	Sequential	37110	28470 ([27798,28858])	23.3% ([22.2,25.1])
Partially-	Baseline	21960	19857 ([19673,20033])	9.6% ([8.8,10.4])
clustered	Sequential	21960	19773 ([19540,19934])	10.0% ([9.2,11.0])
Clustered	Baseline	19950	18489 ([18288,18679])	7.3% ([6.4,8.3])
tissue	Sequential	19950	18546 ([18201,18819])	7.0% ([5.7,8.8])

The average values are shown along with the minimum and maximum ranges across ten simulations. The baseline treatment is more effective or similarly effective compared to the sequential treatment in all samples, but more so in the tissue with a mixed antigen pattern. Sequential treatment yields comparable results only in tissues with clustered antigen expression.

However, the recommendations of sequential administration may not be suitable for all patients. In particular, we propose that injecting the CAR T-cells as early as possible may be a better approach for patients with tissue that displays a mixed pattern. Fewer regions would restrict the CAR T-cells’ movement due to cancer cells with mismatched antigens. To test this hypothesis, we compare the baseline and sequential administration of CAR T-cells in the mixed tissue. In the baseline treatment, we administer 100 CAR T-cells of each type, totaling 300 CAR T-cells, injected on day 0 at the center of the tissue. In the sequential treatment, we keep the total number of CAR T-cells the same; however, IL-13Rα2 CAR T-cells are injected first on day 0, followed by HER2 CAR T-cells on day 7, and EGFR CAR T-cells on day 14. The center slices of the simulated mixed tissue as time progresses are shown in Fig. 5. By comparing the snapshots taken on day 21, we observe that the baseline treatment has eliminated more cancer cells and spread further toward the tissue boundaries. The total number of cancer cells throughout the treatment is plotted on the right. To better illustrate the stochastic variability, the simulation was run three times, and the shaded region in the plot represents the standard deviation. The results indicate that the number of cancer cells decreases more rapidly in the baseline treatment than in the sequential treatment starting from day 0. By the end of the treatment, the total number of cancer cells is smaller in the baseline treatment. Table 1 presents the percentage of tumors eliminated compared to their initial size. In the mixed tissue, we find that the baseline treatment eliminates around 13% more cancer cells than the sequential treatment by the end of the 21-day treatment. Therefore, the sequential treatment should not be considered for highly mixed tissue; instead, it is recommended to administer CAR T-cells at once as early as possible.

Fig. 5. Time-course response of the mixed glioblastoma tissue sample under combinatorial targeting CAR T-cell treatment, comparing the baseline (simultaneous) administration (Day 0) and sequential administration (Days 0, 7, and 14).

The snapshots of the center tissue slice (left) and the total number of cancer cells (right) are shown. The tumor antigens involved in this tissue are IL-13Rα2 (red), EGFR (blue), and HER2 (green). The cancer cells with double and triple antigens expressed are marked in darker colors. The sequential treatment is less effective than the baseline treatment, considering tumor reduction and CAR T-cell infiltration.

In summary, sequential administration is not advisable compared to the baseline simultaneous administration, especially in tissues with less clustered patterns or in those with multiple areas expressing double or triple antigens. The baseline treatment demonstrates significantly better results for mixed tissue. However, for patients with clustered cancer tissue consisting of single antigen-expressing cells, sequential administration may be considered as a potential option, especially if spreading out the dose could offer any benefits. Nevertheless, it is important to note that baseline administration remains more effective in terms of cancer elimination.

Multi-location injection plan

The second treatment strategy we aimed to investigate is a multi-location injection plan. This approach is motivated by the observation that different types of antigen-expressing cancer cells clustered in distinct locations. For instance, in the clustered tissue sample (See Fig. 2), IL-13Rα2-positive and EGFR-positive glioblastoma cells are separated, with IL-13Rα2 cells concentrated in the upper left region of the tissue. In this scenario, injecting CAR T-cells at multiple locations, specifically targeting the corresponding antigens, could lead to better treatment outcomes compared to injecting all CAR T-cells in a single location. This strategy aims to enhance the ability of CAR T-cells to identify and bind to the glioblastoma cells that express matching antigens. To test this hypothesis, we compare the efficacy of the baseline single-location injection and the multi-location injection plans. Similar to our sequential treatment analysis in section 6, we expect treatment efficacy to differ between clustered and mixed antigen patterns. Thus, we compare the two injection plans on the mixed and clustered tissue samples.

We begin by determining the injection locations, which are computed as a weighted average of the positions of cancer cells that express the same antigen, where the weights correspond to the levels of antigen expression. For clustered tissue sample, we compute the injection locations using IL-13Rα2 and EGFR expression levels of glioblastoma cells and use these two centers for the multi-location injection plan. In contrast, all CAR T-cells are injected at the center of the tissue for the baseline injection plan. The time progression of our simulation is shown in Fig. 6, with the injection locations depicted as arrows. Our results indicate that CAR T-cells in the baseline injection plan struggle to spread throughout the tissue, with only the EGFR CAR T-cells successfully locating and eliminating nearby cancer cells with matching antigens. On the other hand, CAR T-cells in the multi-location injection plan are more effective at removing cancer cells due to their injection at strategically chosen locations. The temporal dynamics of cancer cell reduction are shown in Fig. 6b, demonstrating that the multi-location injection plan eliminates both IL-13Rα2 and EGFR cancer cells more rapidly, leading to significantly better results. The overall treatment efficacy is summarized in Table 2, where the average percentage of tumor reduction with the multi-location injection plan is 26.7%, nearly three times greater than the reduction achieved with the baseline injection. Similar results are achieved in the partially-clustered tissue. This outcome supports our hypothesis that treatment can be more effective when CAR T-cells are injected at locations corresponding to the relevant antigens.

Fig. 6. Time-course response of the clustered glioblastoma tissue sample under combinatorial targeting CAR T-cell treatment, comparing the baseline (single-location) injection and multi-location injection plans.

a The snapshots are the center slice of the simulated 3D tissue. The tumor antigens involved in this tissue are IL-13Rα2 (red) and EGFR (blue). The injection locations are depicted as yellow and green arrows for IL-13Rα2 and EGFR CAR T-cells, respectively, in the multi-location injection plan and a black arrow in the baseline injection plan. b The number of all cancer cells, EGFR-positive cells, and IL-13Rα2-positive cells are shown. The multi-location injection plan reduces both EGFR and IL-13Rα2 expressing cancer cells more rapidly than the baseline administration, which results in an overall improved outcome considering the tumor reduction.

Table 2.

Comparison of short-term treatment outcome regarding the percent tumor reduction (%) between the baseline (single-location) injection and multi-location injection plans in the clustered and mixed tissues

	Baseline injection	Multi-location injection
Mixed tissue	36.0% ([34.5,37.2])	48.3% ([46.2,49.3])
Partially-clustered	9.6% ([8.8,10.4])	24.2% ([24.1,25.5])
Clustered tissue	7.3% ([6.4,8.3])	26.7% ([24.9,27.6])

The average values are shown along with the minimum and maximum ranges across ten simulations. The multi-location injection achieves a better result in tumor reduction than the baseline injection in all tissue samples, particularly in the clustered and partially-clustered tissues.

With the success of the multi-location injection plan in the clustered tissue sample, we next explore how this strategy performs in a tissue with a more mixed distribution of antigen-expressing cancer cells. Fig. 7 shows the time progression of the mixed tissue sample under both baseline and multi-location injection plans. In this case, three antigens–HER2, EGFR, and IL-13Rα2–are involved, and we have identified three injection locations, each corresponding to a different type of CAR T-cell. By coincidence, the injection locations for HER2 CAR T-cells and IL-13Rα2 CAR T-cells are close. By day 21, the final snapshot of the tissue shows similar results between the two treatment plans, as mixed tissue contains many cancer cells expressing double and triple antigens, making it easier for any type of CAR T-cells to locate matching targets. Nevertheless, we still observe a slight reduction in cancer cell numbers under the multi-location injection plan compared to the baseline injection plan. At day 21, the percentage of tumor reduction for the mixed tissue is 48.3% under the multi-location injection plan, compared to 36.0% with the baseline injection plan. A summary of these results, comparing the clustered and mixed tissue samples, can be found in Table 2. While the multi-location injection shows a greater tumor reduction in the clustered tissue sample, it also exhibits significant efficacy in the mixed tissue sample. This outcome reinforces the idea that multi-location injections are more effective than the baseline injection, highlighting the importance of tailoring CAR T-cell therapies to the patient-specific heterogeneous pattern of cancer tissue.

Fig. 7. Time-course response of mixed glioblastoma tissue sample under combinatorial targeting CAR T-cell treatment, comparing the baseline (single-location) and multi-location injection plans.

a The snapshots are the center slice of the simulated 3D tissue. The tumor antigens involved in this tissue are IL-13Rα2 (red), EGFR (blue), and HER2 (green). The injection locations are depicted as yellow, purple, and green arrows for IL-13Rα2, HER2, and EGFR CAR T-cells, respectively, in the multi-location injection plan and a black arrow in the baseline injection plan. The results between the baseline and multi-location injection administrations are similar as the injection locations are similar. However, the multi-location injection shows better infiltration of CAR T-cell. b Considering the number of cancer cells in different cancer subpopulations, both treatment plans show similar results, while the total number of cancer cells are less in the multi-location injection plan.

Dose-frequency-dependent treatment plan

The third treatment strategy that we aimed to investigate is a dose-frequency-dependent injection of CAR T-cells, focusing on how dosage affects treatment efficacy. We use a fixed total number of CAR T-cells and compare different injection schedules. For example, we administer larger doses less frequently, while smaller doses are given more often, maintaining the total number of CAR T-cells. The CAR T-cells are introduced at the center of the tissue. Each antigen type receives a dosage of 180 CAR T-cells, summing up to a total of 540 CAR T-cells in the mixed tissue and 360 CAR T-cells in the clustered tissues. We compare three injection plans: high-frequency, medium-frequency, and low-frequency. In the high-frequency plan, we inject 30 CAR T-cells of each antigen type per dose, administering six doses every 3.5 days. For the medium-frequency and low-frequency treatment, we inject 60 and 90 CAR T-cells of each antigen type, with three doses every 7 days and two doses every 14 days, respectively. The simulation runs for 21 days.

Figure 8 shows the progression of the tissue slice and the total number of cancer cells over time using the three dose-frequency-dependent treatment plans for the mixed tissue. The low-frequency treatment demonstrates better efficacy than the other two plans, especially in the early stages when the CAR T-cells are initially injected. We observe a correlation between CAR T-cell dosage and tumor tissue infiltration by CAR T-cells. This is further evidenced by the trend in the cancer cells, where the slope of tumor reduction is steeper with higher doses. The percentage of tumor reduction, presented in Table 3, indicates that the low-frequency treatment eliminates 5% more cancer cells than the other two plans in the mixed tissue. While this is a beneficial effect, the difference in tumor size is modest–ranging from 30% to 35% over the 21-day treatment period. In contrast, the dose-frequency change does not have a significant effect in the clustered tissue samples, as the differences in average outcomes across frequency plans fall within the range of stochastic variability. Overall, the impact of dose frequency is modest and tissue-dependent, with slight benefits observed in mixed tissues but negligible effects in clustered tissues.

Fig. 8. Time-course response of the mixed glioblastoma tissue sample under combinatorial targeting CAR T-cell treatment, studying dose-frequency-dependent administration.

The IL-13Rα2, EGFR, and HER2 CAR T-cells are introduced at the center with time intervals (3.5, 7, and 14 days, from high to low frequency) with corresponding doses (90, 180, and 270 CAR T-cells, respectively). The snapshots of the center tissue slice (left) and the total number of cancer cells (right) are shown. The treatment plan with the strongest dose shows the most tumor reduction, especially at the beginning when the CAR T-cells are first injected.

Table 3.

Comparison of short-term treatment outcome regarding the percent tumor reduction (%) between three treatment plans with different doses and frequencies in the mixed and clustered tissue sample

	High-frequency	Medium-frequency	Low-frequency
Mixed tissue	30.7% ([27.5,31.5])	29.9% ([28.0,32.8])	35.1% ([33.5,36.4])
Partially-clustered	13.9% ([13.3,15.5])	14.5% ([12.2,14.7])	13.0% ([12.5,14.3])
Clustered tissue	20.1% ([18.2,22.5])	21.7% ([18.3,23.0])	19.7% ([18.7,22.9])

The average values are shown along with the minimum and maximum ranges across ten simulations. In the mixed tissue, the low-frequency plan results in a moderately improved tumor reduction compared to the other plans, while outcomes are similar across all plans in the clustered samples.

Long-term combinatorial antigen targeting treatment

So far, we have explored three different treatment strategies based on timing, injection location, and dose frequency. In this section, we will combine the key insights from the previous simulations: (1) multi-location injections can be effective in both clustered and mixed cancer tissues, and (2) larger doses administered at lower frequencies may yield better outcomes in mixed tissue.

Based on these observations, we aimed to integrate these two strategies and evaluate their long-term efficacy. We will compare four treatment plans: a weekly injection plan with CAR T-cells injected at the center (Long-term baseline treatment, Plan 1), a bi-weekly double-dose plan with single-location injection (Plan 2), a weekly injection plan with multiple injection location (Plan 3), and a bi-weekly multi-location injection plan (Plan 4). All plans are simulated on the clustered and mixed tissue samples. CAR T-cells are injected over the first 8 weeks, followed by an additional 8 weeks of monitoring after the final dose. The weekly injections, plans 1 and 3, involve eight doses of 25 CAR T-cells for each type, while bi-weekly injections, plans 2 and 4, consist of four doses of 50 CAR T-cells for each antigen type. All injection plans administer the same total dose of CAR T-cells, which is, 600 CAR T-cells in the mixed tissue and 400 CAR T-cells in the clustered tissues. The multi-location injection of plans 3 and 4 are determined as the weighted center of each subpopulation, as in section 6, while the single-location injection plans 1 and 2 have all CAR T-cells injected at the center.

In Fig. 9, we observed that treatment plan 4 eliminates more cancer cells than the baseline plan 1 for the mixed tissue, resulting in less remaining cancer tissue visible in the tissue slices. The trajectory of total cancer cell numbers shows that treatment plan 4 consistently outperforms the baseline throughout the treatment. After 16 weeks, treatment plan 4 eliminates approximately 14% more cancer cells than the baseline. A similar result can also be seen in the clustered tissue, where treatment plan 4 eliminates 5-8% more cancer cells than the baseline. A summary of these results is provided in Table 4. When comparing the two tested strategies, the improved outcome of plan 4 appears to be primarily driven by the multi-location injection strategy rather than the change in dosing frequency, as the weekly multi-location plan 3 yields results similar to plan 4. In the long term, dosing frequency appears to have less impact than in the short-term outcome. Overall, combining multi-location injection with a lower-frequency plan is more effective than the baseline treatment plan for all mixed and clustered patient tissues with the multi-location strategy contributing more significantly to the improved outcome.

Fig. 9. Time-course response of the mixed glioblastoma tissue under combinatorial targeting CAR T-cell treatment, comparing the baseline treatment (plan 1) and bi-weekly, multi-location injection (plan 4) in the long term.

The snapshots of the center tissue slice (left) and the total number of cancer cells (right) are shown. The treatment plan 4 demonstrates superior outcomes in tumor reduction and CAR T-cell infiltration compared to the baseline plan.

Table 4.

Comparison of long-term outcomes regarding the percent tumor reduction (%) over 16 weeks between the long-term baseline treatment (weekly single-location, Plan 1), bi-weekly single-location treatment (Plan 2), weekly multi-location injection treatment (Plan 3), and bi-weekly multi-location injection treatment (Plan 4)

Plan	Mixed tissue	Partially-clustered	Clustered tissue
1. Long-term baseline	67.5%	73.0%	74.2%
2. Bi-weekly single-location	69.9%	73.2%	74.5%
3. Weekly multi-location	80.5%	76.4%	81.4%
4. Bi-weekly multi-location	81.1%	77.5%	82.6%

The treatment Plan 4 enhances tumor reduction by 5–14% compared to the baseline treatment Plan 1 by the end of the treatment period in all tissue samples. In the long term, the multi-location injection strategy contributed more significantly to the improved outcome than the change in dose frequency.

Discussion

In this paper, we develop a mathematical model to study the combinatorial antigen targeting with multiple CAR T-cell treatments for glioblastoma. Our model is built using PhysiCell, a platform to develop multiscale models of interacting cells within a dynamic 3D microenvironment⁴¹. By modeling the heterogeneous population of glioblastoma cells and multiple CAR T-cell populations with a cell-based model, our model incorporates the heterogeneity of glioblastoma tissue of individual patients to optimize CAR T-cell treatment. We modeled glioblastoma cells that can express any combination of three different antigens – IL-13Rα2, HER2, and EGFR – expressed at a continuous level. The three implemented antigens cover most of the glioblastoma tissue introduced in ref. ⁶. Thus our model can be used to study combinatorial targeting CAR T-cell treatments for general glioblastoma patients.

Using our model, we explored three different treatment strategies – timing, injection location, and dose/frequency – for combinatorial targeting CAR T-cell treatments, comparing these approaches to the baseline treatment, which involves a single injection of all CAR T-cells at the center of the tissue. From the simulation results, we conclude three observations: i) sequential treatment can be significantly less effective in mixed tissue; however, it can be comparable in clustered tissue, ii) multi-location injection treatment tends to be effective across all tested tissues, but to a greater degree in clustered tissue, and iii) low-frequency treatment is more effective than treatments with lower doses and higher frequencies in mixed tissue. In summary, injecting CAR T-cells targeting specific locations of matching antigens can significantly enhance treatment efficacy. Additionally, larger doses can effectively eliminate more cancer cells even with longer intervals between each injection, depending on tissue type. Notably, patients whose tumors exhibit more clustered tissue patterns of antigen expression show more pronounced variability in treatment response, indicating that treatment decisions for such patients require greater precision and individualized assessment. Moreover, for such patients, a sequential injection plan may be a viable option, as it allows the doses to be spread out, potentially reducing patient side effects, such as cytokine storm syndrome and immune effector cell-associated neurotoxicity syndrome^45,46.

Despite the effort that has been made in this work, several important questions remain to be addressed. An imminent future work is to calibrate the model to additional clinical data, in particular, data that captures the distinct killing efficiency of IL-13Rα2, HER2, and EGFR CAR T-cells^42,47,48. Currently, the model is calibrated to in vitro experiment data of IL-13Rα2 CAR T-cell treatment only²⁷. Further experimental data on HER2 and EGFR CAR T-cells will allow for more accurate model calibration and predictions. Another important limitation lies in the classification of cells that do not express any of the three target antigens. In our simulations, these triple-negative cells are treated as healthy, although in reality they could represent either non-targetable cancer cells or normal tissue. Incorporating broader molecular profiling or histopathological annotations in future work could help refine the classification of these cells and improve the biological accuracy. Additionally, our current model does not include patient-specific tumor parameters such as cancer cell proliferation rates or CAR T-cell expansion dynamics. This simplification was made due to the lack of corresponding data in the available dataset. Incorporating tumor growth kinetics and CAR T-cell dynamics—including proliferation, exhaustion, and persistence—calibrated to patient-specific experimental or clinical data, will be important for improving the predictive power and clinical relevance of the model. In addition to model calibration, future work should include a comparison of our antigen-guided multi-location injection strategy with randomly selected injection sites. Such a comparison would help validate the benefit and quantify the impact of spatially distributed delivery. Likewise, a dose sensitivity analysis would provide insights into how variations in CAR T-cell dose influence treatment efficacy across different tissue types. Although these analyses were not included in the current study, they represent important directions for improving robustness and generalizability.

While PhysiCell provides a robust and extensible platform for developing multiscale, agent-based models of cancer dynamics, its general-purpose design introduces several limitations when applied specifically to modeling glioblastoma treatment. PhysiCell does not inherently incorporate glioblastoma-specific biological features, such as anisotropic migration along white matter tracts or infiltrative growth along brain-specific anatomical structures. These features are critical for accurately capturing glioblastoma progression and treatment response. Moreover, our current implementation does not include intracellular signaling or gene regulatory networks that influence CAR T-cell exhaustion and glioma resistance. These mechanisms would require significant custom extensions to the framework. Thus, in future work, we aim to incorporate glioblastoma-specific features, including anisotropic cell migration informed by tractography, detailed microenvironment components, and intracellular regulatory networks. In particular, the tumor microenvironment is highly complex and involves multiple interacting components^49,50. To better capture this complexity, we plan to include various subpopulations of CAR T-cells, such as regulatory T-cells and exhausted T-cells, and details of the microenvironment, including cytokines. These enhancements will enable a more comprehensive analysis of treatment dynamics and potentially guide more effective therapeutic strategies.

Dosage and injection frequency are also critical factors. While our findings suggest that larger doses delivered at longer intervals can improve tumor reduction, future studies must also consider patient safety and tolerability, including risks such as cytokine release syndrome and immune effector cell-associated neurotoxicity syndrome^45,46. Identifying the optimal therapeutic window that balances efficacy and toxicity remains a key clinical challenge. Another limitation of our work that is the scale discrepancy between our simulation and actual glioblastoma tissue. Our study includes 80,000 cells, which is relatively small compared to the actual cell numbers presented in real patient tumors, and the simulated CAR T-cell dosage is correspondingly low. We propose to increase the size of the simulation and include various scenarios, such as surgery before the administration of CAR T-cell therapy. Additionally, the neurosurgical precision for delivering CAR T cells is approximately 3–4mm, determined by the accuracy of catheter placement in the brain using Stealth navigation⁵¹. Given an average tumor diameter of 4cm, this corresponds to about 1/10 of the tumor size. In our simulations, the distance between multi-location injections ranges from 1/10 to 1/2 of the simulated tissue, corresponding to 0.1–0.5mm. While this is currently smaller than the achievable neurosurgical precision, this discrepancy will be addressed when we scale up the simulation. Furthermore, since our multi-location injection study indicates the importance of determining the prioritized treatment area, we plan to explore additional practical approaches, such as combining intratumoral and intraventricular delivery^42,52.

An especially promising direction is to extend our model to study dual-targeting CARs, such as bispecific and tandem CAR T-cell therapies, which have emerged as promising strategies to address antigen heterogeneity and reduce the risk of antigen escape in glioblastoma. Recent preclinical studies and early-phase clinical trials have demonstrated the efficacy of dual-targeting CAR T-cells by targeting both EGFR and IL-13Rα2^53,54 and both HER2 and IL-13Rα2⁵⁵. These findings motivate the extension of our current model to include dual-target recognition, competition between binding sites, and altered activation thresholds, to systematically compare strategies between bispecific CAR T-cells and combinatorial targeting CAR T-cells. In addition, our current model assumes that each CAR T-cell and cancer cell interaction contributes independently to the probability of cancer cell killing. However, this assumption may oversimplify the biological reality, due to factors such as receptor saturation, signaling thresholds, or cooperative killing dynamics. Future versions of the model should explore nonlinear or saturating killing kinetics, as well as the potential for synergistic or competitive effects between CAR T-cells targeting different antigens on the same cell. Furthermore, an important future extension of our model is to incorporate the dynamic downregulation of target antigens on cancer cells in response to CAR T-cell-mediated cytotoxicity. Experimental studies have shown that glioblastoma cells can adaptively escape CAR T-cell targeting through antigen loss or modulation⁵⁶. Incorporating these mechanisms will allow for a more accurate representation of tumor-immune interactions and the emergent dynamics of treatment resistance. Moreover, this could help evaluate the potential advantages of multi-antigen targeting strategies in suppressing immune escape.

Methods

We develop a hybrid partial differential equation and agent-based model of CAR T-cell therapy for glioblastoma based on the open-source multicellular code, PhysiCell⁴¹. The model describes the dynamics and interactions of cells in a three-dimensional microenvironment, with cell phenotypes dependent on the dynamic changes of the tumor microenvironment. It incorporates essential cell behaviors, such as cell cycling, cell death, volume regulation, motility, and cell-cell mechanical interactions. The model is coupled with the bio-transport solver, BioFVM⁵⁷, which efficiently simulates reaction-diffusion PDEs of environmental substrates in 3D.

In our ABM model, we consider four distinct cell types: glioblastoma cells, and three types of CAR T-cells that target the antigens IL-13Rα2, HER2, and EGFR. The PDE model of the tumor microenvironment includes two biochemical substrates: oxygen and an immunostimulatory factor, such as IL-2, for example. Glioblastoma cells and CAR T-cells are modeled using off-lattice agents, with cell volume computed using a dynamical system model that represents changes in liquid and solid fractions within the cell. Each cell can exist in one of three states: proliferating, quiescent, or necrotic—depending on the local oxygen concentration, and the model accommodates cell cycle progression, division, and death accordingly. Cell migration is influenced by cell-cell adhesion, cell-cell repulsion, chemotaxis, and random motility. Glioblastoma cells secrete the immunostimulatory factor, which attracts CAR T-cells to migrate along its gradient. The immunostimulatory factor and oxygen are modeled using reaction-diffusion equations, and these are coupled to the cells in the ABM through the reaction term. The detailed model equations, adapted from the default PhysiCell implementation, are provided in the Supplementary Note S2.

The initial tissue of the three sample patients are shown in Fig. 2. Each computational domain measures 1800 × 1500 × 160 microns and is initialized with approximately 80,000 glioblastoma cells. The CAR T-cell dosage in our simulations was scaled from clinical dosing data based on tumor size. Clinical trials for glioblastoma report doses ranging from 10⁶ to 10⁸ cells, with some up to 10^{10 58}. Based on volume scaling, an average tumor diameter of 4cm and our simulated tissue diameter of 0.1cm yield an estimated range of 16–1,562 CAR T-cells. We therefore use dosages on the order of 100 cells, which fall within this scaled range. The dosing frequency in our model is informed by the recent clinical trials reported in refs. ^27,42, which used weekly injections. In general, other works have administered between one and twelve total doses, with dosing frequencies ranging from once to three times per week⁵⁸. The discretized time step Δt_diff in the simulation is set to 0.01 minute for the reaction-diffusion PDE; Δt_mech in the cell mechanics is set to be 0.1 minute; Δt_cell in the cell volume and cycle/death models is set to be 6 minutes.

ABM interaction model between glioblastoma and CAR T-cells

In the ABM, the glioblastoma cells exhibit specific phenotypes characterized by the expression levels of antigens EGFR, IL-13Rα2, and HER2. We denote the expression levels as a = (a₁, a₂, a₃) ∈ [0, 1]³. A glioblastoma cell is considered to be positive for the i-th antigen if a_i is over some threshold, denoted as a_thr.

The three types of CAR T-cells can form an adhesion and subsequently eliminate the glioblastoma cell if the matching antigen level a_i is over the threshold a_thr. The adhesion between the glioblastoma cells and CAR T-cells is determined by a probability that is linearly dependent on the antigen expression levels as well as the distance between the cells. The adhesion probability is defined as follows:

$$\mathrm{Prob(adhesion)}=\min \left(1,r_{\mathrm{attach}}{s}_{\mathrm{dist}}{\theta }_i\mathrm{\Delta }{t}_{\mathrm{cell}}\right),$$ 1

where r_attach is the CAR T-cell’s rate of forming new cell adhesions, s_dist is the scaled distance between the target glioblastoma cell and the attaching CAR T-cell, and θ_i is the scaled expression level of the i-th antigen on the glioblastoma cell defined as follows:

$${\theta }_i=\left\{\begin{array}{c}\frac{a_i-a_{\mathrm{thr}}}{a_{\max }-a_{\mathrm{thr}}},a_i\ge a_{\mathrm{thr}}\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right)$$ 2

The maximum level of antigens $a_{\max }=100$ represents the maximum value of the antigen expression level from the data. The scaled distance is computed as follows:

$$s_{\mathrm{dist}}=\left\{\begin{array}{c}\frac{d_{\max }-\mathrm{\Delta }d}{d_{\max }-d_{\min }},\mathrm{\Delta }d\le d_{\max }\hfill \\ 0,\mathrm{otherwise},\hfill \end{array}\right)$$ 3

where $d_{\max }$ and $d_{\min }$ denote the maximum and minimum adhesion distance, and Δd denotes the distance between the glioblastoma and CAR T-cell. According to this setup, the adhesion between the glioblastoma cell and i type CAR T-cell is formed only if the distance between two cells is less than $d_{\max }$ and the antigen level a_i exceeds the threshold a_thr. We assume that multiple CAR T-cells can attach to a single glioblastoma cell, and each event is independent.

When two cells are attached, the CAR T-cells attempt to induce apoptosis in the attached glioblastoma cell with the following probability,

$$\mathrm{Prob(inducing\; apoptosis)}=P_i=r_{\mathrm{kill}}{\theta }_i\mathrm{\Delta }{t}_{\mathrm{cell}},$$ 4

where r_kill is the killing rate. Here, we assume that apoptosis probability depends linearly on the antigen expression level. If multiple types of CAR T-cells are attached, each CAR T-cell attempts to induce apoptosis independently. Thus, the overall probability of a glioblastoma cell being eliminated can be computed by using the survival probability P_survive = ∏_i∈I(1 − P_i), where I is the set of attached CAR T-cell antigen index among {1, 2, 3}, and it follows that the overall probability of successful cancer cell elimination is P_kill = 1 − P_survive.

The scheme of the ABM model for the glioblastoma and CAR T-cell interaction are as follows. First, we determine if a CAR T-cell is attached to any glioblastoma cell. If it is attached, the CAR T-cell will attempt to induce apoptosis with a probability given by Eq. (4) and if it is successful, the CAR T-cell proceeds with the killing process. If apoptosis is not induced, it will detach with a probability Prob(detach) = Δt_cell/(T_attach + 10⁻¹⁵), where T_attach is the duration of the attachment. On the other hand, if the CAR T-cell is not attached to a glioblastoma cell, it will search for surrounding cells to attach to with a probability given by Eq. (1). In this case, the CAR T-cell will check all glioblastoma cells within the maximum adhesion distance. A summary flowchart of the interaction model is in the Supplementary Fig. S2.

Parameter calibration

We aim to calibrate the model to the experimental data presented in ref. ²⁷, which includes in vitro data of IL-13Rα2 CAR T-cell treatment of glioblastoma cells. Given the large number of parameters in the model, we first conduct a parameter sensitivity analysis on several key factors in the model: oxygen uptake rate, oxygen decay rate, adhesion strength, repulsion strength, maximum adhesion distance, oxygen diffusion coefficient, CAR T-cell attachment rate to glioblastoma cells, and CAR T-cell killing rate of glioblastoma cells. We find that the attachment rate r_attach and the killing rate r_kill are the most critical parameters affecting tumor apoptosis. Consequently, we focus on calibrating these two parameters to the data from²⁷. The optimal parameters are identified by minimizing the residual sum of squares between the model’s predictions and the experimental data of tumor volume. For the calibration, we consider a range for attachment rate r_attach ∈ [0.5, 5] and the killing rate r_kill ∈ [0.1, 0.2]. The experiments in ref. ²⁷ examine glioblastoma cells mixed with IL-13Rα2-targeted CAR T-cells at different ratios-1:5, 1:10, and 1:20-and with varying antigen expression levels (low, medium, and high). In our simulations, we assign the antigen expression levels a_i to 0.35, 0.65, and 0.95 for low, medium, and high antigen-expressing glioblastoma cells, respectively. The experimental data and the best model fit are shown in Fig. 10, where we find the optimal parameters to be r_attach = 2 and r_kill = 0.15. These values were determined through a grid-based search with a grid size of 0.5 for the attachment rate and 0.1 for the killing rate. The values for the remaining model parameters are estimated from various biological literature sources. Due to the lack of specific data for HER2 and EGFR-targeted CAR T-cells, we assume that the attachment and killing rates for all three types of CAR T-cells (IL-13Rα2, HER2, and EGFR) are identical for the purpose of this study.

Fig. 10. The optimal model calibration using the attachment rate r_attach = 2.0 and the killing rate r_kill = 0.15.

The shown are the experimental data (circle) collected in ref. ²⁷ quantifying the glioblastoma cell population with a dimensionless number referred to as cell-index (CI) and the calibrated model simulation (lines). Columns a–c represent CAR T-cell to cancer cell ratios of 1:5, 1:10, and 1:20, respectively. Rows indicate cancer cell antigen density levels from low to high.

Author contributions

H.C., R.R., M.B., and M.G. conceptualized and designed the study. R.L. and H.C. developed the model, and R.L. developed the code and performed the analyses. R.L., H.C., R.R., M.B., M.G., C.B., and L.F. contributed to the interpretation of results. R.L. and H.C. drafted the manuscript, and R.L., H.C., R.R., M.B., M.G., C.B., and L.F. revised and approved the manuscript. H.C. supervised the overall direction and planning of the study.

Data availability

All of the data supporting the results of this study are available within the paper and its Supplementary Information. The code is available in https://github.com/tony1747/Physicell_Project.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41540-025-00642-7.