Mathematical analysis of metronomic chemotherapy response in metastatic gastrointestinal cancer: identifying critical parameters from clinical data

Abstract

We present the first clinical application of the Schättler mathematical framework for metronomic chemotherapy using data from 30 patients with metastatic gastrointestinal cancer from the COMET trial. The model, comprising three coupled ODEs with 12 parameters representing tumour growth, angiogenesis, and immune response, was successfully fitted to individual patient data through AI-assisted optimization, achieving low fitting errors (median squared error: 0.003 for stable disease, 0.007 for progressive disease) at day 56. The growth control parameter emerged as a potential biomarker, with distinct ranges between stable disease (0.066–0.153) and progressive disease (0.093–0.177), suggesting a critical threshold around . Model parameters showed expected interdependencies reflecting the coupled nature of tumour-microenvironment interactions. While the model demonstrated strong fitting capabilities, prediction of progression-free survival showed moderate correlation (), highlighting the complexity of long-term outcome prediction in metastatic disease. Drug effect parameters were consistently low (0.003–0.09), aligning with the modest clinical benefits observed in recent metronomic chemotherapy trials for metastatic gastrointestinal cancer. This proof-of-concept study establishes the feasibility of applying the Schättler framework to clinical data, with parameter estimates representing exploratory values requiring external validation before clinical implementation. Future studies with larger cohorts and multiple time points will be essential to validate these findings and refine the model for prospective clinical application.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-30446-y.

Introduction

Clinical background

Cancer therapy optimisation remains a critical challenge in clinical oncology. Despite significant advances in treatment options, determining optimal dosing schedules that balance efficacy and toxicity continues to be largely empirical. This challenge stems from the complex interplay between tumour growth and the tumour micro environment where angiogenesis and immune response influence treatment outcomes. Angiogenesis, the formation of new blood vessels, plays a crucial role in tumour progression by providing necessary nutrients and oxygen. Simultaneously, the immune system’s response to both the tumour and treatment can significantly impact therapeutic efficacy. Individual patient variability in these processes further complicates therapeutic optimisation.

Metronomic chemotherapy

Historically, cancer chemotherapy administration has followed established principles based on dose intensity and dose effect, stemming from Skipper’s fundamental work^22,23. However, emerging experimental and clinical evidence challenges the paradigm of ‘more is necessarily better’. Studies have shown that frequently administered low-dose chemotherapy can maintain moderate cytotoxic effects on cancer cells while demonstrating both anti-angiogenic and immune stimulatory properties^3,8. This evidence has led to the development of metronomic chemotherapy – the almost continuous administration of chemotherapeutic agents at significantly lower doses than maximum tolerated dose strategies⁷. Interest in metronomic therapy has grown considerably, particularly due to its favorable efficacy in palliative care and minimal toxicity profile, especially in pediatric and elderly frail patients^4,5. The anti-tumour effects operate through multiple mechanisms: direct cytotoxicity, angiogenesis inhibition, and immune system modulation, with the latter two potentially playing central roles in treatment response.

Mathematical modelling in oncology

Like other complex systems such as climate or financial markets, cancer behaviour can be described and analysed through quantitative mathematical models¹¹. Various approaches have been developed to understand cancer cell proliferation dynamics, from classical models like Gompertz¹² and von Bertalanffy²⁶ to more sophisticated frameworks incorporating multiple biological processes^1,17,25. Contemporary oncology views tumours as complex systems comprising various cancer cell sub-populations with differing therapeutic sensitivities, all existing within a dynamic micro environment. This environment encompasses tumour vasculature, immune system interactions, and structural elements such as fibroblasts and extracellular matrix. However, despite advances in mathematical modelling, there often remains a disconnect between theoretical models and clinical practice in oncology.

Study rationale and objectives

This study aims to bridge this gap by applying a mathematical framework to study tumour response in metastatic gastrointestinal cancers patients receiving metronomic chemotherapy. We analysed data from the COMET trial, a phase II study investigating metronomic chemotherapy in gastrointestinal cancers². Our modelling framework consists of a system of three ordinary differential equations modelling tumour growth, inspired by Schättler and colleagues’ work^15,20. This system aims to capture the essential interactions between tumour and microenvironment under metronomic chemotherapy while maintaining minimal parametrisation for analytical tractability. Using the COMET trial database, we identified specific optimised parameter ranges through numerical methods assisted by artificial intelligence models, separately analysing stable disease (SD) and progressive disease (PD) patients. We then investigated potential relationships between tumour-size response metrics and clinical outcomes. While similar approaches have been applied to non-small-cell lung cancer^9,27, previous studies relied primarily on statistical rather than mechanistic mathematical models.

Methods: trial design and patient population

The study was approved by the Comitato Etico Azienda USL6 Livorno Zona Livornese (approval date 2007-02-13), and registered in the European Clinical Trial Database EudraCT (http://eudract.emea.europa.eu; EudraCT registration number 2007-000065-38, registered 2007-05-04). Patients were informed of the investigational nature of the study, providing their written informed consent. All methods were per- formed in accordance with the Declaration of Helsinki and relevant guidelines and regulations. All the experiments comply with the current Italian laws. The progression free survival (PFS) was the primary endpoint.

A total of 38 patients with histologically confirmed metastatic gastrointestinal cancers were enrolled and evaluated. Eligible patients had undergone surgical removal of their primary tumour and had received at least two prior lines of treatment.

The therapeutic regimen employed a metronomic approach, combining three agents with distinct mechanisms of action:

1. Cyclophosphamide:

   • Day 1: 500 mg/m² intravenously (body surface area-adjusted bolus),
   • Day 2 onwards: 50 mg/day orally (fixed dose).
2. UFT (Tegafur/Uracil):

   • Day 2 onwards: 100 mg twice daily orally.
3. Celecoxib:

   • Day 2 onwards: 200 mg twice daily orally.

The treatment combination was based on preclinical evidence from Nio et al. ¹⁸, demonstrating synergistic effects between cyclophosphamide and UFT, particularly when initiated with a cyclophosphamide bolus.

Treatment response was evaluated using RECIST criteria version 1.1¹⁰ at 8 weeks post-treatment initiation.

For the mathematical modelling analysis, we focused on 2 response categories that emerged from the trial data:

• Progressive Disease (PD): 20% increase in sum of target lesion diameters

• Stable Disease (SD): Neither sufficient shrinkage to qualify for partial or complete response nor sufficient increase to qualify for PD

No partial responses (PR) or complete responses (CR) were observed in the study population, thus the mathematical model was developed and validated specifically for PD and SD categories.

Such binary categorisation provided a clear framework for analysing the differential dynamics of tumour response between these 2 groups. For further details on the population, we refer to the study by Allegrini and colleagues².

Mathematical model description

Model framework

The model is based on a minimally parametrised system of three ordinary differential equations developed by Schättler et al. ²⁰, which we adapted to analyse metastatic response to metronomic chemotherapy. The system has the form

1

2

3

in the unknowns , , , and for given positive parameters , , , b, , d, , , , , , and given treatment function . The time variable t is expressed in days and the zero is set at the treatment’s start.

The model takes into account angıogenic sıgnalling between the tumour and its vasculature and tumour inhibiting effects of tumour-immune system interactions.

The equations combine a model for tumour development under angiogenic signaling formulated by Hahnfeldt et al. in¹³ with a model for tumour-immune system interactions by Stepanova in²⁴. The Cauchy problem is well - posed in octant (see^13,15,20,24).

Key model adaptations

We implemented the following key adaptations.

• Variable p: While in the original model p represented primary tumour volume, in our application it represents the sum of the longest diameters, in millimiters, of metastatic lesions, as the primary tumour had been surgically removed in all patients

• Variables q and r: Approximated using clinically relevant biomarkers:

  • q through vascular endothelial growth factor (VEGF) levels, a validated marker of angiogenesis
  • r through interleukin 2 (IL2) levels, reflecting immune system activity

  Biomarker analyses were performed on plasma samples collected from study participants on days 0, 28, 56, 84, and 112 using Milliplex multiplex immunoassay technology. Comprehensive results of these analyses are pending publication.
• Treatment function u(t):

  4

  The precise choice (4) for u is based on empirical considerations and educated guess from the utilised COMET trial. Notably, such trial employed a hybrid protocol that differs partially from pure metronomic chemotherapy. As described in Section Methods: trial design and patient population, the regimen combined a high-dose intravenous bolus on Day 1 with continuous oral administration from Day 2 onwards. This two-phase structure required modification of the original Schättler model, where u was assumed constant under pure metronomic dosing. The explicit expression (4) was designed to capture these aspects: for the bolus phase, and for the metronomic maintenance. The numerical constants (2.0, 0.65, 100) were empirically fitted to optimise agreement between model predictions and observed clinical outcomes. Critically, these are phenomenological fitting parameters, not quantities derived from mechanistic pharmacokinetic principles. Specifically: the bolus magnitude (2.0) represents the relative intensity of Day 1 administration; the maintenance baseline (0.65) reflects steady-state oral dosing; and the decay constant (100 days) captures progressive therapeutic decline over time. The formulation achieved excellent fit to tumour dynamics (median squared error: 0.003 for SD, 0.007 for PD at Day 56). In this respect, u represents a simplified composite metric of the effective therapeutic pressure on the tumour-microenvironment system over time, aggregating the effects of the three-agent combination (cyclophosphamide, UFT, celecoxib). It does not represent the concentration of a single cytotoxic agent as in the original Schättler framework, but rather the integrated pharmacodynamic impact of the multi-drug regimen. This approach maintains the minimal parameterization philosophy of the Schättler framework while adapting it to the hybrid bolus-plus-metronomic protocol used in the patient’s cohort. It prioritizes biological interpretability and computational tractability over pharmacokinetic precision. Preliminary testing of alternative formulations showed that constant u failed to capture the distinct temporal dynamics between early response (Days 0-28) and later progression patterns observed clinically. A time-averaged formulation was chosen over quasi-periodic representations because tumour response (weeks-months timescale) averages out rapid drug pharmacokinetic fluctuations (hours-days), making daily oscillations negligible for predicting outcomes measured at 4-8-week clinical assessment intervals.

The model’s twelve parameters represent specific biological processes:

• : tumour growth rate;

• : rate of tumour cells elimination by immune system;

• : direct anti-tumour effect of chemotherapy;

• b: rate of tumour-induced vascularisation;

• : natural death rate of endothelial cells;

• d: additional inhibition of vascular decay;

• : chemotherapy’s anti-angiogenic effect;

• : rate of immune system stimulation by tumour presence;

• : immunosuppression coefficient for large tumour burden;

• : baseline immune system activity;

• : natural death rate of immune cells;

• : chemotherapy’s immune-stimulatory effect.

The 12-parameter structure is not our arbitrary choice but represents the established Schättler et al. framework (reference [20]). Schättler et al. explicitly describe this as a “minimally parameterized mathematical model for metronomic chemotherapy”. We made only one modification: adapting u(t) from constant to time-dependent to accommodate the COMET trial’s hybrid protocol, which combines an initial high-dose bolus with continuous low-dose maintenance therapy.

These parameters were optimised separately for SD and PD patients, revealing distinct parametric patterns associated with different clinical outcomes (see Section Results).

Model components

The three coupled Eqs. (1)-(3) account for three fundamental intertwined behaviours.

1. Equation (1): Tumour Growth and Response. This includes

   • the Gompertzian growth term ,
   • the immune-mediated suppression ,
   • the direct chemotherapy effect .
2. Equation (2): Angiogenesis Dynamics. This includes

   • the tumour-induced stimulation term (bp),
   • the natural death and inhibition term ,
   • the anti-angiogenic effect of chemotherapy .
3. Equation (3): Immune System Response. This includes

   • the dual effect of tumour on immune system , and specifically

     • the term accounting for stimulation at small tumour burden,
     • and the term accounting for suppression for large tumour burden,
   • the baseline immune activity ,
   • the natural death rate ,
   • the immune stimulation by chemotherapy .

This modelling framework allows the integration of three key biological processes – tumour growth, angiogenesis, and immune response – while incorporating both direct and indirect effects of metronomic chemotherapy.

Through parameter optimisation, the model demonstrated ability to capture different response patterns corresponding to stable (SD) and progressive disease (PD) patients, as reported in Section Results.

Properties

Let us highlight relevant mathematical aspects of the non-linear dynamical system (1)-(3).

1. Under the assumption of positive initial conditions , solutions remain positive due to the structure of the equations.

2. The system shows local Lipschitz continuity, ensuring existence and uniqueness of local solutions by the Picard-Lindelöf theorem.

3. The term in the first equation requires careful consideration both mathematically and biologically. While this Gompertzian-type term effectively models growth saturation, it introduces a logarithmic singularity when p approaches q. This theoretical limitation is managed in our numerical implementation through appropriate bounds on the ratio p/q.

4. Also, the discontinuity in the treatment function u(t) at requires careful numerical handling, implemented through adaptive step-size control in the numerical integration scheme.

5. We analysed the system’s behaviour by identifying stable equilibrium points for each patient group, characterising the long-term dynamics of tumour growth, angiogenesis, and immune response. The stability of these equilibria was confirmed by examining the eigenvalues of the linearised system at each critical point, with positive values of parameter indicating stable behaviour.

Analytical methods and validation

Parameter optimisation approach

The twelve model parameters were optimised within biologically plausible ranges by minimising an objective functional for each patient. For the j-th patient, we defined the objective functional as the squared difference between model-predicted and clinically observed tumour size at Day 56 (8 weeks):

5

Here:

• is the clinically measured tumour size (in millimeters) for the j-th patient at time t during the clinical trial, with days serving as our reference evaluation point;

• is the vector of twelve model parameters;

• is the model-predicted tumour size at time t for parameter set , obtained as the first component of the numerical solution (p(t), q(t), r(t)) to the system (1)–(3). Initial conditions at were set using patient-specific baseline measurements for the j-th patient: p(0) from the sum of metastatic lesions diameters, while q(0) and r(0) were approximated using VEGF and IL-2 biomarker levels, respectively (see Section Key model adaptations).

This way, (5) defines a function of the twelve parameters , for each patient j. For each patient j, we sought the optimal parameter vector that minimizes subject to biologically plausible box constraints on each parameter. The optimisation was performed using a multi-start Monte Carlo approach with extensive sampling of the parameter space. This strategy ensured robust convergence to stable optimal parameter sets for each patient.

The observed inter-patient variability in optimal parameter values reflects the biological heterogeneity inherent in metastatic disease. Individual patient characteristics—including disease history, age, genetic factors, and prior treatment exposure—influence treatment response in ways not explicitly captured by the model structure. This patient-specific parameter variation is therefore not a limitation but rather an expected feature that allows the model to capture individual response patterns within a unified mathematical framework.

Following individual patient optimisation, we computed group-specific optimal parameter sets by averaging the individual optima separately within the PD and SD patient cohorts. This yielded two representative parameter vectors, and ,at which the model achieves optimal agreement with observed clinical data for each response category within our patient cohort.

For PD patients, we refined the objective functional to incorporate progression-free survival (PFS) information alongside tumour size dynamics:

6

Here, denotes the observed progression-free survival for the j-th patient from clinical records. The model-predicted is computed from the numerical solution as the earliest time t at which progressive disease is detected according to RECIST 1.1 criteria:

7

corresponding to a 20% increase in tumour burden from baseline. This modified form of better captures the full clinical trajectory of PD patients, whereas for SD patients the simpler form (5) sufficed due to the absence of progression events within the observation period.

It is also instructive to report that for each of the two cohorts PD and SD we attempted a parameter optimisation procedure cumulatively across all patients, introducing the cumulative objective functional

8

where the sum runs over all patients in either the PD or SD cohort. However, while minimisation of for the SD cohort yields an optimal parameter string essentially coinciding with (despite the higher computational cost and longer convergence time), the cumulative minimisation for the PD class tends to produce biologically implausible results. Specifically, the optimal parameter string generates individual patient trajectories that converge to nearly identical values of p(56), all of which remain equidistant from their corresponding measured values. This failure suggests fundamental heterogeneity in PD patient dynamics that cannot be captured by a single parameter set, reinforcing the necessity of individual patient optimisation for this cohort.

Numerical optimisation methodology

We utilised the R deSolve package with adaptive step-size control, ensuring numerical stability particularly around the treatment function discontinuity at . A multi-start Monte Carlo approach was employed, using different initial points within the parameter space. Optimisation was conducted independently for each patient while maintaining biological plausibility across the cohort.

Statistics and reproducibility

For statistical analysis, we computed means, standard deviations, and relative errors for each parameter within PD (, excluding one outlier) and SD () groups. Data from 30 patients out of 38 enrolled are reported due to convergence issues in parameter optimisation for the remaining patients.

Data sets are summarised with descriptive statistics including the sample size (n), mean as the measure of centre, and standard deviation as the measure of variability for each parameter group. Parameter ranges (minimum-maximum) are also reported to show the full data distribution.

To assess differences between PD and SD groups, we employed the non-parametric Mann-Whitney U test. This test was selected due to the small sample sizes ( per group) and because preliminary normality testing using the Shapiro-Wilk test indicated that several parameters deviated significantly from normal distributions (p < 0.05), violating the assumptions required for parametric tests such as the t-test. The Mann-Whitney U test does not assume normal data distribution and is appropriate for comparing two independent groups with ordinal or continuous data. All tests were two-tailed with significance levels set at , , and . The alpha level was set at 0.05 for all comparisons.

To investigate parameter relationships, we performed correlation analyses using Spearman’s rank correlation coefficient within each group. Spearman’s correlation was chosen because it does not assume linear relationships or normal data distribution, making it appropriate for our parameter data. Only statistically significant correlations (p ) were retained for interpretation. No adjustments for multiple comparisons were applied to correlation analyses as these were exploratory in nature. Correlation matrices were visualised using hierarchical clustering to identify parameter clusters and potential biological relationships.

For the analysis of the relationship between predicted tumour size changes and progression-free survival, we used Pearson correlation coefficient after confirming approximate normality of the residuals.

All statistical analyses were performed using R version 4.5.0. Individual patient parameter optimisation convergence was assessed through examination of optimisation trajectories, final error values, and gradient norms, with successful convergence defined as gradient norms .

Validation approaches

Our validation strategy consisted of four complementary components.

Numerical validation included analysis of optimisation trajectories to ensure convergence from multiple initial conditions, assessment of final error values to quantify goodness of fit, and verification that gradient norms at convergence satisfied the stringent criterion .

Biological validation confirmed that all optimised parameters remained within biologically plausible ranges established from prior literature and clinical knowledge, with cross-validation of parameter distributions against published studies. Separate validation was performed for SD and PD patient cohorts to account for their distinct biological dynamics.

Sensitivity analysis was conducted by perturbing each parameter by while holding others fixed, investigating parameter correlations to identify potential identifiability issues, and assessing the impact of parameter variations on model predictions at key clinical time points (days 28, 56, and 84).

Finally, we investigated associations between model-predicted tumour size at day 56, p(56), the absolute change in predicted tumour size, , and the clinically observed Progression-Free Survival (PFS).

The optimisation methods were independently implemented by two researchers (AM and MLM) using two different computational approaches to ensure reproducibility. Initial parameter values for all optimisation runs were based on the reference work by Schättler et al. ²⁰.

Artificial intelligence support

The complexity of optimal control problem implementation required sophisticated revision support. We utilised Claude 4.5 Sonnet (Anthropic, 2025), a large language model specifically designed for scientific applications, to assist in several aspects of the research process.

This AI system was instrumental in:

1. code development, including

• implementation of differential equation systems,

• design and optimisation of control strategies,

• development of visualisation routines,

• integration of numerical optimisation methods;

• 2.mathematical modelling, including

• verification of mathematical formulations,

• suggestion of alternative control strategies,

• implementation of toxicity constraints,

• refinement of objective functions;

• 3.analysis support, including

• interpretation of preliminary results,

• identification of numerical instabilities,

• suggestion of parameter adjustments,

• development of comparative analyses.

Claude AI assistance was particularly valuable in rapidly prototyping different control strategies and identifying the most promising approaches for detailed investigation.

All AI suggestions were mathematically verified and clinically validated before implementation. The final optimal control strategy emerged through an iterative process of AI-assisted development and human expert validation.

Results

Mathematical model

Stable equilibrium analysis revealed distinct steady states for PD and SD patient groups, providing a mathematical foundation for understanding treatment response dynamics.

For SD patients, the system stabilised at significantly lower values of tumour burden () and angiogenic factors () compared to PD patients (, ). Importantly, immune response at equilibrium was approximately 2.2-fold higher in SD patients () than in PD patients (), suggesting a critical role of immune function in disease stabilisation.

Both equilibrium points demonstrated mathematical stability, with positive [A] values of 0.8755 for SD and 1.6476 for PD patients. These results quantitatively characterise the underlying biological differences between clinical response categories and offer new insights for therapeutic intervention strategies.

Parameter optimisation and group differences

The optimisation procedure successfully converged for all patients, yielding reliable parameter estimates with notably low relative errors for the PD group (2.1%–7.2%) and higher variability in the SD group (6.5%–48.7%). Table 1 presents the mean values, relative errors, and ranges for all parameters in both groups.

Table 1.

Parameter estimates for PD and SD groups.

Parameter	PD (n = 17)		SD (n = 13)		Sig.
	Mean (Range)	Error (%)	Mean (Range)	Error (%)
Tumour growth	0.137	2.07	0.115	6.46	***
	(0.093 -0.177)		(0.066–0.153)
Immune-tumour	1.220	2.85	1.666	10.85	***
	(0.389–1.660)		(0.406–1.435)
Anti tumour	0.0053	4.25	0.0063	30.05	*
	(0.0027–0.0086)		(0.0027–0.0084)
b Drug accumulation	1.527	2.07	1.122	14.63	***
	(1.065–1.919)		(0.927–1.753)
d Clearance rate	0.033	3.20	0.032	15.47	ns
	(0.019–0.049)		(0.033–0.079)
Anti-angiogenic	0.054	2.95	0.072	9.91	***
	(0.032–0.079)		(0.030–0.080)
Pharmacokinetics	0.0043	5.14	0.0030	48.74	*
	(0.0018–0.0072)		(0.0026–0.0056)
Immune activation	0.0048	5.77	0.0024	34.65	***
	(0.0027–0.0081)		(0.0014–0.0074)
Immune exhaustion	0.0052	7.21	0.0026	25.90	***
	(0.0023–0.0071)		(0.0012–0.0070)
Baseline immune	0.044	3.61	0.045	9.97	ns
	(0.027–0.064)		(0.021–0.079)
Immune decay	1.427	2.22	1.023	16.01	***
	(0.966–1.817)		(0.899–1.648)
Immune-stimulatory	0.058	3.25	0.082	11.53	***
	(0.040–0.091)		(0.021–0.066)

Significance levels: ***, *, ns: not significant. Parameters were obtained through patient-by-patient optimisation with Monte Carlo simulations (up to 80,000 random strings per patient). Error represents parameter variability within each group.

Our incremental optimisation approach employed a patient-by-patient Monte Carlo method with 30–40 randomly selected parameter strings per optimisation run, each within prescribed biological ranges. For PD patients, this process required up to 80,000 random strings to achieve optimal fitting between measured and calculated tumour sizes (p(56)) and progression-free survival (PFS) times.

Individual square discrepancies were consistently low (0.0001–0.4), indicating excellent model fit across patients. The only exception was patient AL, which demonstrated erratic behaviour with higher discrepancy values that might warrant exclusion from analysis.

The Mann-Whitney U test revealed significant differences between PD and SD groups for 10 of 12 model parameters. The most striking differences were observed in tumour growth rate (), immune-tumour interaction (), drug accumulation (b), and immune-related parameters (, , ). Only two parameters showed no significant differences between groups: the clearance rate (d) and immune system baseline activity ().

Notably, PD patients exhibited higher tumour growth rates ( vs 0.115, ), but lower immune-tumour interaction values ( vs 1.666, ). The drug-related parameters revealed that PD patients showed lower direct antitumour effects ( vs 0.0063, ) as well as lower anti-angiogenic ( vs 0.072, ) and immune-stimulatory effects ( vs 0.082, ) compared to SD patients.

The graphs in Fig. 1 show:

1. Tumour volume evolution (p) over time, where SD patient demonstrates disease stabilisation while PD patient shows clear progression;

2. Carrying capacity (q) dynamics, reflecting vascular support with more pronounced increase in PD patient;

3. Immune response (r) changes, showing higher steady-state immune activity in SD versus PD patient.

Initial conditions for SD: , , ; for PD: , , . Final values at for SD: , , ; for PD: , , .

Fig. 1.

Model-predicted tumour dynamics comparing representative SD and PD cases.

The cumulative analysis approach, which attempted to optimise parameters across all patients simultaneously, proved effective for SD patients but failed for PD patients. For PD patients, this approach resulted in unrealistically high cumulative discrepancies (order of ) and produced histories that converged to similar p(56) values across different patients, regardless of their actual measured values. This suggests fundamental heterogeneity in PD patient dynamics that cannot be captured by a single parameter set.

Parameter ranges showed consistent biological patterns (Table 1). For PD patients, ranged from 0.093 to 0.177, while for SD patients, it was notably lower (0.066 to 0.153). Immune-related parameters also showed distinct ranges between groups, with PD patients having higher immune exhaustion (: 0.0023–0.0071 vs 0.0012–0.0070) and immune decay (: 0.966–1.817 vs 0.899–1.648) compared to SD patients.

Parameter correlations

Correlation analysis revealed distinct patterns between PD and SD groups.

The PD group showed stronger and more uniform correlations, with correlation coefficients often approaching , suggesting a more rigid parameter structure. Two main clusters emerged in the PD group: one including , b, and (positively correlated), and another comprising , , , , , and (positively correlated within the cluster but negatively correlated with the first cluster).

The SD group exhibited more moderate correlations with greater variability, suggesting a more flexible parameter structure. Notable correlations included a strong positive relationship between , d, and (correlation coefficients ), and between , , and (correlation coefficients ).

Interestingly, showed significant correlations with multiple parameters in the SD group, while it remained largely independent in the PD group.

These distinct correlation patterns suggest fundamentally different dynamics in the underlying biological processes between PD and SD patients, particularly in the relationship between immune response and drug efficacy parameters.

PFS correlations

Analyzing the entire sample, we observed a significant inverse relationship between the difference of predicted tumour size at day 56 and baseline, and Progression-Free Survival (PFS) (Pearson correlation: ; ).

However, at the individual patient level, the model was unable to accurately predict PFS, suggesting that while the aggregate trend was statistically significant, the predictive power diminished when applied to single patient trajectories.

Discussion

Main aspects and findings

This study’s strengths include the use of prospectively collected clinical trial data, comprehensive biomarker measurements, and sophisticated mathematical modeling with AI-assisted optimisation. The successful model convergence across all patients and concordance with clinical outcomes validate the technical approach.

The application of mathematical model of metronomic chemotherapy response has revealed several key insights into treatment effectiveness and limitations in metastatic gastrointestinal cancer.

The model successfully captures tumour dynamics at 56 days with remarkable accuracy (correlation between predicted and observed : for SD and for PD patients). This technical achievement supports the utility of our mathematical framework for describing short-term tumour behaviour under metronomic schedule of anticancer therapy.

A noteworthy methodological aspect concerns the application of Gompertz-type growth in the first equation of the model. While this modelling approach is traditionally used to describe the growth of a single tumour mass, in our case p represents the sum of metastatic lesion diameters according to RECIST criteria. The predictive success of the model in this context suggests that the aggregate dynamics of metastases effectively follow a Gompertz – type pattern, despite potentially involving lesions at different growth stages.

This finding has several interesting implications: first, it suggests that growth control mechanisms (angiogenesis and immune response) act sufficiently uniformly at the systemic level; second, it supports the model’s robustness even when applied to aggregate measurements; third, the ability of the considered model to accurately fit clinical data offers a potential mathematical explanation for the utility of the RECIST approach. This suggests that using the sum of target lesion diameters reflects modellable biological tumour dynamics, rather than being a purely empirical clinical convention.

A crucial finding emerged from the analysis of drug-related parameters (, , ), representing direct antitumour, anti-angiogenic, and immune-stimulatory effects respectively. These parameters consistently showed low values in both PD and SD groups (–0.009, –0.08, –0.09), which raises a possible hypothesis of limited therapeutic impact across all three mechanisms of action. In PD patients, was particularly low (mean 0.0053), potentially indicating minimal direct antitumour effect. Even in SD patients, where treatment appeared more effective, these parameters remained small (mean ).

These observations may be considered in light of clinical evidence regarding metronomic chemotherapy (metroCT) in metastatic colorectal cancer (mCRC). The MOMA trial¹⁶, which compared maintenance with bevacizumab alone versus bevacizumab plus metroCT after induction with FOLFOXIRI plus bevacizumab, did not demonstrate a significant improvement in PFS with the addition of metroCT. Similarly, while the CAIRO-3 study²¹ showed some benefit with maintenance capecitabine plus bevacizumab, it used considerably higher doses (93.8% of the standard dose intensity) than typical metronomic regimens.

Together, these findings could suggest that the biological mechanisms targeted by metronomic approaches may have inherent limitations in mCRC, which could explain both the modest values of drug effect parameters in our model and the lack of substantial clinical benefit observed in randomised trials.

In this perspective, this mathematical model could be applied to other phase II/III studies where metronomic chemotherapy gave evidence of high rate of complete or partial responses and increase of PFS and OS in treated patients such as in metastatic prostate cancer or oral cancer.

Non-constant parameters reflecting biological heterogeneity

An important methodological consideration concerns the interpretation of the model parameters.

While in mathematical modelling parameters are typically treated as constant values, our approach reveals that the optimal parameter strings vary from patient to patient, albeit within a limited range.

Such variability is not a limitation but rather reflects the biological heterogeneity of the disease and individual response patterns, particularly in the metastatic setting. Metastases may have different microenvironments compared to primary tumours, influencing parameters such as (tumour growth) and (drug efficacy). This might explain why treatments effective for primary tumours often fail in the metastatic context.

We therefore performed descriptive statistics on these parameter distributions, characterising both their central tendencies and their variability across patient subgroups. This approach allowed us to identify meaningful patterns in parameter distributions between SD and PD patients, while acknowledging the inherent heterogeneity in biological responses.

Baseline immune system

The specific relevance of (baseline immune level) deserves particular attention in the metastatic context, as metastases typically interact with the immune system differently than primary tumours.

This parameter might provide a biological window into metastatic ‘immunoediting’, suggesting that metronomic therapies might be more effective when combined with immunotherapeutic approaches specifically in patients with certain baseline immune characteristics.

A key methodological aspect of the model optimisation deserves specific attention. While we could optimise the model parameters using the observed p(56) values as a quantitative target, the situation was different for q(56) and r(56). For these variables, representing angiogenesis and immune response respectively, we lacked direct measurements at day 56, as the approximation through VEGF and IL2 values at this time point would not have been biologically plausible.

Therefore, our approach consisted of a two-step validation: first, optimising the model parameters based on p(56) measurements, and then verifying the biological plausibility of the predicted q(56) and r(56) trajectories.

Immune system dynamics and treatment response

The relationship between immune system dynamics and clinical outcomes deserves specific attention.

Our mathematical model reveals a consistent decline in immune response parameter (r(t)) over time in both SD and PD patients, but with important distinctions in their trajectories. SD patients typically begin with higher baseline immune activity (r(0)) and demonstrate a more gradual decline compared to PD patients, suggesting that the absolute level of immune response may be more critical for treatment outcomes than its overall trend.

A key parameter in our model, , explicitly represents the immunomodulatory effect of metronomic chemotherapy—specifically, how the treatment enhances existing immune responses against the tumour. This parameter showed consistently low values (0.05–0.08) across both patient groups, indicating limited ability of the treatment to sustain immune response over time. Despite this limitation, the higher baseline immune activity in SD patients appears to provide a sufficient buffer to maintain effective tumour control. This observation suggests that therapeutic strategies specifically targeting enhancement of , possibly through combination with immunotherapy agents, might improve outcomes particularly in patients with lower baseline immune activity.

Recent advances in computational immunology¹⁴ have revealed that colorectal cancer progression is governed by complex interactions between tumour cells, immune components, and cytokine networks rather than by individual biomarkers. Our mathematical model parameters reflect this complexity, where differences between PD and SD patients emerge not merely as statistical variations but as representations of distinct biological states. The observed variations in parameters such as (tumour growth control) and (immune-tumour interaction) likely capture fundamental differences in immune-tumour dynamics similar to those identified in cluster analyses of tumour microenvironments.

Particularly noteworthy is the parameter , representing immune activation in response to tumour burden, which varies significantly between patient groups. This mirrors findings showing that T-cell and macrophage populations can have opposing effects on tumour progression depending on the specific immune contexture of the patient. Meanwhile, our treatment efficacy parameters (, , – representing direct cytotoxicity, anti-angiogenic effects, and immune stimulation respectively – demonstrate how therapeutic interventions interact differently with tumour cells, angiogenesis, and immune responses across patient subgroups.

General limitations

This study represents a proof-of-concept demonstration of the Schättler mathematical framework’s first application to clinical metronomic chemotherapy data. As such, the primary objective was to establish feasibility and generate hypotheses rather than provide definitive clinical predictions. The parameter estimates presented should be interpreted as exploratory values derived from initial feasibility analysis, and comprehensive validation on independent external cohorts represents essential future work before any clinical implementation can be considered. Limitations include the single-center design, which may limit generalizability to different patient populations and treatment protocols; the relatively small sample size and the lack of an independent validation cohort, restricting subgroup analyses and limiting validation of the proposed threshold; the 12-parameter model, which approaches identifiability limits with strong correlations (mean ) and high relative errors for some parameters; the phenomenological treatment function u (Section Key model adaptations), which does not capture patient-specific pharmacokinetic variability; the use of VEGF and IL-2 as imperfect proxies for angiogenic and immune dynamics; and moderate correlation between model predictions and long-term progression-free survival (), suggesting that factors beyond the model scope influence progression. Additionally, the proposed threshold requires prospective validation and development of baseline predictors before clinical implementation. However, it is important to emphasize that the 12-parameter structure reflects the established minimally parameterized Schättler framework (reference [20]) rather than an arbitrary modeling choice. This framework represents a careful balance between biological realism and mathematical tractability, extensively validated in theoretical studies. The identifiability challenges observed in our analysis stem primarily from the sparse clinical data structure (30 patients, single primary evaluation timepoint) rather than excessive model complexity. An important distinction must be explicitly recognized between population-level and patient-level prediction. This work establishes that the Schättler framework successfully captures population-level tumour response patterns, as evidenced by excellent agreement between model predictions and observed tumour dynamics (median squared error: 0.003 for SD, 0.007 for PD at day 56). This represents the appropriate validation target for proof-of-concept work demonstrating initial feasibility of the mathematical framework. However, accurate patient-level predictive performance—specifically, reliable individual PFS prediction—represents a fundamentally different challenge. Such individual-level predictions would require incorporating patient-specific clinical features (comorbidities, comprehensive biomarker panels, performance status, genetic profiles) beyond the current population-level mathematical structure, as well as larger cohorts, independent external validation, and prospective clinical studies. These represent essential directions for future work. The parameter estimates should be interpreted as exploratory values from proof-of-concept analysis, requiring validation on larger, independent cohorts before clinical implementation.

Limitations of tumour size in predicting clinical outcomes

The relationship between early tumour dynamics and clinical outcomes proved more complex than initially hypothesised.

While we found a significant inverse correlation between and PFS in the overall population (, ), this relationship weakened when examining PD and SD groups separately ( for PD; for SD). This suggests that early tumour dynamics alone may not be sufficient to predict long-term outcomes in individual patients, particularly in the metastatic setting.

It is worth noting that the groups PD and SD show dynamics that are distinct but not as dramatically different as their clinical outcomes might suggest. This indicates a biological “grey zone” where small parametric variations can lead to divergent clinical trajectories, supporting the notion that response to metronomic chemotherapy in metastatic context may be a threshold phenomenon, with critical parameter values beyond which therapeutic failure occurs.

Such limitation could be due to various factors, including biological heterogeneity of the disease, variability in treatment response, and the complexity of progression mechanisms.

Moreover, this limitation might be intrinsically related to the challenges in PFS assessment and interpretation, as highlighted in scientific literature. As discussed by Bates⁶, PFS remains ‘the endpoint we love and love to hate’, with multiple sources of error that can introduce missing or incomplete data in clinical trials. Bruno and colleagues⁹ point out that the determination of progression is susceptible to subjective factors that can influence the exact time when disease progression is recorded. The 20% increase in tumour size that defines progression has never been proven to actually indicate treatment failure, as this boundary may be crossed despite ongoing tumour growth delay.

This inherent complexity in PFS assessment and prediction is further supported by scientific work on classical mathematical models in oncology. Ghaffari Laleh and coworkers¹¹ analysed six consensus mathematical models on a large dataset of patients undergoing immunotherapy and chemotherapy, showing that even well-established models face significant challenges in predicting individual patient trajectories. In particular, they found that while these models could adequately fit overall population-level patterns, their predictive accuracy for individual patients was more limited, especially for patients with fluctuating responses.

Intrinsic limitations of model optimisation

Interestingly, our finding that increasing the number of random strings in the parameter optimisation does not substantially improve the results suggests that the model has reached an intrinsic limit in its predictive capacity, independent of computational optimisation. This indicates that the residual uncertainty is not due to methodological limitations but reflects the intrinsic biological variability of metastatic processes.

The methodological approach shows interesting parallels with the work of Wang and colleagues²⁷ in NSCLC, where the authors developed a model linking tumour size changes at 8 weeks with survival. Similar to our findings, while their model showed good predictive ability at the population level, individual prediction remained challenging.

However, unlike our study in metastatic gastrointestinal cancer with metronomic chemotherapy, Wang and colleagues²⁷ could validate their model on a large multicenter dataset. Their work highlighted how tumour change at 8 weeks (PTR) can be a useful early biomarker for drug development decisions, but with limitations similar to those we found in our study regarding individual prediction. This concordance suggests that the challenge of individual clinical outcome prediction might be intrinsic to cancer biology, regardless of tumour type and therapeutic regimen.

These findings suggest a need to reconsider the role of metronomic chemotherapy in gastrointestinal cancer.

• Its use might be better suited for maintenance therapy rather than active treatment.

• Combination with other therapeutic approaches might be necessary.

• Better predictive biomarkers are needed to identify potential responders.

• Alternative dosing strategies should be explored to enhance the values of , , .

Future studies should:

1. validate these findings in larger patient cohorts;

2. explore alternative dosing strategies to optimise drug-related parameters;

3. investigate potential biomarkers for patient selection;

4. consider combination approaches that might enhance the therapeutic impact;

5. develop more sophisticated mathematical approaches to integrate multiple biomarker data into single model variables.

In conclusion, this proof-of-concept work successfully captures tumour dynamics and identifies population-level patterns, while also revealing the exploratory nature of current parameter estimates and the challenges in individual patient outcome prediction.

This mathematical model, though currently limited in predictive power at the individual level, establishes a methodological foundation that bridges population-level patterns with individual-specific parameter distributions.

As such, it represents an initial step in the path towards precision medicine approaches in metastatic gastrointestinal cancer, where treatment strategies might eventually be tailored based on patient-specific tumour dynamics and microenvironmental characteristics.

Overall, this work provides both a mathematical framework for understanding treatment response and important insights for optimising therapeutic strategies in metastatic colorectal cancer.

Author contributions

M.L.M. and A.M. contributed equally to this work. M.L.M. conceived the study, designed the analytical approach, performed statistical analyses and biomarker evaluations, interpreted results, coordinated the AI-assisted analysis, and wrote the manuscript. A.M. developed the mathematical framework, adapted the Schättler model for metronomic therapy, implemented optimisation algorithms, conducted mathematical validation, and contributed to manuscript writing. G.B. provided expertise in metronomic chemotherapy mechanisms, contributed to study design and clinical parameter interpretation, and supervised biomarker analyses. P.O. provided clinical oncology expertise, contributed to patient data interpretation and clinical parameter determination, and assisted with manuscript review. G.A. designed and led the COMET clinical trial, provided all clinical data and patient cohort characterization, and contributed to clinical interpretation of results. R.A.K. and G.F. provided scientific oversight, contributed to study conceptualization, supervised the research framework, and contributed to manuscript preparation and review. V.G. provided advanced mathematical modeling expertise, contributed to theoretical framework development and mathematical validation, and assisted with complex optimisation strategies. All authors participated in results interpretation, read and approved the final manuscript.

Funding

A.M. gratefully acknowledges support from the David Flanagan funds at the AUBG, American University Bulgaria, and the Alexander von Humboldt Foundation, Germany. G.F. was funded by an SC1 SCORE grant from NIH (5SC1GM136630). V.G. was partially supported by Gruppo Nazionale per l’Analisi Matematica, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and by Top Global University Project, Waseda University.

Data availability

Individual participant data from the COMET trial (EudraCT: 2007-000065-38) are not publicly available due to privacy and ethical considerations under European Union General Data Protection Regulation (GDPR). De-identified datasets generated and analyzed during the current study may be made available from the corresponding author upon reasonable request, subject to approval by the institutional review board and execution of appropriate data sharing agreements. Requests should include a detailed research proposal outlining the intended use of the data, qualifications of the research team, and measures to ensure participant privacy protection. Additional study documents including the clinical trial protocol and statistical analysis plan may be available upon reasonable request. Data requests will be reviewed within 4-6 weeks of submission.

Computer code

The custom computer code developed for this study includes: (i) R scripts implementing the adapted Schättler mathematical framework for metronomic chemotherapy modeling; (ii) parameter optimisation algorithms using L-BFGS-B with Monte Carlo initialization; (iii) numerical integration routines for the system of ordinary differential equations with adaptive step-size control; (iv) statistical analysis pipelines for biomarker evaluation and correlation analysis; and (v) data visualization scripts for generating all figures presented in this manuscript. The underlying code for mathematical modeling, parameter optimisation, and statistical analyses used in this study is not publicly available but may be made available to qualified researchers on reasonable request from the corresponding author. Requests should include a detailed research proposal outlining the intended use of the code, institutional affiliation, and evidence of appropriate computational resources and expertise to utilize the methods effectively. All analyses were performed using R version 4.5.0 with the following key packages: deSolve for differential equation solving, optimx for parameter optimisation, and standard statistical packages for data analysis and visualization. A representative R script demonstrating the calculation of tumour dynamics and theoretical progression-free survival using pre-optimized parameters is provided as supplementary material (Supplementary File S1: calculate_tumour_dynamics.R). This script implements the Schättler model Eqs. (1-3) and the treatment function u(t) as described in Section Key model adaptations.

Declarations

Competing interests

The authors declare no competing interests.