Table 1 Different types of mathematical modeling approaches and example applications in radiation oncology

Type of differential equations

ODE

Gompertzian growth model integrated with the LQ model for the dynamics of tumor growth and response to radiation [23]

PDE

Oxygen depletion and reoxygenation for FLASH [24]

IDE

Dose calculation utilizing the Boltzmann transport Eq. (25)

SDE

Tumor responses to RT or other radiobiological processes considering dynamics effects [26, 27]

Uncertainty

Deterministic

Deterministic TCP models [28, 29]

Stochastic

Markov chains to model the random nature of cellular responses to radiation in TCP [29, 30]

Type of variables

Continuous

LQ model in radiobiology

Discrete

Agent-based models for tissue mechanics [31] or multiscale diffuse interface models for 3D non-linear tumor growth [32]