A Rainbow Inverse Problem

We consider the radiative transfer equation (RTE) with reflection in a three-dimensional domain, infinite in two dimensions, and prove an existence result. Then, we study the inverse problem of retrieving the optical parameters from boundary measurements, with help of existing results by Choulli and Stefanov. This theoretical analysis is the framework of an attempt to model the color of the skin. For this purpose, a code has been developed to solve the RTE and to study the sensitivity of the measurements made by biophysicists with respect to the physiological parameters responsible for the optical properties of this complex, multi-layered material. Résumé. On étudie l’équation du transfert radiatif (ETR) dans un domaine tridimensionnel infini dans deux directions, et on prouve un résultat d’existence. On s’intéresse ensuite à la reconstruction des paramètres optiques à partir de mesures faites au bord, en s’appuyant sur des résultats de Choulli et Stefanov. Cette analyse sert de cadre théorique à un travail de modélisation de la couleur de la peau. Dans cette perspective, un code à été fait pour résoudre l’ETR et étudier la sensibilité des mesures effectuées par les biophysiciens par rapport aux paramètres physiologiques tenus pour responsables des propriétés optiques de ce complexe matériau multicouche. Introduction Skin is a complex multi-layered media and the most important organ of our body in terms of weight, surface and functionalities. For many years, physicists have tried to understand what physiological components or properties are responsible for its color. The color of an object is defined by a unidimensional curve called the reflectance spectrum, which is the relative energy given back by the object for each wavelength of the visible range, when it is enlighted with a white spot. Physicists have developed a lot of models to link the physiological components of the skin (like, for example, the blood concentration or the diameter of the melanosomes) to its optical properties. Physicists have simulated, by many ways, how light travels into the skin. What has not been theoretically investigated yet, even if very well studied by Magnain and Elias in [2], is the inverse problem of retrieving the physiological parameters from measurements made at the surface of the skin. Before studying the inverse problem, we tried to simulate the direct one, and developed a small Matlab code to do so. This code is proved to be quite satisfying for this purpose, hence we used it to make a sensitivity study of the reflectance curves with respect to the physiological parameters. We went on with the theoretical study of this inverse problem, in a very simplified framework and based on the existing work of Choulli and Stefanov [5]. The paper is organized in the reverse order: theoretical study, then numerical results. 1 Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret, 75013 Paris, e-mail:[email protected] 2 UMPA, ENS Lyon, 46, allée d’Italie 69364 Lyon Cedex 07, e-mail: [email protected] 3 CMLA, Cachan, 61 Avenue du Président Wilson, 94235 Cachan Cedex, e-mail: [email protected] © EDP Sciences, SMAI 2010 ha l-0 05 18 16 4, v er si on 1 16 S ep 2 01 0 2 ESAIM: PROCEEDINGS 1. Modeling 1.1. The radiative transfer equation When light enters an object X , the photons propagate in straight line, unless they are absorbed by the material or scattered (and possibly deviated) by various entities. One classical way to describe the light intensity is the use of a probability density function (p.d.f.): f , that depends on the position x and the velocity v of the photon. The set of all possible directions, V is all or part of the sphere S. The physical interaction with the material is described by: • the absorption coefficient μa (given in m ), which is the number of absorption events per unit length and depend on the position and the direction: μa = μa(x, v) • the scattering coefficient μs = μs(x, v) (also in m ), the same quantity for the scattering • the kernel p = p(x, v, w) which is a probability density with respect to v. It denotes the probability for a photon arriving with the direction w, to get a new direction v after having hit a scatterring center. If the scattering centers are distant enough from one another (compared to the wavelength), the radiative transfer equation (RTE) describes properly the behaviour of the light intensity: v · ∇xf + (μa(x, v) + μs(x, v))f(x, v) = ∫ V μs(x,w)p(x, v, w)f(x,w) dw in X × V (1) 1.2. Geometry and boundary conditions The equation (1) has been written with no internal source of light, and has to be complemented by boundary conditions to model the enlightment of the object. The typical experiment we are interested in is the following: the skin is enlighted from its top, on a large surface, and its color is registered on the same zone. At this scale, the two others dimensions can be considered as infinite. Indeed, the thickness of the whole skin is of the order of 10 m, whereas the skin is much more extended over our body. Hence, we will model our skin sample as a box infinite in the two planar directions. When travelling into the skin, the light will encounter several interfaces, one of them being the epidermisdermis junction. Part of the light will get through it, but the remaining amount will be reflected. Hence, our boundary conditions will be: • a source function f− modeling the enlightment on the top • reflection of part of the light at each interface encountered. 2. Theoretical inverse problem Choulli and Stefanov have already proved in [5] that the parameters can be uniquely determined by surface measurements, under the following assumptions, in the case where the RTE (1) is complemented with Dirichlet boundary conditions. We will conduct the same study with mixed boudary conditions by adding a reflection operator. Before getting into the inverse problem, we have to show existence of the light intensity for the direct problem. 2.1. Notations and functional framework We focus in this study on a single layer, so that the first interface encountered is the bottom of the sample. Hence, the position and velocity spaces are: X =]0, L[×R, V = S. (2) ha l-0 05 18 16 4, v er si on 1 16 S ep 2 01 0 ESAIM: PROCEEDINGS 3