## Section: Scientific Foundations

### Physiological & Clinical research topics

#### The cardiovascular system: a multiscale controlled system

Understanding the complex mechanisms involved in the cardiac pathological processes requires fundamental researches in molecular and cell biology, together with rigorous clinical evaluation protocols on the whole organ or system scales. Our objective is to contribute to to these researches by developing low-order models of the cardiac mechano-energetics and control mechanisms, for applications in model-based cardiovascular signal or image processing.

We consider intrinsic heart control mechanisms, ranging from the Starling and Treppe effects (also called positive staircase effect) on the cell scale to the excitability of the cardiac tissue and to the control by the autonomous nervous system. They all contribute to the function of the heart in a coordinated manner that we want to analyse and assess. For this purpose, we study reduced-order models of the electro-mechanical activity of cardiac cells designed to be coupled with measures available on the organ scale (e.g. ECG and pressure signals). We study also the possibility to gain insight on the cell scale by using model-based multiscale signal processing techniques of long records of cardiovascular signals.

Here are some questions of this kind, we are considering:

- Modeling the controlled contraction/relaxation from molecular to tissue and organ scales.

- Direct and inverse modeling the electro-mechanical activity of the heart on the cell scale.

- Nonlinear spectral analysis of arterial blood pressure waveforms and application to clinical indexes.

- Modeling short-term and long-term control dynamics on the cardiovascular-system scale. Application to a Total Artificial Heart.

*Modeling the controlled contraction/relaxation from molecular to tissue and organ scales.*

We have obtained a controlled constitutive law of cardiac myofibre mechanics aimed at being embedded into 0D or 3D models [4] . This law results from a model of the collective behavior of actin-myosin molecular motors converting chemical into mechanical energy [88] . It is thermodynamically consistent and the resulting dynamics of sarcomeres is consistent with the “sliding filament hypothesis” of A. F. Huxley.

The model in [4] is currently used as the constitutive law for the cardiac tissue in the 3D model of the heart developed in the CardioSense3D project. It is useful for computing stress, strain and the action potential fields when coupled with an electrical model [79] , [134] . Depending upon a small number of parameters having a clear physical meaning, it is well suited for the study of inverse problems as considered in the CardioSense3D project (model-based 3D image processing). In order to check the mathematical consistency of our models, we have considered, in the more simple case of a 1D geometry, the mathematical analysis of the fibre model used in CardioSense3D based on the previous constitutive law [14] ).

*Direct and inverse modeling the electro-mechanical activity of the heart on the cell scale.*

We have revisited the ionic-currents models of cells representing membrane phenomena and calcium dynamics in order to reduce them for signal or image processing applications [97] , [98] , [99] . An objective here, is to obtain invertible (depending upon available measurements) thermodynamically consistent models (the various ATP consumption have to be taken into account). This will allow in particular a better connection with the perfusion models developed in CardioSense3D.

We have studied an intrinsic control effect, represented by the restitution curve associated to a very simple cardiac cell model and estimated by ECG analysis.

For isolated and electrically excited cardiac cells, there is a well known relationship between each action potential duration (APD)
and the preceding diastolic interval (DI) under the name of *restitution curve* .
A similar relationship has been recently revealed between the QT interval and the preceding TQ interval computed from
electrocardiogram (ECG) signals measured at the body surface [13] .
By analogy to the cellular restitution curve, we call this relationship *ECG-based restitution curve* .
To successfully build this curve, the ECG signals must be recorded under some particular conditions.
The isometric Handgrip test has proved to be a good choice for this purpose.
It is also important to delimit the QT interval with a sufficient accuracy.
For that purpose, we use the algorithm for T wave detection, whose robustness and efficiency have been reported in [19] .
In our previous work, the QT interval was obtained by adding a constant to the RT interval
which is easier to delimit [13] . More recently, in order to improve the delimitation of the QT interval,
an algorithm for QRS onset detection has been developed. It is based on the computation of the
envelop signal of the QRS defined with the Hilbert transform, and also on the application of a statistical detection algorithm.
This new algorithm is now used for building ECG-based restitution curves [105] , [106] .

*Nonlinear spectral analysis of arterial blood pressure waveforms and application to clinical indexes*

We have proposed [96] a reduced model of the input-output behaviour of an arterial compartment, including the short systolic phase where wave phenomena are predominant. A more detailed analysis is now available [8] . The objective is to provide basis for model-based signal processing methods for the estimation from non-invasive measurements and the interpretation of the characteristics of these waves. We develop now the corresponding signal processing method and some applications.

This method, based on scattering transform for a one dimensional Schrödinger equation, provides new parameters, related to the systolic and diastolic parts of the pressure. They are compared to the classical blood pressure indexes in four conditions: moderate chronic heart failure, exercise before and after training in high fit triathlets [116] , handgrip isometric exercise and orthostatic tilt test. [117] . In each case these parameters are more significant than the classical ones. Moreover, they bring up new indexes, difficult to measure routinely: we think that the two first invariants might give information about the variation of the stroke volume and the ventricular contractility. At last, the first eigenvalue seems to reflect the baroreflex sensitivity in a certain way. We are now working on the validation of these hypotheses.

*Modeling short-term and long-term control dynamics on the cardiovascular-system scale. *

Our objective is to relate discrete-time (beat-to-beat) cardiovascular signal analysis to models of the cardiovascular and control systems taking into account its multiple feedback loop organisation. This will lead to a model-based signal processing approach for the estimation of the classical arterial-pressure/heart-rate baroreflex sensitivity and of several other discrete-time feedback loop sensitivities of practical interest. It will be also useful for the control of Artificial heart.

In the past we have used time-frequency techniques for these studies (Fourier Transform, spectral gain between the cardiac and blood pressure series, Smooth Pseudo Wigner_Ville Distribution, Complex DeModulation, temporal method of the cardiovascular Sequences). Different situations have been studied: the cardio-respiratory system dynamics in chronic heart failure [122] , [121] , [124] ; the autonomic control of the cardiovascular system during sleep [129] ; the effects of exercise intensity and repetition on heart rate variability during training [94] , [95] , [93] . We will combine these techniques with our new inverse scattering approach. In particular the scattering-based description of cadiovascular signals leads to the definition of new indexes we want to investigate, see paragraph 3.4.1 .

This approach is applied to the control of a Total Artificial Heart.

#### Reproductive system: follicular development & ovulation control

The ovulatory success is the main limiting factor of the whole reproductive process, so that a better understanding of ovulation control is needed both for clinical and zootechnical applications. It is necessary to improve the treatment of anovulatory infertility in women, as it can be by instance encountered in the PolyCystic Ovarian Syndrome (PCOS), whose prevalence among reproductive-age women has been estimated at up to 10%. In farm domestic species, embryo production following FSH stimulation (and subsequent insemination) enables to amplify the lineage of chosen females (via embryo transfer) and to preserve the genetic diversity (via embryo storage in cryobanks). The large variability in the individual responses to ovarian stimulation treatment hampers both their therapeutic and farming applications. Improving the knowledge upon the mechanisms underlying FSH control will help to improve the success of assisted reproductive technologies, hence to prevent ovarian failure or hyperstimulation syndrome in women and to manage ovulation rate and ovarian cycle chronology in farm species.

To control ovarian cycle and ovulation, we have to deeply understand the selection process of ovulatory follicles, the determinism of the species-specific ovulation rate and of its intra- and between-species variability, as well as the triggering of the ovulatory GnRH surge from hypothalamic neurons.

Beyond the strict scope of Reproductive Physiology, this understanding raises biological questions of general interest, especially in the fields of

*Molecular and Cellular Biology. *
The granulosa cell, which is the primary target of FSH in ovarian follicles, is a remarkable cellular model to study the dynamical control of the transitions between
the cellular states of quiescence, proliferation, differentiation, and apoptosis, as well as the adaptability of the response to the same extra-cellular signal according
to the maturity level of the target cell. Moreover, the FSH receptor belongs to the seven transmembrane spanning receptor family, which represent the most frequent
target (over 50%) amongst the therapeutic agents currently available. The study of FSH receptor-mediated signaling is thus not only susceptible to allow the identification
of relaying controls to the control exerted by FSH, but it is also interesting from a more generic pharmacological viewpoint.

*Neuroendocrinology and Chronobiology. *
The mechanisms underlying the GnRH ovulatory surge involve plasticity phenomena of both neuronal cell bodies and synaptic endings comparable to those occurring
in cognitive processes. Many time scales are interlinked in ovulation control from the fastest time constants of neuronal activation (millisecond) to the circannual variations
in ovarian cyclicity. The influence of daylength on ovarian activity is an interesting instance of a circannual rhythm driven by a circadian rhythm (melatonin secretion from
the pineal gland).

*Simulation and control of a multiscale conservation law for follicular cells*

In the past years, we have designed a multiscale model of the selection process of ovulatory follicles, including the cellular, follicular and ovarian levels [12] , [11] . The model results from the double structuration of the granulosa cell population according to the cell age (position within the cell cycle) and to the cell maturity (level of sensitivity towards hormonal control). In each ovarian follicle, the granulosa cell population is described by a density function whose changes are ruled by conservation laws. The multiscale structure arises from the formulation of a hierarchical control operating on the aging and maturation velocities as well on the source terms of the conservation law. The control is expressed from different momentums of the density leading to integro-differential expressions.

Future work will take place in the REGATE project and will consist in:

- predicting the selection outcome (mono-, poly-ovulation or anovulation / ovulation chronology) resulting from given combinations of parameters and corresponding to the subtle interplay between the different organs of the gonadotropic axis (hypothalamus, pituitary gland and ovaries). The systematic exploration of the situations engendered by the model calls for the improvement of the current implementation performances. The work will consist in improving the precision of the numerical scheme, in the framework of the finite volume method and to implement the improved scheme, basing by instance on the current routines designed within the Bearclaw (http://www.amath.unc.edu/Faculty/mitran/bearclaw.html ) academic environment,

- solving the control problems associated with the model. Indeed, the physiological conditions for the triggering of ovulation, as well as the counting of ovulatory follicles amongst all follicles, define two nested and coupled reachability control problems. Such particularly awkward problems will first be tackled from a particular approximation of the density, in order to design appropriate control laws operating on the particles and allowing them to reach the target state sets.

*Connectivity and dynamics of the FSH signaling network in granulosa cells*

The project consists in analyzing the connectivity and dynamics of the FSH signaling network in the granulosa cells of ovarian follicles and embedding the network within
the multiscale representation described above, from the molecular up to the organic level.
We will examine the relative contributions of the G_{s} and arrestin-dependent pathways in response to FSH signal, determine how each pathway controls
downstream cascades and which mechanisms are involved in the transition between different cellular states (quiescence, proliferation, differentiation and apoptosis).
On the experimental ground, we propose to develop an antibody microarray approach in order to simultaneously measure the phosphorylation levels of a large number of signaling
intermediates in a single experiment. On the modeling ground, we will use the BIOCHAM (biochemical abstract machine) environment first at the boolean level, to formalize the network
of interactions corresponding to the FSH-induced signaling events on the cellular scale. This network will then be enriched with kinetic information coming from experimental data,
which will allow the use of the ordinary differential equation level of BIOCHAM. In order to find and fine-tune the structure of the network and the values of the kinetic parameters,
model-checking techniques will permit a systematic comparison between the model behavior and the results of experiments. In the end, the cell-level model should be abstracted
to a much simpler model that can be embedded into a multiscale one without losing its main characteristics.

*Bifurcations in coupled neuronal oscillators. *

We have proposed a mathematical model allowing for the alternating pulse and surge pattern of GnRH (Gonadotropin Releasing Hormone) secretion [6] . The model is based on the coupling between two systems running on different time scales. The faster system corresponds to the average activity of GnRH neurons, while the slower one corresponds to the average activity of regulatory neurons. The analysis of the slow/fast dynamics exhibited within and between both systems allows to explain the different patterns (slow oscillations, fast oscillations and periodical surge) of GnRH secretion.

This model will be used as a basis to understand the control exerted by ovarian steroids on GnRH secretion, in terms of amplitude, frequency and plateau length of oscillations and to discriminate a direct action (on the GnrH network) from an indirect action (on the regulatory network) of steroids. From a mathematical viewpoint, we have to fully understand the sequences of bifurcations corresponding to the different phases of GnRH secretion. This study will be derived from a 3D reduction of the original model.

*Quantification of the follicular vascularization and cell number. *

There is a crucial need for both quantitative and dynamical data on follicular development. Such data may be retrieved from different modalities of ovarian imaging. Within the framework of the REGLO cooperative research initiative, http://www-rocq.inria.fr/who/Frederique.Clement/reglo.html the Asclepios members have been reconstructing a 3-D image of the ovary from a series of 2-D stained histologic images. From this reconstruction, we expect to get statistical (i.e. from a given population of growing follicles) information on the cell number (which corresponds to a follicular level output of the multiscale model), derived from the volume of the granulosa tissue, as well as on the degree of follicular vascularization (which corresponds to a follicular level input in the multiscale model).

In the middle-long term, we intend to design a morphological model of follicular growth. We will base on the analogy of follicle growth with solid tumor growth and on the image-derived data to design the model. In turn, such a model would be very useful in analyzing low signal-to-noise ratio imaging modalities such as ultrasonography.