Ordinary differential equation model of cancer-associated fibroblast heterogeneity predicts treatment outcomes

Abstract

Cancer-associated fibroblasts (CAFs) are key components of the tumor microenvironment (TME). CAF phenotypes are highly heterogeneous and exert anti- and protumorigenic effects. We present a mathematical model that describes cancer-immune-CAF interactions and exploits the heterogeneity of CAF phenotypes to predict cancer progression and treatment response. By simulating multiple treatment options, including targeted monotherapies alone, two different immunotherapies, and a combination of therapies, we have found that CAF composition can impact treatment outcomes, potentially resulting in comparable effectiveness of single-drug treatments and combinatorial approaches or even the ineffectiveness of multicombination therapies. These findings suggest that CAF composition can be a promising indicator, in some cases guiding the choice towards less invasive therapies without compromising effectiveness. Our model indicates that accounting for CAF characteristics might facilitate the matching of targeted treatments, supporting clinical decision-making.

Introduction

The tumor microenvironment (TME) is known to contribute to tumor progression (as reviewed in ref. ¹). In particular, cancer-associated fibroblasts (CAFs) are among the key components of the TME; they release various secretory factors, deposit extracellular matrix components and interact with tumor cells and other cells in the TME (as reviewed in refs. ^2,3). CAFs exhibit significant phenotypic and functional heterogeneity^4–7. Some CAF phenotypes can control the growth of tumor cells by either promoting^8–10 or suppressing cancer progression^11–13. CAFs can also modulate the immune landscape in the TME (as reviewed in refs. ^14,15). Treatment outcomes are also influenced by specific CAF subtypes^16,17. For example, CAFs with cancer-like phenotypes or extracellular matrix-related CAFs with low immune cell infiltration are correlated with poor patient survival. Conversely, CAFs associated with a proinflammatory TME contribute to improved survival in patients with cancer. Therefore, an assessment on the basis of a patient-specific CAF landscape could, in principle, guide more effective therapy choices. Even though previous studies have reported beneficial or detrimental CAF subtypes, the systematic prediction of treatment response and clinical outcomes on the basis of CAF phenotypes for individual patients requires the establishment of tailored algorithms and simulation models.

Mathematical modeling can be useful for predicting treatment outcomes and strategizing effective therapies (as reviewed in refs. ^18–21). For combination therapies, designing a treatment strategy with experimental studies alone is challenging because of the extensive number of possible combinations of multiple drugs with different doses, timings, and orders. Various mathematical modeling approaches have been developed to investigate different types of combination therapies (as reviewed in ref. ²²). In particular, several mathematical models describing cancer-immune dynamics have been developed to propose effective treatment strategies for immunotherapies alone^23–25 or in combination with chemotherapy^26,27.

To understand the role of CAFs in cancer progression and treatment responses, several mathematical and computational models have been developed. For example, Kraut et al. have developed a mechanistic mathematical model composed of a system of ordinary differential equations (ODEs) that captures the dynamics between cancer cells, stromal fibroblasts, and therapeutic agents. Their model is designed to simulate stroma-induced drug resistance and to identify effective treatment strategies within a clinically informed cancer-stroma interaction framework²⁸.

Chowkwale et al. have investigated mesenchymal transformation within the TME by developing a multiscale model²⁹. However, these studies generally consider CAF populations as homogeneous. The impact of CAF phenotypic heterogeneity, a potentially critical determinant of treatment success, has not been investigated thoroughly.

To address this gap, we introduce a novel mathematical model that explicitly integrates the phenotypic diversity of CAFs. This model, constructed using ODEs, allows us to predict how this heterogeneity affects cancer growth and treatment response. We model the complex interactions among cancer cells, CAFs, effector T cells, and regulatory T cells (Tregs), specifically considering four distinct CAF phenotypes: antiimmune, proimmune, anticancer, and procancer (CAF phenotypes are described in detail in Section 2.1). By systematically varying the proportions of these phenotypes in a simulated TME, we investigate cancer growth and responses to targeted therapy and two immunotherapy approaches (PD-L1 inhibitor and Treg inhibitor). On the basis of these simulations, we propose model-informed strategies for optimal treatment tailored to specific CAF phenotypic compositions. To construct and simulate our model, we use a combination of experimentally informed and theoretically motivated parameter values. Some parameters, such as PD-1/PD-L1 binding kinetics and drug decay rates, are drawn from prior biological literature, whereas others particularly those related to CAF phenotype influences are selected from biologically reasonable ranges and calibrated to qualitatively match known cancer-immune-CAF behaviors. This approach enables our model to generate mechanistic insights while remaining grounded in empirical data.

Results

Fibroblast phenotypes

This study focuses on the phenotypic differences in CAFs in the TME. The model assumes that a fixed number of CAFs exist in the TME and uses an indicator function to represent the cell population (F = 1 if CAFs are present in the TME, F = 0 otherwise). To systematically investigate the role of the different CAF phenotypes, we divide them into four categories, antiimmune, proimmune, anticancer, and procancer, on the basis of the experimental and clinical studies described below (Fig. 1).

Fig. 1. CAF phenotypes.

CAF phenotypes are categorized into four groups: antiimmune, proimmune, anticancer, and procancer. The weight parameters (α, β, γ, δ ∈ [0, 1], α + β + γ + δ = 1) assigned to each category vary depending on the specific CAF phenotype.

The procancer phenotype is based on findings that CAFs can produce growth factors and cytokines that promote cancer progression³⁰. For example, CAFs are a key source of Prostaglandin E2 (PGE2), which facilitates cancer growth by activating phosphoinositide 3-kinase (PI3K) and protein kinase AKT pathway³¹. Conversely, we define as anticancer CAFs those phenotypes associated with the restraint of cancer progression, potentially through mechanisms such as the expression of apoptosis-inducing ligands like TRAIL, which has been shown to induce cell death in TRAIL-sensitive cancer cells. For instance, CAFs activated by TNF-α or nucleic acid fragments from apoptotic cells can secrete IFN-β and TRAIL, contributing to a cancer-suppressive microenvironment^12,32.

T cells, or T lymphocytes, are white blood cells that play crucial roles in immune surveillance and cancer control. Among them, effector T cells directly eliminate cancer cells, whereas Tregs suppress immune activity to maintain homeostasis. The antiimmune phenotype of CAFs is defined by their immunosuppressive properties, including the upregulation of immune checkpoints and the recruitment of Tregs^33–35. In particular, CAFs can express PD-L1 and FASL, which bind to PD-1 and FAS receptors on effector T cells, leading to functional exhaustion or apoptosis of these cells, thereby weakening the anticancer immune response³⁶. In contrast, the proimmune phenotype of CAFs is less well characterized; however, emerging evidence suggests that certain subsets of CAFs may support the infiltration and activation of effector T cells within the TME, thereby contributing to cancer suppression^37,38.

We assign four parameters (α, β, γ, δ ∈ [0, 1]) to represent the percentage of each CAF phenotype in the TME. The sum of all the parameters equals 1. This antiimmune effect α indicates that CAFs play a role in binding cancer cells’ programmed death-ligand 1 (PD-L1) to effector T cell programmed cell death protein 1 (PD-1) to inhibit T-cell activity. α represents the recruitment of Treg cells by CAFs in the TME. The percentage of proimmune CAFs, β, represents the reactivation of effector T cells from exhausted effector T cells by CAFs. Furthermore, the anticancer effect (γ) involves cancer cell death by CAFs, and the procancer effect (δ) involves the activation of PI3K in cancer cells by CAFs to stimulate cancer cell growth (Fig.1).

Mathematical model development

We now describe the development of our mathematical model based on the key variables outlined above and in the schematic of Fig. 2. The following variables are introduced:

$$\begin{array}{ccc}\hfill C\left(t\right)& \hfill =\hfill & Cancercelldensity\hfill \\ \hfill P\left(t\right)& \hfill =\hfill & PI3Klevel\hfill \\ \hfill T_a\left(t\right)& \hfill =\hfill & Activatede\mathrm{ff}ectorTcelldensity\hfill \\ \hfill T_e\left(t\right)& \hfill =\hfill & Exhaustede\mathrm{ff}ectorTcelldensity\hfill \\ \hfill T_r\left(t\right)& \hfill =\hfill & RegulatoryTcell\left(Treg\right)density\hfill \\ \hfill \left[L\right]\left(t\right)& \hfill =\hfill & PD-L1level\hfill \\ \hfill \left[R\right]\left(t\right)& \hfill =\hfill & PD-1level\hfill \\ \hfill \left[\overline{L\cdot R}\right]\left(t\right)& \hfill =\hfill & Levelof\left[L\right]-\left[R\right]bindingcomplex\hfill \\ \hfill A_P\left(t\right)& \hfill =\hfill & PI3Kinhibitordensity\hfill \\ \hfill A_T\left(t\right)& \hfill =\hfill & Treginhibitordensity\hfill \\ \hfill A_L\left(t\right)& \hfill =\hfill & PD-L1inhibitordensity\hfill \end{array}$$

Fig. 2. Schematic diagram of the model.

Model diagram describing complex interactions between cancer cells, immune cells (effector T cells, regulatory T cells), and CAFs. Arrows indicate induction and activation. Hammerheads indicate inhibition.

Cancer cell density

Among many mathematical modeling approaches for cancer growth models^39,40, we employ a logistic growth modeling approach to describe the cancer cell dynamics in a constrained TME. We define the cancer cell dynamics as:

$$\frac{dC}{dt}=r\left(1+k\frac{P^{m_1}}{H_P^{m_1}+P^{m_1}}\right)C\left(1-\frac{C}{{\chi }_C}\right)-g_1\frac{T_a}{K_r+g_2{T}_r}C-\left(\gamma F\right)g_{3}C,$$ 1

where r is an intrinsic growth rate of the cancer cells. We assume that the cancer growth rate increases when the PI3K level exceeds a critical threshold (H_P). We define the function representing the rate at which activated effector T cells kill cancer cells as $g_1\frac{T_a}{K_r+g_2{T}_r}C$, where g₁ is the cancer-killing rate by activated effector T cells, K_r is the scaling parameter and g₂ is the suppressive strength of Tregs on the anticancer activity of T cells. This formulation is based on the observation that Tregs suppress the cytotoxic activity of effector T cells^26,41. The anticancer phenotype is modeled through an additional cell death term influenced by CAFs, γF, where γ( ∈ [0, 1]) is the percentage of CAFs with this phenotype in the TME and F indicates CAFs presence (F = 1 if CAFs are present in the TME, F = 0 otherwise). χ_C is the carrying capacity, k is the dimensionless parameter of PI3K-mediated cancer growth, H_P and m₁ are the hill-type function parameters, and g₃ is the cancer death rate of cancer cells in the presence of CAFs.

PI3K activity in cancer cells

The governing equation for the temporal dynamics of PI3K activity in cancer cells is as follows:

$$\frac{dP}{dt}={\lambda }_P+{\lambda }_{PF}\left(\delta F\right)-{\mu }_{P}P-{\omega }_{1}P{A}_P,$$ 2

$$\frac{d{A}_P}{dt}={\lambda }_{AP}-{\mu }_{AP}{A}_P.$$ 3

Here, λ_P is the PI3K activation rate independent of CAFs in cancer cells, λ_PF is the PI3K activation rate by CAFs, and μ_P is the natural decay rate of PI3K activity. The effect of procancer CAFs is modeled as δF, where δ is a percentage of this CAF phenotype ( ∈ [0, 1]) in the TME and F is an indicator of CAFs presence. Additionally, we consider a targeted therapy, the PI3K inhibitor (A_P(t)), which suppresses PI3K activity at a rate of ω₁. The dynamics of the PI3K inhibitor are modeled with an injection rate of λ_AP and a decay rate of μ_AP.

PD-L1 and PD-1 binding kinetics

Next, we model the chemical reactions of ligand PD-L1 and its receptor PD-1. Let [L], [R], and $\left[\overline{L\cdot R}\right]$ denote the levels of PD-L1, PD-1 receptor, and the bound complex of PD-L1 and PD-1, respectively. The association (forward) and dissociation (backward) rate constants are k₁ and k₋₁, respectively.

$$\left[L\right]+\left[R\right]\underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons }}\left[\overline{L\cdot R}\right].$$ 4

The mathematical model describing the dynamics of the ligand, receptor, and complex levels, as outlined in Equation (4), is formulated using the following differential equation:

$$\frac{d\left[L\right]}{dt}=\left(\alpha F\right)(-k_1[L]\cdot [R\left]\right)+k_{-1}\left[\overline{L\cdot R}\right]-{\omega }_2\left[L\right]A_L,$$ 5

$$\frac{d\left[R\right]}{dt}=\left(\alpha F\right)(-k_1[L]\cdot [R\left]\right)+k_{-1}\left[\overline{L\cdot R}\right],$$ 6

$$\frac{d\left[\overline{L\cdot R}\right]}{dt}=\left(\alpha F\right)\left(k_1\right[L]\cdot [R\left]\right)-k_{-1}\left[\overline{L\cdot R}\right],$$ 7

$$\frac{d{A}_L}{dt}={\lambda }_{AL}-{\mu }_{AL}{A}_L.$$ 8

The effect of antiimmune CAFs is modeled as αF, where α is the percentage of CAFs with an antiimmune phenotype in the TME and F is an indicator of the presence of CAFs. Immunotherapy (PD-L1 inhibitor, A_L) is also considered, where λ_AL is the injection rate of the inhibitor, μ_AL is the decay rate of the drug, and ω₂ represents the degradation rate of PD-L1 by the PD-L1 inhibitor in Equation (5).

Effector T cell densities

The governing equations for densities of activated and exhausted effector T cells are as follows:

$$\frac{d{T}_a}{dt}=r_T-a_1\frac{{\left[\overline{L\cdot R}\right]}^{m_2}}{H_C^{m_2}+{\left[\overline{L\cdot R}\right]}^{m_2}}{T}_a+\left(\beta F\right)\left(b_1{T}_e\right),$$ 9

$$\frac{d{T}_e}{dt}=a_1\frac{{\left[\overline{L\cdot R}\right]}^{m_2}}{H_C^{m_2}+{\left[\overline{L\cdot R}\right]}^{m_2}}{T}_a-\left(\beta F\right)\left(b_1{T}_e\right),$$ 10

where r_T is the supply rate of activated effector T cells to the TME, a₁ is the rate at which the activated effector T cells are converted to exhausted effector T cells, H_C is the hill-type parameter, and b₁ is the conversion rate from exhausted effector T cells to activated effector T cells. The effect of proimmune CAFs is modeled as βF, where β( ∈ [0, 1]) is a percentage of the proimmune CAF phenotype in the TME and F is an indicator of the presence of CAFs.

Regulatory T cell density

The governing equations for the density of the Treg density and its inhibitor are as follows:

$$\frac{d{T}_r}{dt}={\lambda }_r+\left(\alpha F\right)\left({\lambda }_{rF}\right)-{\mu }_r{T}_r-{\omega }_3{T}_r{A}_T,$$ 11

$$\frac{d{A}_T}{dt}={\lambda }_{AT}-{\mu }_{AT}{A}_T.$$ 12

Tregs are recruited to cancer sites at a rate of λ_r. CAFs can stimulate Treg recruitment at a rate of λ_rF. We assume that in the TME, the number of Tregs can be naturally reduced at a rate of μ_r. Additionally, we consider immunotherapy that targets Tregs (Treg inhibitor A_T), which reduces the number of Tregs at a rate of ω₃. The dynamics of the inhibitor are modeled with an injection rate of λ_AT and a decay rate of μ_AT.

All state variables (e.g., $C,P,A_P,\left[L\right],\left[R\right],\left[\overline{L\cdot R}\right],A_L,T_a,T_e,T_r,A_T$) are functions of time t, although the time dependency is omitted in the equations for simplicity.

Model parameters

We first nondimensionalize the model. The reference value for each parameter is listed in Table 1, and the detailed process of nondimensionalization is provided in the Methods. The reference dimension represents the unit value of each variable in the model. Our model is parameterized on the basis of literature sources when possible, as shown in Tables 2–3 (see the Supplementary Note for a detailed explanation of some parameterization and Fig. S1). Some parameters have not been measured experimentally or clinically. For these cases, we make a biologically reasonable order of magnitude approximations. To evaluate the robustness of our model predictions and identify key drivers of cancer progression and treatment efficacy, we conduct a sensitivity analysis (see the Methods and Fig. S2). By quantifying how small changes in individual parameters affect the model outputs, this analysis highlights which biological processes and CAF-related effects have the greatest influence on cancer cell dynamics. This approach not only highlights the relative importance of specific mechanisms in the model but also provides insight into which parameters should be prioritized for experimental measurements.

Table 1.

Reference values

Parameter	Description	Dimensional Value	References
t*	Time	1 h
C*	Cancer density	1.0 × 109 cells/cm3	55–59
P*	PI3K level	0.01 μM	60
$T_a^{*}$	Activated effector T cell density	1.0 × 106 cells/cm3	55,61,62
$T_e^{*}$	Exhausted effector T cell density	1.0 × 106 cells/cm3	55,61,62
$T_r^{*}$	Regulatory T cell density	1.0 × 106 cells/cm3	55,61,62
[L]*	PD-L1 level	0.4 μM	63
[R]*	PD-1 level	0.4 μM	63
${\left[\overline{L\cdot R}\right]}^{*}$	Complex level	= [R]*	63
$A_T^{*}$	Treg inhibitor density	8.88 μM	64
$A_L^{*}$	PD-L1 inhibitor density	20 μM	65
$A_P^{*}$	PI3K inhibitor density	1.45 μM	66

An asterisk (e.g., C^*) denotes the dimensional form of a variable, whereas variables without an asterisk (e.g., $\overline{C}$) represent dimensionless quantities used in the simulations. The dimensionless variable is obtained by scaling with respect to a reference value, such that $\overline{C}=C/C^{*}$. Reference values (e.g., C^*) are listed in the table to facilitate this nondimensionalization.

Table 2.

Parameters used in the model

Parameter	Description	Dimensional Value	References
Production rates
r	Growth rate of cancer cells	3.48 × 10−8s−1	42
k	PI3K-mediated growth rate of cancer cell	1*	67
HP	Hill coefficient of cancer cell growth by PI3K	8 × 10−2 μM	Estimated
rT	Supply rate of effector T cells	5.56 × 10−3 cells/cm3s−1	Estimated
χC	Cancer cell carrying capacity	1.0 × 109 cells/cm3	56–59
λP	Source of PI3K	5.56 × 10−7 μMs−1	Estimated
λPF	Stimulation rate of PI3K from CAFs	1.39 × 10−6 μMs−1	Estimated
λr	Recruitment rate of Treg	5.56 cells/cm3s−1	Estimated
λrF	Recruitment rate of Treg by CAFs	27.78 cells/cm3s−1	Estimated
λAT	Injection rate of Treg inhibitor	7.4 × 10−5 μMs−1	Estimated
λAL	Injection rate of PD-L1 inhibitor	5.56 × 10−5 μMs−1	Estimated
λAP	Injection rate of PI3K inhibitor	6.97 × 10−6 μMs−1	Estimated
Degradation/Decay rates
g1	Cancer cells killing rate by activated effector T cell	1.13 × 10−7 s−1	Estimated
g2	Inhibition strength of T cell immune activity by Treg	2*	68
Kr	Inhibition parameter of T cell immune activity	5 × 105 cells/cm3	Estimated
g3	Cancer cells killing rate by CAFs (apoptosis)	2.78 × 10−8 s−1	Estimated
μP	Decay rate of PI3K	8.03 × 10−6s−1	69
μr	Decay rate of Treg	6.03 × 10−6s−1	70
μAT	Decay rate of Treg inhibitor	5.83 × 10−7s−1	71
μAL	Decay rate of PD-L1 inhibitor	5.28 × 10−7s−1	72
μAP	Decay rate of PI3K inhibitor	4.81 × 10−6s−1	73
ω1	Degradation rate of PI3K by inhibitor	1.92 × 10−4 μM−1s−1	Estimated
ω2	Degradation rate of PD-L1 by inhibitor	1.39 × 10−6 μM−1s−1	Estimated
ω3	Degradation rate of Treg by inhibitor	5.24 × 10−5 μM−1s−1	Estimated

Parameters labeled as ‘Estimated’ are not directly measured in experiments but are inferred through simulations to ensure biologically plausible system behavior. These estimations are based on known biological timescales, qualitative observations, or values reported in related literature. Dimensionless values are marked in *.

Table 3.

Parameters used in the model (Continued from Table 2)

Parameter	Description	Dimensional Value	References
Conversion parameters of T cell
a1	Conversion rate of activated effector T cell to exhausted effector T cell	1.67 × 10−4s−1	Estimated
b1	Conversion rate of exhausted effector T cell to activated effector T cell	5.56 × 10−5s−1	Estimated
HC	Hill coefficient in T cell conversion	1.2 μM	Estimated
Binding parameters
k1	Association rate	1.2 × 10−3 μM−1s−1	63
k−1	Dissociation rate	1.8 × 10−3s−1	63
CAF phenotypic parameters
α	Antiimmune rate	[0, 1]*	Varied throughout the study
β	Proimmune rate	[0, 1]*	Varied throughout the study
γ	Anticancer rate	[0, 1]*	Varied throughout the study
δ	Procancer rate	[0, 1]*	Varied throughout the study

Parameters labeled as ‘Estimated’ are not directly measured in experiments but are inferred through simulations to ensure biologically plausible system behavior. These estimations are based on known biological timescales, qualitative observations, or values reported in related literature. Dimensionless values are marked in ^*.

Initial conditions

The initial cancer cell density is set to 0.01 (C(0) = 0.01), representing a small cancer burden at the onset of the simulation. The initial density of activated effector T cell is 0.1 (T_a(0) = 0.1), allowing for a minimal preexisting immune presence. For PD-1/PD-L1 binding kinetics, both the ligand and receptor levels are initialized to 10 ([L](0) = 10, [R](0) = 10), allowing for a maximal complex level of 10. The PI3K level is initialized at 0.3 (P(0) = 0.3) to ensure dynamic responsiveness to CAF-derived stimuli. All other variables are initialized to zero. The system of ODEs is numerically solved using MATLAB’s ODE45 solver.

CAF phenotypes modulate cancer growth

In this section, to investigate the impact of the CAF phenotype on cancer growth, we simulate cancer growth for 27 months under various TME conditions. This time frame is chosen because the cancer growth rate estimated through parameter fitting is relatively low (Fig. S1); thus, a longer simulation period is required to observe meaningful changes in cancer dynamics, while ensuring that cancer cell density does not exceed the carrying capacity under protumorigenic conditions. In our model, TME conditions are defined by varying the phenotypic composition of CAFs using four parameters: the antiimmune rate (α), proimmune rate (β), anticancer rate (γ), and procancer rate (δ). These parameters represent the relative functional tendencies of the CAFs and are systematically varied under the constraint α + β + γ + δ = 1.

Prior to the cancer growth simulation, we check the local dynamics of PD-1 and PD-L1 (Fig. S3). In scenarios where all four CAF phenotypes are equally present (α = β = γ = δ), we observe increased cancer growth compared with that in a TME devoid of any CAFs (all CAF phenotypic parameters = 0). The activity of cytotoxic T cells is significantly reduced, resulting in a subdued immune response (Fig. 3A). Notably, exhausted effector T cells are absent when CAFs are not present. This is because, in our model, exhaustion of activated effector T cells occurs only through CAF-mediated antiimmune mechanisms, such as increased PD-1/PD-L1 complex formation. Without CAFs, this pathway is not triggered; thus, activated effector T cells do not become exhausted. This design emphasizes the role of CAFs in driving immune evasion via the PD-1/PD-L1 axis. Additionally, the recruitment of Tregs increases in the presence of CAFs (Fig. 3B), and the activity of PI3K in cancer cells is elevated in the presence of CAFs compared with that in the absence of CAFs (Fig. 3B). Overall, our findings indicate that the presence of CAFs (α = β = γ = δ) promotes the growth of cancer cells (Fig. 3C).

Fig. 3. CAFs promote the growth of the cancer cell.

A Density of activated effector T cells at 27-month in the absence (blue) and presence (orange) of CAFs (left), where the value is normalized to the maximum density of activated effector T cells without CAFs. The density of exhausted effector T cells at 27 months in the absence (blue) and presence (orange) of CAFs. The density of exhausted effector T cells is normalized to the maximum density of exhausted effector T cells with CAFs. B (left): The density of recruited Tregs at 27 months in the absence (blue) and presence (orange) of CAFs. The density is normalized to the maximum density of Tregs with CAFs. (right): The level of PI3K in cancer cells at 27 months in the absence (blue) and presence (orange) of CAFs. The level is normalized to the maximum level in cancer cells with CAFs. Note that the left and right panels are normalized with respect to different reference values. This design choice is made to highlight the relative changes in each context rather than to directly compare their absolute magnitudes. C Temporal evolution of cancer cell densities in the absence (CAF-; blue line) and presence (CAF+; red line) of CAFs.

Figure 4 shows the impact of CAFs on cancer progression within the TME using various parameter values for the four influences of CAF phenotypes. As the percentage of antiimmune CAFs α increases, the PD-1/PD-L1 binding level also increases (Fig. 4A), leading to a steep decrease in the T_a/T_r ratio, an indicator of T cell immunity (Fig. 4B), which increases the cancer cell density (Fig. 4C). On the other hand, as the proportion of proimmune CAFs (β) increases, the T_a/T_r ratio increases, and the cancer cell density decreases (Fig. 4D). As expected, increasing the anticancer CAF phenotype (γ) results in a reduction in the cancer cell density (Fig. 4E). An increasing procancer CAF phenotype (δ) leads to an increase in cancer cell density (Fig. 4F).

Fig. 4. Effects of individual CAF phenotypes on cancer growth.

A–C Effect of antiimmune CAFs (α ∈ [0, 0.75], β = 0.25, γ = 0, and δ = 0). A Binding levels normalized to the level where α = 0.75. B Ratio of activated effector T cells to Treg (T_a/T_r) normalized to the level where α = 0. C Cancer cell density normalized to the density where α = 0.75. D Normalized cancer cell density with varying proimmune CAF percentage (β ∈ [0, 0.75]) and other fixed CAF phenotypes (α = 0.25, γ = 0, and δ = 0). E Normalized cancer cell density with varying anticancer CAF percentage (γ ∈ [0, 0.9]) and other fixed CAF phenotypes (α = 0.1, β = 0, and δ = 0). F Normalized cancer cell density with varying procancer rate (δ ∈ [0, 0.4]) and other fixed CAF phenotypes (α = 0.25, β = 0, and γ = 0.35). In this figure, the sum of the phenotype parameters (α + β + γ + δ) is not constrained to 1. This relaxation is intentional to evaluate the individual impact of each phenotype on cancer dynamics in isolation. All densities and ratios are reported at the 27-month simulation time point. The x-axes represent dimensionless CAF phenotype parameters.

We conduct simulations of cancer cell density growth over a 27-month period using varying compositions of the four CAF phenotypes (α, β, γ, δ ∈ [0.07, 1], α + β + γ + δ = 1) in the TME (Fig. 5A). Increases in both the proimmune rate (β) and anticancer rate (γ) lead to a small cancer cell density (<0.1). Cancer growth accelerates with increasing procancer rate (δ) in the TME (Fig. 5B, arrow). An increase in the anticancer rate (γ) inhibits cancer growth (Fig. 5C), and similarly, an increase in the proimmune rate (β) also suppresses cancer growth (Fig. 5D). Immune cell activity in the TME is also influenced by CAF phenotypes. Higher proimmune rate (β) promotes sustained activation of effector T cells in the TME, leading to a high ratio of activated effector T cell density to Treg density, and thus more robust T cell-mediated cancer cell density control. Since increasing the antiimmune rate (α) promotes Treg recruitment, the ratio decreases with increasing antiimmune rate (α; Fig. S4). This CAF-mediated immune modulation can thus have an impact on both cancer cell density growth and immunotherapy outcomes.

Fig. 5. 3D plot of the effect of CAF heterogeneity on cancer progression.

A Cancer cell density plotted over the full space of CAF phenotypic compositions where α, β, γ, δ ∈ [0.07, 1] under the constraint α + β + γ + δ = 1. B Cross-section of (A) in the β - γ plane, fixing α = 0 to examine the joint effect of proimmune (β) and anticancer (γ) CAFs. C Cross-section in the α - γ plane with β = 0, isolating the interaction between antiimmune (α) and anticancer (γ) CAFs. (D) Cross-section in the β - α plane with γ = 0, showing the interplay between proimmune (β) and antiimmune (α) CAFs. Note: In panel, the δ value increases implicitly as α, β, or γ decrease under the constraint α + β + γ + δ = 1. The upward arrow (`δ↑') is used to visually emphasize this reciprocal increase of the procancer CAFs rate in this setting.

Phenotype of CAFs affects immunotherapy outcomes

CAF phenotypes can affect immunotherapy outcomes. For example, our 27-month simulations of two immunotherapy strategies (PD-L1 inhibitor and Treg inhibitor) demonstrate that, depending on the CAF phenotype, one treatment may effectively suppress cancer growth, whereas the other may not (Fig. 6). If cancer cells reside in a TME with 100% proimmune CAFs (α = 0, β = 1, γ = 0, and δ = 0), a Treg inhibitor can significantly inhibit cancer cell growth (Fig. 6A) owing to the effective restraint of Treg recruitment by therapy in combination with the sustained activity of cytotoxic T cells by proimmune CAFs. However, the effect of a PD-L1 inhibitor is negligible in this TME. The PD-L1 inhibitor can restrain cancer cell density by approximately 20% when cancer cells reside in a different TME with a proimmune knockout (i.e., α = γ = δ = 1/3; Fig. 6B). Unlike the proimmune TME, the Treg inhibitor does not have any effect on cancer growth.

Fig. 6. Modulation of immunotherapy outcomes by CAF phenotypes.

A Cancer cell density normalized to the maximum cancer burden under three treatment conditions with proimmune CAFs. No treatment (CAF+), a Treg inhibitor (CAF+Treg-), or a PD-L1 inhibitor (CAF+PDL-). The cancer burden is compared at three different time points (10, 20, and 27 months). B Cancer cell density normalized to the maximum cancer burden under three treatment conditions with a proimmune KO TME (antiimmune, anticancer, and procancer CAFs (1/3 each)).

Next, we consider more diverse compositions of CAF phenotypes by varying proportions of each phenotype (α, β, γ, δ ∈ [0, 1], α + β + γ + δ = 1). With these varying compositions, we simulate cancer growth until the cancer cell density increases 2.5 times from the initial density and then simulate treatment with a PD-L1 inhibitor or Treg inhibitor for 6 months. To quantify treatment outcomes, we compare the cancer cell densities before and after therapy. We compute the cancer burden change (CBC), which is a measure of how much the cancer cell density changes relative to the density at the start of therapy (CBC = (Cancer cell density after treatment - Cancer cell density at the start of treatment)/Cancer cell density at the start of treatment × 100%). The CBC is calculated as a function of the four CAF phenotype rates (Figs. 7 and 8).

Fig. 7. Cancer burden change (CBC) following PD-L1 inhibitor treatment across diverse CAF phenotypic compositions.

Each point on the plot represents a unique combination of the four CAF phenotype parameters: antiimmune (α), proimmune (β), anticancer (γ), and procancer (δ), with values ranging from [0, 1], under the constraint that α + β + γ + δ = 1. The color scale indicates the percentage CBC at the end of the 27-month simulation compared with the baseline value before PD-L1 inhibitor treatment. Positive CBC values (in warm colors, such as yellow to red) represent cancer growth, whereas negative CBC values (in cool colors, such as blue) represent cancer cell density reduction. The upward arrow (`δ↑') is used to visually emphasize this reciprocal increase of the procancer CAFs rate in this setting.

Fig. 8. Cancer burden change (CBC) following Treg inhibitor treatment across diverse CAF phenotypic compositions.

Each point on the plot represents a unique combination of the four CAF phenotype parameters: antiimmune (α), proimmune (β), anticancer (γ), and procancer (δ), with values ranging from [0, 1], under the constraint that α + β + γ + δ = 1. The color scale indicates the percentage CBC at the end of the 27-month simulation compared with the baseline value before Treg inhibitor treatment. Positive CBC values (in warm colors, such as yellow to red) represent cancer growth, whereas negative CBC values (in cool colors, such as blue) represent cancer cell density reduction. The upward arrow (`δ↑') is used to visually emphasize this reciprocal increase of the procancer CAFs rate in this setting.

The cancer cells are generally resistant to PD-L1 inhibitor (CBCs >0%, as shown in Fig. 7, warm colors such as yellow to red), except in several cases where the CAF phenotypes are mostly anticancer. In a TME dominated by anticancer CAF phenotypes (γ = 0.8), the CBC ranges from − 40% to 30%. In this case, the therapy can reduce cancer cell density to 40%. When the level of anticancer CAFs is moderate (γ = 0.4), the CBC of PD-L1 inhibitor exceeds 30% in most cases, except when procancer CAFs are absent (δ = 0), where the CBC decreases to less than 0. When anticancer CAFs are absent (γ = 0), the therapy fails to reduce the cancer cell density in all cases. Notably, the absence of procancer CAFs (δ = 0) leads to a reduced CBC, a pattern also evident in the γ − α cross-sectional view of Fig. 8 (see Fig. S5).

In our simulations, the Treg inhibitor is more effective than the PD-L1 inhibitor (Fig. 8). We observe more CAF phenotype compositions that lead to reduced cancer cell density. A more than 70% reduction can be achieved with the most favorable CAF phenotypes (e.g., a combination of high anticancer and low procancer CAFs). For example, therapy can reduce the cancer burden by approximately 70%, with a greater abundance of anticancer CAFs (γ = 0.8) and no procancer CAFs (δ = 0; Fig. 8 cross-section plot, with γ = 0.8). Moreover, an increase in the number of proimmune CAFs (β) contributes to the effectiveness of the treatment. We also observe cancer cell density reduction in some cases where the TME lacks antiimmune CAFs (i.e., CAFs with proimmune, anticancer, and procancer effects only; Fig. S6).

The complex treatment response pattern highlights the complex interaction between CAFs and cancer cells under immunotherapy. Although either PD-L1 inhibitor or Treg inhibitor can be highly effective under favorable TME conditions, which depend on specific CAF phenotypes, their overall effectiveness is limited by the complex and dynamic interactions within the TME. Understanding these interactions is essential for improving the use of these therapies.

PI3K inhibitor treatment outcomes are associated with CAF phenotypes

To better understand how different treatment modalities interact with CAF phenotypes in the TME, we first focus on immunotherapy strategies, including PD-L1 inhibitor and Treg inhibitor. These therapies aim to restore or enhance anticancer immune responses by targeting immune checkpoint pathways or Tregs suppression. In contrast, PI3K inhibitor acts directly on intracellular signaling pathways in cancer cells to reduce their proliferation. By examining immunotherapy responses before PI3K-targeted treatments, we aim to elucidate how immune modulation via CAFs-immune interactions influences treatment outcomes, and subsequently assess how these effects differ from those of intracellular pathway inhibition.

In this section, we investigate the effect of the phenotypic heterogeneity of CAFs on the outcomes of targeted therapies. With varying CAF phenotype compositions, we simulate cancer growth until the cancer cell density increases 2.5 times from the initial value and then simulate treatment with the PI3K inhibitor for 6 months. The CBC is calculated as a function of the CAF phenotype composition (α, β, γ, δ), as shown in Fig. 9. The CBC of PI3K inhibitor is varied from −58.5% to 88.9%. For any cancer cell density reduction (CBC < 0%), at least 40% of CAFs need to be anticancer CAF phenotypes (γ) in the TME. A more than 50% cancer cell density reduction by treatment can be achieved if cancer cells reside in a TME composed predominantly of anticancer CAFs (γ ≥ 0.8) and some proimmune CAFs (β).

Fig. 9. Cancer burden change (CBC) following PI3K inhibitor treatment across diverse CAF phenotypic compositions.

Each point on the plot represents a unique combination of the four CAF phenotype parameters: antiimmune (α), proimmune (β), anticancer (γ), and procancer (δ), with values ranging from [0, 1], under the constraint that α + β + γ + δ = 1. The color scale indicates the percentage CBC at the end of the 27-month simulation compared with the baseline value before PI3K inhibitor treatment. Positive CBC values (in warm colors, such as yellow to red) represent cancer growth and treatment failure, whereas negative CBC values (in cool colors, such as blue) represent cancer cell density reduction and treatment success. The upward arrow (`δ↑') is used to visually emphasize this reciprocal increase of the procancer CAFs rate in this setting.

Model-guided therapy considering CAF phenotypes

The model simulations demonstrate that CAF phenotypes can modulate anticancer therapy outcomes. The same therapy can be effective in one TME with particular CAF phenotypes but not in another TME with different CAF phenotypes. We seek to identify an effective treatment strategy on the basis of CAF phenotypes. To this end, we conduct various treatment simulations, including three different monotherapies (a PI3K inhibitor, a Treg inhibitor, or a PD-L1 inhibitor), a combination of two (a Treg inhibitor+PI3K inhibitor, a PI3K inhibitor+PD-L1 inhibitor, a Treg inhibitor+PD-L1 inhibitor), and a combination of three therapies (a PI3K inhibitor+Treg inhibitor+PD-L1 inhibitor). The treatment is started when the cancer cell density reaches 2.5 times the initial density. The treatment duration is 6 months. We consider a treatment strategy effective if it has reduced the cancer cell density by more than 25% following drug administration.

Figure 10 shows the model-guided treatment strategies for each CAF phenotype. Monotherapy alone is sufficient to reduce the cancer cell density in the TME with several CAF phenotypes. The combination of a Treg inhibitor with a PI3K inhibitor is effective for several compositions of CAFs (Fig. 10, red box). The combination of a Treg inhibitor with a PD-L1 inhibitor is effective for a few cases of CAF phenotypes (Fig. 10, orange box). Combining all three drugs is required for at least 25% cancer cell density reduction for many CAF phenotypes, including low anticancer, moderate to high levels of antiimmune, and low to moderate levels of proimmune CAFs (Fig. 10, green box). We also find several cases where even the combination of all three drugs is not effective (Fig. 10, black box).

Fig. 10. Treatment strategy.

Various treatment strategies are tested across diverse CAF phenotypic compositions (α + β + γ + δ = 1). Each box represents treatment strategies where the corresponding treatment results in a cancer cell density reduction exceeding 25%. We use distinct color families to visually differentiate single-agent therapies from combination therapies. For example, single-agent therapies are represented in cooler hues (blue), whereas combination of two therapies are displayed in warmer tones (orange, red), and the combination of all three therapies is shown in green.

Impact of CAF phenotypic heterogeneity on treatment efficacy under diverse cancer growth conditions

Thus far, we have assumed a single cancer cell growth rate. However, in reality, this rate can vary between patients. To address this variability and better capture the uncertainty inherent in cancer growth rates, we vary key model parameters to simulate a range of treatment outcomes. Specifically, we sample 500 growth rates (r ∈ [7.03e − 5, 1.5e − 3]) from a standard normal distribution, reflecting a 95% confidence interval centered on the median cancer cell growth rate⁴². Additionally, to capture the inherent stochasticity and interpatient variability in CAF phenotypes, we randomly generate 500 sets of CAF phenotype rates-α, β, γ, and δ-within the constraint α + β + γ + δ = 1.

We construct a dataset of 500 randomly generated CAF phenotype combinations and cancer growth rates, reflecting patient-to-patient heterogeneity in both TME composition and cancer proliferation dynamics. Using the dataset, we simulate 5 different treatment scenarios: no treatment (control), PI3K inhibitor (AP+), PD-L1 inhibitor (AL+), Treg inhibitor (AT+), and combination (AP+AL+AT+), as shown in Fig. 11. We define treatment failure (cancer progression) as the point when the cancer cell density increases 50-fold from the initial density and show Kaplan–Meier curves to represent the probability of remaining progression-free over time under different treatment and CAF phenotype conditions.

Fig. 11. Kaplan–Meier progression-free survival curves for 500 parameter sets.

A Progression-free survival curve (solid line) and 95% confidence intervals (dotted line) for each treatment. Control: no treatment; AP+: PI3K inhibitor; AL+: PD-L1 inhibitor; AT+: Treg inhibitor; and AP+AL+AT+ combination of three inhibitors. B Survival curves (solid lines) and 95% confidence intervals (dotted lines) under PI3K inhibitor and Treg inhibitor, highlighting those who respond well to Treg inhibitor therapy (CAF2; α = 0, β = 0.1, γ = 0.9, and δ = 0) and those who do not (CAF1; α = 0.2, β = 0, γ = 0, and δ = 0.8).

Under no treatment conditions, more than 70% of the simulated cases have progressed after 10 months, and approximately 90% have progressed after 27 months. Monotherapy with either a PD-L1 inhibitor or a Treg inhibitor slightly improves the probability (1% increase with a PD-L1 inhibitor & 3% increase with a Treg inhibitor). Notably, the application of a PI3K inhibitor alone significantly increases progression-free survival, increasing survival to 21%. The PI3K inhibitor strikingly improves early survival outcomes, increasing the 10-month survival rate to 60%, which is almost double the survival rate observed without any treatment (33.2%). Combining all three therapeutic agents–PD-L1 inhibitor, Treg inhibitor, and PI3K inhibitor–offers the most dramatic improvement. This combined treatment approach increases the 10-month survival probability to 72.2% and increases the 27-month survival probability to 60.8%.

Our previous simulations reveal that Treg inhibitor can lead to a wide range of treatment responses depending on the specific CAF phenotype (Fig. 8). In this simulation, we choose two CAF phenotypes, one that shows the least efficacy under inhibitor therapy (CAF1 in Fig. 8, where α = 0.2, β = 0, γ = 0, and δ = 0.8) and another that exhibits the highest efficacy (CAF2 in Fig. 8, where α = 0, β = 0.1, γ = 0.9, and δ = 0). In TMEs characterized by CAF1, the median survival time under Treg inhibitor treatment is only 6 months, whereas under PI3K inhibitor treatment, it extends to 11 months, indicating a more favorable outcome. However, for the CAF phenotype that responds best to Treg inhibition (CAF2), at 27 months, PI3K inhibitor treatment results in a survival rate of 38.8%, whereas Treg inhibitor treatment results in a higher survival rate of 66.4%. This finding suggests that while Treg inhibitor may take longer to manifest its full potential, it can provide superior survival benefit compared with that of PI3K inhibitor, particularly in patients whose CAF phenotype is optimally aligned with the mechanism of action of Treg inhibitor.

Discussion

Here, we develop a mathematical model based on ODEs, incorporating 11 state variables and 33 parameters, to simulate the impact of CAF phenotypic heterogeneity on cancer cell growth and treatment responses. By integrating different CAF phenotypes into the model, we are able to capture their distinct influences on cancer progression. CAFs with cancer cell growth-promoting phenotypes or immunosuppressive phenotypes increase the cancer cell growth rate. In contrast, an increased presence of anticancer or proimmune CAF phenotypes counteracts these effects, leading to suppressed cancer growth. These findings reflect the complex interplay between stromal and immune components in the TME and are consistent with recent studies showing that specific CAF subtypes are associated with worse survival outcomes in non-small cell lung cancer patients⁴³.

PI3K inhibitor responses vary significantly depending on the CAF phenotypic composition. This finding supports recent experimental findings that colorectal cancer cells exhibit increased resistance to PI3K pathway inhibitors when exposed to fibroblast-conditioned media⁴⁴. Similarly, our model predicts variability in immunotherapy responses, with more pronounced variability observed for a Treg inhibitor than for a PD-L1 inhibitor.

The broad variability in treatment response highlights the critical need for a tailored therapeutic approach. Currently, the immune landscape is considered to predict and identify patients responsive to immunotherapy; however, our findings indicate that CAFs play an important role in predicting the outcome of immunotherapy⁴⁵.

Model-driven effective treatment strategies concerning specific CAF phenotypes suggest that a significant (at least 25%) cancer burden reduction can be achieved through either single-drug treatments or a combination of two involving PD-L1 inhibitor, Treg inhibitor, and PI3K inhibitor. This finding is particularly important, as monotherapies are often associated with reduced side effects, and identifying patients in which monotherapy is predicted to be as effective as combinatorial therapy might improve their quality of life, minimizing adverse events⁴⁶. Notably, when the proportion of anticancer CAFs exceeds 60%, our model predicts that monotherapy alone can yield outcomes comparable to those of combination therapies. Some cancers in the TME with low to moderate levels of anticancer effects in combination with moderate levels of antiimmune CAFs require a combination of all three factors to reduce the burden. Conversely, a combination of high antiimmune, low proimmune, and low anticancer CAFs results in cancers that are resistant even to combination therapy with three drugs. These findings highlight the potential for CAF phenotype analysis to inform optimal therapy selection and underscore the possibility of targeting CAFs directly to overcome treatment resistance⁴⁷. These predictions are experimentally testable. Recent advances in single-cell profiling techniques, such as multiplex immunofluorescence (mIF) and mass cytometry, have allowed the phenotypic composition of CAFs to be quantified at single-cell resolution in patient biopsy samples^48,49. Moreover, multiplex fluorescence immunohistochemistry (mFIHC) is a clinically applicable method for profiling CAF phenotypes in situ⁵⁰. The quantitative outputs from these technologies can be directly incorporated into our mathematical modeling as input parameters, enabling individualized predictions of treatment outcomes on the basis of cancer-specific CAF profiles. In addition, model predictions can be validated in preclinical settings using 3D tumor organoid systems or patient-derived xenograft (PDX) models, which recapitulate the complex stromal context of human tumors. Taken together, these experimental approaches provide a translational bridge between in silico modeling and clinically actionable precision oncology strategies.

Furthermore, this modeling approach may be translated into clinical practice by leveraging mFIHC, which enables high-resolution spatial profiling of CAF phenotypes in patient biopsy samples. The quantitative outputs of mFIHC can be used as input parameters for our mathematical model, allowing the prediction of individual treatment outcomes on the basis of the unique CAF composition within each TME. This approach supports the integration of model-driven simulations into clinical decision-making pipelines and paves the way for CAF-informed precision oncology strategies⁵⁰.

This study also investigates the potential effects of variations in cancer cell growth rates on treatment outcomes. Considering the various parameter sets, the predicted progression-free survival under the PI3K inhibitor is significantly longer than that under either the PD-L1 inhibitor or Treg inhibitor. This variability suggests that both cancer-intrinsic factors and microenvironmental influences contribute to therapeutic success. Progression-free survival is, however, significantly affected by the CAF phenotype. In one CAF phenotype condition, progression-free survival under the Treg inhibitor is high, with almost 70% not progressing after 27 months of treatment, which is greater than that under the PI3K inhibitor (40% not progressing). In another CAF phenotype condition, only 10% does not progress under the Treg inhibitor after 27 months of treatment. These findings demonstrate the potential of CAF characterization for better personalized therapy. They also suggest that variability in CAF composition may be responsible for unexpected differences in treatment outcomes between patients.

Our current work presents a deterministic mean-field model describing cancer cell dynamics interacting with immune cells and CAFs with various phenotypes. This model can be extended to specific cancer types by integrating the relevant underlying biology of each cancer type. Several other factors, such as stochasticity and spatial heterogeneity, may affect treatment outcomes and can be investigated in future studies. In our mathematical model, we treat the population of CAFs as static owing to their long half-life, assuming stability within the TME. Although CAF populations are modeled as static parameters, we introduce variability in CAF phenotype parameters to mimic the effects of dynamic population changes. This approach allows for the exploration of phenotypic variations and their impacts on therapeutic outcomes, maintaining model simplicity while providing insights into cancer progression. However, this simplification may overlook specific transient dynamics that could be crucial under certain therapeutic conditions. Future models may include dynamic CAFs population and phenotypic variability to further assess their roles under different conditions.

An additional advantage of our mathematical modeling approach is that it enables the tracking of cancer and immune dynamics continuously across time. This approach allows for the assessment of treatment responses and microenvironmental effects at any desired time point, providing temporal resolution that may not be feasible in clinical or in vivo settings. However, a notable limitation of our model lies in its lack of spatial considerations, as it is constructed on the basis of ODEs. Consequently, the model does not account for spatial heterogeneity or cell–cell interactions that can be critical in shaping therapeutic outcomes, particularly in complex cancer architectures. Our modeling approach can be extended to incorporate spatial aspects using PDEs and agent-based modeling, facilitating the investigation of local microenvironmental cues, CAFs positioning, and the spatial coordination of immune–cancer interactions with increased fidelity.

Our results provide a theoretical understanding of the impact of CAF heterogeneity on cancer progression and treatment outcomes. Our model results emphasize the importance of CAF phenotypes in determining treatment outcomes. We also provide a rationale for personalized treatment choices that are finely tuned to the specific characteristics of the TME. Our model can contribute to the refinement of predictive models aimed at optimizing targeted therapy matching in precision oncology to ultimately maximize efficacy and minimize side effects. However, we acknowledge that the clinical application of this approach requires accurate identification of CAF phenotypic composition in individual patients. While this is not yet routinely feasible in current clinical settings, recent advances in single-cell sequencing and spatial profiling technologies offer promising avenues for capturing patient-specific CAF heterogeneity^49,51. As such data become increasingly available, our model may serve as a computational platform to guide treatment design and hypothesis generation and can be used in synergy with experimental and clinical research to advance personalized cancer therapy.

Methods

Nondimensionalization

To reduce the complexity of the model and facilitate generalization across different systems, we nondimensionalize all state variables and parameters. Each dimensional variable is scaled using a reference value that reflects a biologically meaningful magnitude. For instance, the cancer cell density C is rescaled using its carrying capacity χ_C, such that the dimensionless form $\overline{C}=C/C^{*}$ lies in [0,1]. Similarly, binding-related quantities (ligands, receptors, and complexes) are normalized with respect to their maximal expected levels to maintain dimensionless dynamics. This approach allows us to examine relative changes and dynamics within biologically feasible ranges, independent of absolute units, which are often unavailable in vivo studies. Asterisks (^*) are used in tables and figures to distinguish dimensional quantities from their nondimensional counterparts. We nondimensionalize the variables and the parameters in the model equation as follows:

$$\begin{array}{c}\overline{t}=\frac{t}{T},\overline{C}=\frac{C}{C^{*}},\overline{P}=\frac{P}{P^{*}},{\overline{A}}_P=\frac{A_P}{A_P^{*}},\overline{\left[L\right]}=\frac{\left[L\right]}{{\left[L\right]}^{*}},{\overline{\left[R\right]}}_a=\frac{\left[R\right]}{{\left[R\right]}^{*}},\overline{\left[\overline{L\cdot R}\right]}=\frac{\left[\overline{L\cdot R}\right]}{{\left[\overline{L\cdot R}\right]}^{*}},\hfill \\ {\overline{A}}_L=\frac{A_L}{A_L^{*}},{\overline{T}}_a=\frac{T_a}{T_a^{*}},{\overline{T}}_e=\frac{T_e}{T_e^{*}},{\overline{T}}_r=\frac{T_r}{T_r^{*}},{\overline{A}}_T=\frac{A_T}{A_T^{*}},\overline{r}=rT,{\overline{H}}_P=\frac{H_P}{P^{*}},\overline{k}=k,\hfill \\ {\overline{\chi }}_C=\frac{{\chi }_C}{C^{*}},{\overline{g}}_1=g_{1}T,{\overline{K}}_r=\frac{K_r}{T^{*}},{\overline{g}}_2=g_2,{\overline{g}}_3=g_{3}T,{\overline{\lambda }}_P=\frac{{\lambda }_{P}T}{P^{*}},{\overline{\lambda }}_{PF}=\frac{{\lambda }_{PF}T}{P^{*}},\hfill \\ {\overline{\mu }}_P={\mu }_{P}T,{\overline{\omega }}_1={\omega }_1{A}_T^{*}T,{\overline{\lambda }}_{AP}=\frac{{\lambda }_{AP}T}{A_P^{*}},{\overline{\mu }}_{AP}={\mu }_{AP}T,{\overline{k}}_1=k_1{\left[R\right]}^{*}T,{\overline{k}}_{-1}=k_{-1}T,\hfill \\ {\overline{\omega }}_2={\omega }_2{A}_L^{*}T,{\overline{\lambda }}_{AL}=\frac{{\lambda }_{AL}T}{A_L^{*}},{\overline{\mu }}_{AL}={\mu }_{AL}T,{\overline{r}}_T=\frac{r_{T}T}{T^{*}},{\overline{a}}_1=a_{1}T,{\overline{H}}_C=\frac{H_C}{{\left[\overline{L\cdot R}\right]}^{*}},\hfill \\ {\overline{b}}_1=b_{1}T,{\overline{\lambda }}_r=\frac{{\lambda }_{r}T}{T^{*}},{\overline{\lambda }}_{rF}=\frac{{\lambda }_{rF}T}{T^{*}},{\overline{\mu }}_r={\mu }_{r}T,{\overline{\omega }}_3={\omega }_3{A}_P^{*}T,{\overline{\lambda }}_{AT}=\frac{{\lambda }_{AT}T}{A_T^{*}},\hfill \\ {\overline{\mu }}_{AT}={\mu }_{AT}T,\overline{\alpha }=\alpha ,\overline{\beta }=\beta ,\overline{\gamma }=\gamma ,\overline{\delta }=\delta .\hfill \end{array}$$ 13

Then, the governing equations in a dimensionless form are:

$$\frac{d\overline{C}}{d\overline{t}}=\overline{r}\left(1+\overline{k}\frac{{\overline{P}}^{m_1}}{{\overline{H}}_P^{m_1}+{\overline{P}}^{m_1}}\right)\overline{C}\left(1-\frac{\overline{C}}{{\overline{\chi }}_C}\right)-{\overline{g}}_1\frac{{\overline{T}}_a}{{\overline{K}}_r+\overline{g_2}{\overline{T}}_r}\overline{C}-\overline{\gamma }F\overline{g_3}\overline{C}.$$ 14

$$\frac{d\overline{P}}{d\overline{t}}=\overline{{\lambda }_P}+\overline{\delta }{\overline{\lambda }}_{PF}F-{\overline{\mu }}_P\overline{P}-{\overline{\omega }}_1{\overline{A}}_P,$$ 15

$$\frac{d{\overline{A}}_P}{d\overline{t}}={\overline{\lambda }}_{AP}-{\overline{\mu }}_{AP}{\overline{A}}_P.$$ 16

$$\frac{d\overline{\left[L\right]}}{dt}=-{\overline{k}}_1\overline{\alpha }F\overline{\left[L\right]}\cdot \overline{\left[R\right]}+{\overline{k}}_{-1}\overline{\left[\overline{L\cdot R}\right]}-{\overline{\omega }}_2\overline{\left[L\right]}{\overline{A}}_L,$$ 17

$$\frac{d\overline{\left[R\right]}}{d\overline{t}}=-{\overline{k}}_1\overline{\alpha }F\overline{\left[L\right]}\cdot \overline{\left[R\right]}+{\overline{k}}_{-1}\overline{\left[\overline{L\cdot R}\right]},$$ 18

$$\frac{d\overline{\left[\overline{L\cdot R}\right]}}{d\overline{t}}={\overline{k}}_1\overline{\alpha }F\overline{\left[L\right]}\cdot \overline{\left[R\right]}-{\overline{k}}_{-1}\overline{\left[\overline{L\cdot R}\right]},$$ 19

$$\frac{d{\overline{A}}_L}{d\overline{t}}={\overline{\lambda }}_{AL}-{\overline{\mu }}_{AL}{\overline{A}}_L,$$ 20

$$\frac{d{\overline{T}}_a}{d\overline{t}}={\overline{r}}_T-\overline{a_1}\frac{{\overline{\left[\overline{L\cdot R}\right]}}^{m_2}}{{\overline{H}}_C^{m_2}+{\overline{\left[\overline{L\cdot R}\right]}}^{m_2}}{\overline{T}}_a+\overline{\beta }{\overline{b}}_{1}F{\overline{T}}_e,$$ 21

$$\frac{d{\overline{T}}_e}{dt}={\overline{a}}_1\frac{{\overline{\left[\overline{L\cdot R}\right]}}^{m_2}}{{\overline{H}}_C^{m_2}+{\overline{\left[\overline{L\cdot R}\right]}}^{m_2}}{\overline{T}}_a-\overline{\beta }{\overline{b}}_{1}F{\overline{T}}_e,$$ 22

$$\frac{d{\overline{T}}_r}{d\overline{t}}={\overline{\lambda }}_r+\overline{\alpha }{\overline{\lambda }}_{rF}F-{\overline{\mu }}_r{\overline{T}}_r-{\overline{\omega }}_3{\overline{T}}_r{\overline{A}}_T,$$ 23

$$\frac{d{\overline{A}}_T}{d\overline{t}}={\overline{\lambda }}_{AT}-{\overline{\mu }}_{AT}{\overline{A}}_T.$$ 24

Sensitivity analysis

We perform a sensitivity analysis of model parameters. Specifically, we utilize a model sensitivity analysis method developed by Krischner’s group, a general Latin Hypercube Sampling (LHS) scheme, and Partial Rank Correlation Coefficient (PRCC)⁵². The Latin Hypercube Sampling (LHS) scheme is a method used to efficiently sample input parameter spaces by ensuring that the entire range of each parameter is evenly represented. It allows for broad and systematic exploration of parameter combinations with fewer samples compared to simple random sampling. The Partial Rank Correlation Coefficient (PRCC) is a statistical technique used to evaluate the strength and direction of the relationship between an input variable and an output, while controlling for the effects of other variables. PRCC is based on rank correlations, making it suitable for detecting monotonic relationships even when the underlying relationship is nonlinear. This sensitivity analysis method has been used in several mathematical modeling studies^53–55.

We investigate the sensitivity of densities of cancer cells (C), activated effector T cells (T_a), exhausted effector T cells (T_e), regulatory T cells (T_r), level of PD-L1 ([L]), PD-1 ([R]), complex ($\left[\overline{L\cdot R}\right]$) and PI3K (P) to these parameters at several time points. We choose a range for each of these parameters of interest and divide the ranges into 10,000 subintervals of uniform length for the calculation of the corresponding partial rank correlation coefficient (PRCC) value. A PRCC value is a real number in the interval [−1, 1] with the sign, where the sign indicates an increase (+) or decrease (-) of the given variable in response to an increase in the parameter at a given time.

Figure S2 shows the PRCC values of all variables at t = 1, 30, 810 days. The density of cancer cells is positively correlated to r, k, g₂, a₁, α, λ_RF, λ_R, k₁, λ_P, λ_PG, δ but negatively correlated to g₁, g₃, r_T, β, μ_R, k₋₁, μ_P, γ, b₁. Especially, some parameters related to immune activity (r_T, β, b₁) show low PRCC absolute values on day 1, indicating a weak correlation with cancer cell density, but as time progresses, the absolute values of PRCC increase, demonstrating a stronger relevance.

Author contributions

Conceptualization: Junho Lee, Eunjung Kim; Formal analysis: Junho Lee; Funding acquisition: Eunjung Kim; Investigation: Junho Lee, Eunjung Kim; Methodology: Junho Lee; Supervision: Eunjung Kim; Visualization: Junho Lee; Writing – original draft: Junho Lee, Eunjung Kim; Writing – review and editing: Eunjung Kim. All authors have reviewed the manuscript.

Data availability

The authors declare that the estimated parameter values supporting the numerical simulations of this study are available within the paper.

Code availability

The relevant codes were developed using MATLAB and are publicly available at https://github.com/Junho-Lee37/CAF. The sensitivity analysis of our model parameters is carried out by utilizing MATLAB codes available from the website of Denise Kirschner’s Lab: http://malthus.micro.med.umich.edu/lab/usadata/.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41540-025-00578-y.