Exploring a model-based analysis of patient derived xenograft studies in oncology drug development

Xenograft data

Data from a study by Gao et al. (2015) which analysed 1000 PDX models across 59 treatments and controls was collected. The data was then grouped creating 59 two arm studies involving a control and treatment arm (hence a ‘study’ refers to a different treatment where, within that study, there was 2 arms, a treatment and control arm). This data was then truncated to either 14, 21 or 28 days, to mimic typical lengths of xenograft studies, for analysis.

Empirical approach: un-paired, two-tailed T-test

On the final day of the study, day 14, 21 or 28, an un-paired, two-tailed t-test between the control and treated volumes was conducted with a p-value and %TGI reported.

Model-based approach: likelihood ratio-test

The time-series from the animals used within the empirical approach were used for the following model-based analysis. By assuming that the tumour is spherical we first converted tumour volume to radius using: ${\displaystyle R={\left(\frac{3V}{4\pi }\right)}^{\frac{1}{3}}}$

where V is the volume of the tumour and R the radius.

Next, we fitted the following model, whose mechanistic derivation can be found in Mistry, Orrell & Eftimie (2018), Orrell & Mistry (2019) and Mayneord (1932) and in the Supplemental Information, to the combined two arm data-set:

${\displaystyle R_{ij}=\left(a_i+b_i{t}_{ij}\right)\left(1+e_{ij}\right)}$ ${\displaystyle log\left(a_i\right)\sim N\left({\mu }_1,{\sigma }_1^2\right)}$ ${\displaystyle log\left(b_i\right)\sim N\left({\mu }_2,{\sigma }_2^2\right)}$ (1)${\displaystyle e_{ij}\sim N\left(0,{\sigma }_3^2\right).}$

where R_ij is the observed radius of xenograft i at time j, a_i is the value of the radius at time 0, time of randomisation, for xenograft i, b_i is the rate of growth of the radius for xenograft i, t_ij is the time-point for xenograft i at time j at which the observation was recorded and e_ij is the residual error for xenograft i at time j. Note, that a proportional error model was used as the variability in tumour size grew over time i.e., we have heteroscedastic variance (see Supplemental Information for more details). We then modified the distribution of b_i and introduced a population treatment effect parameter c to account for difference between control and treated growth rates in the following way

${\displaystyle b_i\sim N\left({\mu }_2+c{T}_i,{\sigma }_2^2\right)}$ (2)${\displaystyle T_i=\left\{\begin{array}{c}0\text{if control}\hfill \\ 1\text{if treated}\hfill \end{array}\right.}$

and re-fitted the model to the data. Since the models considered are nested the likelihood ratio-test (LRT) was used to assess if adding a treatment effect improved model fit to the data over a model with no treatment effect.

In addition to regressing against radius an analysis regressing against volume was also considered. The model used for the volume analysis was as follows:

${\displaystyle V_{ij}=\frac{4\pi {\left(a_i+b_i{t}_{ij}\right)}^3}{3}\left(1+e_{ij}\right)}$ ${\displaystyle log\left(a_i\right)\sim N\left({\mu }_1,{\sigma }_1^2\right)}$ ${\displaystyle log\left(b_i\right)\sim N\left({\mu }_2,{\sigma }_2^2\right)}$ (3)${\displaystyle e_{ij}\sim N\left(0,{\sigma }_3^2\right).}$

where V_ij is the observed radius of xenograft i at time j. The assessment of treatment effect was conducted in a similar way as stated above.

In addition to using mixed-effects we also conducted a naïve pooled data modelling approach to assess treatment effect i.e., using just fixed-effects. This analysis was conducted using the radius model only.

All parameter estimation for the mixed-effect models was conducted using the saemix library in R. Parameter estimation for the pooled data approach was done using the minpack.LM library in R.