Building mean field ODE models using the generalized linear chain trick & Markov chain theory

Abstract

The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0, ∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

Impact of variants of concern on SARS-CoV-2 viral dynamics in non-human primates

“…We assumed a delay of 3 days for the immune response to take place to account for the differentiation and proliferation of the immune response (29). We modelled this delayed immune effector compartment using the Linear Chain Trick (LCT) assuming an Erlang distribution with j = 20 transition compartment and a mean time spent in those compartment of τ = 3 d -1 (30). This number of compartments allowed us to shift the distribution of the time spent in the transition’s states from an exponential to a normal distribution.…”

“…This number of compartments allowed us to shift the distribution of the time spent in the transition's states from an exponential to a normal distribution. The equations for the transfer compartments are written as follows: (30) In the following only the compartment 𝐹 20 will serve as the effector for the action of the immune system. The transfer rate parameter 𝑔 is then written as 𝑗 𝜏 and fixed to 6.67 d -1 and the loss rate of the final effector 𝑑 𝐹 is fixed to 0.4 d -1 (28).…”

Analysis and modeling of cancer drug responses using cell cycle phase-specific rate effects

“…2A ). We incorporated a “linear chain trick “ into our model, which creates similarly-distributed time delays in the cell cycle phase durations through a mean-field system of ODEs 29 . Additionally, we simplified the model by sharing parameters that were not drug specific, such as the number of cell cycle subphases and the initial fraction of cells in G 1 phase.…”

“…With improved reporter strategies 42 , we may be able to further subdivide these phases into constituent parts, which could help to localize the effect of a drug to a more specific portion of one cell cycle phase. Generalizations of the linear chain trick could be used to account for both subphases of varying passage rates, as well as heterogeneity in the rates of passage, such as would arise through cell-to-cell heterogeneity 29 . While the subdivisions within each cell cycle phase are phenomenological, it is tempting to imagine they represent mechanistic steps within each phase.…”

Can Systems Biology Advance Clinical Precision Oncology?

“…The simplest form of differential equations are ODEs. These models use the mean field approximation and can accurately describe the time evolution of a biochemical systems if the number of molecules of each species is large enough [ 123 ]. ODEs relate a function (or several functions) with its (their) derivative.…”