Section: Application Domains

Medical applications of modeling, signal analysis and control

- 3D cardiac modeling for personalized medicine. Our main contribution to Inria's collective efforts in this field (project-teams Asclepios, MACS, REO, Sisyphe) is the so-called “Bestel-Clément-Sorine” model of contraction of cardiac muscle [39] , at the origin of the 3D electromechanical direct and inverse modeling of the heart at Inria. This model is based on ideas originating from the kinetic equation theory, used to model, on the molecular scale, the controlled collective behavior of actin-myosin nanomotors at the root of muscle contraction. The classical Huxley model was recovered on the sarcomere scale by using moment equations and a controlled constitutive law on the tissue scale was obtained using the same type of scaling techniques. The model, now embedded in heart simulators, is used in various studies [51] , [3] , [54] , [53] .

- Semiclassical analysis of cardiovascular signals. This work began with the article [41] and the PhD of M. Laleg-Kirati [47] , [46] , [49] . The theory and a validation of a new method of blood pressure analysis are now published [17] , [48] .

The main idea is to consider a signal $x\to y\left(x\right)$ to analyze as the multiplication operator $\phi \to y\phi$ on some function space, and to analyze it as a potential. The signal is represented by the spectrum of an associated Schrödinger operator, combined with a semi-classical quantification: $-h^2{\displaystyle \frac{d^2}{d{x}^2}}-y\left(x\right)$ with $h>0$ small. For signals looking as “superpositions of bumps” (e.g. the systolic pulse, the dichrotic notch for the arterial pulse pressure), this leads to some kind of nonlinear Fourier analysis [17] . The spectral parameters associated with the arterial pressure can be useful cardiovascular indices, e.g. for noninvasive blood flow estimation [48] . In the arterial pressure case, this is equivalent to approximate the traveling pressure pulse by a N-soliton solution of a Korteweg-de Vries (KdV) equation  [41] and using ideas similar to the Lax pair representation of $N$-solitons and proof technique for the weak dispersion limit of KdV. A striking result is that an $N$-soliton is a very good representation of the arterial pressure waveform for values of $N$ as small as $N=3$. The representation of pulse-shaped signals is parsimonious, having only $2N$ parameters [55] .

- Multiscale signal analysis of cardiovascular signals: collaboration with Julien Barral (former member of Sisyphe) and partners of the ANR project DMASC. The starting point was the common idea that “A Healthy Heart Is a Fractal Heart”. We have developed a method to test the existence of scale laws in signals and applied it to RR signals: the heart rate is not always fractal or even multifractal in an Healthy Heart [19] .

- Modeling and control of CARMAT Total Artificial Heart. This TAH has been implanted for the first time in a patient in Dec 2013. We have contributed to this industrial project since 2008 on modeling and control questions during the post-doc of Karima Djabella (now engineer at CARMAT), Frédéric Vallais and the two-year contract for supervising Julien Bernard (CARMAT control engineer). It was an opportunity to exploit some results on the baroreflex control [42] or heart rate variability during exercise [40] .

- Glycemic control in Intensive Care Units (ICUs): Blood glucose is a key biological parameter in ICU since the study of van den Berghe et al [60] who demonstrated decreased mortality in surgical intensive care patients in association with tight glycemic control (TGC), based on intensive insulin therapy. In their work, however, there was only one ICU and the protocol was not formalized. Trying to decrease mortality in standard ICUs by using computer aided glycemic control is still a challenge. Previous studies have failed because of high rates of severe hypoglycemia. The last one was NICE-SUGAR [57] with a 2% increase in mortality (death ratio from any cause within 90 days after randomization compared between control and TGC patients). In cooperation with Pierre Kalfon (Intensive Care, Hospital of Chartres) and in the framework of a CIFRE contract with a small medtech company LK2 (Tours, France), we have studied the origins of these failures and proposed more robust control algorithms tuned using a database of representative “virtual patients”. See [44] and the PhD of A. Guerrini, [43] . A first version of the controller has been tested in a large clinical study CGAO-REA [14] .