Continuation of homoclinic orbits in the suspension bridge equation: a computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation $u""+\beta u" + e^u-1=0$ for all parameter values $\beta \in [0.5,1.9]$. For each $\beta$, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.Comment: 37 pages, 6 figure

Modélisation de l'écoulement sanguin et du transport de molécules dans la microcirculation sanguine cérébrale : impact des occlusions capillaires dans la maladie d'Alzheimer

Le système microvasculaire est un acteur essentiel du fonctionnement cérébral. Il est en effet responsable de l’approvisionnement des cellules en oxygène et glucose ainsi que de l’évacuation des déchets métaboliques comme le dioxyde de carbone. Ce système est composé d’une multitude de petit vaisseaux appelés artérioles, veinules et capillaires, qui sont entourés de tissu cérébral. Ces vaisseaux forment un immense réseau qui étend ses ramifications à travers tout le cerveau. A cause de son rôle prépondérant dans l’homéostasie cérébrale le système microvasculaire est impliqué dansde nombreuses pathologies, allant de l’accident vasculaire cérébral aux maladies neurodégénératives. Ces dernières décennies ont été marquées par des avancées significatives dans le domaine de l’imagerie du vivant (e.g. la microscopie multi-photonique) qui ont permis l’observation du système microvasculaire cérébral avec un niveau de précision sans précédent. Ces techniques génèrent cependant de grandes quantités de données qu’il est difficile d’analyser sans outils théoriques adaptés. C’est pourquoi, dans cette thèse, nous développons des modèles capables de décrire l’écoulement sanguin ainsi que le transport de soluté au sein de vastes réseaux microvasculaires anatomiques. La principale difficulté dans la résolution de tels problèmes, vient de la taille de ces réseaux. En effet, même s’ils ne représentent qu’une fraction du système microvasculaire, ils sont composés de plusieurs dizaines de milliers de vaisseaux et possèdent des géométries complexes. Il est donc inenvisageable de résoudre l’écoulement sanguin et le transport de soluté par le biais de méthodes classiques comme les volumes finis ou les éléments finis. Afin de surmonter cette difficulté, nous combinons une approche réseau de pores avec des méthodes de changement d’échelles (prise de moyenne volumique et développements asymptotiques) et des fonctions de Green. Cela nous permet de simplifier à la fois la description de l’écoulement sanguin et du transport de soluté tout en restant cohérent avec la physique sous-jacente. Pour nous assurer de la pertinence de ces simplifications nous validons systématiquement nos modèles en les comparant à des mesures in vitro et in vivo si elles existent et à des solutions analytiques de référence sinon. Une fois validés, nous utilisons nos modèles afin d’élucider le rôle joué par le système microvas- culaire aux stades précoces de la maladie d’Alzheimer. En effet, il a été récemment montré qu’une baisse du débit sanguin cérébral était le premier marqueur quantitatif de la maladie. Simultanément, nos collaborateurs, les professeurs Schaffer et Nishimura de l’université de Cornell, ont observé chez les souris malades qu’une faible proportion (2%-4%) des capillaires étaient obstrués par des globules blancs. En conséquence ils ont injecté un anticorps inhibant l’adhésion de ces derniers. Les vaisseaux se sont alors débloqués, entraînant une augmentation du débit sanguin ainsi qu’une amélioration des capacités cognitives chez les souris malades. Si l’on suppose qu’après l’injection le débit sanguin retrouve sa valeur de référence, on peut estimer que les occlusions capillaires réduisent de 20 % à 30 % le débit sanguin. Une si faible proportion de capillaires obstrués peut-elle avoir un impact aussi important sur le débit sanguin cérébral ? Il est difficile de répondre simplement à cette question en se fiant uniquement à l’expérience puisqu’il est quasiment impossible d’isoler un tel phénomène in vivo que ce soit chez la souris ou chez l’humain. Pour contourner ce problème nous utilisons nos modèles et simulons numériquement l’impact de ces occlusions sur le débit sanguin. Nous trouvons que 2% à 4% d’occlusions capillaires conduisent à une baisse de débit pouvant aller jusqu’à 12%, faisant de ces occlusions un mécanisme important dans l’apparition de la maladie d’Alzheimer. Pour finir, nous quantifions leurs conséquences sur les échanges moléculaires

Mathematical modelling with Bayesian inference to quantitatively characterize therapeutic cell behaviour in nerve tissue engineering

Cellular engineered neural tissues have significant potential to improve peripheral nerve repair strategies. Traditional approaches depend on quantifying tissue behaviours using experiments in isolation, presenting a challenge for an overarching framework for tissue design. By comparison, mathematical cell–solute models benchmarked against experimental data enable computational experiments to be performed to test the role of biological/biophysical mechanisms, as well as to explore the impact of different design scenarios and thus accelerate the development of new treatment strategies. Such models generally consist of a set of continuous, coupled, partial differential equations relying on a number of parameters and functional forms. They necessitate dedicated in vitro experiments to be informed, which are seldom available and often involve small datasets with limited spatio-temporal resolution, generating uncertainties. We address this issue and propose a pipeline based on Bayesian inference enabling the derivation of experimentally informed cell–solute models describing therapeutic cell behaviour in nerve tissue engineering. We apply our pipeline to three relevant cell types and obtain models that can readily be used to simulate nerve repair scenarios and quantitatively compare therapeutic cells. Beyond parameter estimation, the proposed pipeline enables model selection as well as experiment utility quantification, aimed at improving both model formulation and experimental design

An in vitro study of highly confined blood flows: From single bifurcations to 2D-networks

Since the very first observations of microvascular networks in small animals by Jean-Marie Poiseuille in the XVIIIth century, the blood microcirculation has been extensively studied. One of the most striking feature highlighted by the French physicist is the highly heterogeneous distribution of the red blood cells (RBCs) throughout microvessel networks. Despite the intimate link between local RBC concentration (also called hematocrit) and surrounding tissue oxygenation, the coupling between microvascular architecture and hemodynamics is still poorly understood. In vivo experiments provide data on spatio-temporal distribution of RBC concentration and velocities within a given microvascular network, but are limited to dilute regimes or highly confined flows, where the RBCs are arranged in single files and are therefore individually discernible [1]. Also, in such conditions, shape and diameters of the vessels cross-section is not precisely known. Along with physiological feedbacks, these uncertainties might be sources of errors. In vitro experiments can overcome such issues, inherent to the living [2]. Yet, providing reliable quantitative data at the scale of the blood microcirculation is challenging since the vessel diameters range from ~3 to ~10 μm, which is of the same order as the RBCs', or even smaller, and studies are often limited by the microfabrication process. Our study aims at providing in vitro quantitative data on the distribution of the RBCs in geometries of increasing complexity: from diverging bifurcations to 2D-channel networks, with squared cross-section (WxW, W=5, 10, 20 μm). First, we have developed a calibration method that allows us to measure the hematocrit in situ, i.e. directly in the channel of interest, for a broad range of concentrations. Alongside with the hematocrit profiles, we are able to measure the RBC velocity profiles and thus deduce the RBC flow rate. By making simple assumptions on the suspending fluid, we deduce also the total blood flow rate. As a result, we have performed a parametric study of the phase separation (PS) effect, i.e. the non-proportional distribution of RBCs between the two daughter branches of a simple divergent bifurcation, and compared our results with the only in vivo empirically derived PS law [3]. Finally, we have designed 2D honeycomb networks and compared our experimental network perfusion with numerical simulations. For these networks, we show that the correlation between hematocrit and blood flow rate depends on the confinement of RBCs. References [1] Desjardins et al., Neurobiol. Aging, 37, 1947-1955 (2014) ; [2] Roman et al., Biomicrofluidics, 10, 034103 (2016) ; [3] Pries et al., Microvascular Res., 38, 81-101 (1989)

Modelling solute transport in the brain microcirculation: is it really well mixed inside the blood vessels?

Most network models describing solute transport in the brain microcirculation use the well-mixed hypothesis and assume that radial gradients inside the blood vessels are negligible. Recent experimental data suggest that these gradients, which may result from heterogeneities in the velocity field or consumption in the tissue, may in fact be important. Here, we study the validity of the well-mixed hypothesis in network models of solute transport using theoretical and computational approaches. We focus on regimes of weak coupling where the transport problem inside the vasculature is independent of the concentration field in the tissue. In these regimes, the boundary condition between vessels and tissue can be modelled by a Robin boundary condition. For this boundary condition and for a single cylindrical capillary, we derive a one-dimensional cross-section average transport problem with dispersion and exchange coefficients capturing the effects of radial gradients. We then extend this model to a network of connected tubes and solve the problem in a complex anatomical network. By comparing with results based on the well-mixed hypothesis, we find that dispersive effects are a fundamental component of transport in transient situations with relatively rapid injections, i.e. frequencies above one Hertz. For slowly varying signals and steady states, radial gradients also significantly impact the spatial distribution of vessel/tissue exchange for molecules that easily cross the blood brain barrier. This suggests that radial gradients cannot be systematically neglected and that there is a crucial need to determine the impact of spatio-temporal heterogeneities on transport in the brain microcirculation

VITAE : VIrTual brAin pErfusion

VITAE is an ERC funded software project aimed at providing full brain simulations of cerebral blood flow and solute exchange between blood and the neural tissue. The endgoal is to understand fine scale interactions between the architecture of the microvascular network in the brain and its functions (blood supply, oxygen and nutrient delivery, waste removal). This may indeed help unveil potential causes of cerebral disease like Alzheimer’s Disease. In the actual state of the art, full scale brain simulations are something new. First, acquiring input anatomical data of the blood vessel network is difficult and is an active domain of research. Next, simulation by itself is a CPU intensive Computational Fluid Dynamic problem requiring both inversion of large matrices and manipulation of large amounts of data. The current milestone is capable of running pressure resolution in a full mouse brain composed of about 5 millions of microvessels in one second on 1024 processor cores. The software written in C++ fully supports parallelized IO and graph partitioning to optimize the placement of vertices and reduce computing times. The next challenge is to run simulations taking the complex behavior of blood into account, which requires to run the pressure solver from one hundred to several thousand times. This will require to improve significantly the convergence time. Acknowledgements: ERC Funded Project: Proof of Concept (PoC), ERC-2018-PoC A. Sauvé, J.-D. Julien, M. Berg, M. Peyrounette, P. Elyakime, Y. Davit, M. Pigou, S. Lorthoi

Persistently elevated levels of sST2 after acute coronary syndrome are associated with recurrent cardiac events

Purpose Higher soluble ST2 (sST2) levels at admission are associated with adverse outcome in acute coronary syndrome (ACS) patients. We studied the dynamics of sST2 over time in post-ACS patients prior to a recurrent ACS or cardiac death. Methods We used the BIOMArCS case cohort, consisting of 187 patients who underwent serial blood sampling during one-year follow-up post-ACS. sST2 was batch-wise quantified after completion of follow-up in a median of 8 (IQR: 5-11) samples per patient. Joint modelling was used to investigate the association between longitudinally measured sST2 and the endpoint, adjusted for gender, GRACE risk score and history of cardiovascular diseases. Results Median age was 64 years and 79% were men. The 36 endpoint patients had systematically higher sST2 levels than those that remained endpoint free (mean value 29.6 ng/ml versus 33.7 ng/ml, p-value 0.052). The adjusted hazard ratio for the endpoint per standard deviation increase of sST2 was 1.64 (95% confidence interval: 1.09-2.34; p = 0.019) at any time point. We could not identify a steady or sudden increase of sST2 in the run-up to the combined endpoint. Conclusion Asymptomatic post-ACS patients with persistently higher sST2 levels are at higher risk of recurrent ACS or cardiac death during one-year follow-up

The 5-year data of the DESIR cohort

Funding Information: Acknowledgements the dESIr cohort was sponsored by the département de la recherche Clinique et du développement de l’Assistance publique–Hôpitaux de paris. this study is conducted under the umbrella of the French Society of rheumatology and InSErM (Institut national de la Santé et de la recherche Médicale). the database management is performed within the department of epidemiology and biostatistics (professor paul Landais, d.I.M., nîmes, France). An unrestricted grant from pfizer was allocated for the 10 years of the follow-up of the recruited patients. the authors would like to thank the different regional participating centres: pr Maxime dougados (paris – Cochin B), pr André Kahan (paris - Cochin A), pr olivier Meyer (paris - Bichat), pr pierre Bourgeois (paris - La pitié Salpetrière), pr Francis Berenbaum (paris - Saint Antoine), pr pascal Claudepierre (Créteil), pr Maxime Breban (Boulogne Billancourt), dr Bernadette Saint-Marcoux (Aulnay-sous-Bois), pr philippe Goupille (tours), pr Jean-Francis Maillefert (dijon), dr xavier puéchal, dr Emmanuel dernis (Le Mans), pr daniel Wendling (Besançon), pr Bernard Combe (Montpellier), pr Liana Euller-Ziegler (nice), pr philippe orcel, dr pascal richette (paris - Lariboisière), pr pierre Lafforgue (Marseille), dr patrick Boumier (Amiens), pr Jean-Michel ristori, pr Martin Soubrier (Clermont-Ferrand), dr nadia Mehsen (Bordeaux), pr damien Loeuille (nancy), pr rené-Marc Flipo (Lille), pr Alain Saraux (Brest), pr Corinne Miceli (Le Kremlin Bicêtre), pr Alain Cantagrel (toulouse), pr olivier Vittecoq (rouen). the authors would also like to thank UrC-CIC paris Centre for the coordination and monitoring of the study. Contributors All authors contributed and finally approved the current manuscript. Competing interests none declared. Patient consent obtained. ethics approval Comitte de protection des personnes Ile de France III. Provenance and peer review not commissioned; externally peer reviewed. Open Access this is an open Access article distributed in accordance with the Creative Commons Attribution non Commercial (CC BY-nC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited and the use is non-commercial. See: http://creativecommons.org/ licenses/by-nc/4.0/ © Article author(s) (or their employer(s) unless otherwise stated in the text of the article) 2017. All rights reserved. no commercial use is permitted unless otherwise expressly granted. Publisher Copyright: © Article author(s) (or their employer(s) unless otherwise stated in the text of the article) 2017. All rights reserved.Objective To estimate sacroiliac joint radiographic (X-SIJ) progression in patients with axial spondyloarthritis (axSpA) and to evaluate the effects of inflammation on MRI (MRI-SIJ) on X-SIJ progression. Methods X-SIJ and MRI-SIJ at baseline and after 2 and 5 years in patients with recent onset axSpA from the DESIR cohort were scored by three central readers. Progression was defined as (1) the shift from non-radiographic (nr) to radiographic (r) sacroiliitis (by modified New York (mNY) criteria) or alternative criteria, (2) a change of at least one grade or (3) a change of at least one grade but ignoring a change from grade 0 to 1. The effects of baseline inflammation on MRI-SIJ on 5-year X-SIJ damage (mNY) were tested by generalised estimating equations. Results In 416 patients with pairs of baseline and 5-year X-SIJ present, net progression occurred in 5.1% (1), 13.0% (2) and 10.3% (3) respectively, regarding a shift from nr-axSpA to r-axSpA (1), a change of at least one grade (2) or a change of at least one grade but ignoring a change from grade 0 to 1 (3). Baseline MRI-SIJ predicted structural damage after 5 years in human leukocyte antigen-B27 (HLA-B27) positive (OR 5.39 (95% CI 3.25 to 8.94)) and in HLA-B27 negative (OR 2.16 (95% CI 1.04 to 4.51)) patients. Conclusions Five-year progression of X-SIJ damage in patients with recent onset axSpA is limited but present beyond measurement error. Baseline MRI-SIJ inflammation drives 5-year radiographic changes.publishersversionpublishe