Exploratory personalized radiobiological modeling of bystander and immune effects to inform SFRT‐SBRT scheduling

Abstract

Background

Spatially fractionated radiation therapy (SFRT) demonstrates clinical efficacy against bulky tumors, but optimal treatment scheduling remains empirical. The technique's heterogeneous dose distribution triggers complex biological effects—including bystander signaling and immune activation—that are not captured by conventional dose‐response models.

Purpose

This study aims to develop a proof‐of‐concept computational framework to simulate tumor and immune responses during combined Lattice radiotherapy and SBRT) as an initial step toward patient‐specific modeling.

Methods

A four‐compartment ordinary differential equation (ODE) model was established to simulate tumor and lymphocyte dynamics, integrating Gompertz tumor growth kinetics, direct radiation‐induced cell killing, and indirect biological effects mediated by intercellular signaling and immune activation. Bystander signaling was described by reaction–diffusion equations modeling spatial propagation from high‐ to low‐dose regions. Immune responses were modeled with coupled lymphocyte‐tumor equations, with lymphocyte dose exposure estimated using the HEDOS model. Model parameters were derived from literature and fitted to tumor volume and absolute lymphocyte count (ALC) data from four non‐small cell lung cancer (NSCLC) patients treated with SFRT and SBRT.

Results

The model demonstrated feasibility in reproducing tumor volume (normalized root‐mean‐square error [NRMSE]: 0.071–0.197) and ALC dynamics (NRMSE: 0.017–0.385). Simulations revealed substantial inter‐patient heterogeneity in the estimated contributions of direct radiation and immune‐mediated effects, with immune‐mediated killing exceeding direct radiation in 2 patients. Combined SFRT‐SBRT achieved superior tumor control over either modality alone in our simulations. Notably, the optimal SFRT‐SBRT interval appeared patient‐specific: extending the interval to 2 months improved outcomes in some patients, while others showed limited benefit, depending on the balance between treatment‐induced cell kill and tumor regrowth.

Conclusions

We developed a radiobiological model that simulates tumor and lymphocyte dynamics under combined SFRT–SBRT regimens, providing a preliminary framework for exploring personalized SFRT‐SBRT scheduling. These findings warrant further prospective validation in larger patient cohorts before clinical translation.

1. INTRODUCTION

Spatially fractionated radiation therapy (SFRT) is a radiation technique characterized by non‐uniform dose distributions, in contrast to the homogeneous patterns used in conventional radiotherapy. ¹ The concept of SFRT was first introduced by Kohler in 1909 to reduce skin toxicity while delivering higher therapeutic doses to tumor sites during kilovoltage x‐ray treatment. ² , ³ It was first implemented using a steel grid (GRID) that created alternating high (“hot spots”) and low (“cold spots”) dose regions, and later, GRID therapy delivered by a linear accelerator proved effective for treating large and challenging tumors with high response rates. ⁴ , ⁵ Advances in SFRT technology have expanded its modalities to Lattice therapy (LRT), minibeam radiation therapy (MBRT), ⁶ , ⁷ and microbeam radiation therapy (MRT). ⁸ While MRT remains in the preclinical stage, GRID and LRT have been successfully applied in clinical practice. LRT represents a three‐dimensional extension of conventional GRID therapy, in which multiple high‐dose regions (spherical vertices planned to receive 15–20 Gy) are interleaved with low‐dose zones. ⁹ , ¹⁰ High doses are confined to the tumor, sparing surrounding normal tissue. Often combined with conventional radiotherapy or stereotactic body radiotherapy (SBRT), LRT has been applied to several clinical indications, including cervical squamous cell carcinoma, ovarian cancer, and non‐small cell lung cancer (NSCLC), achieving promising local control with minimal toxicity. ¹⁰ , ¹¹

The heterogeneous dose distribution in SFRT elicits complex radiobiological effects beyond direct cell killing. ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ Investigated mechanisms include cellular‐level bystander effects, ¹⁷ tissue‐level tumor microenvironment modulation, ¹⁸ and systemic immune regulation. ¹⁴ At the cellular level, the bystander effect refers to DNA damage and mutations in unirradiated cells caused by molecular signals from neighboring irradiated cells, which has been confirmed in experiments involving direct cell contact or exposure to conditioned medium from irradiated cells. ¹⁷ , ¹⁹ , ²⁰ , ²¹ In addition, SFRT can stimulate systemic immune responses, including abscopal effects and localized immune cell infiltration. ²² , ²³ Doses above 10 Gy can induce immunogenic tumor cell death, releasing tumor antigens that activate anti‐tumor immunity. ¹⁴ Moreover, SFRT has the potential to enhance infiltration of immune cells, such as CD8+ T cells and macrophages, and promote activation of antigen‐presenting cells. ²⁴ The magnitude of the radiobiological effect is influenced by cell type, spatial‐temporal modulation, ²⁵ , ²⁶ and dosimetric parameters. ²⁷ , ²⁸

Previous studies have primarily employed the equivalent uniform dose (EUD) model to quantify tumor control under nonuniform dose distributions, which mainly capture direct radiation‐induced cell killing while neglecting critical biological effects that contribute to tumor cell death in low‐dose regions. ¹ To address this limitation, several computational models have been developed for SFRT. McMahon et al. proposed a radiation‐induced intercellular signaling model based on experimental cell data to quantify bystander effects on survival fraction. ²⁹ Cho et al. developed a mathematical model of SFRT‐induced immune responses in mice to compare SFRT with homogeneous dose distributions. ³⁰ Yang et al. introduced a predictive model integrating dynamic carrying capacity and proliferation to simulate tumor response under SFRT regimens. ³¹ Nevertheless, a comprehensive understanding of these radiobiological mechanisms and their therapeutic benefit in clinical practice remains incomplete.

To better characterize the radiobiological effects of clinical SFRT regimens combined with SBRT, this study establishes a patient‐specific modeling framework that integrates bystander signaling and immune activation mechanisms. The proposed radiobiological model employs a system of ordinary differential equations (ODEs) to describe tumor cell proliferation, radiation‐induced cell death, and lymphocyte‐mediated immune responses, allowing simulation of tumor and lymphocyte dynamics under various treatment regimens. Reaction‐diffusion equations are employed to estimate bystander signaling. Patient‐specific parameters are estimated by fitting the model to clinically observed tumor volume and absolute lymphocyte count (ALC) data using a nonlinear least‐squares optimization approach. Quantitative analysis of radiobiological effects and simulations across different treatment regimens could provide new insights to inform the design of personalized SFRT‐SBRT strategies for optimal therapeutic efficacy.

2. METHODS AND MATERIALS

2.1. Patient data

This retrospective study analyzed four patients with NSCLC who received SFRT‐based radiotherapy alone at the University of Kansas Medical Center between 2022 and 2024. The treatment regimen consisted of LRT and SBRT, both delivered using Volumetric Modulated Arc Therapy (VMAT). LRT comprised a single 20 Gy fraction delivered to 6–15 spherical high‐dose vertices (≈0.5 cm³ each) within the tumor. SBRT was subsequently delivered to the entire tumor in 4–5 fractions totaling 20–32 Gy. Three patients (P1–P3) received the combined LRT‐SBRT protocol, with SBRT administered 1–2 weeks after LRT, while one patient (P4) received LRT alone. Clinical data included planning CT images, contoured target volumes and organs‐at‐risk (OARs), three‐dimensional dose distributions for both regimens, and weekly ALC measurements during and after treatment. Tumor volumes were delineated by experienced medical physicists on CT or cone beam computed tomography (CBCT) images. Treatment details for all patients are listed in Table 1.

TABLE 1.

Treatment‐related characteristics of non–Small cell lung cancer patients treated with combined spatially fractionated radiotherapy (SFRT) and stereotactic body radiotherapy (SBRT). The peak‐to‐valley dose ratio (PVDR) was calculated as the ratio of the mean peak dose to the mean valley dose.

	Clinical case number
Treatment‐related Parameter	P1	P2	P3	P4
SFRT
Prescription dose (Gy)	20	20	20	20
Fractionation	1	1	1	1
Number of vertices	15	7	6	10
Peak‐to‐valley dose rate	3.83	3.13	3.35	3.00
Total volume of vertices (cm3)	7.44	2.99	2.99	4.62
SBRT
Prescription dose (Gy)	32	30	20	–
Fractionation	4	5	4	–
Gross tumor volume (cm3)	520.6	153.77	67.4	238.6
Time interval between SFRT and SBRT (days)	7	13	7	–
Numbers of tumor volume and ALC measurements	9	9	18	12
Time to last available follow‐up (days)	88	100	428	181

2.2. Model assumptions and equations

To quantitatively simulate the radiobiological effect induced by SFRT, a series of coupled ODEs was constructed to describe the dynamic interactions among tumor cells, lymphocytes, and SFRT‐induced biological responses. The system comprises four key cellular populations: (1) Tumor cells (T); (2) Deceased tumor cells (D), which are irradiated cells that have lost proliferative capacity and are awaiting clearance; (3) Tumor‐infiltrating lymphocytes (TILs, I); and (4) Circulating lymphocytes (L). Figure 1 depicts the primary interactions between tumor and lymphocyte populations in response to SFRT.

FIGURE 1.

Schematic overview of the radiobiological mechanisms in response to spatially fractionated radiotherapy (SFRT) in the model, with arrows indicating stimulatory and suppressive effects (Created with https://BioRender.com). The model comprises four cell compartments: tumor cells (T), deceased tumor cells (D), tumor‐infiltrating lymphocytes (I), and circulating lymphocytes (L). It accounts for direct radiation‐induced damage to tumor cells and circulating lymphocytes, as well as indirect radiobiological effects, including intercellular signaling and immune responses.

In our model, the SFRT‐induced immune response is decomposed into two temporally distinct components: immediate lymphocyte recruitment occurring right after SFRT, and delayed lymphocyte activation driven by immunogenic tumor cell death, which typically manifests 1–2 weeks post‐treatment. ²² , ³²

Tumor growth dynamics were characterized using a Gompertz model, ³³ , ³⁴ in which cellular proliferation follows an exponential pattern constrained by environmental carrying capacity. Tumor cell loss was attributed to both radiation‐induced cytotoxicity and immune‐mediated cell killing. At doses above 10 Gy, SFRT can trigger immunogenic cell death, releasing tumor‐associated antigens that stimulate intratumoral lymphocytes and enhance tumor eradication. ¹⁴ Deceased tumor cells were assumed to be cleared from the system at a constant rate. The total tumor volume was defined as the sum of viable tumor cells and residual deceased cells. Equations (1) and (2) describe the temporal evolution of viable tumor cells (T) and deceased tumor cells (D):

$$\frac{dT}{dt}=\stackrel{tumorgrowth}{\overbrace{\mu T\ln \left(\frac{K}{T}\right)}}-\stackrel{SFRT-induced\mathrm{immune}\mathrm{activation}}{\overbrace{F\left(\left(t-t_{SFRT}\right)-\mathrm{\Delta }t\right){\omega }_1\frac{T}{g+T}I}}$$ (1)

$$\frac{dD}{dt}=-\stackrel{\mathrm{decay}}{\overbrace{cD}}$$ (2)

In Equation (1), μ denotes the exponential growth rate, K denotes the maximum tumor volume that can be carried by the tissue, F(t) is a time‐dependent weighting function representing the activation of immune function induced by the stimulation of SFRT. In this study, we used Δt = 9 days, representing the interval between SFRT delivery and the onset of delayed lymphocyte activation due to immunogenic tumor cell death. ${\omega }_1$ denotes the interaction between tumor cells and infiltrating lymphocytes, and the interaction of lymphocytes with tumor cells is expressed using the empirical Monod form (T/(T + g)), also known as the Michaelis‐Menten term, with g being the half‐saturation constant. This form explains the saturation effect due to an increase in tumor volume and a decrease in the fraction in contact with lymphocytes. ³⁵ In Equation (2), c represents the decay rate at which deceased tumor cells are cleared from the tumor microenvironment.

Immune cells were categorized according to their spatial distribution into circulating lymphocytes within peripheral blood and TILs within the tumor microenvironment. A fraction of circulating lymphocytes was assumed to be recruited to the tumor site in response to antigens released from viable and deceased tumor cells, thereby contributing to the TILs population. Lymphocyte recruitment induced by SFRT was modeled as a short‐term response after SFRT. The temporal dynamics of tumor‐infiltrating lymphocytes (I) and circulating lymphocytes (L) were described by Equations (3) and (4), respectively:

$$\frac{dI}{dt}=\stackrel{transfer}{\overbrace{bL}}-\stackrel{decay}{\overbrace{fI}}+\stackrel{SFRT-inducedrecruitment}{\overbrace{F\left(t-t_{SFRT}\right){\omega }_2\frac{D}{g+D}I}}$$ (3)

$$\begin{array}{ccc}\hfill \frac{dL}{dt}& =& {\omega }_3\stackrel{recruitment}{\overbrace{\frac{T+D}{g+T+D}L}}+\stackrel{supply}{\overbrace{s}}-\stackrel{decay}{\overbrace{fL}}-\stackrel{transfer}{\overbrace{bL}}\hfill \\ & & +F\stackrel{SFRT-inducedrecruitment}{\overbrace{\left(t-t_{SFRT}\right){\omega }_2\frac{D}{g+D}L}}\hfill \end{array}$$ (4)

In Equations (3) and (4), b denotes the rate of lymphocyte influx into the tumor microenvironment, s denotes the supply of circulating lymphocytes by the immune system, f denotes the rate of decay of I and L over time, ${\omega }_2$ denotes the recruitment of lymphocytes due to radiation, F(t), also a time‐dependent function, denotes the time at which the radiation‐induced recruitment of lymphocytes takes place, and ${\omega }_3$ denotes the recruitment of lymphocytes due to the release of immunity by tumor cells and inactivated tumor cells. Saturation kinetics are assumed similar across all tumor‐immune interaction terms, with a uniform g factor. The model assumes that the number of tumor‐directed effector lymphocytes increases in response to tumor antigen presentation, which is proportional to the number of T and D.

To account for the SFRT‐induced immune activation processes in the mathematical framework, a time‐dependent weighting function F(t) was introduced into Equations (1), (3), and (4). Drawing on existing evidence for both early immune cell recruitment and longer‑term effector responses following SFRT, ³⁶ , ³⁷ , ³⁸ we modeled the sustained effector phase (long‐term) via a sigmoid‑type term and the recruitment phase (short‐term) via a Gaussian‐type term, as mechanistic assumptions:

$$F\left(t\right)=\left\{\begin{array}{c}Sigmoid\left(t\right),longterm\\ Gaussian\left(t\right),shortterm\end{array}\right.$$ (5)

$$Sigmoid\left(t\right)=\frac{1}{1+e^{-\left(t-4\right)}}$$ (6)

$$Gaussian\left(t\right)=e^{\frac{-{\left(t-2\right)}^2}{8}}$$ (7)

In Equation (1), the sigmoid‐type F(t), gradually increasing from 0 to 1 for t ≥ 0, was used to represent the gradual enhancement of intratumoral lymphocyte cytotoxicity occurring approximately 5–9 days after SFRT. When F(t) approaches 1, maximal immune activation is achieved. In contrast, in Equations (3) and (4), F(t) follows a Gaussian distribution, rising sharply to a peak and then returning to baseline, reflecting short‐term lymphocyte recruitment immediately following radiotherapy.

2.3. Radiotherapy response

The dynamics of these cell populations are modulated by two radiotherapy regimens: SFRT (20 Gy in 1 fraction) and SBRT (20–32 Gy in 4–5 fractions). SFRT induced tumor cell death through both direct radiation damage and indirect radiobiological effects arose from intercellular signaling and immune activation. Immune activation and lymphocyte recruitment induced by SFRT were incorporated into the differential equations. The direct and signaling‐mediated killing effects were modeled through transient changes in T, D, and L at discrete time points.

Physical radiation kill and signal‐mediated kill were treated as independent processes and combined multiplicatively (additively on the log scale). The direct dose effect on tumor cells followed the linear‐quadratic (LQ) model, while the spatial and temporal dynamics of intercellular signals were computed from the reaction–diffusion equation described in Section 2.4. The killing effect of the signal on tumor cells is calculated by linear response equation:

$$S{F}_{dose}=\exp \left(-\alpha d-\beta d^2\right)$$ (8)

$$S{F}_{signal}=\exp \left(-\sigma \tau \right)$$ (9)

$$S{F}_{total}=\exp \left(-\alpha d-\beta d^2-\sigma \tau \right)$$ (10)

where σ denotes the sensitivity coefficient of tumor cells to the bystander signaling concentration exceeding the threshold, and $\tau$ represents the exposure duration of tumor cells to the signaling.

High‐dose vertices in LRT are treated as peak‐dose regions, while the remaining tumor volume represents valley‐dose regions. The cell survival rates of the two regions were calculated separately and then linearly superimposed. Circulating lymphocytes, which are highly radiosensitive, ¹² were modeled to decline rapidly following irradiation. In contrast, direct radiation‐induced depletion of TILs was assumed to be negligible, based on evidence that tumor‐resident T cells can survive clinically relevant radiation doses and maintain antitumor functionality. ³⁹ The dose to circulating lymphocyte during LRT was estimated using the HEDOS model ⁴⁰ for lung cancer, followed by a linear dose‐response model to compute lymphocyte survival. The update over a single time step at RT timepoint was expressed as:

$$\begin{array}{ccc}& & T\left(t^{+}\right)=T\left(t^{-}\right)\hfill \\ & & ·\left[V_{peak}·e^{-{\alpha }_T{d}_p-{\beta }_T{d}_p^2-\sigma {\tau }_p}+\left(1-V_{peak}\right)·e^{-{\alpha }_T{d}_v-{\beta }_T{d}_v^2-\sigma {\tau }_v}\right]\hfill \end{array}$$ (11)

$$D\left(t^{+}\right)=T\left(t^{-}\right)-T\left(t^{+}\right)+D\left(t^{-}\right)$$ (12)

$$L\left(t^{+}\right)=L\left(t^{-}\right)·e^{-{\alpha }_L{d}_{L1}}$$ (13)

where α_T and β_T denote the radiosensitivity parameters of tumor cells to the radiation and bystander signaling concentration, d _p and d_v denote the peak and valley doses in LRT, respectively, ${\tau }_p$ and ${\tau }_v$ are the average exposure time of tumor cell in peak and valley regions, α_L denotes the radiosensitivity parameter of the circulating lymphocytes, and d_L1 denotes the dose of the circulating lymphocytes in the LRT. For model parsimony, σ was assigned the same numerical value as α_T, though we acknowledge this is a pragmatic simplification rather than a biological equivalence. Future work with richer data could treat σ as an independent parameter.

During SBRT, tumor cell survival is modeled with the LQ model, whereas circulating lymphocyte survival is computed using a linear dose‐response model.

$$T\left(t^{+}\right)=T\left(t^{-}\right)·e^{-{\alpha }_T{d}_{RT}-{\beta }_T{d}_{RT}^2}$$ (14)

$$D\left(t^{+}\right)=T\left(t^{-}\right)\left(1-e^{-{\alpha }_T{d}_{RT}-{\beta }_T{d}_{RT}^2}\right)+D\left(t^{-}\right)$$ (15)

$$L\left(t^{+}\right)=L\left(t^{-}\right)·e^{-{\alpha }_L{d}_{L2}}$$ (16)

where d _RT represents the average dose to the tumor in SBRT, and d_L2 denotes the circulating lymphocyte dose in SBRT calculated by the HEDOS model. Description and references of the radiobiological model parameters are provided in Table 2.

TABLE 2.

Description of radiobiological model parameters used to simulate tumor and immune dynamics under combined spatially fractionated radiotherapy (SFRT) and stereotactic body radiotherapy (SBRT). ⁴⁵ , ⁴⁶ , ⁷⁴ , ⁷⁵ , ⁷⁶ , ⁷⁷ , ⁷⁸ , ⁷⁹

Source	Parameter	Description	Treatment	Reference
Fixed	μ (d−1)	Tumor growth	No	Li et al. 1
	K (cells)	Tumor‐carrying capacity	No	–
	c (d−1)	Deceased tumor cell decay rate	No	Marks 2
	b (d−1)	Lymphocyte influx rate	No	Silverstone 3
	f (d−1)	Lymphocyte decay rate	No	Mohiuddin et al. 4
	${\alpha }_T/{\beta }_T$ (Gy)	Tumor – LQ model	SFRT/SBRT	Mohiuddin et al. 5
	s (d−1)	Lymphocyte regeneration	No	Dilmanian et al. 6 ; Prezado 7
	g (cells)	Geometric saturation constant	No	Dilmanian et al. 6 ; Prezado 7
Fitted	${\alpha }_T$ (Gy−1)	Tumor radiosensitivity– LQ model	SFRT/SBRT	Mohiuddin et al. 5
	${\alpha }_L$ (Gy−1)	Lymphocyte radiosensitivity – Linear model	SFRT/SBRT	Slatkin et al. 8
	${\omega }_1$ (d−1)	Tumor‐killed lymphocyte efficiency	SFRT	Dilmanian et al. 6 ; Prezado 7
	${\omega }_2$ (d−1)	SFRT‐induced lymphocyte recruitment	SFRT	Dilmanian et al. 6 ; Prezado 7
	${\omega }_3$ (d−1)	Tumor‐lymphocyte recruitment rate	No	Dilmanian et al. 6 ; Prezado 7

2.4. Intercellular signaling calculation

Irradiated cells can secrete cytokines and chemokines that mediate communication with neighboring unirradiated cells, thereby influencing their biochemical behavior. To simulate intercellular signaling induced by LRT, we calculated the three‐dimensional bystander signaling distribution by a reaction‐diffusion equation. ²⁹ , ⁴¹ It accounts for both local generation and decay of signaling molecules, as well as their spatial diffusion from high‐dose vertices (peaks) to adjacent low‐dose regions (valleys). When a voxel received a radiation dose d and satisfied the condition t < γd, the signal concentration within that voxel was assumed to instantaneously reach its maximum value, denoted as ρ_max, which can be expressed as:

$$\rho \left(x,y,z,t\right)=\left\{\begin{array}{c}{\rho }_{max},ift<\gamma d\hfill \\ \rho \left(x,y,z,t\right),otherwise\hfill \end{array}\right.$$ (17)

where ρ represents the local signal concentration at spatial coordinates (x, y, z) and time t, ρ_max is the peak concentration, d is the local radiation dose, and γ is a constant specific to the tumor cell type, derived from in vitro experiments.

The tumor volume was discretized into a three‐dimensional grid of voxels, each assumed to have uniform signal concentration. The diffusion of signaling molecules between neighboring voxels follows the standard diffusion equation, while signal decay occurs at a rate determined by the decay coefficient λ:

$$\frac{d\rho }{dt}=\theta {\nabla }^2\rho -\lambda \rho$$ (18)

where θ denotes the diffusion coefficient, ∇² is the Laplacian operator, and λ is the decay constant of the signaling molecules.

To reduce computational cost, the 3D dose distribution was resampled to a voxel spacing of 4 × 4 × 4 mm³ prior to numerical simulation. Intercellular signaling dynamics were simulated using a discrete time‐stepping scheme, starting from the onset of radiation delivery (t₀ ) with a temporal resolution of 1 min. The simulation proceeded until intercellular signaling ceased and the maximum signal concentration dropped below a predefined response threshold. For each voxel, the strength of the bystander effect was quantified as the accumulated duration τ during which the local signal concentration ρ exceeded the response threshold ρ_τ . The threshold was defined as a fixed fraction of the maximum signal amplitude, ρ_τ /ρ_max = 0.21.

Model parameters were derived from experimental data in human large‐cell lung cancer H460 cells ²⁹ and applied to patient‐specific NSCLC simulations. Under the assumption of isotropic molecular diffusion in tissue, the diffusion coefficient θ was selected such that the resulting characteristic diffusion distance, ($r=\sqrt{\theta /\lambda }$), was consistent with experimentally reported tissue‐scale radiation‐induced bystander effect propagation distances (approximately 0.2–1.0 mm). ⁴² , ⁴³ All parameter values and units are listed in Table 3.

TABLE 3.

Description of parameters in the reaction–diffusion equations used for intercellular signaling calculations. ²⁹ , ⁴¹

Parameter	Description	Value	Reference
$\mathit{\gamma }$ (min/Gy)	Signal generation coefficient	132	Wu et al. 9
$\mathit{\lambda }$ (min−1)	Decay coefficient	0.019	Wu et al. 9
$\mathit{\theta }$ (mm2/min)	Diffusion coefficient	0.0079	Iori et al. 10
${\mathit{\rho }}_{\mathit{t}}/{\mathit{\rho }}_{\mathit{max}}$	Cell response threshold	0.21	Wu et al. 9
r (mm)	Diffusion distance	0.64	Wu et al. 9

2.5. Parameter fitting

Baseline cell populations were initialized as follows: T was estimated from the tumor volume with a density of 10⁹ cells per cc; ⁴⁴ D was set to zero; I was assumed to constitute 4% of the tumor cell population; and L were set based on the ALC with a total blood volume of 5L. Model parameters were estimated using nonlinear least‐squares optimization within biologically plausible bounds derived from the literature (Table 2). Given the limited number of data points per patient, five key parameters were selected for fitting: three immune‐related parameters (ω₁, ω₂, ω₃) governing immune activation and lymphocyte recruitment, and two radiosensitivity parameters (α_T, α_L) for T and L compartments. Model calibration was performed using the Python package “lmfit,” minimizing the sum of squared residuals (SSR) between simulated and observed tumor volumes and ALC values:

$$r_{tumor,i}=\frac{V_{sim,i}-V_{obs,i}}{V_{obs,i}},r_{ALC,j}=\frac{L_{sim,j}-L_{obs,j}}{L_{obs,j}}$$ (19)

$$SSR=\sum _{i=1}^N{r}_{tumor,i}^2+\sum _{j=1}^M{r}_{ALC,j}^2$$ (20)

where V _obs,i denote the observed tumor volume at time t _tumor,i, L _obs,j denote the observed ALC at time t _ALC,j; corresponding model‐simulated values V _sim,i and L _sim,j were obtained by interpolation, N and M denote the number of tumor volume and ALC observations, respectively.

2.6. Model evaluation and parameter sensitivity analysis

Model performance was evaluated using the normalized root mean square error (NRMSE) and the coefficient of determination (R²). The NRMSE and R² between simulated and observed values were computed as:

$$NRMSE=\frac{\sqrt{\frac{1}{N}\sum {\left(y_{obs}-y_{sim}\right)}^2}}{\max \left(y_{obs}\right)-\min \left(y_{obs}\right)}$$ (21)

$$R^2=1-\frac{\sum {\left(y_{obs}-y_{sim}\right)}^2}{\sum {\left(y_{obs}-{\overline{y}}_{obs}\right)}^2}$$ (22)

where y _obs and y _sim represent the observed and simulated data points, respectively.

Sensitivity analysis was conducted on patient‐specific parameters (α_T, α_L, ω₁, ω₂, ω₃). Each parameter was independently perturbed by ± 10% from its estimated value, while all other parameters were held fixed. For each perturbation, the model was re‐evaluated to obtain the predicted tumor volume at the final follow‐up time point. The resulting change in predicted tumor volume relative to the original simulation was used to quantify parameter sensitivity. The magnitude of this change was computed as follows:

$$\mathrm{\Delta }\left(\%\right)=\left(\frac{V_{perturbed}-V_{original}}{V_{original}}\right)\times 100$$ (23)

3. RESULTS

We present our results in four parts. First, we quantify the three‐dimensional distribution of intercellular signaling induced by LRT, which is incorporated as an input to the model. Second, we assess the model's ability to reproduce patient‐specific tumor and lymphocyte dynamics through parameter estimation. Third, we use the fitted model to examine the relative contributions of direct radiation damage and indirect biological effects. Finally, we investigate how variations in treatment scheduling affect model‐predicted outcomes in combined SFRT–SBRT regimens.

3.1. Intercellular signaling distribution

Before fitting the radiobiological model to patient data, we first needed to quantify the spatial and temporal characteristics of bystander signaling induced by LRT. This signaling represents an indirect killing mechanism that extends beyond regions receiving direct radiation, potentially enhancing tumor cell death in low‐dose valleys. Figure 2 illustrates the physical dose distributions and corresponding intercellular signaling maps for three NSCLC patients treated with VMAT‐based LRT. The signaling maps display the total time during which the local signal concentration exceeded the predefined response threshold within the target volume.

FIGURE 2.

Representative dose and intercellular signaling distributions for three non‐small cell lung cancer patients treated with spatially fractionated radiotherapy (SFRT). Panels (a–c) show the physical dose distributions of lattice radiotherapy, and panels (d–f) show the corresponding intercellular signaling maps. The signaling maps represent the voxel‐wise accumulated duration (minutes) during which the signaling variable ρ exceeds the predefined threshold ρ_t (ρ_t/ρ_max = 0.21). All images correspond to the central axial slice of the resampled 3D distributions, with a voxel size of 4 × 4 × 4 mm³.

Reaction–diffusion simulations demonstrated signal propagation into low‐dose regions. The physical peak‐to‐valley dose ratio (PVDR) ranged from 3.00 to 3.84, while the corresponding signal intensity ratio ranged from 2.47 to 2.98. Notably, signal molecules persisted at biologically relevant concentrations in valley regions for extended durations. This sustained signaling exposure suggests that bystander effects contribute to tumor killing beyond physical dose alone, providing a mechanistic rationale for the clinical efficacy of LRT's heterogeneous dose distribution. The spatial extent of propagation was associated with the dose gradient and the inter‐vertex spacing. Temporal signal profiles were then converted to days and incorporated as signaling input into the radiobiological model, enabling patient‐specific fitting.

3.2. Model fitting results

Model fitting results for all patients are presented in Figures 3 and S1. The NRMSE between simulated and observed tumor volumes ranged from 0.071 to 0.197, indicating that the model effectively captured tumor response dynamics. Lymphocyte predictions showed comparable performance (NRMSE: 0.017–0.385), except for patient P3, whose rapid week‐to‐week fluctuations in ALC were not well captured, likely reflecting biological variability beyond radiation‐driven effects (Figure S1b). Figure 3c,f demonstrates strong agreement between simulated and observed values (Pearson's r = 0.998).

FIGURE 3.

Model fitting to tumor volume and absolute lymphocyte count (ALC) dynamics for Patients 1 and 2 treated with spatially fractionated radiation therapy (SFRT) followed by SBRT. (a,d) Simulated and measured tumor volumes. (b,e) Simulated and measured ALC trajectories. (c,f) Normalized measured versus model‐predicted tumor volumes and ALC. Vertical dashed black lines indicate SFRT delivery, and green dashed lines indicate SBRT fractions.

Fitted parameter values are summarized in Table 4. Tumor radiosensitivity (α_T) ranged from 0.009 to 0.057, consistent with literature values (0.0242), ⁴⁵ while lymphocyte radiosensitivity (α_L) ranged from 1.71 × 10⁻⁴ to 2.00, in agreement with previous reported estimates. ⁴⁶ , ⁴⁷ Substantial inter‐patient heterogeneity was observed in immune parameters: patient P4 exhibited the highest immune activation (ω₁) and recruitment (ω₂) coefficients, while patient P1 showed the lowest ω₁ value, suggesting markedly different immunological responses to SFRT despite receiving similar treatment regimens.

TABLE 4.

Summary of the optimal parameters (best‐fit value ± standard error) and corresponding fitting results for each NSCLC patient.

Source	Parameter	P1	P2	P3	P4
Fixed	μ (d−1)	0.008	0.008	0.008	0.008
	K (cells)	9 × 1011	9 × 1011	9 × 1011	9 × 1011
	c (d−1)	0.053	0.053	0.053	0.053
	b (d−1)	0.097	0.097	0.097	0.097
	f (d−1)	0.033	0.033	0.033	0.033
	${\alpha }_T/{\beta }_T$ (Gy)	12.8	12.8	12.8	12.8
	s (d−1)	1.47 × 10 8	1.47 × 10 8	1.47 × 10 8	1.47 × 10 8
	g (cells)	10 11	10 11	10 10	10 11
Fitted	${\alpha }_T$ (Gy−1)	0.015 ± 0.010	0.009 ± 0.041	0.057 ± 0.040	0.015 ± 0.057
	${\alpha }_L$ (Gy−1)	1.84 ± 0.22	1.36 ± 2.62	2.00 ± 7.52	1.71 × 10−4
	${\omega }_1$ (d−1)	0.047 ± 0.27	0.300 ± 0.654	0.105 ± 0.018	0.397 ± 0.130
	${\omega }_2$ (d−1)	0.046 ± 0.066	1.002 × 10−5	0.003	0.835 ± 3.68
	${\omega }_3$ (d−1)	0.137 ± 0.0026	0.200 ± 0.068	0.119 ± 0.011	0.113 ± 0.010
Tumor volume	R 2	0.942	0.960	0.789	0.741
	NRMSE	0.087	0.071	0.147	0.197
ALC	R 2	0.997	0.912	−0.072	0.734
	NRMSE	0.017	0.107	0.385	0.180

Sensitivity analysis revealed patient‐specific effects of model parameters on tumor volume (Table S1). Tumor radiosensitivity (α_T) showed a moderate and consistent impact, with ± 10% perturbations altering tumor volume by 1.6–10.6%. Lymphocyte radiosensitivity (α_L) had minimal effects in P1 and P3 but reached ∼9% in P4, reflecting variability in lymphocyte dynamics. Among immune parameters, ω₃ showed the largest influence, especially in P2, while ω₁ and ω₂ had modest effects. These findings highlight that tumor response is generally dominated by α_T, but immune parameter effects are highly patient‐specific, underscoring the value of individualized modeling.

3.3. Analysis of radiobiological effects

A key strength of this mechanistic model is its ability to decompose overall tumor response into direct radiation damage and indirect immune‐mediated killing. To illustrate this, we analyzed patient P1 (Figure 4). Following SFRT (day 17), tumor cells declined rapidly due to direct radiation effects and bystander signaling. Subsequently, SFRT‐induced immune activation generated a sustained response that by 3 months exceeded the initial radiation damage (Figure 4c), underscoring the potential importance of indirect mechanisms.

FIGURE 4.

Model‐simulated dynamics of each cell compartment and corresponding radiobiological effects for Patient 1. (a–b) Dynamics of tumor cells and deceased tumor cells. (c) Tumor volume response to SFRT and SBRT, including direct and immune‐mediated damage. (d–e) Dynamics of tumor‐infiltrating lymphocytes and circulating lymphocytes. (f) ALC response, showing direct damage and SFRT‐induced recruitment.

Lymphocyte dynamics showed a contrasting pattern. ALC initially increased after LRT, consistent with immune recruitment, but declined sharply after SBRT due to radiation‐induced killing (Figure 4d). This depletion was greater than that observed with LRT alone (Figure 4f), suggesting a trade‐off between additional tumor dose and immune preservation. TILs followed a similar trend (Figure 4e). These observations motivated us to investigate whether optimizing the SFRT–SBRT interval could improve outcomes.

3.4. Effects of varying SFRT‐SBRT time intervals

Current practice typically schedules SBRT 1–2 weeks after SFRT, though the biological rationale remains empirical. Previous studies suggest immune activation peaks within 1–2 weeks post‐SFRT, raising the possibility that extending the interval might allow fuller immune‐mediated tumor killing before the second radiation course.

We first assessed the model's predictive capability by simulating three baseline scenarios: SFRT alone, SBRT alone, and combined SFRT‐SBRT at actual clinical intervals (Figure 5). In patient P1, SFRT alone produced transient tumor reduction with minimal lymphocyte depletion, whereas SBRT alone improved tumor control but caused substantial lymphocyte loss. The combined regimen (1‐week interval) achieved superior tumor control, consistent with synergistic effects.

FIGURE 5.

Simulated tumor volume and ALC changes under SFRT‐alone, SBRT‐alone, and combined SFRT‐SBRT regimens for Patient 1.

We then simulated SFRT–SBRT combinations with intervals of 1 week, 2 weeks, 1 month, and 2 months (Figure 6). Results revealed marked inter‐patient heterogeneity. Extending the interval could enhance immune‐mediated killing, though overall tumor reduction depended on the balance between treatment‐induced cell kill and tumor regrowth. For patient P1 (weak immune activation, ω ₁ = 0.047), tumor response was dominated by direct SBRT‐induced damage. For P2, extending the interval (e.g., to 2 months) improved tumor control at day 150, suggesting that additional time for immune activation may enhance efficacy. For P4, however, the benefit of delaying SBRT was limited, as larger tumor burden and rapid regrowth outweighed potential immune gains.

FIGURE 6.

Simulated tumor volume dynamics and corresponding radiobiological effects under varying time intervals (1 week, 2 weeks, 1 month, 2months) between SFRT and SBRT across all patients. Black and gray dash lines indicate the start days of SFRT and SBRT, respectively.

Shorter intervals (1–4 days) yielded negligible differences in tumor dynamics compared to the 1‐week interval (Figure S2). Treatment sequencing (SFRT→SBRT vs. SBRT→SFRT) showed substantial inter‐patient variability (Figure S3): Patient 4 benefited more from SBRT→SFRT, Patient 2 from SFRT→SBRT, while Patients 1 and 3 showed similar responses under all conditions. To account for measurement uncertainty, we incorporated ± 10% variation in initial tumor volume (Figure S4), which demonstrated robustness of tumor response trends across intervals and perturbations while highlighting patient‐specific variability. Overall, these findings suggest pronounced inter‐patient heterogeneity, indicating that the optimal strategy may warrant individualized consideration in future studies.

4. DISCUSSION

This study proposes a proof‐of‐concept radiobiological modeling framework that integrates intercellular signaling and immune activation to investigate the biological mechanisms of SFRT. Formulated using ODEs, the model demonstrates feasibility in reproducing temporal changes in tumor volume and ALC in a small cohort of NSCLC patients. It further enables exploratory simulations of tumor control under monotherapy and combination regimens, as well as variable treatment intervals, offering a preliminary quantitative basis for investigating optimal SFRT sequencing with conventional radiotherapy or SBRT. However, we acknowledge that these findings are hypothesis‐generating and require validation in larger, prospective studies.

Indirect cytotoxic effects in the model arise from bystander signaling and immune activation, both modulated by physical dose. Signal intensity and retention are governed by the PVDR, with higher PVDRs enhancing diffusion to low‐dose regions. SFRT doses above 10 Gy can induce immunogenic cell death and activate antitumor immunity, whereas excessive peak doses can deplete circulating lymphocytes and limit immune responses. Optimal SFRT design requires balancing PVDR and lymphocyte preservation to stimulate immunity while minimizing lymphocyte depletion. These predictions align with experimental observations linking lower valley doses to enhanced tumor response. ²⁷ , ²⁸

Evidence from clinical and modeling studies suggests that supplementing large‐fraction SFRT with open‐field radiotherapy can enhance tumor control, ⁴⁸ , ⁴⁹ which is consistent with our simulations comparing SFRT‐only and combined regimens. SFRT's spatially heterogeneous dose distribution can induce systemic immune effects, including localized immune infiltration and abscopal responses. By simulating dynamic tumor and lymphocyte changes, the model quantifies SFRT‐induced immune activity, represented by the immune‐activation parameter ${\omega }_1$ and ${\omega }_2$. SFRT has been reported to upregulate PD‐L1 and increase activated CD4⁺ and CD8⁺ T cells, suggesting potential synergy with immune checkpoint inhibitors. ²² SFRT may enhance systemic antitumor immunity through the release of cytokines and soluble mediators such as IL‐6, IL‐8, TGF‐β, and TNF‐α, thereby potentiating the effects of immune checkpoint inhibitors. ⁵⁰ The model can be extended to simulate immunotherapy by converting ${\omega }_1$ into a dynamic function, ⁵¹ and chemotherapy effects can be incorporated via a dose‐response term, enabling optimization of combination therapy timing and dose.

Previous surveys ¹² indicate that while SFRT is most commonly delivered prior to conventional external beam radiation therapy, alternative sequencing strategies, including interdigitated schedules or SFRT delivered after conventional radiation, are used in a notable proportion of cases. These observations suggest that both varying intervals and flexible sequencing are compatible with current clinical practice, provided that careful consideration is given to cumulative dose, normal tissue tolerance, and patient‐specific biological factors. ⁵² In practice, individualized planning based on tumor characteristics, patient condition, and treatment logistics is essential to safely implement these strategies.

Several limitations should be noted. The model was fitted to data from only four NSCLC patients, restricting the assessment of parameter robustness and generalizability. Some parameters exhibited relatively large standard errors, reflecting limited identifiability due to the sparse longitudinal measurements available. Future studies with more frequent longitudinal assessments and larger patient cohorts are needed to improve parameter identifiability and enhance model robustness. While the fitted parameters retain biological relevance, the near‐zero α_L estimated for Patient P4 likely reflects ALC dynamics dominated by immune recruitment and activation rather than intrinsic lymphocyte radioresistance. Given the limited data, we interpret these values as effective parameters that collectively describe the observed trajectories within our modeling framework, offering biologically plausible hypotheses to be tested in future studies with larger cohorts and more frequent sampling. For example, ω ₁ is a model‐derived parameter representing tumor‐killed lymphocyte efficiency, and no established single biomarker directly corresponds to this integrated functional measure. Future prospective studies incorporating longitudinal immune monitoring (e.g., cytokine dynamics, T‐cell subsets) could help validate its biological interpretation. In addition, several parameters were fixed to literature‐derived values to avoid overfitting given the limited data, though these values may vary across patients and could shift the fitted estimates. Future studies with richer data are needed to quantify the impact of fixed parameter uncertainty. Moreover, validation in diverse patient populations, tumor types, and fractionation schedules is needed to confirm the observed relationships among dose, immune activation, and tumor response.

Incorporation of prospective clinical data, including immunological biomarkers, could further enhance biological realism. Future refinements may consider SFRT‐specific planning factors, such as PVDR, ⁵³ , ⁵⁴ , ⁵⁵ lattice positioning, ⁵⁶ , ⁵⁷ , ⁵⁸ , ⁵⁹ vertex size, ⁶⁰ , ⁶¹ , ⁶² , ⁶³ EUD, ⁶⁴ and treatment site, ⁶⁵ to optimize plan quality and delivery. Beyond biological modeling, integration with advanced SFRT modalities, including proton, ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ carbon‐ion, ⁷⁰ and FLASH‐based approaches, ⁷¹ , ⁷² , ⁷³ could enable modality‐specific enhancement of radiobiological effects.

5. CONCLUSION

This study developed a personalized radiobiological model that integrates bystander signaling and immune activation to simulate tumor and lymphocyte responses under combined SFRT‐SBRT regimens. The model demonstrated feasibility in capturing key tumor and lymphocyte dynamics and provided preliminary estimates of the contributions of direct radiation damage and immune‐mediated effects. Simulations suggested that the combined SFRT‐SBRT regimen may achieve superior tumor control compared with either modality alone within our modeling framework. Notably, our simulations indicated that the optimal interval between SFRT and SBRT may be patient‐specific, potentially depending on the balance of direct radiation effects, immune‐mediated synergy, and tumor regrowth. Overall, this framework offers mechanistic insight into SFRT efficacy and provides a preliminary quantitative basis for designing personalized radiotherapy strategies, though further validation in larger prospective cohorts is needed before clinical application.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

Supporting information

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