The transit compartment model: The truth behind n!
Mohammed H. Cherkaoui-Rbati
Roche Pharma Research and Early Development, Roche Innovation Center Basel, Basel, Switzerland
Objectives:
The transit compartment (TC) model was first introduced by Yu et al. [1], and popularized by Savic et al. [2] to better describe drug absorption delay as a compromise between simpler model (e.g. lag-time model) and more complex model (e.g. physiological-based absorption pharmacokinetics models). Yu et al. compared the TC model to the dispersion model, which is considered more physiological, but difficult to implement, while having the same number of parameters. They also concluded that both models describe data equivalently. Additionally, Savic et al. showed that when n increases to infinity, the TC model becomes equivalent to a lag-time model. In this poster, we explain how mathematicians link the TC model to the dispersion model and what a pharmacometrician can learn from it.
Methods:
Mathematicians have developed multiple tools to numerically solve partial differential equations (PDEs), such as the dispersion equation or the advection equation (which is equivalent to the dispersion equation with a null dispersion term). One of the tools consists in discretizing in space PDEs to transform the PDE into a series of ordinary differential equations (ODEs) for which multiple solvers are then available. In one spatial dimension, one could describe a discretization as space divided into multiple compartments and applying various flux between them. To link the TC model to the dispersion, we applied the upwind scheme on the advection equation, which is a simple and stable discretization scheme with interesting properties. We then compared the numerical solution of the advection equation with the exact solution of the TC and dispersion models in a special case.
Results:
When using the upwind discretization on the advection equation, the eye of the mathematician will immediately recognize the TC model. However, the mathematician also knows that the upwind scheme is a diffusive scheme, i.e. that numerical diffusion is artificially added to the solution of the advection equation, and that this diffusion coefficient is equal to 1/[2(n+1)MTT]. Therefore, the solution of the TC model of n compartments is the numerical approximation of the exact solution of the dispersion equation with a dispersion coefficient of 1/[2(n+1)MTT] (Note: Mathematically diffusion and dispersion are modeled the same way but the physical cause/meaning is different).
Conclusions:
We showed that estimating n, as it is done with the TC model for absorption, is numerically equivalent to estimating a dispersion coefficient with the dispersion model. Therefore, one could use either and would obtain similar results.
So, is there an advantage to use the dispersion model? The answer is yes, not because it is superior in describing the data, but because of three main reasons. The first reason is to express the mechanistic process explicitly, i.e. that dispersion occurs during the absorption process. Secondly, the schematic representation of PDEs is more elegant than TC models; i.e. representing an intestine by a tube is more intuitive than by a series of n compartments (e.g. the ACAT model would visually benefit of it). Finally, it can simplify model selection by fixing n to minimize numerical error using a non-diffusive scheme (e.g. a flux limiter scheme) while focusing the estimation on the dispersion parameter and still being able to simulate with multiple dose administration.
It is not to say that there are no disadvantages, such as the complexity in selecting the appropriate discretization scheme, which is problem dependent, or the risk that the required n leads to a high number of ODEs and consequently increases computation time.
To conclude, pharmacometricians would benefit in learning more about PDEs to better manipulate the mathematical objects they use, even if at the end we disguise our dispersion model into a TC one.
References:
[1] L.X. Yu, J.R. Crison, G.L. Amidon, Compartmental transit and dispersion model analysis of small intestinal transit flow in humans, International Journal of Pharmaceutics, 140 (1), 111–118, 1996, https://doi.org/10.1016/0378-5173(96)04592-9.
[2] R.M. Savic, D.M. Jonker, T. Kerbusch et al., Implementation of a transit compartment model for describing drug absorption in pharmacokinetic studies, Journal of Pharmacokinetics and Pharmacodynamics, 34, 711–726, 2007, https://doi.org/10.1007/s10928-007-9066-0.
