Oxygenation and spatial heterogeneity shape radiotherapy protocol ranking through phenotypic adaptation

The gist Oxygenation and spatial heterogeneity shape radiotherapy protocol ranking through phenotypic adaptation.

Model Framework

The study develops a phenotypically structured partial differential equation (PS-PDE) model to integrate spatial oxygen dynamics with continuous phenotypic adaptation to hypoxia and radiation, allowing for a systematic comparison of radiotherapy protocols under various constraints. The cell dynamics are described by the PS-PDE where the reaction term models proliferation influenced by oxygen availability and phenotypic adaptation. Radiotherapy response is modeled using a phenotype- and oxygendependent extension of the linear–quadratic (LQ) formulation, where parameters depend on both oxygen concentration and phenotype. The biologically effective dose (BED) is defined as BED(P) = Nf d / (1 + d/((α/β)H)), which is used to constrain protocols under a common normal-tissue toxicity constraint.

Uniform Oxygenation Effects

Under spatially uniform oxygen supply, the analysis reveals that alternative fractionation schedules provide little improvement over standard-of-care protocols in normoxic conditions. However, under moderate hypoxia, a distinct class of protracted schedules with longer inter-fraction intervals substantially increases time-to-progression, in some cases by up to twofold. This regime-dependent benefit is consistent with a shift in the balance between reoxygenation and selection for resistant phenotypes.

Phenotypic Adaptation and Reoxygenation

The interplay between reoxygenation and phenotypic selection is crucial, as the response to radiotherapy is shaped by this interaction. Under normoxic conditions, the maximal-TTP protocol induces only a limited phenotypic shift, while the standard-of-care protocol promotes tumor reoxygenation while delivering more densely spaced radiation fractions, thereby redistributing the population toward higher-resistance trait values. Under hypoxic conditions, this compensation mechanism weakens, leading to earlier relapse of the standard-of-care protocol after reoxygenation.

Impact of Toxicity Constraints and Intermediate Oxygenation

The therapeutic efficiency frontier is identified as the subset of protocols achieving at least 80% of the maximum predicted TTP. The analysis shows that the benefit of alternative fractionation schedules is not a monotone function of oxygen availability, with gains being non-monotone and reaching their largest values in the intermediate hypoxic range. This identifies moderate hypoxia as the regime where fractionation and timing have the largest impact.

Role of Spatial Heterogeneity

The assumption of spatially uniform supply neglects geometric heterogeneity, and identical radiotherapy schedules may produce substantially different timesto-progression depending on the geometry of oxygen delivery. The Gini coefficient is used as a compact descriptor of spatial heterogeneity, where Gini(m) = 0 corresponds to a perfectly uniform oxygen source. Spatial heterogeneity induces a marked increase in variability, and in hypoxic conditions, for highly heterogeneous configurations (large Gini coefficient), the gain can decrease significantly and may even become negative.

Conclusion

Overall, the results indicate that intermediate hypoxia is the regime in which fractionation and timing have the largest impact, as oxygen is sufficiently low to reshape phenotypic selection but not so low that radiation efficacy is globally suppressed. The framework provides a mechanism by which effective radiosensitivity and fractionation response can vary across oxygenation regimes.

How it works

The PS-PDE model couples tumor-cell dynamics, oxygen diffusion–consumption, and oxygen- and phenotype-dependent radiotherapy response. Cell density evolution is governed by the equation ∂tn(t, x, u) = Dp ∂2un(t, x, u) + Ds ∂2xn(t, x, u) + R(t, x, u) − T(t, x, u). The reaction term R(t, x, u) defines the net proliferation rate of the tumor cell population as a function of oxygen availability and phenotypic adaptation.

Radiotherapy is modeled through T(t, x, u) = α(O(t, x), u) d + β(O(t, x), u) d2, where the coefficients α and β depend on oxygen concentration and phenotype. The OER formulation rescales the delivered dose as d/OER(O), which increases with oxygenation, effectively modulating radiosensitivity.

Key Findings Summary

1. Uniform oxygen supply leads to a relatively flat TTP landscape in normoxia but a structured high-performance region under moderate hypoxia.

2. The ranking of fractionation schedules changes with oxygenation, with hypoxia making timing substantially more relevant.

3. Phenotypic adaptation involves a trade-off where phenotypes less resistant to radiotherapy and less adapted to hypoxia exhibit higher proliferative potential.

4. The benefit of alternative fractionation schedules is non-monotone with respect to oxygen availability, peaking in the intermediate hypoxic range.

5. Spatial heterogeneity introduces substantial variability in TTP gains, meaning protocols selected under uniform assumptions can be compromised by unfavorable spatial arrangements.

Implications for Protocol Selection

The study suggests that "geometry-aware selection may reduce the variability of TTP gains across heterogeneous configurations and, more importantly, may help exclude spatial regimes in which protocols selected under uniform assumptions yield negative gains relative to the SoC". The model offers a mechanism where oxygen acts both directly through radiosensitivity modulation and indirectly through the selection of phenotypes with different proliferative and radiotherapy-response traits. Future developments may incorporate optimal control formulations in which fractionation schedules are optimized under explicit tumor control and normal-tissue objectives.

Model Component Analysis

The model component analysis reveals that the structure of the therapeutic efficiency frontier is modulated by tissue tolerance, where Under severe hypoxia, protocol selection becomes strongly governed by normal-tissue responsiveness. The extension to intermediate oxygenation levels clarifies that the benefit of schedule adaptation is non-monotone with respect to oxygen availability.

Supporting Information Details

The simulation details specify a spatial domain of Lx = 2 cm and a detection threshold ΓRT = 1.3 · 108 cells. The parameter estimates are informed by empirical literature, and the crowding coefficient κ is calibrated such that r0 + γO - κK = 0, where K is the upper-bound carrying capacity. The Gini coefficient quantifies spatial polarization of oxygen supply, with Gini(m) = 0 corresponding to a perfectly uniform oxygen source. The supporting figures provide complementary material on the influence of oxygen concentration and phenotypic state on radiosensitivity parameters. The analysis of schedule-domain robustness shows that qualitative conclusions are robust to the choice of explored inter-fraction interval range.

References

[1] Hanahan D and Weinberg RA. The hallmarks of cancer. Cell 2000; 100:57–70.

1. Phenotypic Adaptation Modeling The AI system can now incorporate continuous phenotypic adaptation to hypoxia and radiation by modeling cell density as a function of a continuous trait variable (u), allowing it to simulate how phenotypes select from fully hypoxia- and radiation-sensitive cells (u = 0) to fully resistant ones (u = 1).

2. Spatial Heterogeneity Analysis The system can evaluate the impact of oxygen distribution geometry by using a phenotypically structured partial differential equation (PS-PDE) model where the oxygen concentration O(t, x) is spatially dependent, and it can quantify this heterogeneity using a Gini coefficient G(m) to determine if the geometry of oxygen delivery rather than to changes in the overall oxygen supply alters outcomes.

3. Regime-Dependent Protocol Selection The AI can identify that treatment benefit is not constant, enabling it to select protocols based on the environment, as indicated by findings that intermediate hypoxia emerging as the regime in which fractionation and timing have the largest impact.

4. Toxicity Constraint Integration The system can systematically compare protocols under various constraints by enforcing a common limit, such as requiring BED(P) ≤ BED(PSoC), allowing it to test alternative fractionation schedules under a common normal-tissue toxicity constraint.

5. Geometric Robustness Assessment The AI can assess the reliability of protocol selection by evaluating gains in heterogeneous settings, noting that the uniform-source frontier remains qualitatively informative but that spatial heterogeneity introduces substantial variability in treatment outcomes, suggesting it can flag configurations where protocols selected under simplified assumptions yield negative gains relative to the SoC.

Tumor response to radiotherapy is strongly influenced by oxygen availability and phenotypic heterogeneity, yet their combined impact on the relative performance of fractionation schedules remains unclear. Here, we develop a mathematical model that integrates spatial oxygen dynamics with continuous phenotypic adaptation to hypoxia and radiation, and use it to systematically compare radiotherapy protocols under a common normal-tissue toxicity constraint. Under spatially uniform oxygenation, we find that alternative fractionation schedules provide little improvement over standard-of-care protocols in normoxic conditions. Under moderate hypoxia, however, a distinct class of protracted schedules with longer inter-fraction intervals substantially increases time-to-progression, in some cases by up to twofold. This regime-dependent benefit is consistent with a shift in the balance between reoxygenation and selection for resistant phenotypes. When oxygen delivery is spatially heterogeneous, treatment outcomes depend strongly on the geometric organization of oxygen sources. Even with identical total oxygen supply, different spatial configurations lead to large variability in time-to-progression and can alter the relative ranking of radiotherapy protocols. These results show that radiotherapy effectiveness is not an intrinsic property of a treatment schedule alone, but emerges from its interaction with tumor microenvironmental structure and evolutionary dynamics. Incorporating both spatial heterogeneity and phenotypic adaptation may therefore be important for the consistent evaluation and design of fractionation strategies in heterogeneous tumors.