# Ribba et al., (2012). A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy.

March 2014, model of the month by *Maciej Swat*

Original model: BIOMD0000000521

The paper by Ribba et al. (2012) [1, BIOMD0000000521] exemplifies a remarkable example of a model developed in close collaboration between modellers and clinicians for which the data has been collected for over 10 years. At the time of publication, it was the first model that successfully described the time course of tumour growth inhibition for patients with low-grade gliomas (LGG) as consequence of chemotherapy and radiotherapy.

The model is based on the observation that after a termination of PCV chemotherapy, LGGs often continue to shrink in volume for an extended period of time, ranging from months to years. The hypothesis explaining this phenomenon assumes a certain delay in the action of chemotherapy on non proliferating cells. This is in line with the known cell-cycle non-specific mechanism of action of the PCV regimen related agent.

PCV stands for a drug cocktail, i.e. a mixture of three chemotherapeutic components:

- drug 1: procarbazine
- drug 2: 1-(2-chloroethyl)-3-cyclohexyl-l-nitrosourea
- drug 3: vincristine

Figure 2The prediction model consists of a coupled pharmacokinetic model (PK) and tumour growth model implemented as a system of ordinary differential equations (ODE) and an algebraic equation. Figure taken from [1].

Figure 4Observed MTD with model prediction using subject specific parameter values for three typical patients for each treatment type. Figure taken from [1].

Figure 1The figure shows typical MTD curves, which are characterised by 4 different phases: phase of slow growth of tumour before the treatment, MTD decrease as the result of the treatment begin, a prolonged phase of decrease after treatment termination and final regrowth phase. Figure taken from [1].

Figure 3The schematic view of the mathematical model. See symbol explanation in text. Figure taken from [2].

Based on this hypothesis, a tumour growth inhibition (TGI) model Figure 1 was formulated which deals with quiescent and proliferating tumour tissue as variables. The latter one is assumed to be the primary target to the treatment. After the successful application of this model to PCV patients, it has been also applied to patient data from temozolomide (TMZ) and radiotherapy (RT) related trials. 24 PCV patients datasets have been collected between 1994-2005, 24 TMZ patients datasets between 2000-2005 and 25 RT datasets in the same period of time. The main readout obtained during the treatment was the mean tumour diameter (MTD) in [mm] Figure 2, estimated from MRI pictures.

To analyse the data, a non-linear mixed effect model (NLME) was applied which allows parallel analysis of all patients simultaneously providing both the population and individual parameter estimates. The model prediction part consists of:

- pharmacokinetics model (PK) expressed with one ODE
- tumour growth model implemented as a system of ordinary differential equations (ODE); one equation for each of the considered tissues: proliferating cells, P, nonproliferative tissue, Q, and damaged quiescent cells, QP
- total size of the tumour, expressed as one algebraic equation, see Figure 3

All model parameters and two unknown initial conditions are assumed to be log-normally distributed. Note that the SBML encoded model cannot, so far, store such distribution information. Also any trial design relevant information (dosing times and amounts, arm size, observation data) usually necessary to perform a population analysis is out of scope.

The model successfully reproduces Figure 4 the tumour size dynamics in LGG patients under two different chemotherapy regimens and radiotherapy. It handles both the slow growth of LGGs and the prolonged response of the tumour upon termination of the treatment.

## Bibliographic references

- Ribba et al. A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. Clin Cancer Res. (2012) Sep 15;18(18):5071-80.