A comparison of ordinary differential equation and agent-based approaches to PKPD modelling in oncology
Van Thuy Truong1,2, Paul G. Baverel3, Grant D. Lythe2, Paolo Vicini1,4, James W. T. Yates5,6, Vincent F. S. Dubois1
1 Clinical Pharmacology and Quantitative Pharmacology, Clinical Pharmacology and Safety Sciences, AstraZeneca, Aaron Klug Building, Granta Park, Cambridge, CB21 6GH, UK 2 Department of Applied Mathematics, University of Leeds, Leeds, UK, 3 Roche Pharma Research and Early Development, Clinical Pharmacology, PharmaceuticalSciences, Roche Innovation Center Basel F. Hoffmann-La Roche Ltd, Switzerland 4 Confo Therapeutics, Technologiepark 94, 9052 Ghent (Zwijnaarde), BelgiumOncology RD, AstraZeneca, Chesterford Research Park, Little Chesterford, Ca 5 DMPK, Early Oncology,Cambridge, CB10 1XL, UK 6DMPK, IVIVT, RD Research, GSK, Gunnels Wood Road, Stevenage, Hertfordshire, SG1 2NY, United Kingdom
Introduction
Mathematical models in oncology aid in the design of drugs and understanding of their mechanisms of action by simulation of drug bio-distribution, drug effects, and interaction between tumour and healthy cells. The traditional approach in pharmacometrics is to develop and validate ordinary differential equation models to quantify trends at the population level. In this approach, time-course of biological measurements is modelled continuously, assuming an homogenous population. Another approach, agent-based models focus on the behaviour and fate of biological entities at the individual level which subsequently could be summarized to reflect the population level. Heterogeneous cell populations and discrete events are simulated, and spatial distribution can be incorporated.
Objectives
- Comparing agent-based model (ABM) and an ordinary differential equation (ODE) model for a tumour efficacy model inhibiting the pERK pathway
- Highlighting strengths, weaknesses, and opportunities of each approach
Methods
A simple example of a PKPD-ABM of anti-cancer treatment with the MEK inhibitor
cobimetinib targeting the RAF/MEK/ERK pathway is given. This signalling pathway is initiated by binding of growth factors which activates receptor tyrosine kinases and leads to a reaction cascade resulting in activation of MEK and phosphorylation of ERK. As a consequence, cellular responses are cell proliferation and survival. Cobimetinib aim to counteract this by decreasing ERK phosphorylation downstream signalling cascade [1, 2]. The ABM simulates the behaviour of tumour cells as agents during treatment with cobimetinib. The time course and effect of the administered drug are included in the model conditions. Cell death and division occur on the microscopic scale and a summary of cell behaviour provides the total number of tumour cells on the macroscopic scale. The model is implemented in the Python programming language. A PKPD model previously published by Wong et al [2] implemented in ODE was used to simulate drug concentration time-course and the phosphorylation of ERK. To account for the PK, an hybrid approach is considered, with PK being governed by an ODE system and combined with an tumour ABM to determine the interactions of cells in the presence of drug concentrations. To compare the behaviour of the hybrid PKPD-ABM tumour model, a PKPD-ODE tumour model is implemented to highlight the relative strengths and limitations of each method.
Results
We successfully implemented the MEK inhibitor case study in an ABM and ODE model.
In the ABM, agents are tumour cells, each with a different levels of pERK. Depending on a cell’s current pERK expression, it can either die, stay quiescent or divide on the microscopic scale, translating into an emerging tumour growth dynamic on the macro-level scale. In the ODE model, all cells have the same pERK expression level and constitutes a homogenous population. Stochasticity is introduced by drawing parameters from probability distributions to model between-subject variability.
Overall, ODE and ABM provided similar time-course of pERK and tumour cell dynamics at the population and individual level following 1 mg/kg once daily of cobimetinib given orally. After 7 days the treatment, the mean reduction in baseline tumour burden was 22.5% with a standard deviation (SD) of 12.0% in the ODE model while it was estimated at 30.7% (SD=10.2%) in the ABM. However, ABM allowed us to track individual cell fate over time on the microscopic level while ODE remained limited in mechanistic insights.
Simulations with ABM are more intuitive than ODE since they recapitulate biological processes, allowing complex biological systems to be described in sub-scale components (molecular, cellular, tissue, organism) with inherent emerging behaviour outputs.
On the other hand, systems of ODEs are well-suited for simulating processes that can be approximated as homogeneous, well-mixed systems and are best suited for traditional pharmacometrics analyses with sufficient data (population PK and PD models, physiology-based PK models), or for simplistic theoretical PKPD models. For quantitative clinical pharmacology models, ODEs could also be implemented to recapitulate complex biological systems but would rely on extensive model assumptions, including parameter distributions [4].
Conclusion
In summary, ABMs can provide more detailed insights into complex biological systems and are often complemented with ODEs in hybrid multiscale models. Both methodologies have their strengths and weaknesses, depending on context and purpose. With the advent of more single-cell experiments, gene expression, and spatial transcriptomics, and other technological advances in imaging we anticipate that the use of ABMs in discovery and drug development will increase with direct applications in PKPD and pharmacometrics to help elucidate dose scheduling and rationalize combination strategy in oncology and other therapeutic areas.
References[1] Li Y, Dong Q, Cui Y. Synergistic inhibition of MEK and reciprocal feedback networks for targeted intervention in malignancy. Cancer Biol. Med.. 2019 Aug;16(3):415.
[2] Wong H, Vernillet L, Peterson A et al. Bridging the gap between preclinical and clinical studies using pharmacokinetic–pharmacodynamic modeling: an analysis of GDC-0973, a MEK inhibitor. Clin. Cancer Res.. 2012 Jun 1;18(11):3090-9.
[3] Solovyev A, Mi Q, Tzen YT, Brienza D, Vodovotz Y. Hybrid equation/agent-based model of ischemia-induced hyperemia and pressure ulcer formation predicts greater propensity to ulcerate in subjects with spinal cord injury. PLoS Comput Biol. 2013 May 16;9(5):e1003070.
[4] Milberg O, Gong C, Jafarnejad M, Bartelink IH, Wang B, Vicini P, Narwal R, Roskos L, Popel AS. A QSP model for predicting clinical responses to monotherapy, combination and sequential therapy following CTLA-4, PD-1, and PD-L1 checkpoint blockade. Sci. Rep. 2019 Aug2;9(1):1-7.