ESTRO 2024 - Abstract Book

S4986

Physics - Radiomics, functional and biological imaging and outcome prediction

ESTRO 2024

We present a hierarchical Bayesian model to predict the loss in eGFR relative to baseline at a time, t, months following treatment. The model considers the effects of age, healthy kidney volume and mean BED (Biologically Effective Dose) to the healthy kidney. A hierarchical model is used to allow each patient to have a specific values of model parameters which are in turn drawn from global distributions representing the effect of each clinical variable. This technique also allows for the incorporation of systematic uncertainties such as those in the α/β ratio.

Material/Methods:

The model was trained using 30 patients with primary RCC (Renal Cell Carcinoma) treated at a single institution. Patients were treated with either 26Gy/1# or 42Gy/3# using either Cyberknife or a linear accelerator. The patients had between 6- and 42-months follow-up. The model assumed that following treatment with radiotherapy the eGFR subsequently declines over time. We described this with the relationship,

eGFR(t) = eGFR0(A + (1-A)exp(-Bt))

where t is the time since treatment in months and eGFR0 is the baseline value prior to treatment. A is the asymptotic value of the eGFR, relative to baseline, a long time after treatment and should be between 0 and 1. B is the rate at which the eGFR declines and must be a positive value. A and B were in turn modelled as a function of patient age, healthy kidney volume and mean BED to the healthy kidney. A and B were described by the sigmoid function

F(age, mean BED, volume) = n/(1–exp(β0 + age β1 + volume β2 + mean BED β3))

For A, n was set to one, enforcing the requirement that eGFR cannot increase in the long run. For B, n was set to two as preliminary analysis suggested this was far larger than any plausible rate coefficient.

Figure 1 shows the structure of the realised model. The prior distributions for the regression coefficients were represented by normal distributions. The parameters A and B were given Gamma distributed priors as they are constrained to be positive. A uni-modal parameterisation was used as in Wang et al. [3], to guarantee ‘U’ shaped posteriors were not possible. The mode was set by the sigmoid term. The variance was assigned a beta distributed prior. This parameterisation enforces a positive modal value for A but permitted individual samples to exceed one, this was necessary as individual data did exceed the baseline value and needed to be explicable by the model. Three patients with missing baseline eGFR data were included in the analysis with values imputed as Gamma distributed random variables with mean and variance estimated from subsequent measurements. The posterior densities were sampled via MCMC (Markov Chain Monte Carlo) using PyMC5 [4].