Mathematical modelling

Mathematical models are routinely used to represent biological processes, and computers can be used to simulate the absorption, distribution and elimination of chemical compounds. We offer physiologically-based and population-based models to simulate compound distribution for application in areas such as (but not limited to): simulation of clinical trials, prediction of blood levels following environmental exposure, evaluation of formulation and dosing strategies to improve drug delivery.

Drug/compounds distribution can be quantitatively investigated through computational approaches, utilising data from clinical studies to provide a Top-down description of pharmacokinetics and its variability in populations (i.e population pharmacokinetic modelling, popPK) or integrating drug specific in vitro data in models to predict Bottom-up pharmacokinetics in populations of virtual patients (i.e physiologically based pharmacokinetic modelling, PBPK).

popPK modelling aims to investigate the main PK variables (such as clearance, volume of distribution and constant of absorption) and to identify the sources of variability in concentrations in a population of patients.

This approach allows the characterisation of variability by identifying factors of demographic, anthropometric, physiological, environmental, or drug-related origin that can affect the distribution of drugs and compounds. This is possible through the development of mono or multi compartmental mathematical and nonlinear mixed-effects model that can provide a quantitative evaluation of key PK variables. popPK methods are emerging as an pivotal part of drug development having a very relevant impact on preclinical and clinical studies, as well as for post-marketing investigations for the optimisation of therapeutic strategies. The main advantage of this approach is the possibility of including sparse data collected from different and unbalanced dosing strategies.

We can provide the development of flexible and customised popPK models for the characterisation of drug/compound distribution in humans and pre-clinical species, for the investigation of drug-drug interactions, therapy optimisation and management of clinical scenarios in patients as described in our recent publications: Dickinson et al. Clin Pharmacol Ther. 2015 Oct;98(4):406-16.; Schipani et al. J Acquir Immune Defic Syndr. 2013 Jan 1;62(1):60-6.; Dickinson L, et al. Antimicrob Agents Chemother. 2015 Oct;59(10):6080-6. Schipani A, et al Antivir Ther. 2012;17(5):861-8.


The mathematical description of anatomical, physiological and molecular processes defining compound distribution is called physiologically-based pharmacokinetic (PBPK) modelling for drugs or physiologically-based toxicokinetic (PBTK) modelling for other chemicals.

A broad range of compound-specific in vitro data, obtained through experimental approaches, is included in a mathematical description of compound distribution. Different routes of administrations and exposure can be simulated taking into account the dynamic interaction between solubility, activity of transporters, transmembrane permeability, local compound metabolism and chemical degradation. The distribution of compounds in organs is represented through an interconnected network of differential equations representing blood flow and the diffusion of molecules into tissues. In vitro intrinsic metabolism rate is quantified using cell lines expressing known concentrations of metabolic enzymes or hepatic and intestinal microsomes, and subsequently scaled up to organ clearance considering local enzyme expression and other tissue-specific factors.

Compound distribution is highly variable between subjects and virtual individuals can be simulated considering specific anatomical and physiological factors that differ between populations and sub-populations. Changes in organ size and other anatomical characteristics have been correlated with demographic variables in anthropometric studies, and multifactorial equations have been defined to generate anatomical and physiological parameters and their inter-individual variability. Through this approach it is possible to generate a virtual (but realistic) description of the anatomical characteristics of individuals and therefore obtain a representative evaluation of the expected exposure variability in populations.

The EMA (Guideline on the investigation of drug interactions) and FDA regulatory guidelines were recently updated to include PBPK modelling at different stages of drug development with the purpose of dosing finding, drug-drug interactions, and dose adjustment in special populations. Substitution with PBPK modelling has a major positive impact on use of pre-clinical species. The integration of in vitro data in PBPK models offers the possibility to run simulated pharmacokinetic studies in virtual pre-clinical species, reducing the number of doses as well as the number of animals in each experimental group. Additionally, the approach would also favour bridging from pre-clinical species to humans by taking into account physiological and anatomical differences between species.

We have applied physiologically based models for the investigation of numerous clinical scenarios such as drug interactions (Siccardi M, et al In Silico Pharmacol. 2013 Mar 1;1:4. and Siccardi M, et al .Conference on Retroviruses and Opportunistic Infections (CROI), 23-26 Feb 2015, Seattle, WA, USA.) as well as novel formulations (Rajoli RK  et al Clin Pharmacokinet. 2015 Jun;54(6):639-50 and  McDonald TO et al Adv Healthc Mater. 2014 Mar;3(3):400-11., simulating drug distribution in population of virtual patients.