CONCLUDING REMARKS

level of shear stress, until it reaches a steady state value. Yao et al. [70] linked the imbal-ance of the qì in traditional Chinese medicine to the imbalimbal-ance of the nutrients and waste exchanges between blood and interstitial fluid.

1.4 Concluding remarks

From a mathematical modeling point of view, the remaining problems arise :

(1) How can the physical stress that serves as stimuli during acupuncture needling be studied?

During moxibustion therapy, a local temperature field is developed by directly or indirectly applying a burning moxa stick on the skin at acupoints. The temperature contours have been predicted in the case of direct moxibustion and needle moxi-bustion therapy [63]. The computed temperature decreases rapidly away from the heating point.

During acupuncture needling, does the mechanical stress field created by the in-sertion of a needle behave in the same way ? Shear stress and interstitial pressure are the two main physical parameters needed to measure the mechanical stress field developed by the motions of the needle. Mastocytes are considered as one of the primary actors in acupuncture therapy. Degranulation of mastocytes can occur un-der high shear stress induced by interstitial flow. In the interstitial fluid models, such as (1.7), studied in [67, 68], shear stress on mastocytes varies with interstitial pressure. The interstitial pressure is considered as a boundary condition and is thus essential to predict the flow. In the mastocyte degranulation dynamic model (1.10), studied in [53], shear stress on mastocytes is considered as a parameter.

One of the main purposes of chapter 3 is to give a positive answer to the previous question and provide some tools to predict numerically the shear stress on masto-cytes and pressure of the interstitial flow by extending the studies of the Brinkman model with a moving needle.

(2) How can the response of mastocytes to external physical stimuli be modeled?

The models of mastocyte degranulation dynamics [53] and more recently mastocyte-nerve interaction [71] have been studied. These models strengthen the idea raised by experimentation that mastocytes could play a primary role at the acupoint. The models give a clearer insight on the mechanisms of mechanotransduction and de-granulation processes induced by acupuncture and the interaction of released nerve stimulants with nerve endings. However, it is not clear how the observed abundance of mastocytes at acupoint could be a factor in the effects of acupuncture. More gen-erally, to our knowledge, the evolution of the spatial distribution of mastocytes in response to physical stimuli has yet to be studied. Recent experiments [72] have been designed to exhibit the temporal and spatial dynamics of mastocytes, blood

1.4. CONCLUDING REMARKS

vessels and nerves in stimulated acupoints by comparing the initial and final states of the components at the acupoint.

The main purpose of chapter 4 is to give a first model of the spatial evolution of mastocytes in response to physical stimuli. The model integrates magnitude and the spatial range of physical stimuli to which the mastocytes respond by releasing chemoattractants, nerve stimulants, and endocrine stimulants. The transport equa-tion of mastocyte density is derived from the random (brownian) diffusion of the mastocytes with a bias directed by the released chemoattractant concentration. In that way, the model can describe the dynamics of mastocytes and make it easier to understand the role of the abundance of mastocytes near acupoints. The model reveals that mastocyte chemotactic recruitment plays a key role in the long lasting physiological effects of acupuncture.

Chapter 2 FreeFem++

Contents

2.1 Introduction . . . 50 2.2 FreeFem++ and its interpreted language . . . 50 2.2.1 The syntax . . . 50 2.2.2 Meshing tools and mesh exemples . . . 51 2.2.3 Finite element method . . . 60 2.3 Solving problems in FreeFem++ . . . 64 2.3.1 Evolution problem . . . 64 2.3.2 Incompressible Navier-Stokes equation . . . 66 2.3.3 Moving domain problem in computational fluid dynamics . . . 69 2.4 Concluding remarks . . . 76

Abstract

The second chapter of this study is designed to briefly introduce the software FreeFem++

developed by Hecht [1]. It includes details on the tools needed to solve the differential system presented in this thesis within the finite element method. Parts of this chapter are from courses and presentations given within the FreeFem++ Activity Group of the Taiwan Society of Industrial and Applied Mathematics (TWSIAM Activity Group : FreeFem++.

http://homepage.ntu.edu.tw/~twhsheu/twsiamff++/freefem.html.

Accessed January 26, 2015.). A full documentation of FreeFem++ can be found on the FreeFem++ website (http://www.freefem.org/ff++/. Accessed January 26, 2015.).

2.1. INTRODUCTION

2.1 Introduction

FreeFem++ is an open source partial differential equation solver developed by Frédéric Hecht, in collaboration with Olivier Pironneau, Jacques Morice, Antoine Le Hyaric and Kohji Ohtsuka, in the Laboratory Jacques-Louis Lions (LJLL) of University Pierre et Marie Curie (Paris, France). FreeFem++ runs on Unix, Windows and MacOS platforms.

FreeFem++ allows its users to solve problems which admit a weak formulation. Users can write their own algorithms in FreeFem++ scripts to solve 2D and 3D PDEs and visualize their results within the software. Freefem++ includes the following modules: triangu-lar finite element spaces, including discontinuous FE spaces; automatic mesh generator;

mesh adaptation; fast linear solvers; MPI (Message Passing Interface) tools for parallel computing.

2.2 FreeFem++ and its interpreted language

2.2.1 The syntax

FreeFem++ language is an interpreted language based on the C++ object oriented pro-gramming language. FreeFem++ script instructions are executed directly by the FreeFem++

software. FreeFem++ syntax embeds the C++ syntax. For instance, the syntax for the loops for and while or the conditional statements if/else is the same as C++ with the use of the brackets { } to define blocks. As in C++, the user can declare and define types of variables. FreeFem++ basic types real, int, and bool correspond in C++

to double, long, and bool, respectively. The declared variables can be manipulated with most of the usual C++ operators. A special feature of the FreeFem++ language is that some variable types, such as the mesh type mesh, the finite element space type fespace, or the variational formulation type varf, are relevant to the finite element method. Examples of types used in FreeFem++ are given in script 2.1.

1 x,y, z / / Cartesian coordinates

2 N .x, N .y, N . z / / Normal vector components

3 i n t k = 1 0 ; / / integer

4 r e a l a = 2 . 5 ; / / real

5 b o o l b = ( a < 3 . ) ; / b o o l e a n

6 r e a l [ i n t] a r r a y ( k ) ; / / array of k real elements

7 a r r a y [ ] [ 5 ] ; / / 6th value of the array

8 mesh Th ; / / 2d mesh

9 mesh3 Th3 / / 3d mesh

10 f e s p a c e Vh ( Th ,P1) ; / / finite element space

11 Vh u=x; / / finite element function

12 Vh3< complex > uc = x+ 1 . i ∗y; / / complex finite element function