This work aims at quantifying the effect of inherent uncertainties from cardiac output on the sensitivity of a human compliant arterial network response based on stochastic simulations of a reduced-order pulse wave propagation model. A simple pulsatile output form is used to reproduce the most relevant cardiac features with a minimum number of parameters associated with left ventricle dynamics. Another source of significant uncertainty is the spatial heterogeneity of the aortic compliance, which plays a key role in the propagation and damping of pulse waves generated at each cardiac cycle. A continuous representation of the aortic stiffness in the form of a generic random field of prescribed spatial correlation is then considered. Making use of a stochastic sparse pseudospectral method, we investigate the sensitivity of the pulse pressure and waves reflection magnitude over the arterial tree with respect to the different model uncertainties. Results indicate that uncertainties related to the shape and magnitude of the prescribed inlet flow in the proximal aorta can lead to potent variation of both the mean value and standard deviation of blood flow velocity and pressure dynamics due to the interaction of different wave propagation and reflection features. Lack of accurate knowledge in the stiffness properties of the aorta, resulting in uncertainty in the pulse wave velocity in that region, strongly modifies the statistical response, with a global increase in the variability of the quantities of interest and a spatial redistribution of the regions of higher sensitivity. These results will provide some guidance in clinical data acquisition and future coupling of arterial pulse wave propagation reduced-order model with more complex beating heart models.
We give an unified framework to solve rough differential equations. Based on flows, our approach unifies the former ones developed by Davie, Friz-Victoir and Bailleul. The main idea is to build a flow from the iterated product of an almost flow which can be viewed as a good approximation of the solution at small time. In this second article, we give some tractable conditions under which the limit flow is Lipschitz continuous and its links with uniqueness of solutions of rough differential equations. We also give perturbation formulas on almost flows which link the former constructions.This theory was very fruitful to study stochastic equations driven by Gaussian process which is not covered by the Itô framework, like the fractional Brownian motion [13,26]. More generally, the rough path framework allows one to separate the probabilistic from the deterministic part in such equation and to overcome some probabilistic conditions such as using adapted or non-anticipative processes.Recently, the ideas of the rough path theory were extended to stochastic partial differential equations (SPDE) with the works of [21,22] which have led to significant progress in the study of some SPDE. This theory also found applications in machine learning and the recognizing of the Chinese ideograms [10,24].Since the seminal article [25] by T. Lyons in 1998, several approaches emerged to solve (1). They are based on two main technical arguments: fixed point theorems [20,25] and flow approximations [2,12,14,16,19]. In particular, the rough flow theory allows one to extend work about stochastic flows, which has been developed in '80s by Le Jan-Watanabe-Kunita and others, to a non-semimartinagle setting [4].The main goal of this article is to give a framework which unifies the approaches by flow and pursue further investigations on their properties and their relations with families of solutions to (1).A flow is a family of maps t , u from a Banach space to itself such that ,˝, " , for any ď ď . Typically, the map which associates the initial condition to the solution of (1) has a flow property. The existence of a such flow heavily depends on the existence and uniqueness of the solution. However, it was proved in [7,8] and extended to the rough path case in [6] that when non-uniqueness holds, it is possible to build a measurable flow by a selection technique. In this article we are interested by the construction of a Lipschitz flows.The main idea to build the flow associated is to find a good approximation , of , when |´| is small enough. We iterate this approximation on a subdivision " t ď 﨨¨ď ď u of r , s by setting , :" ,˝¨¨¨˝, . Ifdoes converges when the mesh of goes to zero, , the limit is necessarily a flow This computation is similar to the ones of numerical schemes as Euler's methods of different order [11]. Moreover, this idea is found among the Trotter's formulas for bounded or unbounded linear operators which allows to compute the semi-group of the sum of two non-commutative operators only knowing the semi-groups associated to e...
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The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were inspired by the rough path theory. We take a pedagogical approach to facilitate the understanding of this new theory. We recover results of the rough path theory with the regularity structure framework. Hence, we show how to formulate a fixed point problem in the abstract space of modelled distributions to solve the rough differential equations. We also give a proof of the existence of a rough path lift with the theory of regularity structure.
Kupffer cells (KCs) are of bone‐marrow origin. After liver transplantation, recipient KCs are supposed to replace donor KCs. On the other hand, KCs are currently hypothesized to play a major immunogenic role in acute liver allograft rejection. In the present study, we investigated the immunogenic role of KCs in acute rat liver allograft rejection. For this purpose, we depleted the donor KCs using intravenous injection of liposome‐encapsulated dichloromethylene diphosphonate (DMDP) in the fully allogenic ACI‐to‐LEW rat liver transplantation model. Kupffer cell depletion was confirmed using monoclonal antibodies ED2. In a first set of experiments, graft survival was evaluated, as were body weight and serum bilirubin changes, after the transplantation. Graft survival time showed no difference between the groups (treated, 12.5 ± 0.92 days; control, 11.9 ± 0.80 days). Body weight and serum bilirubin changes were similarly affected in both groups. In a second set of experiments, recipients were killed on day 6 after the transplantation, and rejection was histologically graded from 0 to 4. All grafted livers were judged as grade 3 regardless of treatment. ED2 staining showed KCs repopulation in both untreated and the dichloromethylene diphosphonate treated livers. The results of the present study provide evidence that KCs do not play an important immunogenic role in acute liver allograft rejection of the rat.
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