Begell House Inc.
Critical Reviewsâ„˘ in Biomedical Engineering
CRB
0278-940X
24
2-3
1996
Concepts, Properties, and Applications of Linear Systems to Describe Distribution, Identify Input, and Control Endogenous Substances and Drugs in Biological Systems
73-139
10.1615/CritRevBiomedEng.v24.i2-3.10
Davide
Verotta
Department of Biopharmaceutical Sciences, and Pharmaceutical Chemistry, Box 0446, University of California San Francisco, San Francisco CA 94143-0446; Department of Epidemiology and Biostatistics, Box 0446, University of California San Francisco, San Fran
deconvolution
least-squares
regularization
Bayesian estimation
constraints
model selection
bootstrap
population
bioavailability
control
The response at time t (R(t)) of a (causal linear time invariant) system to an input A(t) is represented by:
R(t)=∫t0A(τ)K(t-τ)dτ
where K(t) is called the unit impulse response function of the system, and the integration on the right side of the equation (above) is called the convolution (from the latin cum volvere: to intertwine) of A(t) and K(t). The system described by this equation is at zero (initial conditions) when t = 0. Although it does not even begin to describe the incredible variety of possible responses of biological systems to inputs, this representation has large applicability in biology. One of the most frequently used applications is known as deconvolution: to deintertwine R(t) given a known K(t) (or A(t)) and observations of R(t), to obtain A(t) (or K(t)). In this paper attention is focused on a greater variety of aspects associated with the use of linear systems to describe biological systems. In particular I define causal linear time-invariant systems and their properties and review the most important classes of methods to solve the deconvolution problem, address. The problem of model selection, the problem of obtaining statistics and in particular confidence bands for the estimated A(t) (and K(t)), and the problem of deconvolution in a population context is also addressed, and so is the application of linear system analysis to determine fraction of input absorbed (bioavailability). A general model to do so in a multiinput-site linear system is presented. Finally the application of linear system analysis to control a biological system, and in particular to target a desired response level, is described, and a general method to do so is presented. Applications to simulated, endocrinology, and pharmacokinetics data are reported.
Modeling of Cardiac Electrophysiological Mechanisms: From Action Potential Genesis to its Propagation in Myocardium
141-221
10.1615/CritRevBiomedEng.v24.i2-3.20
Alain L.
Bardou
Institut National de la Sante et de la Recherche Medicale, Cardiologie Theorique, INSERM U66, CHU Pitie-Salpetriere, 91 Bd. de I'Hopital, F-75674 Paris Cedex 14, France
Pierre M.
Auger
Universite Claude Bernard Lyon 1, URA CNRS 2055, 43 Bd. du 11 Novembre, F-69622 Villeurbanne Cedex, France
Pierre J.
Birkui
Laboratoire de Cardiologie theorique, Hopital Broussais, Secteur Jaune, Bat. INSERM (Pte 9, 5eme Et.), 96 rue Didot, F-75674 Paris Cedex 14, France.
Jean-Luc
Chasse
Universite Claude Bernard Lyon 1, URA CNRS 2055, 43 Bd. du 11 Novembre, F-69622 Villeurbanne Cedex, France
cardiac electrophysiology
cardiac modeling
reentry
arrhythmias
fibrillation.
The aim of the present paper is to describe the different attempts at modeling cardiac electrophysiological mechanisms, mainly at the membrane and cellular level, from action potential genesis to its propagation in myocardium. The Hodgkin and Huxley model describing the nervous action potential's theoretical reconstruction is first recalled, for it represents the basic model for a large part of cardiac action potential models. These models (Beeler and Reuter, Van Capelle and Durrer, Luo and Rudy) are then successively studied as their main applications by diverse authors. Varied approaches, like the Fitzhugh-Nagumo model (derived from the Bonhoeffer-Van der Pol model of oscillatory systems) or cellular automata models applied to the study of ventricular activation wave propagation and diseases associated with its perturbation, are then presented and discussed. Other, different approaches, such as general studies of excitable media, are evoked.
This paper concludes with a critical evaluation of these different methods of electrophysiological cardiac modeling and of the main domains in which they led to significant results and in which they appear able to generate future perspectives.