Multiscale modeling reveals synergy between CCL19 and PD-1 blockade in reshaping the TNBC microenvironment

Abstract

Triple-negative breast cancer (TNBC) presents a major clinical challenge owing to its immunosuppressive tumor microenvironment, target scarcity, and poor therapeutic response. Recently, the combination therapy of immune checkpoint blockade and CCL19 has shown significant efficacy in TNBC. To systematically unravel the synergistic mechanisms between CCL19 and anti-PD-1, we developed a mathematical model by integrating cellular and molecular scales to capture essential tumor-immune interactions and predict the dynamics of tumor evolution under various therapies. In this study, we proposed three quantitative indicators: (1) the tumor relative volume index (TRVI), (2) the therapeutic efficacy discrepancy index (TEDI), and (3) the immune heterogeneity treatment response index (IHTRI). Our model validated that the immunostimulatory effect of CCL19 in synergizing with anti-PD-1, and revealed that this synergy is highly modulated by individual baseline immune heterogeneity. Notably, our analysis identified (CTLs × CCL19)/PD-L1 as a novel dynamic biomarker combination with significant predictive (AUC = 0.86) and prognostic value (log-rank p= 0.019). Finally, virtual clinical trials revealed that administering anti-PD-1 therapy prior to CCL19 injection draws more significant clinical benefits in TNBC. Collectively, this study provides a theoretical foundation for elucidating the synergistic mechanism between CCL19-mediated immunostimulation and anti-PD-1 therapy.

Introduction

Triple-negative breast cancer (TNBC), a molecularly distinct breast cancer subtype, is defined by the absence of estrogen receptor (ER), progesterone receptor (PR), and human epidermal growth factor receptor 2 (HER2) expression¹. This subtype accounts for 10–15% of all breast cancer cases and is characterized by aggressive behavior and poor outcomes, demonstrating a 5-year survival rate of 65–77% and a 30% risk of recurrence/metastasis within three years of diagnosis². TNBC treatment remains largely chemotherapy-dependent, despite its limited efficacy and substantial toxicity³. Given these limitations, immune checkpoint blockade (ICB) targeting the PD-1/PD-L1 axis has emerged as a promising therapeutic alternative⁴. Approximately 40%–60% of TNBC tumors express PD-L1 at high levels, supporting the rationale for inhibiting the PD-1/PD-L1 axis as a promising therapeutic strategy against TNBC⁵, but fewer than 20% of patients derive clinical benefit from current immunotherapies⁶. The low response rate has driven efforts to develop combination therapies addressing tumor immune evasion mechanisms, such as T cell exhaustion or immunosuppressive microenvironments.

Recent studies highlight the therapeutic and prognostic potential of CCL19 in combination with anti-PD-(L)1 for TNBC, attributed to its unique immunomodulatory properties⁷. As a homeostatic chemokine, CCL19 binds CCR7 on T cells to drive naïve T cell recruitment into the tumor microenvironment (TME) and facilitate tertiary lymphoid structure (TLS) formation, thereby potentiating immune surveillance⁸. These immunomodulatory properties make CCL19 a promising synergistic partner for anti-PD-(L)1 therapy⁹. For instance, a preclinical study demonstrated that plasmid DNA-encoded CCL19 combined with anti-PD-(L)1 induced high IFN-γ secretion from immune cells and significantly suppressed growth and metastasis in melanoma and colon carcinoma models¹⁰. Recent studies further confirmed that intratumoral CCL19 delivery synergized with anti-PD-1 therapy, increasing CD8+ T cell infiltration and prolonging survival in TNBC¹¹. Moreover, multi-center cohort studies identify CCL19 as an independent predictive biomarker for anti-PD-1 response in TNBC patients¹².

To quantitatively dissect such complex combination therapies involving multi-cellular, multi-target, and cross-scale interactions, traditional experimental approaches often face challenges of high costs. In this context, multiscale mathematical modeling has become a powerful tool for deciphering intricate tumor-immune interactions and elucidating cross-scale biological networks^13,14. A systematic model by Lai et al.¹⁵ decoded the therapeutic efficacy of combining the Bromo- and Extra-Terminal inhibitor and ICB through multicellular/cytokine cascade integration. Focusing on the ER+ breast cancer subtype, He et al.¹⁶ proposed a comprehensive model integrating ER signaling and cell cycle regulation. The model quantitatively elucidated the synergistic effects of ICB and endocrine therapeutic agents, providing a mathematical basis for optimizing clinical combination therapy. Furthermore, considering endothelial cells, oxygen, and VEGF, Mohammed’s team¹⁷ systematically unveiled a bidirectional regulatory network of adipocyte-mediated metabolic reprogramming with angiogenic abnormalities in the TME of breast cancer, opening new avenues for the development of therapeutics targeting hypoxic ecological niches.

Notably, the Quantitative Systems Pharmacology (QSP) framework abstracts biological systems into physiologically interconnected compartments—such as tumors, lymph nodes, and peripheral blood^18,19. This approach mathematically captures cross-compartmental dynamics of drugs and immune cells, establishing a robust platform for simulating immune cell trafficking, drug spatial distribution, and tumor-immune interactions across tissues^20,21. By integrating these capabilities, QSP provides essential foundations for virtual clinical trials while serving as a powerful tool for biomarker discovery and treatment optimization^21,22. For instance, Arulraj et al.²² developed a QSP-based computational platform to simulate immunotherapy responses in TNBC. Their virtual clinical trials demonstrated that combining the proportion of M1 macrophages with the dynamic ratio of regulatory T cells (Tregs) to cytotoxic T lymphocytes (CTLs) served as a near-perfect predictive biomarker. Similarly, Li et al.²³ employed modeling to elucidate the competitive binding kinetics and quantify synergy between a fibroblast growth factor receptor 3 inhibitor and ICB. Furthermore, they developed a quantitative cancer-immunity cycle model, revealing that tumor-infiltrating CD8+ CTL density acts as a core predictor of metastatic colorectal cancer progression, while dynamic changes in the CD4+ T helper 1 (Th1)/Tregs ratio were closely associated with survival outcomes²⁴. These groundbreaking advances underscore the powerful capability of multiscale modeling in decoding the dynamic evolution of biomarkers and optimizing combination strategies. However, in the context of combination therapy of CCL19 and anti-PD-1 for TNBC, a systematic understanding of the cross-scale coupling mechanisms in the immune regulatory network remains unclear, particularly in terms of constructing dynamic prognostic biomarker panels and their potential clinical translational implications.

In the current study, we established a multi-scale framework focused on TME to quantitatively dissect the synergistic mechanism between CCL19-mediated immune enhancement and PD-1 blockade in TNBC. By integrating computational systems biology approaches, the aim of this study is to elucidate the pivotal synergistic effects of CCL19 in combination with anti-PD-1 to reprogram the TME and provide a robust theoretical framework for clinical translation. With regards to this, we introduced three innovative metrics to precisely quantify the dynamic crosstalk between immune effector functions and tumor control. The indicator of tumor relative volume index (TRVI) could be used to evaluate dynamic treatment response before and after treatment under identical regimens. The indicator of therapeutic efficacy discrepancy index (TEDI) could be used to quantify efficacy differences between distinct combination strategies. The indicator of immune heterogeneity treatment response index (IHTRI) is introduced to assess the impact of baseline immune status on combination therapy outcomes. Furthermore, we discovered a novel nonlinear biomarker combination (CTLs × CCL19)/PD-L1 with reliable predictive performance (AUC = 0.86). Subsequently, the stochastic survival simulation on the basis of the death probability function (DPF) confirmed its significant prognostic value for long-term survival. More than that, virtual clinical trial results indicated that an anti-PD-1 prioritized sequential administration strategy could amplify therapeutic efficacy. Collectively, this study establishes a comprehensive theoretical framework for mechanistic elucidation, dosage optimization, and precision stratification in TNBC combination therapy.

Results

Multiscale tumor-immunity interactions framework

The interactions between tumor cells, immune cells, and cytokines constitute a complex dynamic regulatory network. Given that the dynamics of cellular-scale processes typically occur on a significantly faster temporal scale than those at the protein molecular scale, integrating these dynamics presents a fundamental challenge. Mathematical modeling serves as a powerful tool for cross-scale dynamic analysis, enabling the integration of slower-scale (molecular) dynamics into models of faster-scale (cellular) behavior. To address this, we described distinct functional cell populations and their interactions with protein molecules, as depicted in Fig. 1.

Fig. 1. Dynamic tumor-immune interaction network in TNBC.

Cell populations with different functions are represented with different colored blocks. The symbol system is defined as follows: → (solid arrow) indicates intercellular promotion process; ⊣ (horizontal arrow) denotes intercellular inhibitory processes; ⇢ (dashed arrow) represents cell polarization/chemotaxis process. The circular area shows PD-1-PD-L1 complex formation. Cytokine-mediated promotional effects are highlighted in green text, whereas inhibitory processes mediated by cytokines are marked in red text.

This intricate network initiates with antigen presentation and immune priming (orange module in Fig. 1). Dendritic cells (DCs), as the most potent antigen-presenting cells (APC), prime adaptive immunity through cross-presentation of tumor-associated antigens to naïve T cells, a process enhanced by their secretion of IL-12^25,26. This triggers the Adaptive Immune Response (teal module in Fig. 1): naïve CD8+ T cells differentiate into CTLs that mediate direct tumor cell killing, which is primarily dependent on the secretion of IFN-γ^27,28. Meanwhile, mature DCs could recruit CTLs to tumor sites via CCL19-dependent chemotaxis. Th cells, which are derived from naïve CD4+ T cells, can be polarized by IL-12 but suppressed by IL-10 and TGF-β²⁹. However, these two types of cells are actively suppressed by the PD-1-PD-L1 axis, thereby subverting adaptive immune surveillance. Regulatory T cells (Tregs), pivotal regulators of immune homeostasis, further drive TME immunosuppression through secretion of inhibitory cytokines (IL-10, TGF-β)²⁷. The innate immune cell population (blue module in Fig. 1) provides the first line of anti-tumor defense, with natural killer (NK) cells playing a pivotal role^25,30. Monocytes are a heterogeneous population of innate immune cells that can functionally polarize into two phenotypically distinct sub-populations of M1 and M2 macrophages in response to different stimuli. Specifically, TGF-β and M-CSF promote the M1 to M2 transition, while TNF-α and IL-12 facilitate the M2 to M1 transition²⁷. Particularly, as a main component of TME in TNBC, cancer-associated adipocytes (CAAs) actively promote tumor progression³¹ (light purple module in Fig. 1).

Based on the aforementioned multiscale dynamic regulatory network of tumor-immune interactions, we have developed a systematic mathematical model (Section “Multiscale mathematical modeling formulation”).

Calibration of tumor evolutionary dynamics and global sensitivity analysis

Calibration of tumor evolutionary dynamics

To calibrate our model, we utilized in vivo experimental data from Wu et al.¹¹, which monitored tumor progression in a syngeneic mouse model of TNBC. In their experiment, female BALB/c mice (6–8 weeks old) were orthotopically implanted with 4T1 or 66c14 TNBC cells into the fourth mammary fat pad. Animals were divided into four treatment groups: (1) vehicle control, (2) intratumoral recombinant CCL19, (3) intraperitoneal anti-PD-1 antibody, and (4) the combination of CCL19 and anti-PD-1. Each group contained n = 6 mice. Tumor volumes were measured with calipers and calculated as 0.5 × length × width²; data are presented as the mean ± standard error of the mean (SEM). We converted these measurements into absolute tumor cell counts using the relation: N_cells = V × ρ, where ρ = 8 × 10⁴ cells/mm³ denotes the assumed cellular density of solid tumors²³. The derived cell counts were integrated into our ordinary differential equation (ODE) system, enabling numerical simulation of tumor-immune dynamics across all therapeutic conditions. Solutions were computed using the Fourth-Order Runge-Kutta method to resolve the temporal evolution of the TME.

We identified the model parameters by fitting the simulated tumor progression dynamics to the experimental data obtained from the 66c14 syngeneic model (training set), as described in the Methods. Optimal parameter estimates were derived by maximizing the coefficient of determination (R²) against the corresponding experimental measurements (see Supplementary Tables S1 and S2 for details). The model demonstrated strong fitting performance on the training set, with R² values exceeding 0.81 across all treatment groups and reaching above 0.93 for the control and CCL19 monotherapy groups. The mean absolute percentage error (MAPE) ranged from 15.05% to 46.73%, indicating good agreement between simulated and observed tumor volumes.

To validate the model’s generalizability and robustness, we performed an independent test using data from the 4T1 syngeneic model (testing set), while keeping all parameters fixed at their training-set estimates. The model successfully predicted tumor dynamics in the testing set, achieving R² values between 0.738 and 0.896 across treatment groups, with MAPE values ranging from 21.6% to 48.9%. Notably, the combination therapy group showed the highest predictive accuracy (R² = 0.896, MAPE = 21.6%). To further assess model stability, we performed 1000 simulations with ±5% random parameter fluctuations around the fitted values. The resulting prediction curves remained tightly clustered around the nominal trajectory (see Fig. 2), demonstrating low sensitivity to parameter uncertainty.

Fig. 2. The results of data calibration based on multiscale model simulations.

All calibrated parameter values are documented in Supplementary Tables S1 and S2. A–D show the model fits (solid lines) alongside the experimental measurements (points with error bars representing SEM) for each treatment group in the training set (66c14 model). A Control group: anti-PD-1 parameters (α₂, d_Y, γ_Y) and CCL19-related parameters (d_X, γ_X) were disabled. B CCL19 monotherapy group: anti-PD-1 parameters (α₂, d_Y, γ_Y) were disabled. C Anti-PD-1 monotherapy group: CCL19-related parameters (d_X, γ_X) were disabled. D Combination therapy group: Displays the scenario where all parameters are active, reflecting the combined effects of both treatments. E–H present the corresponding independent validation on the testing set (4T1 model), where the same fixed parameters were used to generate predictions (solid lines). The shaded bands indicate the range of 1000 simulations with ±5% random parameter fluctuations, illustrating model robustness. R² and MAPE values are reported for each panel.

Simulation of immune dynamic responses

To characterize the temporal immune dynamics underlying therapeutic synergy, we analyzed the evolutionary trajectories of other cellular components across regimens. Key results revealed that (Supplementary Figs. S1 (Training set) and (Testing set) S2), during the treatment period, the combination therapy group significantly altered the dynamics of several immune cell populations compared to monotherapy and the control group. Specifically, CTLs, Th, and M1 macrophages exhibited substantial increases in the later stages of treatment. Furthermore, the modulation of drug concentrations elicited periodic fluctuations in key immunomodulatory cytokines, including IL-2, TNF-α, and IFN-γ. This cytokine oscillation pattern likely mediated the subsequent emergence of rhythmic density variations in CTLs and Th cells, a phenomenon that became apparent from the sixth day of treatment onwards (Supplementary Figs. S3 (Training set) and (Testing set) S4).

Global sensitivity analysis

To validate the stability and robustness of our model framework, we performed a comprehensive global sensitivity analysis encompassing all 52 estimated parameters (Supplementary Table S3). Using Latin hypercube sampling (LHS), we generated 1000 parameter sets by perturbing each parameter within ±20% of its baseline calibrated value. Each set was integrated into the ODE system to simulate tumor cell dynamics over a 22-day combination therapy period. The output was quantified by the TRVI, which captures parameter-induced changes in tumor burden (see Section “Quantitative metrics”). Sensitivity was then assessed via the partial rank correlation coefficient (PRCC) between each parameter and TRVI, with statistical significance defined at p < 0.05.

Based on the magnitude of the PRCC values, we categorized parameter sensitivity as follows: ∣PRCC∣ < 0.2 (non-sensitive), 0.2 ≤ ∣PRCC∣ < 0.4 (low), 0.4 ≤ ∣PRCC∣ < 0.6 (moderate), and ∣PRCC∣ ≥ 0.6 (high). From the full set of 52 parameters, we subsequently selected 16 key parameters for further in-depth analysis. This selection was based on a combination of their high sensitivity rank (primarily moderate to high PRCC values) and their established biological relevance in immune-tumor crosstalk (Supplementary Table S4).

Sensitivity analysis of these 16 key parameters revealed that several exhibited a positive correlation with TRVI (Fig. 3A): the proliferation rate of tumor cells (β_c, PRCC = 0.9031), the half-saturation concentration of CCL19 for recruitment of CTLs (K_L19, 0.6913), the equilibrium constant for the formation of the PD-1-PD-L1 complex (α₁, 0.1693), and the initial density of monocytes (M₀, 0.1803). Conversely, negative correlations were observed for: the killing rate of tumor cells by CTLs (${\eta }_{T_c,C}$, –0.6378); the number of bystander CTLs attracted by CCL19 ($T_C^{*}$, –0.4417); the recruitment rate of CCL19 for CTLs (${\xi }_{T_c,L19}$, –0.4335); the death rate of tumor cells (d_C, –0.3666); the killing rate of tumor cells by NK cells (η_N,C, –0.3189); the initial density of immature DCs (D₀, –0.3057); the constant source of NK cells (N^*, –0.2289); the killing rate of tumor cells by Th cells (${\eta }_{T_h,C}$, –0.2088); the half-saturation concentration of the PD-1-PD-L1 complex (K_P, –0.1516); the initial density of naïve CD8+ T cells (T₈₀, –0.1186); and the equilibrium constant for the binding process of PD-1 with anti-PD-1 drug (α₂, –0.1010). Moreover, the correlation between the initial density of naïve CD4+ T cells (T₄₀, PRCC = 0.0088) and tumor burden was not statistically significant (p > 0.05). Among these parameters, β_c, K_L19, ${\eta }_{T_c,C}$, and ${\xi }_{T_c,L19}$ exhibited a strong linear relationship with tumor burden after combination therapy (the correlation trend is shown in Fig. 3B).

Fig. 3. Global sensitivity analysis.

A PRCC values of 16 key parameters selected based on their biological relevance and sensitivity ranking from the full set of 52 parameters. Statistical significance was assessed at the 0.05 level. A sample size of 1000 for each parameter is taken based on the LHS approach with a uniform probability distribution. B Correlation trends between highly sensitive parameters and TRVI, where TRVI values were log-transformed to a standard scale. Note: Full parameter descriptions are provided in Supplementary Table 3.

Virtual cohort analysis reveals dynamic heterogeneity in immune cell subset

To quantify the impact of inter-individual heterogeneity on therapeutic response, we constructed a virtual cohort comprising 500 samples using LHS based on our validated multiscale tumor-immune dynamical model (workflow schematized in Fig. 4A). A key step in this process was to define biologically plausible variation ranges for model parameters to simulate inter-individuals differences. The design of these parameter ranges was informed by prior experimental calibration and global sensitivity analysis, ensuring they reflect a balance between capturing true biological variability (wider ranges) and maintaining the structural robustness of the model (narrower ranges)^20,23,32. Based on their PRCC values (Fig. 3A and Supplementary Table S3) and biological roles, we categorized them into four distinct groups, each assigned a specific perturbation range justified as follows:

• Tumor evolution parameters: β_c, ${\eta }_{T_c,C}$, d_C, η_N,C, and ${\eta }_{T_h,C}$. Most parameters exhibit medium to high sensitivity, particularly the tumor cell proliferation rate β_c (PRCC = 0.9031). To maintain system stability while accommodating natural variation in core kinetic processes, this category is assigned a narrow perturbation range of ±10%.

• Baseline immune signature parameters: M₀, D₀, T₈₀, T₄₀, and N^*. These parameters exhibit minimal sensitivity and largely capture inter-individual differences in initial immune cell states. To better simulate genetic and microenvironmental heterogeneity in subjects, a larger perturbation range of ±50% is applied.

• Drug immune checkpoint function parameters: α₁, K_P, and α₂. These govern the dose-response dynamics of therapeutic targeting and exhibit low sensitivity. Accordingly, they are assigned a perturbation range of ±30%.

• Chemokine signaling parameters: These include K_L19, $T_C^{*}$, and ${\xi }_{T_c,L_{19}}$, which regulate CCL19-mediated T-cell recruitment and positioning. Their PRCC values all exceed 0.4 (with K_L19 reaching 0.6913), indicating medium to high sensitivity, albeit somewhat lower compared to the “tumor evolution parameters”. Therefore, a slightly wider perturbation range of ±15% was applied.

Fig. 4. Heterogeneity in innate/adaptive immune cell dynamics within various virtual cohorts.

A Schematic representation of virtual cohorts generated through parameter heterogeneity. Heterogeneity ranges (see Supplementary Table 3, Range 2) were applied to parameters to simulate biological variability. Five hundred virtual samples were modeled per treatment regimen, with tumor-immune dynamics tracked across cohorts. Heterogeneity in immune cell densities of CTLs (Tc, B), Th cells (Th, C), Tregs (Tr, D), M1 macrophages (M1, E), M2 macrophages (M2, F), NK cells (NK, G) in four virtual cohorts at the treatment endpoint (day 22).

Further details regarding the construction of the virtual cohort are provided in Supplementary section “Supplementary text for virtual cohort”.

Heterogeneity in immune cell dynamics across various virtual cohorts

To characterize the temporal evolution of the immune response under each therapy, we tracked the dynamics of innate and adaptive immune cell populations at key time points. A comprehensive analysis of these dynamics is presented in Supplementary Figs. S5 and S6. For clarity and to focus on the final therapeutic outcome, we analyzed the simulated data at the treatment endpoint (day 22). This analysis revealed significant differential effects of the therapies on specific immune cell subsets compared to the control group, as detailed in the box plots of Fig. 4B–G. In adaptive immune system dynamics, CCL19 specifically enhances CTLs activation, increasing CTLs counts 2.20-fold versus controls (p < 0.001; Fig. 4B), while anti-PD-1 elevates Th cell density to 1.81-fold (p < 0.001; Fig. 4C) through its essential role in Th cell activation. Combination therapy synergistically amplifies these effects, further increasing CTLs to 2.19-fold and Th cells to 1.54-fold of control levels (p < 0.001), while concurrently suppressing immunosuppressive Tregs to 0.74-fold (p < 0.001; Fig. 4D). Concurrently in innate immunity, CCL19 dramatically increased M1 macrophages to 1.90-fold of controls (p < 0.001), while Anti-PD-1 monotherapy induced a modest 1.18-fold increase (p < 0.001; Fig. 4E). No statistically significant differences in M2 macrophage and NK cell density were observed across any of the treatment groups (p > 0.05; Fig. 4F, G).

These results established that combination therapy achieves superior immunomodulation by simultaneously enhancing adaptive effector cells and suppressing immunosuppressive populations, demonstrating mechanistic synergy between CCL19 and anti-PD-1 in reprogramming the TME. Specifically, CCL19 drives this synergy through enhancing adaptive anti-tumor immunity via CTLs proliferation and promoting monocyte polarization to M1 macrophages, while anti-PD-1 plays a critical role in activating Th cells.

Individual baseline immune features determine treatment response in virtual cohorts

In this study, we posit that the baseline activity of innate immunity (N^* and M₀) combined with adaptive immune precursor reserves (T₄₀ and T₈₀) collectively define the quantitative basis of baseline immune competence in virtual subjects. To systematically investigate how baseline immunity influences treatment response, we introduced the IHTRI (Section “Quantitative metrics”). By randomly selecting a virtual subject while fixing other parameters, we modulated its baseline immune components (N^*, M₀, T₄₀, T₈₀) and analyzed IHTRI dynamics under combination therapy. As presented in Fig. 5, NK cell density (N^*) and naïve CD8+ T cell density (T₈₀) exhibited negative correlations with IHTRI. Higher baseline densities of these populations corresponded to lower IHTRI values, indicating enhanced treatment sensitivity. Naïve CD4+ T cell density (T₄₀) demonstrated a non-monotonic association, where both low and high baseline densities correlated with elevated IHTRI values. This suggested that either extreme may compromise therapeutic efficacy during combination therapy. Elevated monocyte levels were associated with reduced treatment sensitivity, potentially through fostering immunosuppressive TME. Notably, these correlation patterns persisted across all treatment regimens (Supplementary Fig. S7). We also found that IHTRI exhibited differential sensitivity to baseline variations depending on therapy: anti-PD-1 monotherapy showed greater IHTRI volatility than the CCL19 group. Moreover, combination therapy demonstrated the highest sensitivity to baseline fluctuations, exhibiting amplified IHTRI shifts compared to controls or CCL19 monotherapy.

Fig. 5. The impact of individual immune baseline heterogeneity on the outcomes of combination therapy.

The density of all baseline immune components (N^*, M₀, T₄₀, T₈₀) was systematically varied between 0 and 2 times based on the reference value. Heatmap intensity reflects therapeutic efficacy, with red regions indicating poor treatment response and blue regions denoting favorable outcomes. The color bar represents the IHTRI index.

Key immunomodulatory molecules heterogeneity modulates treatment efficacy in virtual cohorts

Beyond cellular immune components, we examined the influence of baseline heterogeneity in key immunomodulatory molecules CCL19 and PD-L1 on therapeutic efficacy. In the combination therapy cohort, virtual subjects were stratified into quartiles based on Day 5 (pre-treatment) expression levels of these molecules, with corresponding TRVI distributions shown in Fig. 6.

Fig. 6. Impact of pre-treatment biomarker levels on the efficacy of combination immunotherapy in a virtual cohort.

A Scatter plot showing the relationship between baseline CCL19 expression and post-treatment TRVI in the virtual cohort (n = 500). Data are stratified into quartiles (Q1: lowest; Q4: highest). Differences across CCL19 quartiles were statistically significant (Kruskal–Wallis test: χ² = 107.552, p < 0.001). B Scatter plot showing the relationship between baseline PD-L1 expression and post-treatment TRVI. Data are stratified into quartiles (Q1: lowest; Q4: highest). Differences across PD-L1 quartiles were statistically significant (Kruskal–Wallis test: χ² = 22.033, p < 0.001). In both panels, lower TRVI indicates better therapeutic efficacy.

We observed a progressive decrease in TRVI with increasing baseline CCL19 quartiles (Fig. 6A). Subjects in the highest CCL19 quartile (Q4) exhibited greater tumor reduction (mean TRVI: 15.8 ± 5.75) compared to those in the lowest quartile (Q1; mean TRVI: 26.8 ± 9.79). This suggests that higher pre-treatment CCL19 levels enhance combination immunotherapy efficacy. Similarly, elevated baseline PD-L1 expression correlated with improved treatment response (Fig. 6B). The highest PD-L1 quartile (Q4) showed lower TRVI values (mean: 17.3 ± 7.95) than the lowest quartile (Q1; mean: 21.9 ± 8.58), indicating that pre-treatment PD-L1 may mark pre-existing anti-tumor immune activity.

Identification of biomarker combination for prediction and prognosis

Biomarkers for predicting tumor burden

To evaluate the potential of our multiscale model for identifying biomarkers, we performed statistical analysis on model-simulated data. Firstly, virtual subjects from four treatment groups were stratified into low tumor burden (favorable response) and high tumor burden (poor response) cohorts based on the median tumor volume at day 30. Secondly, key immune features and molecular concentrations measured at day 22 served as predictive variables in a random forest classification model to predict tumor burden at day 30. Finally, model reliability was assessed through 1000 permutation tests, with important biomarkers identified via Gini index ranking. Across all four treatment groups, CTLs, PD-L1, and CCL19 consistently emerged as the most significant predictors (Fig. 7), each showing the highest Gini indices (p < 0.001). The out-of-bag (OOB) error rates were 6.4% for the Control group, 7.4% for CCL19 monotherapy, 11.2% for anti-PD-1 monotherapy, and 8.4% for Combination therapy.

Fig. 7. Random forest feature importance across treatment regimens.

All variables were ranked according to their importance based on the Mean Decrease in Gini, with higher values indicating a greater contribution to the prediction of treatment response. The out-of-bag (OOB) error was used to quantify model accuracy, with values reflecting differences in predictive ability between different treatment regimens. Notes: asterisks (*) indicate that the feature importance values are statistically significant after performing 1000 replacement tests, whereas crosses (×) indicate non-significant features.

Building on the key variables identified by random forest modeling, we employed receiver operating characteristic (ROC) curves to evaluate the efficacy of discrimination between individual biomarkers and combinations of biomarkers. The area under the curve (AUC) was calculated to compare predictive performance across different biomarker configurations. The combined biomarker index, (CTLs × CCL19)/PD-L1, demonstrated significantly enhanced predictive accuracy across all four treatment groups compared to any single marker alone, with all AUC values substantially exceeding 0.85 (Fig. 8A). This consistent improvement confirms that integrating CTLs, PD-L1, and CCL19 into a multi-analyte panel provides a superior predictive capability for classifying tumor burden at day 30 based on day 22 immune profiles.

Fig. 8. Predictive and prognostic discrimination ability of biomarkers.

A Predictive performance of biomarkers evaluated by ROC analysis. B Stratification by joint biomarker index (optimized threshold) reveals significant survival disparity between high/low groups (log-rank p < 0.05). The risk tables show subject counts at key timepoints.

Biomarkers for distinguishing prognosis

To systematically assess the prognostic value of key biomarkers for survival outcomes in combination therapy, we developed DPF (Section “Quantitative metrics”) based on dynamic tumor burden trajectories. Extending the observation period to 35 days through virtual experimentation, we applied DPF to generate survival states and calculate survival times for all 500 virtual samples under the combination treatment. Kaplan–Meier analysis revealed significant survival stratification based on biomarker expression (Fig. 8B). Concretely, we stratified subjects by day 22 expression levels of four biomarkers: (CTLs × CCL19)/PD-L1, CCL19, PD-L1, and CTLs by using Youden index optimized thresholds from ROC analysis. Subsequent log-rank testing of Kaplan–Meier curves demonstrated significantly improved survival prognosis for subjects with elevated (CTLs × CCL19)/PD-L1 ratios (p = 0.019). In contrast, CCL19 alone (log-rank p = 0.27), CTLs alone (log-rank p = 0.053) and PD-L1 alone (log-rank p = 0.26) did not yield statistically significant survival differences. These findings indicate that while CTLs, PD-L1, and CCL19 are each relevant classifiers for tumor burden, their integration into a multi-analyte signature provides the most robust prognostic discrimination for survival following combination therapy.

Dosing holiday and optimization strategies to enhance combination therapy efficacy

The aforementioned results demonstrate that the combination regimen synergistically modulates the effects of CCL19 and anti-PD-1 on the TME. To further amplify this synergistic effect, we employed the virtual cohort from the previous section to simulate tumor evolution dynamics under six distinct combination therapy strategies, aiming to elucidate the impact of drug administration protocols on treatment response (Fig. 9A). The dosage and administration schedule used in the experiments by Wu et al.¹¹ served as the baseline regimen (Baseline). All other regimens maintained the same total drug dose as the baseline while varying the timing and individual dose of each administration. The specific implementation details are as follows:

1. Baseline: Both anti-PD-1 and CCL19 were administered starting on day 6. Anti-PD-1 was injected twice a week, while CCL19 was injected three times a week.

2. CCL19-Priority Injection (CCL19-P): CCL19 administration remained identical to the Baseline regimen. Anti-PD-1 administration was delayed until the second injection (day 10), followed by injections every 3 days.

3. Anti-PD-1-Priority Injection (Anti-PD-1-P): Anti-PD-1 administration was identical to the Baseline regimen. CCL19 administration was delayed until the second injection (day 9), followed by injections every 2 days.

4. Stepwise Decreasing Dose (SDD): Anti-PD-1 was administered 5 times as in Baseline, with doses sequentially set at 2-, 1-, 0.5-, 0.36- and 0.36-fold relative to the baseline dose. CCL19 was administered 6 times at the Baseline time points, with doses sequentially set at 2-, 1.5-, 1-, 0.43-, 0.43- and 0.43-fold relative to the baseline dose.

5. Low-Frequency High-Dose (LFHD): Anti-PD-1 and CCL19 were administered weekly (days 6 and 13) at the maximum average dose per injection.

6. High-Frequency Low-Dose (HFLD): Both anti-PD-1 and CCL19 were administered daily starting on day 6 for 19 consecutive days at the minimum single dose.

Fig. 9. Virtual clinical trial elucidates impact of drug administration protocols on treatment response.

A Schematic design of virtual clinical trial protocols. The size of the solid circle represents the current dose as a multiple of the single dose of the baseline regimen. B Dynamic trajectory of tumor evolution across different drug administration regimens. C Differences in efficacy of different regimen designs compared to the baseline regimen. D, E Combination efficacy map relative to the drug-free/baseline regimen across varying drug doses. Notes: Warmer colors indicate superior tumor suppression. Pentagrams (⋆) denote baseline dosing.

We simulated the temporal evolution of tumor cell density across six dosing regimens, dividing the experimental timeline into three consecutive phases: a drug-free phase (days 0–5), a drug administration phase (days 6–22), and a drug-elimination & observation phase (days 23–35) (Fig. 9B). During the drug-free phase, tumor progression was identical under all regimens. In the drug administration phase, tumor growth in the LFHD, HFLD, and SDD protocols was slower than in the Baseline group. In contrast, the Anti-PD-1-P regimen initially showed limited efficacy relative to Baseline, but subsequently suppressed tumor growth. Meanwhile, tumor growth in the CCL19-P group remained largely comparable to that in the Baseline group. During the drug clearance and observation phase, SDD, LFHD, and HFLD regimens displayed rapid tumor progression. In comparison, the Anti-PD-1-P regimen in particular demonstrated a gradual reduction in tumor burden after an initial increase.

Quantitative comparison of tumor volume changes at the end of treatment between baseline and the other five strategies by the TEDI (Fig. 9C) showed that the Anti-PD-1-P regimen was the most optimal strategy (TEDI = –7%), achieving significant tumor reduction. The CCL19-P regimen yielded a TEDI value comparable to that of the baseline, indicating similar tumor control efficacy. In contrast, the remaining regimens, notably LFHD (TEDI = 40%), resulted in markedly poorer outcomes, with tumor volume exhibiting a clear upward trend.

To further elucidate the impact of drug dose on the synergistic efficacy of CCL19 and anti-PD-1, we defined the synergistic efficacy functions P(γ_X, γ_Y) and Q(γ_X, γ_Y), which were used to measure the reduction of tumor volume at the endpoint of the combined therapy with respect to that in the no-drug control and baseline regimen, respectively. The results showed that the synergistic efficacy of the combination was nonlinearly enhanced with increasing drug dose compared with the drug-free control and baseline regimens (Fig. 9D, E). Specifically, a double dose enhanced synergistic efficacy by nearly 95% relative to the drug-free control group and by 90% relative to the baseline regimen. This suggests a diminishing marginal effect of increasing drug dose on efficacy.

Model robustness and extension to chemotherapy-immunotherapy

To evaluate the generalizability and clinical relevance of our framework, we extended our model to incorporate paclitaxel (PTX) chemotherapy—a cornerstone of first-line TNBC treatment. The extended model (see Supplementary section “Extended Model of Chemotherapy and Immunotherapy”) integrates PTX pharmacokinetics and direct cytotoxicity on tumor and immune cells. It was independently calibrated and validated using longitudinal tumor volume data from the EO771 syngeneic TNBC model in mice (Supplementary Fig. S8). Simulations in this setting successfully recapitulated the core dynamic outcomes across all treatment arms (Supplementary Fig. S9). Relative to the control group, PTX monotherapy markedly suppressed tumor growth but impaired lymphocyte preservation. This reflects the key clinical trade-off: robust tumor debulking is associated with concurrent immunotoxicity. Notably, the PTX plus anti-PD-1 combination not only achieved more potent tumor control but also partially restored lymphocyte preservation, indicating that anti-PD-1 can mitigate certain immunosuppressive effects of chemotherapy. Furthermore, the simulated differential expression patterns of CCL19 and PD-L1 across treatment groups align with their recognized roles as immune-modulatory biomarkers, demonstrating the model’s utility in mechanistically linking treatment perturbations to biomarker dynamics.

Discussion

TNBC is characterized as an immunogenic, “immuno-hot” tumor subtype, marked by substantial immune cell infiltration. Paradoxically, this immunogenicity fails to translate into adequate clinical responses to ICB. Emerging evidence suggests that the chemokine CCL19, through its potent immunomodulatory properties, may effectively convert this “hot but ineffective” TME into a therapeutically responsive state. Simultaneously, it may serve as a promising predictive biomarker for immunotherapy efficacy in TNBC^11,33. Therefore, our study employs a multi-scale mathematical modeling approach to systematically investigate the synergistic effect between CCL19 and PD-1 blockade in TNBC treatment. Our findings not only confirm the pivotal role of CCL19 in enhancing immune infiltration but also elucidate the impact of drug dosage, administration sequence, and individual immune heterogeneity on therapeutic efficacy, providing a theoretical framework for clinical translation.

CCL19 functions as an essential modulator of immune responses by orchestrating the migration and positioning of T cells within secondary lymphoid organs and inflammatory sites⁹. Previous experimental results revealed that CCL19 is a potent adjuvant for anti-tumor immunity, an effect attributed to its ability to enhance CD8+ T cell recruitment³⁴ and polarize monocytes toward pro-inflammatory M1 macrophages over immunosuppressive M2 phenotypes¹⁰. Notably, our data reveal a robust correlation between CCL19-based immunomodulation and the recruitment of M1 macrophages and CD8+ T cells (Fig. 4). This recruitment signifies a potent antitumor immune response within the tumor, where M1 macrophages promote inflammation and antigen presentation, while CD8+ T cells execute direct cytotoxic killing⁹. However, the differentiation of naïve CD8+ T cells into fully functional CTLs critically depends on CD4+ T cell help. Anti-PD-1 immunotherapy addresses this requirement by counteracting CD4+ T cell exhaustion and sustaining its activity, thereby enabling robust antitumor immunity. Our study successfully captured and quantitatively resolved the complex network of dynamic interactions within TME during the combination of these two therapies by constructing a multiscale model. These findings demonstrate the power of multiscale modeling in unveiling fundamental biological mechanisms and highlight its potential for clinical application in optimizing and personalizing cancer immunotherapy strategies.

The baseline immune state exhibits substantial heterogeneity across individuals, reflecting divergent intrinsic capacities to mount effective antitumor responses upon therapeutic challenge³⁵. In this paper, baseline adaptive immunity could be reflected in naïve T cell abundance; a higher baseline frequency of naïve T cells may indicate greater “immune reserve,” enabling robust activation and differentiation into tumor-specific effector T cells after immunotherapy^36,37. Monocytes and NK cells are key components of the innate immune system that determine immune surveillance and the formation of an inflammatory microenvironment in the early stages of a tumor³⁸. Their densities within TME thus serve as indicators of baseline innate immune function. On the basis of these assumptions, we quantitatively analyzed the nonlinear modulation of baseline immune status on the efficacy of combination therapy using the IHTRI defined in Methods. The results showed that individuals with good baseline immune function (higher naïve CD8+T cell reserve and NK cell activity) were an advantageous group for combination therapy, whereas individuals with higher baseline monocyte density may have reduced efficacy. This variation may be attributed to the heterogeneity of monocyte differentiation, which has been shown to increase the risk of drug resistance by differentiating monocytes to immunosuppressive M2 macrophages^39,40. Notably, the status of naïve CD4+T cells exhibits a unique U-shaped regulatory effect. That means an insufficient density of these cells may compromise the efficacy of combination therapy⁴¹, while an excessive density tends to promote the over-differentiation of Tregs. Therefore, stratifying treatment based on baseline immune state is expected to further optimize the efficacy of combination therapy. For example, for the immunocompetent group, CCL19 combined with PD-1 blockade can be preferentially recommended to maximize the synergistic effect and treatment benefit. In patients with unfavorable factors such as high monocyte or naïve CD4+T imbalances, this suggests the need to carefully evaluate or explore potentiation strategies in combination with other therapies (e.g., direct targeting of M2 macrophages⁴⁰ or modulation of Treg differentiation⁴² therapies) to overcome potential resistance mechanisms.

Our analysis revealed a significant negative correlation between baseline levels of CCL19 and PD-L1 with tumor burden (TRVI). This suggests that their high expression may cooperatively establish an immune microenvironment barrier that inhibits tumor progression. High expression of PD-L1 is commonly regarded as a marker of immune evasion⁴³. However, it also often indicates the presence of pre-activated T cells within the TME^9,44. Therefore, elevated PD-L1 can, to some extent, be considered an indicator of an “immune-inflamed” or “hot tumor” phenotype. As previously described, CCL19 serves as the central chemokine that guides the homing of CCR7+ lymphocytes to tumor sites, providing the foundation for the formation of TLS and the spatial coordination of immune cells⁴⁵. TLS are ectopic lymphoid tissues that form within tumors and provide an ideal microenvironment for T and B cell interaction, activation, and differentiation⁴⁶. Their presence is generally associated with a more favorable prognosis and enhanced anti-tumor immunity⁴⁴. Although our computational model does not explicitly simulate the three-dimensional architecture of TLS, the incorporation of a CCL19-mediated chemotaxis term successfully captures its core function in spatially coordinating immune cell recruitment and reshaping the TME.

Multiscale modeling and computational simulation identified CCL19 concentration, CTLs density, and dynamic PD-L1 changes as key determinants. We therefore developed the composite biomarker (CTLs × CCL19) / PD-L1 to quantify the balance between immune activation and suppression. This signature demonstrated significantly higher predictive power than any individual component, achieving an AUC of 0.86 (Fig. 8A). Its strong performance aligns with emerging evidence that multidimensional biomarkers more accurately capture the dynamics of the immune microenvironment²². Survival analysis confirmed its ability to stratify long-term survivors (Log-rank p < 0.05; Fig. 8B). Notably, this combination of biomarkers demonstrates significant clinical translational potential and provides new ideas for clinical stratification of treatment. Circulating CCL19 is highly correlated with intratumoral levels and can be conveniently detected in both tumor and blood samples¹¹. Furthermore, PD-L1 expression is a well-established biomarker in immunotherapy and can now be quantitatively assessed in liquid biopsies through circulating tumor cells⁴⁷ or exosome-based assays⁴⁸. When integrated with peripheral CTLs quantification via flow cytometry⁴⁹, this triad enables comprehensive immune monitoring while minimizing invasive procedures.

Our data highlighted the immunostimulatory effect of CCL19 therapeutics to synergize with anti-PD-1 to augment effective antitumor T cell immunity in TNBC. However, key translational questions regarding drug delivery remain, particularly the optimal dosing regimen and sequence of therapy, which must be addressed through rigorous clinical trials to ensure efficacy and patient safety. To bridge this gap, we employed computational modeling to simulate various administration protocols, providing preclinical optimization insights while reducing experimental costs. Our simulations indicated that the Anti-PD-1-priority sequence (TEDI = –7%) showed the greatest reduction among all regimens, while the CCL19-priority sequence yielded comparable efficacy to the baseline. In contrast, intermittent high-dose strategies such as LFHD (TEDI = 40%) were associated with increased tumor volume, suggesting potential adverse effects under the simulated conditions. Mechanistically, preferential use of anti-PD-1 may reverse T-cell exhaustion and remodel the TME, thereby promoting IFN-γ secretion. This cascade of actions enhanced antigen presentation by DCs and stimulated the release of endogenous CCL19, establishing a self-reinforcing “release-recruitment-reactivation” feedback loop. In contrast, although CCL19 promotes immune cell infiltration, persistent PD-1-PD-L1 axis suppression maintains functional impairment in T cells, thereby limiting cytotoxicity and abrogating anti-tumor efficacy. Intermittent dosing strategies (SDD and LFHD) have demonstrated a transient therapeutic advantage during the active phase of treatment, with rapid tumor progression after cessation of treatment. This temporal response pattern implies that pulsed high-dose exposure may induce compensatory PD-L1 upregulation in tumor cells, promoting T-cell anergy⁵⁰ and driving adaptive immune evasion through chronic antigenic stimulation⁵¹. These results indicate that excessive dose escalation may accelerate the evolution of treatment resistance while yielding diminishing therapeutic returns. Consequently, high-dose-burst regimens should be avoided to mitigate resistance risk.

Chemotherapy remains the cornerstone of systemic treatment for TNBC, and its combination with ICBs has been shown to significantly improve pathological complete response rates and survival benefits⁵². However, treatment-related toxicities, particularly hematological adverse events, continue to pose a major clinical challenge. It is reported that approximately half of the patients receiving combination therapy require chemotherapy dose reduction or discontinuation due to toxicity, which has been observed to adversely impact the likelihood of achieving a favorable response⁵³. Our extended mathematical model successfully quantifies this critical efficacy-toxicity trade-off. Simulations demonstrate that PTX effectively suppresses tumor growth while reducing lymphocyte counts. When combined with anti-PD-1, the model not only reproduces enhanced tumor control but also captures a partial restoration of lymphocyte preservation, validating the potential of immunotherapy to counteract chemotherapy-induced immunosuppression. We posit that this synergistic enhancement is partly attributable to chemotherapy-induced immunogenic cell death (ICD). Research⁵⁴ indicates that certain chemotherapeutic agents can trigger ICD, causing dying tumor cells to release antigens and danger signals, thereby potentiating antigen presentation and subsequent T-cell activation. Our model mechanistically incorporates this process, providing a quantitative framework to understand how ICD can transform a cytotoxic effect into an immunogenic one, reshape the tumor microenvironment, and ultimately enhance combination therapy efficacy. Furthermore, the model successfully simulates the dynamic pattern of CCL19 being suppressed in the chemotherapy group yet relatively maintained in the combination therapy group. This aligns with the clinical consensus¹¹ that CCL19 serves as a predictive biomarker specifically for immunotherapy response, not for chemotherapy, thereby strengthening the clinical relevance of our model.

In this paper, we propose a multiscale modeling-driven precision immunotherapy framework designed to address the therapeutic dilemma of the “hot but ineffective” TME in TNBC. The framework provides a systematic reference for achieving durable clinical remission in TNBC by integrating sequential dosing optimization, individualized stratification of therapy, and dynamic biomarker monitoring. However, several important limitations remain in the present work, which should be addressed in future research.

First, the method for generating the virtual patient cohort has inherent limitations. This study follows the current mainstream paradigm in virtual clinical trials, which defines parameter perturbation spaces by integrating global sensitivity analysis and biological plausibility constraints to simulate a heterogeneous cohort. However, the setting of perturbation ranges still lacks a unified, objective standard. Future accumulation of high-dimensional biological data (e.g., from single-cell sequencing or spatial transcriptomics) and longitudinal clinical data may provide direct empirical distributions for parameterization. This could enhance the clinical representativeness and predictive power of virtual cohorts.

Second, the model inadequately captures spatial and clonal heterogeneity within tumors. Although we adopted a classical ODE framework and introduced inter-individual heterogeneity through parameter sampling, the model still treats the tumor within each virtual individual as a spatially homogeneous unit. PD-L1 and CCL19 expression are represented by uniform, time-dependent concentrations. This simplification fails to explicitly describe the spatial distribution, competition, and evolutionary dynamics of tumor subclones with different immunophenotypes (e.g., high vs. low PD-L1). Consequently, the model may overestimate the overall synergistic effect of combination therapy in highly heterogeneous tumors and likely underestimates the diluting effect of low-immunogenicity subclones on treatment response. Future work urgently needs to develop a multi-compartment, multi-clonal modeling framework to directly simulate clonal competition, spatial heterogeneity, and the ensuing therapeutic resistance, thereby more precisely defining the boundary conditions under which synergy prevails or diminishes.

Third, the temporal scale of the model focuses on short-term treatment responses and does not encompass long-term evolution and acquired resistance. The simulation period (approximately 30 days) primarily captures the initial kinetic response to therapy. It does not incorporate adaptive evolution of tumor cells under sustained treatment pressure, such as through mutation, epigenetic remodeling, or signaling pathway reprogramming. Therefore, the model has limited ability to predict long-term efficacy evolution, selection of resistant clones, or disease recurrence.

Finally, the extended component of the model combining chemotherapy with immunotherapy remains preliminary. Although this part demonstrates model extensibility within standardized treatment scenarios and reveals potential dynamic mechanisms of chemo-immunotherapy synergy, its parameter calibration relies on limited pre-existing data. It has not been systematically validated in heterogeneous cohorts based on real-world populations.

Methods

Multiscale mathematical modeling formulation

The tumor immune response is intrinsically a dynamic, multi-compartment process involving the local tumor site, peripheral lymphoid organs, and the circulatory system. Fully simulating such a system would require capturing complex processes such as cell migration, differentiation, and signal transduction across compartments, necessitating the introduction of a large number of inter-compartment transport parameters that are difficult to estimate accurately from experimental data. An excess of parameters not only significantly increases model complexity but also leads to severe parameter non-identifiability issues, whereby distinct parameter sets can produce similar system outputs, thereby undermining the reliability and predictive value of the model.

To maintain model interpretability and analytical tractability while focusing on the core immune-tumor interactions within the TME, this study develops a TME-centered multiscale dynamical model. This model simplifies multi-compartment processes—such as the origin and homing of peripheral immune cells—into time-varying inputs or implicit initial conditions for the TME, while placing the modeling emphasis on key dynamic processes inside the TME, including tumor cell proliferation, immune-cell-mediated killing, suppressive interactions, and chemokine-directed spatial redistribution of cells.

Based on the assumptions presented in Fig. 1, we developed a multi-scale modeling strategy integrating key biological events across divergent temporal scales. At the cellular level, the framework quantifies proliferation, apoptosis, migration, and interactions between tumor and immune cells, as detailed below.

DCs (D)

As principal APC, DCs undergo activation upon encountering tumor-derived neoantigens, subsequently maturing to prime T cell responses against specific antigens²⁶. We use the Michaelis-Menten term $\frac{C}{K_C+C}$ to describe how tumor antigen concentration regulates DCs activation, capturing saturation dynamics under limited antigen availability, as expressed:

$$\frac{dD}{dt}=\underset{\mathrm{activation}\mathrm{by}\mathrm{tumor}}{\underbrace{{\lambda }_{D,C}{D}_0\frac{C}{K_C+C}}}-\underset{\mathrm{death}\mathrm{of}\mathrm{DCs}}{\underbrace{d_{D}D}},$$

where immature DCs originate from a constant source (D₀) and undergo activation at a rate (λ_D,C) proportional to tumor-derived antigen concentration. K_C denotes the half-saturation constant for tumor antigen-dependent DC activation, and d_D represents the death rate of DCs.

CTLs (T_c)

We modeled their activation process from naïve CD8+ T cells using the Michaelis-Menten equation, reflecting its dependence on IL-12-induced differentiation and IL-2-mediated clonal expansion⁵⁵, while also incorporating antagonism by the immunosuppressive cytokines IL-10 and TGF-β⁵⁶. However, the engagement of the PD-1 receptor (P) on CTLs with its ligand PD-L1 (L) expressed on tumor cells induces T cell dysfunction. Anti-PD-1 antibodies (Y), a class of immune checkpoint inhibitors, block this interaction by binding to PD-1, thereby preventing PD-1/PD-L1-mediated signaling and T cell dysfunction. Following the derivation by Li et al.²³, we represent this inhibition using the function F(P, L, Y) (see Eq. (9)). More importantly, our model incorporates CTLs homing to the tumor site, which we designed to be critically dependent on the chemokine CCL19. The efficiency of this chemokine-driven infiltration is described by a Hill function: ${\chi }_{T_c,L_{19}}\frac{L_{19}^n}{K_{L_{19}}^n+L_{19}^n}{T}_c^{*}$, where $T_c^{*}$ represents the recruitable pool of CTLs. This equation captures the half-saturation of CTLs recruitment to increasing CCL19 concentration (L₁₉)⁷. Therefore, we developed the following model:

$$\begin{array}{c}\frac{d{T}_c}{dt}=\left\{\underset{\mathit{activation}\mathit{by}\mathit{IL}-12}{\underbrace{{\lambda }_{T_c,I_{12}}{T}_{80}\frac{I_{12}}{K_{I_{12}}+I_{12}}}}\times \underset{\mathit{inhibition}\mathit{by}\mathit{IL}-10\mathit{and}\mathit{TGF}\mathit{-}\mathrm{\beta }}{\underbrace{\frac{1}{1+I_{10}/K_{T_c,I_{10}}}\times \frac{1}{1+T_{\beta }/K_{T_c,T_{\beta }}}}}+\underset{\mathit{proliferation}\mathit{by}\mathit{IL}\mathit{-}2}{\underbrace{{\lambda }_{T_c,I_2}{T}_c\frac{I_2}{K_{I_2}+I_2}}}\right\}\hfill \\ \times \underset{\mathrm{i}\mathrm{n}\mathrm{h}\mathrm{i}\mathrm{b}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{b}\mathrm{y}\mathrm{P}\mathrm{D}-1-\mathrm{P}\mathrm{D}-\mathrm{L}1}{\underbrace{F(P,L,Y)}}+\underset{\mathit{recruitment}\mathit{of}\mathit{CCL}\mathit{19}}{\underbrace{{\chi }_{T_c,L_{19}}\frac{L_{19}^n}{K_{19}^n+L_{19}^n}{T}_c^{*}}}-\underset{\mathit{death}\mathit{of}{\text{T}}_{\text{c}}}{\underbrace{d_{T_c}{T}_c}},\hfill \end{array}$$

fhow have computational models where ${\lambda }_{T_c,I_{12}}$ denotes the IL-12-dependent activation rate; T₈₀ represents the number of naïve CD8+ T cell; $K_{I_{12}}$, $K_{T_c,I_{10}}$, $K_{T_c,T_{\beta }}$, and $K_{I_2}$ are half-saturation constants for IL-12, IL-10, TGF-β, and IL-2 respectively. The parameter ${\lambda }_{T_c,I_2}$ quantifies the IL-2-mediated CTLs proliferation rate. The function F(P, L, Y) models PD-1/PD-L1-mediated immunosuppression; $d_{T_c}$ defines the death rate of CTLs.

Th cells (T_h)

The cytokine IL-12 drives the differentiation of naïve CD4+ T cells (T₄₀) polarization into Th, thereby potentiating CTLs-mediated tumor elimination²⁹. This pro-inflammatory response is counterbalanced by immunosuppressive cytokineswhile IL-10 and TGF-β^56,57. Additionally, activated CD4+ T cells maintain their homeostasis by secreting IL-2⁵⁸. Most importantly, Th cell activity is also inhibited by ICB, which requires the inclusion of a PD-1-PD-L1 inhibitory term (F(P, L, Y)), and we characterized this cellular dynamic as:

$$\begin{array}{c}\frac{d{T}_h}{dt}=\left\{\underset{\mathrm{activation}byIL\mathit{-}\mathit{12}}{\underbrace{{\lambda }_{T_h,I_{12}}{T}_{40}\frac{I_{12}}{K_{I_{12}}+I_{12}}}}\times \underset{\mathrm{inhibition}byIL\mathit{-}10\mathrm{and}TGF\mathit{-}\beta }{\underbrace{\frac{1}{1+I_{10}/K_{T_h,I_{10}}}\times \frac{1}{1+T_{\beta }/K_{T_h,T_{\beta }}}}}+\underset{\mathrm{proliferation}byIL\mathit{-}2}{\underbrace{{\lambda }_{T_h,I_2}{T}_h\frac{I_2}{K_{I_2}+I_2}}}\right\}\hfill \\ \times \underset{\mathit{inhibition}\mathit{by}\mathit{PD}\mathit{-}1-\mathit{PD}\mathit{-}\text{L}1}{\underbrace{F(P,L,Y)}}-\underset{\mathit{death}\mathit{of}{\text{T}}_{\text{h}}}{\underbrace{d_{T_h}{T}_h}},\hfill \end{array}$$

where ${\lambda }_{T_h,I_{12}}$ denotes the activation rate of naïve CD4+T cells differentiating into Th cells under IL-12 stimulation, and ${\lambda }_{T_c,I_2}$ represents the proliferation rate of Th cells in response to IL-2.

Tregs (T_r)

In contrast to Th cell polarization, naïve CD4+ T cells can differentiate into immunosuppressive Tregs characterized by Foxp3 expression. TGF-β promotes Foxp3 upregulation and Treg generation⁵⁹, while IFN-γ inhibits this process⁶⁰, as modeled by:

$$\frac{d{T}_r}{dt}={\underbrace{{\lambda }_{T_r,T_{\beta }}{T}_{40}\frac{T_{\beta }}{K_{T_{\beta }}+T_{\beta }}}}_{\mathrm{activation}\mathrm{by}\mathrm{TGF}-\beta }\times {\underbrace{\frac{1}{1+I_{\gamma }/K_{T_r,I_{\gamma }}}}}_{\mathrm{inhibition}\mathrm{by}\mathrm{IFN}-\gamma }-{\underbrace{d_{T_r}{T}_r}}_{\mathrm{death}\mathrm{of}{\mathrm{T}}_{\mathrm{r}}},$$

where ${\lambda }_{T_r,T_{\beta }}$ denotes the stimulation rate of naïve CD4+ T cells by TGF-β; $K_{T_r,I_{\gamma }}$ denotes the half-saturation concentration of IFN-γ; and $d_{T_r}$ is the death rate of Tregs.

Macrophage cells (M₁ and M₂)

Monocytes exhibit remarkable heterogeneity and plasticity, dynamically polarizing into pro-inflammatory M1 macrophages and anti-inflammatory M2 macrophages in response to microenvironmental signals.

$$\begin{array}{c}\frac{d{M}_1}{dt}=\underset{\mathrm{activation}\mathrm{by}{\mathrm{I}}_{\gamma }}{\underbrace{{\lambda }_{M_1,I_{\gamma }}{M}_0\frac{I_{\gamma }}{K_{I_{\gamma }}+I_{\gamma }}}}+\underset{{\mathrm{M}}_2\mathrm{to}{\mathrm{M}}_1\mathrm{activated}\mathrm{by}\mathrm{TNF}-\alpha \mathrm{and}\mathrm{IL}-12}{\underbrace{{\beta }_2{M}_2\left[\frac{T_{\alpha }}{K_{T_{\alpha }}+T_{\alpha }}+\frac{I_{12}}{K_{I_{12}}+I_{12}}\right]}}\hfill \\ -\underset{{\mathrm{M}}_1\mathrm{to}{\mathrm{M}}_2\mathrm{activated}\mathrm{by}\mathrm{TGF}-\beta \mathrm{and}\mathrm{M}-\mathrm{C}\mathrm{S}\mathrm{F}}{\underbrace{{\beta }_1{M}_1\left[\frac{T_{\beta }}{K_{T_{\beta }}+T_{\beta }}+\frac{M_c}{K_{M_c}+M_c}\right]}}-\underset{\mathrm{death}\mathrm{of}{\mathrm{M}}_1}{\underbrace{d_{M_1}{M}_1}},\hfill \end{array}$$

$$\begin{array}{c}\frac{d{M}_2}{dt}=\underset{\mathrm{activation}\mathrm{by}\mathrm{IL}-10}{\underbrace{{\lambda }_{M_2,I_{10}}{M}_0\frac{I_{10}}{K_{I_{10}}+I_{10}}}}+\underset{{\mathrm{M}}_1\mathrm{to}{\mathrm{M}}_2\mathrm{activated}\mathrm{by}\mathrm{TGF}-\beta \mathrm{and}\mathrm{M}-\mathrm{CSF}}{\underbrace{{\beta }_1{M}_1\left[\frac{T_{\beta }}{K_{T_{\beta }}+T_{\beta }}+\frac{M_c}{K_{M_c}+M_c}\right]}}\hfill \\ -\underset{{\mathrm{M}}_2\mathrm{to}{\mathrm{M}}_1\mathrm{activated}\mathrm{by}\mathrm{TNF}-\alpha \mathrm{and}\mathrm{IL}-12}{\underbrace{{\beta }_2{M}_2\left[\frac{T_{\alpha }}{K_{T_{\alpha }}+T_{\alpha }}+\frac{I_{12}}{K_{I_{12}}+I_{12}}\right]}}-\underset{\mathrm{death}\mathrm{of}{\mathrm{M}}_1}{\underbrace{d_{M_2}{M}_2}},\hfill \end{array}$$

where ${\lambda }_{M_1,I_{\gamma }}$ and ${\lambda }_{M_2,I_{10}}$ denote the polarization rates of monocytes polarized to M1 and M2 phenotypes by IFN-γ and IL-10, respectively; β₁ and β₂ denote the conversion rate of M1 → M2 and M2 → M1 macrophages; $K_{T_{\beta }}$ and $K_{M_c}$ are the half-saturation concentrations of TNF-α and M-CSF, respectively; and $d_{M_1}$ and $d_{M_2}$ are the death rates of M1 and M2 macrophages, respectively.

NK cells (N)

IL-2 produced by T cells, enhances NK cell activation and proliferation⁶¹; this process is counteracted by TGF-β⁶². We model NK cell dynamics as:

$$\frac{dN}{dt}={\underbrace{N^{*}}}_{\mathrm{innate}\mathrm{NK}\mathrm{cells}}+{\underbrace{{\lambda }_{N,I_2}N\frac{I_2}{K_{I_2}+I_2}}}_{\mathrm{proliferation}\mathrm{by}\mathrm{IL}-2}\times {\underbrace{\frac{1}{1+T_{\beta }/K_{N,T_{\beta }}}}}_{\mathrm{inhibition}\mathrm{by}\mathrm{TGF}-\beta }-{\underbrace{d_{N}N}}_{\mathrm{Death}\mathrm{of}\mathrm{N}},$$

where N^* denotes the recruitment rate of NK cells; ${\lambda }_{N,I_2}$ denotes the proliferation rate of IL-2 on NK cells; $K_{N,T_{\beta }}$ is the half-saturation concentration of TGF-β to inhibit the proliferation of NK cells; and d_N is the death rate of NK cells.

CAAs (A)

CAAs support tumor progression via metabolic symbiosis, particularly under hypoxia³¹. Their logistic growth is modeled as:

$$\frac{dA}{dt}=\underset{\mathrm{logistic}\mathrm{growth}\mathrm{of}\mathrm{adipocytes}\mathrm{cells}}{\underbrace{{\lambda }_{A}A\left(1-\frac{A}{K_A}\right)}}-\underset{\mathrm{death}\mathrm{of}\mathrm{A}}{\underbrace{d_{A}A}},$$

where λ_A denotes the proliferation rate of adipocytes; K_A represents the environmental holding capacity of adipocytes; and d_A denotes the death rate of adipocytes.

Tumor cells (C)

Tumor cell proliferation is commonly modeled using logistic growth kinetics, a process stimulated by adipocytes⁶³. Conversely, innate and adaptive immune components, including NK, Th cells and CTLs, exert antitumor effects through direct cytotoxicity and antigen-specific recognition^28,64. The dynamic is captured by the following equation:

$$\begin{array}{c}\frac{dC}{dt}=\underset{\mathrm{promoted}\mathrm{by}\mathrm{A}}{\underbrace{{\beta }_c(1+{\alpha }_{A}A)\left(1-\frac{C}{C_{\max }}\right)C}}-\underset{\mathrm{killed}\mathrm{by}\mathrm{N}}{\underbrace{{\eta }_{N,C}NC}}-\underset{\mathrm{killed}\mathrm{by}{\mathrm{T}}_{\mathrm{h}}}{\underbrace{{\eta }_{T_h,C}{T}_{h}C}}\hfill \\ -\underset{\mathrm{killed}\mathrm{by}{\mathrm{T}}_{\mathrm{c}}}{\underbrace{{\eta }_{T_c,C}{T}_{c}C}}-\underset{\mathrm{death}\mathrm{of}\mathrm{C}}{\underbrace{d_{c}C}},\hfill \end{array}$$

where β_c denotes the net proliferation rate of the tumor; $C_{\max }$ is the maximum environmental holding capacity of the tumor cells; η_N,C, ${\eta }_{T_h,C}$, ${\eta }_{T_c,C}$ is the killing rate of the tumor cells by NK cells, Th cells, and Tc cells, respectively; and d_c indicates the death rate of tumor cells.

Meanwhile, at the molecular scale, we track the concentration dynamics of cytokines (e.g., interferons, chemokines) through the following modeling framework, where parameters such as half-life and rate of production define their biological processes at the fast scale.

IL-2 (I₂)

IL-2 is primarily produced by mature CD4+ T cells⁶⁵. This dynamic is represented in the model as:

$$\begin{array}{c}\hfill \frac{d{I}_2}{dt}={\delta }_{I_2,T_h}{T}_h-{\mu }_{I_2}{I}_2.\end{array}\left(1\right)$$

IL-10 (I₁₀)

IL-10 is produced by Tregs and M2 macrophages⁶⁶, and may also be secreted by tumor cells under certain circumstances⁶⁷. Therefore,

$$\begin{array}{c}\hfill \frac{d{I}_{10}}{dt}={\delta }_{I_{10},C}C+{\delta }_{I_{10},T_r}{T}_r+{\delta }_{I_{10},M_2}{M}_2-{\mu }_{I_{10}}{I}_{10}.\end{array}\left(2\right)$$

IL-12 (I₁₂)

IL-12 mainly comes from activated DCs⁶⁸ and partly from M1 macrophages' secretion⁶⁹. The equation is given by:

$$\begin{array}{c}\hfill \frac{d{I}_{12}}{dt}={\delta }_{I_{12},D}D+{\delta }_{I_{12},M_1}{M}_1-{\mu }_{I_{12}}{I}_{12}.\end{array}\left(3\right)$$

TNF-α (T_α)

TNF-α is produced primarily by M1 macrophages, but can also be produced by mature Th cells and CTLs⁷⁰. This process is derived as:

$$\begin{array}{c}\hfill \frac{d{T}_{\alpha }}{dt}={\delta }_{T_{\alpha },T_h}{T}_h+{\delta }_{T_{\alpha },T_c}{T}_c+{\delta }_{T_{\alpha },M_1}{M}_1-{\mu }_{T_{\alpha }}{T}_{\alpha }.\end{array}\left(4\right)$$

IFN-γ (I_γ)

CTLs and Th cells are the main paracrine source of IFN-γ during the adaptive immune response⁷¹, which formulated as:

$$\begin{array}{c}\hfill \frac{d{I}_{\gamma }}{dt}={\delta }_{I_{\gamma },T_h}{T}_h+{\delta }_{I_{\gamma },T_c}{T}_c-{\mu }_{I_{\gamma }}{I}_{\gamma }.\end{array}\left(5\right)$$

TGF-β (T_β)

Tumor cells can actively secrete TGF-β to promote immune escape and metastasis⁶⁷, and Tregs can inhibit effector T cell function through sustained secretion of TGF-β in immunosuppressive microenvironments⁵⁶. Therefore,

$$\begin{array}{c}\hfill \frac{d{T}_{\beta }}{dt}={\delta }_{T_{\beta },C}C+{\delta }_{T_{\beta },T_r}{T}_r+{\delta }_{T_{\beta },M_2}{M}_2-{\mu }_{T_{\beta }}{T}_{\beta }.\end{array}\left(6\right)$$

M-CSF (M_c)

M-CSF is a major regulator of M2 macrophage proliferation and differentiation, which is mainly produced by tumor cells⁷². We obtain:

$$\begin{array}{c}\hfill \frac{d{M}_c}{dt}={\delta }_{M_c,C}C-{\mu }_{M_c}{M}_c.\end{array}\left(7\right)$$

CCL19 (L₁₉)

CCL19 is a leukocyte chemokine secreted mainly by mature DCs and tumor cells⁷³. This process is mathematically described by:

$$\frac{d{L}_{19}}{dt}={\delta }_{L_{19},D}D+{\delta }_{L_{19},C}C-{\mu }_{L_{19}}{L}_{19}.\left(8\right)$$

For equations from (1)–(8), δ_a,b is the rate at which class a cytokines are produced by class b cells, and μ_a is the rate at which this cytokine decays. Applying the quasi-steady state approximation, we set $\frac{d_{cytokine}}{dt}=0$ to obtain analytical expressions for fast-equilibrating molecular species. This formulation enables efficient numerical implementation while preserving the biological interpretability of cytokine regulation in the tumor microenvironment. The quasi-steady state approximation is justified by the typically faster timescales of molecular dynamics compared to cellular population changes.

Pharmacokinetic modeling

CCL19 pharmacokinetics

The chemokine CCL19 enhances antitumor immunity by mediating CTLs’ recruitment to tumor sites. Our model accounts for dual CCL19 sources: endogenous secretion (primarily from DCs and tumor cells) and exogenous therapeutic administration. Building upon³², we model the CCL19 injection dose as instantaneous concentration boosts at administration times. So, the dynamical equation of CCL19 concentration is given by:

$$\frac{dX}{dt}=\widehat{X}\left(t\right)-d_{X}X,$$

where d_X is the degradation rate of CCL19 and $\widehat{X}$ is the administration rate given by a segmented function dependent on time t:

$$\widehat{X}\left(t\right)=\sum _{i=1}^n{\gamma }_X\cdot \delta \left(t-{\tau }_i\right),$$

where γ_X is the effective concentration per injection, $\delta \left(t-{\tau }_i\right)$ is the Kronecker delta function defined as:

$$\delta \left(t-{\tau }_i\right)=\left\{\begin{array}{cc}1,\hfill & t={\tau }_i\hfill \\ 0,\hfill & t\ne {\tau }_i\hfill \end{array}\right),$$

where τ_i is defined as the administration schedule of CCL19, thus, $\widehat{X}$ would equal to γ_X under treatment, and to zero for drug holiday.

We assume that the half-life of CCL19 is very short, so that the dynamics in Eq. e is in a quasi-steady state.

$$\frac{dX}{dt}\approx 0\Rightarrow X\left(t\right)=\frac{1}{d_X}\sum _{i=1}^n{\gamma }_X{e}^{-d_X(t-{\tau }_i)}\delta (t-{\tau }_i).$$

Hence, the total CCL19 bioactivity at time t is:

$$L_{19}\left(t\right)=\underset{\mathit{endogenous}}{\underbrace{\frac{{\delta }_{L_{19},D\left(t\right)}}{d_X}}}+\underset{\mathit{exogenous}}{\underbrace{X\left(t\right)}}.$$

Anti-PD-1 pharmacokinetics

Due to the expression of PD-L1 on the surface of tumor cells being higher than others, we introduce a new parameter ϵ_C to regulate the scale of expression level. Thus, the concentration of PD-L1 on the T_h, T_c, and tumor cells' surface could be derived as:

$$L={\rho }_L(T_h+T_c+{ϵ}_{C}C),$$

where ρ_L is the expression rate of PD-L1 on cell surface. Accordingly, we assume that the PD-1 is mainly expressed on the surface of T_h and T_C cells, with the expression rate denoted as ρ_P. The concentration of PD-1 on the T_h and T_c cells' surface could be formulated as:

$$P={\rho }_P(T_h+T_c).$$

Thus, the immune inhibitory function mediated by PD-1/PD-L1 interaction is defined through competitive binding kinetics²³:

$$F(P,L,Y)=\frac{1}{1+P_L/K_P},P_L=\frac{{\alpha }_{1}PL}{1+{\alpha }_{1}L+{\alpha }_{2}Y},\left(9\right)$$

where P_L represents the PD-1-PD-L1 concentration in the case of competition between anti-PD-1 and PD-L1, K_P is the immune checkpoint inhibition constant, α₁ and α₂ are the equilibrium constants for the binding process of PD-1 to PD-L1 and anti-PD-1, respectively.

For anti-PD-1 antibody pharmacokinetics, we followed a pulsed administration scheme based on established protocols¹¹. The temporal dynamics of anti-PD-1 antibody concentration Y(t) follow:

$$\frac{dY}{dt}=\hat{Y}\left(t\right)-d_{Y}Y,$$

here, d_Y is the clearance rate of drug Anti-PD-1, $\widehat{Y}$ is the administration rate given by a segmented function dependent on time t:

$$\widehat{Y}\left(t\right)=\sum _{j=1}^m{\gamma }_Y\cdot \delta \left(t-{\tau }_j\right),$$

where γ_Y is the effective level of the source of Anti-PD-1 under treatment, $\delta \left(t-{\tau }_j\right)$ is the Kronecker delta function defined as

$$\delta \left(t-{\tau }_j\right)=\left\{\begin{array}{c}1,t={\tau }_j\hfill \\ 0,t\ne {\tau }_j\hfill \end{array}\right),$$

where τ_j is defined as the administration schedule of Anti-PD-1. Thus, $\widehat{Y}$ would equal γ_Y under treatment, and to zero for a drug holiday.

Drug dosage conversion

To align drug dosage units with the model, this study employs the conversion formula proposed by Li et al.²³:

$$\gamma (\mathrm{nmol}/\text{L})=\xi \times \frac{m\left(\mathrm{g}\right)}{V\left(\mathrm{L}\right)\cdot M_{\text{mol}}(\mathrm{g}/\text{mol)}}\times 10^9,$$

where γ represents the drug concentration, m is the administered mass per mouse, V is the mouse volume, M_mol is the drug’s molar mass, and ξ is the equivalent conversion factor for mouse experiments. Following parameter settings in Li et al.²³, this study uses ξ = 9.1 and V = 2 × 10⁻²L.

Dose conversion for CCL19: According to the GeneCards database⁷⁴, the molecular mass of CCL19 is 10993 g/mol. The experimental protocol by Wu et al. (2023) involved: administering CCL19 via injection three times per week to 6 mice bearing 4T1 tumors, with a single dose of 0.5 μg (i.e., 5 × 10⁻⁷g). Substituting the parameters into the formula yields its concentration:

$${\gamma }_X=9.1\times \frac{5\times 10^{-7}}{2\times 10^{-2}\times 10993}\times 10^9=20.69\mathit{nmol}/\mathrm{L}.$$

Dose conversion for Anti-PD-1: The experiment used a dose of 10 mg/kg. Assuming a mouse mass of 0.02 kg, the single administered mass is 2 × 10⁻⁴ g. The molar mass of this drug is 150 kDa (i.e., 1.5 × 10⁵ g/mol). Substituting into the formula gives:

$${\gamma }_Y=9.1\times \frac{2\times 10^{-4}}{2\times 10^{-2}\times 1.5\times 10^5}\times 10^9=606.7\mathit{nmol}/\mathrm{L}.$$

Parameter estimation

The multi-scale model we developed incorporates a substantial number of parameters, which describe biological processes spanning multiple scales from molecular interactions to population-level cellular dynamics. Since most of these parameters cannot be directly measured, their values were estimated through model calibration against experimental data. To ensure biological plausibility, parameter intervals were defined based on established literature and mechanistic principles. Each parameter θ_i was constrained to a biologically reasonable range:

$${\mathrm{\Omega }}_i=[{\theta }_i^{\min },{\theta }_i^{\max }]$$

For example, the initial numbers of immune cells in experimental mice typically fall within the range of 10⁷ to 10⁹ cells^23,75,76. Accordingly, we searched for optimal parameter values within this interval. The activation rates were constrained between 0 and 20 day^−115,76. The tumor cell killing rate was bounded between 1 × 10⁻¹¹ and 1 × 10⁻⁷cells⁻¹day^−177–79. Natural apoptosis rates for most cell types typically range from 0.1 to 0.3 day^−115,23. The half-saturation constants for cytokines generally lie between 0.01 and 10 ng⋅mL⁻¹²³. The proliferation rate of tumor cells is generally estimated between 0.2 and 0.5 day^−177,80.

For the cytokine production rate δ_xy, we referred to^15,23,81, which gives the magnitude between 10⁻¹⁰ and 10⁻⁶ng⋅mL⁻¹⋅day⁻¹⋅cell⁻¹. The apoptosis rate of cytokines μ_y was obtained using the half-life as $d_y=\frac{ln2}{t_{1/2}}$ where t_1/2 represents the half-life of the cytokine. Comprehensive details on the ranges of parameter values are available in Supplementary text for Table S1 and S2 .

Subsequently, across the entire parameter space Θ = Ω₁ × Ω₂ × ⋯ × Ω_n, we employed LHS to generate 10,000 parameter vectors, ensuring uniform coverage in the high-dimensional space. For each parameter vector θ^(j), the goodness of fit between the corresponding model solution and experimental data was evaluated using the coefficient of determination R²:

$$R^2\left(\mathit{\theta }\right)=1-\frac{{\sum }_{i=1}^m{\sum }_{k=1}^K{\left(y_i^{\mathrm{s}\mathrm{i}\mathrm{m}}(t_k;\mathit{\theta })-y_i^{\exp }\left(t_k\right)\right)}^2}{{\sum }_{i=1}^m{\sum }_{k=1}^K{\left(y_i^{\exp }\left(t_k\right)-{\overline{y}}^{\exp }\right)}^2}$$

here, $y_i^{\exp }\left(t_k\right)$ denotes the observed value of the ith variable at time point t_k, $y_i^{sim}(t_k;\mathit{\theta })$ is the corresponding simulated value, and ${\overline{y}}^{\exp }$ represents the global mean of the experimental data. Thus, the parameter estimation problem was formulated as the following optimization task:

$${\mathit{\theta }}^{*}=\arg \underset{\mathit{\theta }\in \mathit{\Theta }}{\max }{R}^2\left(\mathit{\theta }\right)$$

We conducted three sets of parameter simulations, each consisting of 10,000 independent runs corresponding to distinct parameter combinations. The first set represented the no-treatment control group, where all drug-related parameters (such as dosage and consumption rate of drug) were set to zero. The second and third sets simulated the CCL19 monotherapy and Anti-PD-1 monotherapy conditions, respectively. In each corresponding group, we configured the drug dosage and estimated other drug-specific parameters. Finally, all estimated parameters were incorporated into the combination therapy group to run simulations and generate simulated curves under the combined treatment condition, allowing us to evaluate the goodness-of-fit.

The numerical solution of the model equations was obtained using a fourth-order Runge-Kutta method with a time step of Δt = 0.01. The initial cell densities used in the model were given by the vector:

$${\mathbf{X}}_0=(3.1,9.5,6.21,8.04,0.2,3.0,1.0,0.01,0.005)\times 10^8,$$

where the components correspond to DCs, Th, Tregs, CTLs, NK, M1, M2, CAAs and tumor cells respectively. All parameter values and estimation results could be found in Supplementary Tables S1 and S2.

Quantitative metrics

Tumor relative volume index (TRVI)

To quantify the dynamic evolution of tumors under the same treatment regimen, we define the TRVI to capture the dynamic changes in tumor volume under therapeutic interventions, with the following expression:

$$\text{TRVI}=\frac{V_{end}-V_{strat}}{V_{strat}},$$

where V_end is the terminal tumor volume, and V_start is the initial volume. A higher value of this metric indicates poorer treatment efficacy, while a lower value signifies better treatment efficacy and greater tumor reduction.

Immune heterogeneity treatment response index (IHTRI)

To systematically analyze the impact of individual heterogeneity in baseline immune status on treatment response, we propose the IHTRI to quantify the change in tumor volume under combination therapy relative to the untreated group:

$${\text{IHTRI}}_{\left(i,j\right)}=\frac{V_{end}^{\left(i,j\right)}-V_{end}^{\left(i_0,j\right)}}{V_{end}^{\left(i_0,j\right)}},$$

where i = (T₄₀, T₈₀), (N^*, M₀) models the individual heterogeneity in baseline immune status, representing different combinations of adaptive and innate immune capacities; i₀ represents the baseline parameter values under different treatment regimens; $j=${Control, CCL19, Anti-PD-1, Combination}, denotes different treatment regimens.

The IHTRI quantifies the degree of change in tumor volume relative to the baseline immune state under different combinations of specific immune and innate immune parameters (i) and treatment regimens (j). A value greater than 0 indicates that the current immune profile i contributes to treatment resistance, while a value less than 0 suggests that immune status i enhances treatment sensitivity.

Therapeutic efficacy discrepancy index (TEDI)

To quantify the efficacy difference between two distinct treatment regimens, we define the TEDI:

$$\text{TEDI}=\frac{V_{end}^a-V_{end}^b}{V_{end}^b},$$

where $V_{end}^a$ represents the tumor volume after treatment with regimen a, and $V_{end}^b$ represents the tumor volume after treatment with regimen b. Using regimen b as the reference baseline, a higher TEDI value indicates that regimen a is less effective, while a lower value indicates that regimen a is more effective and achieves greater tumor reduction.

Optimization function for combination therapy dosing

To explore dose optimization strategies for combination therapy, we define an absolute effect difference function, P(γ_X, γ_Y), to quantify the difference in terminal tumor volume between the combination therapy and no-treatment scenarios:

$$P({\gamma }_X,{\gamma }_Y)=\frac{V_{\mathit{end}}({\gamma }_X,{\gamma }_Y)-V_{\mathit{end}}(0,0)}{V_{\mathit{end}}(0,0)},$$

where V_end(γ_X, γ_Y) represents the tumor volume at treatment completion under given doses of CCL19 (γ_X) and Anti-PD-1 (γ_Y), and V_end(0, 0) denotes the terminal tumor volume in the untreated group. P(γ_X, γ_Y) > 0 indicates tumor progression, signifying that the combination therapy performs no better than no treatment. P(γ_X, γ_Y) < 0 reflects relative tumor volume reduction compared to the untreated group. Increasing $\mid P({\gamma }_X,{\gamma }_Y)\mid$ corresponds to enhanced efficacy, providing an objective metric for evaluating the absolute therapeutic effect.

Additionally, we define a Relative-to-Baseline Effect Difference Function, Q(γ_X, γ_Y), to assess the difference in terminal tumor volume between the optimized combination dose and the baseline therapeutic dose (as used in studies like²³):

$$Q({\gamma }_X,{\gamma }_Y)=\frac{V_{\mathit{end}}({\gamma }_X,{\gamma }_Y)-V_{\text{end}}({\gamma }_X^{*},{\gamma }_Y^{*})}{V_{\mathit{end}}({\gamma }_X^{*},{\gamma }_Y^{*})}$$

where $V_{end}({\gamma }_X^{*},{\gamma }_Y^{*})$ is the terminal tumor volume under the baseline dose regimen. This metric quantifies the marginal therapeutic effect of dose optimization: Q(γ_X, γ_Y) > 0 suggests inferior efficacy of the optimized dose (potentially due to toxicity or antagonistic effects). Q(γ_X, γ_Y) < 0 indicates therapeutic gain from optimization. Increasing $\mid Q({\gamma }_X,{\gamma }_Y)\mid$ signifies greater optimization benefit, offering quantitative guidance for dose adjustment.

Survival time prediction

We employ a tumor burden-dependent DPF to stochastically simulate survival times for 50 in silico mice under combination therapy. Following Li et al.²⁴, the mortality probability density function for mice is defined as:

$$P_i\left(x_i\left(t\right)\right)=\frac{1}{1+\exp \left(-\frac{x_i\left(t\right)-{\mu }_i}{{\sigma }_i}\right)},$$

where $x_i\left(t\right)=\frac{C_i\left(t\right)}{K_i}$ is the normalized tumor cell density for the ith virtual sample at simulation day t; K_i is the maximum environmental carrying capacity for tumor cells in the ith sample; μ_i and σ_i are shape parameters generated via LHS to model inter-sample heterogeneity (refer to Li et al.²⁴).

During the simulation period t ∈ [0, 35] days: At each time step, a uniform random number p ∈ (0, 1) is generated daily. Mortality is recorded if p < P_i(x_i(t)), with survival recorded otherwise. Survival time and mortality status are tracked for each virtual sample.

Author contributions

J.Z.L. and L.W. designed the research and supervised the project. C.J.G., J.L., L.D., and C.H.L. performed research and analyzed data. C.J.G. and C.H.L. drafted and finilized all manuscript materials, including the main text, figures, and tables. All authors provided critical feedback and helped shape the research and the final manuscript.

Data availability

All data generated and analyzed during this study are included in this article. The datasets generated during the current study are available through the codes in the following link: https://github.com/gaochunjie98gg/code_MultiscaleODE_CCL19.

Code availability

All simulation and analysis code has been deposited in a permanent GitHub repository and is publicly accessible at: https://github.com/gaochunjie98gg/code_MultiscaleODE_CCL19. For any additional inquiries regarding this work, please contact the corresponding author (Lei Wang).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41540-026-00695-2.