# Pharmacokinetics model

## Model description:

The pharmacokinetics model is a two compartment model that embodies the ligands of the macrophage mannose receptor, and it is represented mathematically as a system of differential equations of the form:

\begin{align*} \dot{x}_1 &= \alpha_1(x_2-x_1) - \dfrac{k_av_mx_1}{k_ck_a+k_cx_3+k_ax_1},\\ \dot{x}_2 &= \alpha_2(x_1-x_2),\\ \dot{x}_3 &= \beta_1(x_4-x_3) - \dfrac{k_av_mx_3}{k_ck_a+k_cx_3+k_ax_1}, \\ \dot{x}_4 &= \beta_2(x_3-x_4), \end{align*}

where $x_1(0) = c_0$, $x_2(0)=0$, $x_3(0) = \gamma x_4(0)=0$, $x_1$ represents the enzyme concentration in plasma, $x_2$ its concentration in compartment, $x_3$ is the plasma concentration of the mannosylated polymer that acts as a competitor of glucose oxidase for the mannose receptor of macrophages, and $x_4$ is the concentration of the same competitor in the part of the extravascular fluid of the organs accessible to this macromolecule.

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## Publication details:

 Title Structural Identifiability of Systems Biology Models: A Critical Comparison of Methods Publication Type Journal Article Year of Publication 2011 Authors Chis, Oana-Teodora, Banga Julio R., and Balsa-Canto Eva Secondary Authors Jaeger, JohannesEditor Journal PLoS ONE Volume 6 Issue 11 Start Page 1 Pagination 1-16 Date Published 10/2011 ISSN 1932-6203 Abstract Analysing the properties of a biological system through in silico experimentation requires a satisfactory mathematical representation of the system including accurate values of the model parameters. Fortunately, modern experimental techniques allow obtaining time-series data of appropriate quality which may then be used to estimate unknown parameters. However, in many cases, a subset of those parameters may not be uniquely estimated, independently of the experimental data available or the numerical techniques used for estimation. This lack of identifiability is related to the structure of the model, i.e. the system dynamics plus the observation function. Despite the interest in knowing a priori whether there is any chance of uniquely estimating all model unknown parameters, the structural identifiability analysis for general non-linear dynamic models is still an open question. There is no method amenable to every model, thus at some point we have to face the selection of one of the possibilities. This work presents a critical comparison of the currently available techniques. To this end, we perform the structural identifiability analysis of a collection of biological models. The results reveal that the generating series approach, in combination with identifiability tableaus, offers the most advantageous compromise among range of applicability, computational complexity and information provided. DOI 10.1371/journal.pone.0027755