Environmental stress level to model tumor cell growth and survival

Abstract: Survival of living tumor cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the "environmental stress level". It is an immeasurable auxiliary variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to quantify uncertainties and calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. The calibration results of both models are compared and the quality of fit is quantified. While predictions of both models show good agreement with the data, there is indication that the model considering the stress level yields a better fitting. The proposed modeling approach offers a flexible and extendable framework for considering systems with additional environmental factors affecting the cell dynamics.

• The paper introduces an environmental stress level (ESL) framework that integrates multiple environmental factors into tumor cell growth and survival modeling.
• The study employs a robust mathematical framework using ODEs, Bayesian inversion, and Sequential Monte Carlo calibration on liver cancer cell data.
• Analysis reveals that the ESL model, while matching traditional nutrient-based scaling under simple conditions, offers extensibility for multifactor environmental influences.

Environmental Stress Level Modeling of Tumor Cell Growth and Survival

Introduction

The paper "Environmental stress level to model tumor cell growth and survival" (2201.06985) advances mathematical modeling for in vitro tumor cell dynamics by formalizing the concept of an "environmental stress level" (ESL) as an auxiliary, immeasurable variable. This construct provides a principled framework to collectively quantify the influence of multiple environmental factors—such as oxygen, nutrient saturation, drug concentrations, and mechanical forces—on cell proliferation, death, and movement. The work rigorously contrasts the ESL-based approach with conventional rate scaling based on nutrient availability. Model calibration leverages Bayesian inversion and Sequential Monte Carlo (SMC), capitalizing on time-resolved population data from liver cancer cell cultures across varied nutrient saturations.

Mathematical Framework

The study formalizes three ODE systems:

• Model $\mathcal{M}_{\text{opt}}$: Tumor cell growth under optimal nutrients, employing generalized logistic growth with capacity $K$, proliferation rate $\rho$, death rate $\lambda$, and inhibition parameter $m$.
• Model $\mathcal{M}_S$: Nutrient-dependent cell dynamics with proliferation and starvation rates directly scaled by nutrient saturation $S$. Hill-type influence functions $\delta^{+}(S)$ and $\delta^{-}(S)$, parameterized by a threshold $\kappa$, mediate rate modulation.

  Figure 1: Qualitative behavior of the Hill type scaling functions~$\delta^{+}(S)$ and~$\delta^{-}(S)$.

• Model $\mathcal{M}_\eta$ (ESL-based): Proliferation and death rates are scaled via the dynamic environmental stress level $\eta(t)$, which evolves in response to environmental variables. Here, only nutrient saturation is considered explicitly, leveraging the modular ESL ODE to encapsulate recovery and stress induction via sensitivity parameter $\alpha_\eta$ and Hill functions as before.

Through theoretical investigations and mathematical analysis, all models guarantee positivity and boundedness of solutions (steady states and stability are characterized for all models), supporting biological feasibility.

Data Acquisition and Noise Modeling

Experimental calibration used fluorescence-based cell viability assays, correlating signal intensity to viable cell density, with datasets spanning five FBS concentrations. Calibration integrates multiplicative noise, validated empirically as gamma-distributed, ensuring realism in measurement uncertainty and protecting against spurious negative values.

Figure 2: Left: Probability density function $f_\varepsilon$ of $\varepsilon\sim\Gamma(1/\sigma^2, 1/\sigma^2)$ for different $\sigma^2$ and illustrative percentiles; Right: 90% uncertainty range around a sample model solution.

Bayesian Inversion and SMC Calibration

Bayesian parameter estimation is framed under comprehensive priors for all biologically relevant parameters, aligning hyperparameter choices with prior biological knowledge. SMC is employed to sample the complex posterior induced by sequential, time-resolved cell viability measurements. The process cycles through time slices and initial seeding densities, compositing evidence across 480 measurements for robust inference.

Figure 3: Schematic description of data utilization (D1–D5) for incremental SMC model calibration.

Statistical Validation and Model Selection

Calibration outputs include marginal posteriors and associated variances for all parameters. ESL model $\mathcal{M}_\eta$ introduces an additional sensitivity parameter $\alpha_\eta$, exhibiting substantial variance due to limited identifiability—primarily impacting model response in the earliest timepoints or significant environmental transitions. Quality of fit is measured using cumulative distribution-based validation metrics, complemented by evidence-based model selection (Bayes factor). The Bayes factor (log-scale) generally trends in favor of the ESL model, especially in early calibration phases, despite increased model complexity.

Figure 4: Logarithm of the Bayes factor $\mathcal{Z}_{\eta:S}$ over sequential SMC calibration steps. Error bars indicate 95% confidence intervals across SMC replicates.

Numerical Results and Analysis

Both ESL-based and direct-scaling models fit the experimental data closely, with overlapping or indistinguishable parameter posteriors outside ESL-specific parameters. Deviations in calibration are primarily attributable to measurement anomalies at particular time points ($t \approx 3$ days), generating outliers and large uncertainty contributions. Model solutions remain consistent, with validation metrics approximating unity across datasets.

Figure 5: Time evolution of the average model solution $V$ for each initial density $V_0$ compared to the scaled data median.

The stress sensitivity parameter $\alpha_\eta$ is found to be large, resulting in rapid attainment of steady-state $\eta$ (typically within one day), effectively reducing the dynamic ESL model to steady-state scaling analogous to direct rate modulation.

Figure 6: Time evolution of the environmental stress level $\eta(t)$ for various nutrient saturations, illustrating rapid convergence to steady-state.

Correlations among model parameters are elucidated via scatter plots: strong positive correlation between proliferation and natural death rates, and moderate negative correlation between carrying capacity and logistic inhibition parameter, confirming model structural expectations.

Figure 7: Marginal and pairwise posterior distributions for parameters, color-coded by model; selected pairwise scatterplots and regression lines for parameters with significant correlation.

Model Validation and Limit Analysis

Under optimal nutrient conditions, steady-state analysis and comparison with extended-duration data validate the model's reliability across growth regimes. Both ESL and direct-scaling approaches maintain predictive accuracy, though neither captures population overshoot phenomena seen in the biological data, exposing limitations inherent to logistic-based models without explicit time-lag or resource-sensing mechanisms.

Figure 8: Time evolution of the average model solution $V$ for the "limit model" $\mathcal{M}_{\text{opt}}$ versus median scaled validation data over 21 days.

Implications and Future Directions

The ESL framework, though only modestly superior in the current nutrient-only scenario, provides a structurally modular and extensible approach for integrating additional environmental influences—e.g., oxygen tension, drug exposure, mechanical stress—within a single variable. This enhances model interpretability and parameter inference, especially as multi-factor and spatial complexities are introduced (partial differential equation extension for motility and chemotaxis-like behaviors). The methodological infrastructure is readily adaptable to next-generation experimental systems, such as organ-on-a-chip platforms, affording realistic, controllable microenvironments for model validation and calibration.

Conclusion

The environmental stress level approach offers a mathematically rigorous, data-calibrated formalism for collective modeling of environmental influences on tumor cell dynamics. While simple experimental conditions yield nearly equivalent descriptive quality to direct scaling, the ESL stands out for theoretical and practical extensibility. As experimental sophistication grows and environmental complexity increases, the ESL methodology is poised to provide critical insights and flexible model architectures for predictive oncology and systems biology.