IV-044 Van Thuy Truong

Stochastic pharmacodynamics: Drug effects on the time to extinction of a heterogeneous tumour cell population

Van Thuy Truong 1,2, Paolo Vicini 3, James Yates 4, Vincent Dubois 2 and Grant Lythe 1

1 School of Mathematics, University of Leeds, Leeds LS2 9JT, UK. 2 Clinical Pharmacology and Quantitative Pharmacology, AstraZeneca, Granta Park, Cambridge, CB21 6GH, UK. 3 Confo Therapeutics, Technologiepark 94, 9052 Ghent (Zwijnaarde), Belgium. 4 DMPK, IVIVT, RD Research, GSK , Gunnels Wood Road, Stevenage, Hertfordshire, SG1 2NY, United Kingdom.

Introduction: 

PK-PD models are often based on ordinary differential equations (ODEs) which are useful for ease of analysis but they lack two important aspects of reality: stochasticity and heterogeneity. Heterogeneity is important in many contexts: protein expression can vary from cell to cell; bacteria vary in their susceptibility to antibiotics [4]. Events that occur at random times can impact disease progression and treatment effects. To address those limitations, we propose a stochastic model. Stochastic models have the advantage that many outcomes are possible even if all conditions and initial states are given [5]. Additionally, elimination of the last cell in a population is a natural endpoint that is not available in deterministic models.

Objectives: 

We developed stochastic equations to describe the time to extinction of a heterogeneous population under drug influence and find characteristics of a tumour population to predict outcomes of drug treatment.

Methods: 

We developed and analysed a stochastic model of an idealised heterogeneous tumour-cell population treated with a drug, where each cell has a different value of an attribute linked to survival. In the model, heterogeneity originates in the initial conditions: each cell’s starting k value is chosen randomly between 0 and 1. The effect of the drug, during a dose or doses, is to decrease each cell’s k value with timescale 1/δ. Under the drug’s effect, the cell population changes in size and distribution; those cells with k < 0.25 are in the death pool, while those with k > 0.5 are in the division pool. The survival or death probability of a typical cell, and survival or extinction of the whole population, is calculated. The mean time to extinction depends on the logarithm of the initial number of cells; the distribution of extinction times has a characteristic Gumbel form.

We analysed multiple dose treatment where the cells are allowed to recover between the cycles and divide. Using the method, characteristics of the population such as a critical division rate for uncontrolled growth or a successful treatment can be identified. Cell death and cell division determine the ultimate fate of the population after repeated rounds of drug dose and recovery.

Results: 

We derived formulas for

-the survival probability of single cell depending on its initial k value,

-the time to arrive in death pool,

-the constant or k dependent death rate,

-the stochastic description of survival and death of individual cells,

-the extinction time of a cohort of tumor cells as a logarithmic relationship depending on the population size,

-the difference equation to describe cell fate during multiple dose treatment,

-the critical value for death rate to population extinction during multiple dose treatment.

Conclusions: 

Models of the effect of a drug on tumour cells using ordinary differential equations [7, 6] often have the advantage of easy implementation and analysis, but they do not naturally capture stochasticity or heterogeneity.

With the advance of molecular biology and the development of therapies that target intracellular signaling pathways [3, 8, 10], it is more important to consider heterogeneity of target cells. Biological heterogeneity also manifests itself in variable susceptibility to antibiotic treatments [1, 2].

Agent-based models overcome many shortcomings of simpler deterministic models because cells and their interactions are governed by stochastic rules, but they often require high computational power and running times and have large parameter spaces [9].

Here, based on a published agent-based model [9] where a heterogeneous can- cer cell population is treated with a MEK inhibitor, we use stochastic modelling and analysis as a bridge between those different types of models.

References:

  1. [1]  Nathalie Q Balaban et al. “Bacterial persistence as a phenotypic switch”. In: Science 305.5690 (2004), pp. 1622–1625.
  2. [2]  Liselot Dewachter, Maarten Fauvart, and Jan Michiels. “Bacterial hetero- geneity and antibiotic survival: understanding and combatting persistence and heteroresistance”. In: Molecular cell 76 (2019), pp. 255–267.
  3. [3]  Renaud Felten et al. “New biologics and targeted therapies in systemic lu- pus: from new molecular targets to new indications. A systematic review”. In: Joint Bone Spine 90 (2023), p. 105523.
  4. [4]  Orit Gefen and Nathalie Q Balaban. “The importance of being persistent: heterogeneity of bacterial populations under antibiotic stress”. In: FEMS microbiology reviews 33.4 (2009), pp. 704–717.
  5. [5]  Itziar Irurzun-Arana et al. “Beyond deterministic models in drug discovery and development”. In: Trends in pharmacological sciences 41.11 (2020), pp. 882–895.
  6. [6]  Helen Moore. “How to mathematically optimize drug regimens using op- timal control”. In: Journal of Pharmacokinetics and Pharmacodynamics 45 (2018), pp. 127–137.
  7. [7]  Helen Moore and Natasha K Li. “A mathematical model for chronic myel- ogenous leukemia (CML) and T cell interaction”. In: Journal of Theoret- ical Biology 227.4 (2004), pp. 513–523.
  8. [8]  E William St Clair. “Novel targeted therapies for autoimmunity”. In: Cur- rent opinion in immunology 21 (2009), pp. 648–657.
  9. [9]  Van Thuy Truong et al. “Step-by-step comparison of ordinary differential equation and agent-based approaches to pharmacokinetic-pharmacodynamic models”. In: CPT: Pharmacometrics and Systems Pharmacology 11.2 (2022), pp. 133–148.
  10. [10]  Apostolia-Maria Tsimberidou. “Targeted therapy in cancer”. In: Cancer chemotherapy and pharmacology 76 (2015), pp. 1113–1132.

Reference: PAGE 32 (2024) Abstr 11085 [www.page-meeting.org/?abstract=11085]

Poster: Methodology - New Modelling Approaches