I would like to thank all members of the REO team: they have always been available for scientific discussions. In the first part of the thesis we present a simplified model of fluid-structure interaction that includes autoregulation.
Motivation
Retinal blood autoregulation has recently been modeled on 0D networks [AHS+13] and on 1D networks [CM15]. In [ACS08, AHS+13], the authors investigated the biological factors that induce the contraction of smooth muscle cells.
Contents and manuscript organization
The method is designed for the modeling of multi-physics systems where one, or more, compartments can be replaced by a reduced representation of the corresponding Steklov-Poincaré operator. The approach is based on the low-rank decomposition of the Steklov-Poincaré operator that represents the action of one of the compartments.
My contributions
As an interesting numerical side effect, the presence of fibers makes the model less sensitive than others to strong variations or inaccuracies in wall curvatures. The main focus of this study is the simulation of autoregulation of blood flow in retinal arteries.
Structure model
The total thickness of the structure is denoted by hs and its density is denoted by ρs. In the following, for the sake of simplicity, the overall contribution of the structure will be denoted as:
Fluid-structure coupling
However, in view of the application that motivated this work (autoregulation of hemodynamics, see Chapter 3), it is important to calculate the flow variation caused by wall dynamics. Letv, q, χ,wbe tests functions defined in appropriate functional spaces according to the boundary conditions of the problem.
Numerical discretization
In this way, the motion of the structure was implicitly incorporated as a boundary condition for the fluid problem. This is particularly important for accuracy in calculating the normal component of the velocity and thus the displacement field.
Numerical testcases
In Fig.2.5 the displacement in the longitudinal direction in the middle of the flat region fort= 0.012 is shown. In Fig.2.6 the same curves are reported for the displacement in the non-flat region.
Application
From a practical point of view, it is interesting to note that these fluid structure results were obtained at a computational cost similar to a fluid problem. It would be interesting to compare the results with those provided by more complex approaches, as was done in [CDQ14] for other simplified models.
Appendix: derivation of the nonlinear elastic energy
Appendix: Details on the time discretization of the boundary condition 32
Autoregulation consists of an active change in the diameter of the artery in response to a change in mean perfusion pressure. In this part, the behavior of the structure is similar to that found in the first test case.
Fluid-structure coupling: main modelling assumptions
Two conditions must be satisfied at the fluid-structure interface Γt: the continuity of the velocity and the continuity of the stress. Since the structural displacement is assumed to be parallel to the normal direction, the equations for the continuity of the velocity for all x ∈ Γ are u(I −n⊗n)|x+η(x)n(x) ) = 0 and u·n|x+η (x)n(x) =∂tη.
Modelling the vessel wall dynamics
In this section, a model that describes the behavior of the smooth muscle cells (SMC) is examined. A chemical state model of the smooth muscle cells has been proposed by Hai and Murphy [HM88]. The consequence of Eq. (3.15) is the appearance, in the balance of the normal forces on Γ, of a force term.
Autoregulation and pressure feedback
For the sake of clarity, we now summarize the model derived in the previous sections. We estimate ζ using the mean values of the incoming pressure over the different cardiac cycles. For simplicity, all these compartments are assumed to share the same values for the resistances (Rprox and Rdistal) and the capacitance (C).
Numerical simulations
To compare the results of our simulation with the experimental data presented in [RGSP85], we calculate the average value (in time, over a cardiac cycle) of the blood velocity in the center of different sections of the artery along the network. The selected points of the network are depicted in Fig.3.7, and a comparison of the data is given in Fig.3.8. The results of the model are within the same range of values as the experimental data.
Limitations and conclusions
We briefly introduce the anatomy of the eye and then discuss the state of the art in modeling the human eye. Theuvea denotes the area of the eye that includes the choroid, hunis and ciliary body. The iris, the colored part of the eye, is the last part of the uvea and begins at the ciliary body and enters the anterior cavity.
Fluid dynamics modelling in the eye
The outflow rate of the trabecular pathway depends on the pressure drop between the IOP and the pressure in the episcleral vein. A review of the various studies that aim to quantitatively estimate the flow can be found in [McL09]. Buoyancy forces are the main driving mechanism of the flow (see e.g. [CGDF02,FG06]).
Models of eye mechanics
An improvement on this work [GMJ11] consists of a more detailed description of the area surrounding the optic nerve, taking into account the peripapillary sclera and lamina cribrosa. The model is solved by exploiting the symmetries in the geometry and the tension in the center of the lamina is used as an external force acting on the vessel wall of the central retinal artery. The goal is to understand the effects of changes in IOP on the hydraulic resistance of the central retinal artery.
Models of the eye as a global system
We show some preliminary results on the poro-elastic model of the choroid, on the flow in the anterior cavity and on a two-compartment model of the choroid and the vitreous. The model reproduces the mechanics and the impact of hemodynamics on the various structures. For now, we are working on a geometry of the eye that includes multiple compartments such as the choroid, the vitreous, and the sclera.
Multi-domain and multi-physics modelling of the eye
Despite being a rough simplification of reality, this is already an extremely complex model to handle. The aim of this study is to begin to tackle the problem of providing a comprehensive model of the eye. ΩCS Corneoscleral shell, thick elastic shell, almost incompressible, material properties change in different parts of the domain;.
Modelling aqueous humor
The temperature was assigned to the interface with cornea with T = 35◦ and on the lowest interfaces with the lens, the zonules and the ciliary body: T = 37◦. At the interface with the iris and the part of ciliary body near the cornea, homogeneous Neumann boundary conditions were imposed on the temperature. In agreement with the literature, we recover two different configurations of aqueous humor recirculation, depending on the direction of gravity.
Poro-elastic model of the choroid
The second equation of the Biot's system is the balance of the liquid mass which reads as follows. We remember here the expression for the time derivative of the increase in liquid mass as it will be useful in the following. Which is the case of the processes where the liquid is maintained at a given constant pressure p = ¯p+p0.
Choroid coupled with the vitreous
First, the Stokes problem in ΩV is solved by using the choroid velocity of the previous iteration as the Dirichlet datum. With this approach it is now possible to assign the time derivative of the choroid displacement as a Dirichlet datum for the vitreous compartment. The boundary conditions and parameters of the choroid are the same as in the test case of section 5.4.2.
Conclusions, on-going and future work
The idea developed in the next chapter is to replace such a compartment with a reduced representation of the corresponding Steklov-Poincaré operator. A Reduced Order Representation of the Poincaré-Steklov Operator: An Application to Coupled Multiphysics Problems. In the online phase, only the limited representation of the operator is needed to take into account the influence of the external problem on the main system.
Introduction
An approximation of the Poincaré-Steklov operator via a Padé expansion was detailed in [LP10] for the study of the vibrations in fluid-structure couplings. In the present work, a low rank decomposition of the Poincaré-Steklov operator is calculated by a Reduced-Order Modeling method. The output of P2 at the interface is used to obtain a low rank analysis of the Poincaré-Steklov operator.
Problem formulation
The coupled problem described above can be solved using the Domain Decomposition (DD) method. A complete review and detailed treatment of domain decomposition methods can be found in [QV99,TW05,SBG04,MQ89]. Other coupling schemes can be considered, such as Neumann-Neumann and Robin-Robin schemes.
Outline of the method
In special (but meaningful) geometric parameters, the basis coincides with the eigenfunctions of the Poincaré-Steklov operator (see Appendix 6.7). Online updating of the database is similar in spirit to that proposed in [PW15, AZW15]. This is a reduced eigenvalue problem for representing Poincaré-Steklov eigenfunctions in Laplace-Beltrami ones (ie Uij).
Numerical Analysis
The second point can be proved by considering the approximation properties of the Laplace-Beltrami eigenfunctions (see e.g. [CHQZ07,APV15]). The proof of the convergence for the inverse map is more delicate since the map causes loss of regularity. The speed of solving the problem P2 is expected to be very high.
Numerical Experiments
Therefore, this test is a good benchmark to examine the performance of the method both in. The total cost in terms of number of solved problems P2 is the sum of the two. The final time for the simulation was T = 0.2 for a total number of time steps equal to: 1108.
Conclusions and perspectives
Appendix
Publications
In AppendixAwe have attached a procedure on a retinal imaging tool that we developed for the reconstruction of the geometries used in the first two chapters.
Implementation
We attached a proceedings of the EMBC 2015 conference (the 37th Annual International Conference of the IEEE Engineering in Medicine and Biology Society).
