Mathematical modeling insights into improving CAR T cell therapy for solid tumors with bystander effects

CAR T-cell therapies have transformed treatment for several hematologic malignancies, yet their efficacy against solid tumors remains limited. Key barriers include tumor antigen heterogeneity, poor trafficking and infiltration, and immunosuppressive tumor microenvironments. Antigen heterogeneity is particularly problematic because CARs target only antigen-expressing cells, so efficacy declines when tumors contain antigen-negative subclones. An important question is whether CAR T therapy elicits bystander effects, including indirect killing via innate cells and antigen spreading that activates endogenous CD8 T cells against non-targeted antigens. The study aims to mathematically model CAR T therapy in solid tumors with mixed antigen-positive and antigen-negative populations, quantify how bystander effects influence outcomes, identify sensitive parameters, and evaluate strategies to improve efficacy, including enhancing bystander cytotoxicity.

Evaluating bystander effects experimentally is challenging and requires immunocompetent models to capture endogenous immune responses; many preclinical studies use immunodeficient mice that cannot assess bystander phenomena. Quantifying bystander effects further requires precise antigen profiling, controlled mixtures of antigen-positive and negative tumors, and accurate readouts. Prior mathematical models have addressed CAR T therapy dynamics in hematologic contexts, CAR T proliferation and exhaustion, and effects of conditioning regimens, but none have specifically examined bystander effects in solid tumors. Experimental work has shown that low-dose cyclophosphamide can induce bystander effects and cure heterogeneous tumors, and mechanisms such as FasL–Fas mediated killing and antigen spreading have been implicated. The present study fills a gap by constructing and calibrating a model to in vivo data of mixed-antigen solid tumors to study bystander mechanisms and treatment strategies.

Model structure: An ordinary differential equation model tracks four cell populations within the tumor microenvironment (units in mm^3): antigen-positive tumor cells (T_pos), antigen-negative tumor cells (T_neg), CAR T cells (C), and bystander cells (B). Tumor subpopulations grow logistically with distinct proliferation rates and a shared carrying capacity. Antigen-positive cells are killed by both CAR T cells and bystander cells; antigen-negative cells are killed only by bystander cells. Cell killing uses ratio-dependent lysis terms with an exponent controlling dependence on effector-to-target ratios and a steepness parameter. CAR T and bystander populations have natural death, are inhibited by interactions with tumor cells, and are recruited via Michaelis-Menten-type terms limited by an immune carrying capacity. CAR T administration is modeled as a pulsed source on injection days. Equations: The system includes logistic growth for T_pos and T_neg toward K1 with rates r1 and r2, ratio-dependent kill terms for CAR T and bystander effects on T_pos, ratio-dependent bystander kill on T_neg, CAR T and bystander dynamics with natural death (gamma_C, gamma_B), recruitment limited by log((B + C)/K2) with maximum recruitment rates (mu_C, mu_B), and inhibition terms (omega_C, omega_B). The CAR T input v_C(t) is nonzero only on injection days with dose I_C. Data and calibration: The model is calibrated to in vivo data (Klampatsa et al., Mol. Ther. Oncol. 2020) using mesothelioma tumors with 100%, 90%, and 75% mesothelin-positive cell mixtures. Assumptions: 10^6 cells ≈ 1 mm^3; only 1% of infused CAR T cells reach the tumor, setting I_C = 0.1 mm^3 per dose for the two-dose regimen given on days 4 and 6 post-inoculation; initial conditions at treatment initiation: total tumor ≈ 50 mm^3 with T_pos(0) = 50 p, T_neg(0) = 50 (1 − p), C(0) = 0, B(0) = 0.1, where p is the antigen-positive fraction. Parameters r1 and K1 are fit to 100% control data (no therapy), then r2 is fit to 90% and 75% controls; CAR T parameters l, d_c (max CAR T kill), and mu_c (max CAR T recruitment) are fit using two-dose CAR T data without bystander effects; bystander parameters d_b (max bystander kill) and mu_b (max bystander recruitment) are calibrated using low-dose cyclophosphamide (CTX) plus CAR T data that elicit cures at 90% and 75% antigen positivity. Objective function: root mean square error between simulated and observed total tumor volume; optimization via MATLAB fmincon. The ODEs are solved with MATLAB ode45 (Dormand–Prince) with piecewise integration between discontinuous dose times, step size Δt = 0.05 days, over t ∈ [3, 30] days. Sensitivity analysis: Global variance-based sensitivity (Sobol indices) is computed using sparse grid interpolation (spinterp) in MATLAB on a 32,000-point interpolant over specified parameter ranges; outputs are squared deviation between simulated tumor size and data at day 9 for 90% and 75% antigen-positive cases with CTX. First-order and total Sobol indices identify parameters with largest effects. Virtual patients: A control set of parameter triplets (r1, r2, K1) is generated to fall within a 0.5 relative error band to control data across 100%, 90%, and 75% cases, yielding 14,000 acceptable sets with ranges r1 ∈ [0.12, 0.24] day^-1, r2 ∈ [0.14, 0.28] day^-1, K1 ∈ [1e6, 9e6] mm^3. Two VP cohorts are then formed by fixing these and sampling the remaining 13 parameters uniformly around literature/baseline values: Cohort-1 uses [0.25×, 1.5×] baseline (primary analysis), Cohort-2 uses [0.7×, 1.3×] (supplement). Treatment outcomes at day 30 are classified as: Responder (R) if T_total < 2 mm^3, Partial Responder (PR) if 2 ≤ T_total < 50 mm^3, and Non-Responder (NR) if T_total ≥ 50 mm^3. Statistical analysis: Nonparametric Kruskal–Wallis tests assess differences of parameter distributions across R, PR, NR with significance threshold p = 0.05; effect sizes reported via eta-squared (η^2) with Cohen’s guidelines; significant parameters undergo Dunn’s post-hoc pairwise comparisons. Empirical cumulative distribution functions and box plots visualize group differences. Baseline parameters: Fitted or literature-sourced values are reported (e.g., r_pos = 0.18 day^-1, r_neg = 0.21 day^-1, K = 5.1×10^6 mm^3, l = 1.56, mu_c = 0.6 day^-1, d_c = 0.41 day^-1, s = 0.305, gamma_c = 2.93×10^-2 day^-1, k ≈ 2.019×10^-2 day^-2, omega_C and omega_B from literature, mu_b = 0.3 day^-1, mu_g ≈ 0.89 day^-1, b = 1.4×10^-2 mm^3 day^-1, gamma_b = 7×10^-3 day^-1, beta ≈ 3.42×10^-8 mm^3 day^-1).

- Sensitivity analysis: For outputs defined as day-9 tumor size deviation from data, first-order Sobol indices highlight r1 and l as most influential (S[r1] ≈ 0.167, S[l] ≈ 0.143). Total Sobol indices indicate strong interaction contributions for r1 (S_T[r1] up to ≈ 0.85) and notable importance of l, d_c, and mu_c, among others. - Dose-response: Increasing CAR T cell dose alone does not ensure favorable outcomes. When antigen-positive fraction is about 70% or lower, escalating CAR T dose does not convert non-responders into responders, suggesting limited benefit of dose escalation without bystander effects. A threshold behavior is observed where benefits plateau beyond certain doses. - Virtual patient outcomes: Across 14,000 VPs (Cohort-1) at 75% antigen positivity under a standard regimen, outcomes were R: 23.53%, PR: 22.68%, NR: 53.79%. Mean trajectories show that in R and PR groups, CAR T expansion and bystander engagement reduce tumor burden, whereas in NR they are insufficient to halt growth. - Statistical discrimination of parameters: Kruskal–Wallis tests identify parameters with significant differences across R, PR, NR. The exponent in tumor lysis and the maximum bystander killing rate show highly significant p-values and large effect sizes, distinguishing responders from non-responders. Parameters such as r1, r2, mu_c, d_c, s, k, b, and K2 are significant with smaller effect sizes. Post-hoc Dunn’s tests show most significant differences occur between R and NR groups. - Enhancing bystander cytotoxicity: Systematically increasing the maximum bystander killing rate by 10%, 20%, 30%, and 40% (holding other parameters constant) monotonically increases the responder proportion relative to baseline, indicating that boosting bystander cell cytotoxicity can substantially improve outcomes, more so than increasing CAR T dose. - Model-data agreement: The model reproduces control tumor growth across antigen-positivity levels (MREs ~0.08–0.21), captures two-dose CAR T outcomes (tumor elimination at 100% positivity; slowed but not cured at 90% and 75% without bystander effects), and fits CTX-induced bystander cures at 90% and 75% positivity.

The model suggests that limited efficacy against heterogeneous solid tumors arises when bystander effects are insufficient, even if CAR T cells expand and traffic to tumors. This aligns with experimental findings that bystander effects require endogenous CD8 T cells and are not driven purely by CAR T persistence. Sensitivity and VP analyses indicate that improving the cytotoxic potential of bystander cells and the lysis nonlinearity can shift outcomes from NR to R, whereas increasing CAR T dose alone offers diminishing returns below roughly 70% antigen positivity. These insights support strategies that augment bystander mechanisms (e.g., antigen spreading, innate activation, FasL–Fas mediated killing) or co-therapies that activate endogenous immunity (e.g., low-dose cyclophosphamide). The framework also points to future model extensions: incorporating regulatory T cells and CTX effects on Treg dynamics, cytokine-mediated innate activation (IFN-γ, TNF-α), dendritic cell cross-presentation to drive CD8 priming and epitope spreading, and spatial PDE formulations to capture intratumoral heterogeneity and diffusion of mediators. Despite data limitations and cross-species parameter sourcing, the model provides mechanistic hypotheses and guides experiment design to quantify bystander contributions and optimize combination regimens.

This work presents, to our knowledge, the first mathematical model focused on bystander effects in CAR T therapy for solid tumors with mixed antigen expression. Calibrated to in vivo data, the model shows that CAR T dose escalation alone does not overcome antigen heterogeneity, whereas enhancing bystander cell cytotoxicity markedly improves response rates. Global sensitivity and virtual patient analyses identify key drivers (lysis nonlinearity and bystander killing capacity) and quantify outcome distributions. The study supports therapeutic strategies that amplify bystander mechanisms and outlines model-driven directions for future research, including incorporating Tregs and CTX effects, cytokine and innate immune pathways, dendritic cell cross-presentation, and spatial modeling. Further experimental data, particularly in immunocompetent and spatially resolved settings, will refine parameters and validate predicted intervention benefits.

- Data availability: Calibration relies on a specific immunocompetent mouse model with limited time points; bystander quantification is challenging and may not generalize across tumor types or antigens. - Parameter sourcing: Some parameters are synthesized from diverse literature (mouse and human, various CAR constructs and cancers). Although sensitivity analysis guided the use of literature values for less influential parameters, cross-species and context differences may limit predictive scope. - Model structure: The ODE framework neglects spatial heterogeneity (infiltration gradients, antigen distribution) and simplifies complex immune interactions (e.g., cytokines, innate subsets, dendritic cell priming, Tregs) into aggregate bystander terms. - Dosing and trafficking assumptions: Assumes 1% of infused CAR T cells reach the tumor and uses ratio-dependent lysis; alternative trafficking efficiencies or kill kinetics could alter quantitative predictions. - Statistical interpretation: Nonparametric tests detect distributional differences but do not specify causal mechanisms; effect sizes and VP sampling ranges influence detected significance.