Mathematical and physical systems biology: application to pharmacokinetic drug-drug interactions and tumour growth

In this thesis, a multi-scale approach is provided to a pharmacokinetic and a pharmacodynamic problem. The first part of this research provides a realistic mathematical physiological model of the liver to predict drug drug interactions (DDIs). The model describes the geometry of a lobule (liver unit) and integrates the exchange processes, diffusion and active transport, between the hepatocytes and the blood and possible drug-drug interactions such as; reversible inhibition, mechanistic based inhibition (MBI) and enzyme induction. The liver model is subsequently integrated into a PBPK model with 7 compartments (artery blood, venous blood, gut, liver, kidney, lung, rest of the body). To assess the efficiency of the model to predict DDIs, 77 clinical DDI studies were compared to the model. These 77 clinical studies represent 5 victim drugs (midazolam, simvastatin, triazolam, cerivastatin and nifedipine) and 30 perpetrator drugs. The reversible inhibition, MBI and induction parameters for the majority of the perpetrators were estimated with in vitro experiments and adjusted for the human liver size. The PK parameters, such as clearance and absorption rate, and the physiological parameters were obtained from the literature. The DDIs were measured as the ratio of the AUC (Area Under the Curve of the blood concentration) or the ratio of the maximum concentration Cmax of the victim drug administered with and without the perpetrator drug. The predicted ratios were compared with the clinical observation by calculating the geometric fold error GMFE. The GMFE for AUCratio and Cmax,ratio were calculated to be 1.54 and 1.34, respectively. Moreover, the PBPK model excluding the gut compartment under-predicts both inhibition (lower AUCratio) and induction (higher AUCratio) which strongly suggests that the gut DDI component can not be neglected for accurate clinical prediction. However, the static combined model by Fahmi et al. [1, 2] without the gut component fortuitously predicts the clinical AUCratio better than inclusion with the gut component. To conclude, the model predicts DDIs relatively well as it is in the lower range of errors reported in the literature (1.47-2.00 [1, 2]). Moreover, the model is able to predict the pharmacokinetics of drugs and provides a dynamic description of the DDIs, such as the enzyme level and spatial distribution within a lobule. Furthermore, the perpetrator dose regimen can be changed or the error in the in vitro parameters can be integrated to observe their influences on the AUC ratio. The second part of this research explored the Warburg effect in a avascular tumour growth model incorporating a cell shedding term to account for tumour shrinkage. The tumour model was based on an extension of the Ward and King model [3], where two sub-populations; living cells and dead cells are considered. Three diffusion equations for glucose, lactate and the drug are considered and included into the model for growth rate, natural death rate and a death rate due to the drug. The simulation of the model without a drug shows similar behaviour to the original model by Ward and King despite the presence of the shedding term and predicts an extracellular pH of 6.8. However, when a drug treatment is added, the model is able to simulate the shrinkage of the tumour unlike the original model. Moreover, two scenarios with a basic, neutral and acidic drug were explored, assuming similar efficiency at physiological pH to assess the effect of changes in the extracellular pH. Acidic or weak base drugs seem to be more efficient in low pH environment as the fraction of neutral form is greater and therefore more drug is available to cross the cell membrane to reach its target