Page 23 - Control Theory in Biomedical Engineering

Modeling and control in physiology 11

models are very useful in physiological systems, where stochasticity and ran-
dom variables have an important impact in real applications. In recent
research, environmental and demographic stochasticity are the two main
types of stochasticity. Several physiological systems in literature are described
by stochastic models such as the plasma membrane system (Sato et al., 2019)
and gastric emptying system (Yokrattanasak et al., 2016).
A parametric model is a finite-dimensional model of statistical models.
Specifically, a parametric model is a family of probability distributions that
has a finite number of parameters. A model is "nonparametric" if all the
parameters are in infinite-dimensional parameter spaces (Bickel and Dok-
sum, 2001). The reader can find many examples of physiological systems
written in parametric and nonparametric models in Marmarelis (2004).

2.4 Structural identifiability
Mathematical models in physiology are generally described by a set of con-
sistent differential equations and a set of physiological parameters to be esti-
mated in an accurate way. However, systems in physiology are naturally
characterized by poor observability. Observability is a modeling property
that describes the possibility of inferring the internal state of a system from
observations of its output. Indeed, the possibility for the clinician to observe
practically and quantify the relevant phenomena occurring in the body
through clinical tests is very limited due to complexity of dynamics of such
systems due to the high number of interacting and unmeasured variables.
Systems in physiology are also characterized by poor controllability due
to the limited capacity of such systems to drive the state of the system by
acting on some control variables. All these factors may severely hinder
the practical identifiability of these models, i.e., the possibility to estimate
the set of parameters from clinical data. Identifiability is a structural property
˚
of a model introduced in Bellman and Astr€om (1970) that defines the
amount of useful information that can be generated from the output (clinical
data for physiological systems). Hence, the importance of designing clinical
protocols that allow for estimating the model parameters in the quickest and
more reliable way. In fact, structural identifiability becomes a particular case
of observability if the parameters are considered as constant state variables.
Identifiability has been largely studied by researches and the reader can find
several survey papers in this framework (Pia Saccomani et al., 2003; Bouba-
ker and Fourati, 2004; Chis et al., 2011; Kabanikhin et al., 2016; Villaverde
and Barreiro, 2016; Bezzo and Galvanin, 2018; Villaverde, 2019) where