The inverse problem for time-harmonic diffuse optical tomography: stability and reconstruction in anisotropic media

Diffuse Optical Tomography (DOT) is a promising medical imaging technique, which shows potential to be a less expensive, more versatile and a more comfortable alternative to already established imaging modalities, such as magnetic resonant imaging (commonly referred to as MRI). In time-harmonic DOT, low frequency infra-red light is delivered to the surface of the object being examined (usually the brain or breast) and the light propagation (i.e., the photon density) is measured, also on the surface of the object. These measurements are then used to reconstruct the interior, spatially dependent, optical properties (absorption and scattering coefficients - µa and µs respectively) of the object.

From a mathematical point of view, the object of interest is a bounded domain Ω ⊂ R n , n ≥ 3 with Lipschitz boundary ∂Ω. Light propagation through Ω is modelled by the diffusion equation, obtained by applying the so-called P1-approximation to the Radiative Transfer Equation. The measurements are modelled by the so-called Dirichlet-to-Neumann map and the DOT inverse problem addressed here is the stable determination of (the derivatives of any order of) µa = µa(x), x ∈ Ω, when µs = µs(x), x ∈ Ω is assumed to be known and the medium Ω is anisotropic. The anisotropic nature of the diffusion tensor in our DOT forward model is encompassed by the real, matrix-valued function B = B(x), x ∈ Ω, which can be obtained using other imaging modalities such as MRI.

Our stability results are at the boundary of Ω, ∂Ω, and are of Hölder type. They rely on the construction of novel singular solutions of the DOT forward operator, having an isolated singularity at the centre of a ball. As the DOT forward operator considered here is a second-order partial differential operator with complex coefficients, under suitable conditions, it is equivalent to a strongly elliptic real system. We construct singular solutions for such a system, extending the results in Journal of Differential Equations, 84 (2): 252- 272 for the single elliptic equation in the context of the companion Calder´on’s inverse conductivity problem.

The Hölder stability estimates of the derivatives Dhµa, of any order h, are then obtained first for media Ω having an isotropic layer near the boundary. Here we provide two intervals of variability for low and high frequencies, for which our stability results hold true. The intervals of variability are expressed quantitatively in terms of the a-priori assumptions made on our DOT inverse problem. The stability estimates are then extended to fully anisotropic media. Here our stability results hold true in the low frequency regime only, with some preliminary results also with local boundary measurements. The upper bound for the low frequencies is not expressed quantitatively in terms of the a-priori assumptions here.

We compliment these analytical results with 2-dimensional reconstructions of both µa and µs in anisotropic domains, represented by squares of size 1 cm × 1 cm and 2 cm × 2 cm. We investigate the effect the frequency of the light and the presence of anisotropy have on the photon density, providing some important insights into the underlying physics of the problem. We then apply these insights to chose suitable values for the frequency and anisotropy matrix B in our reconstructions, which we obtain using Dirichlet, Neumann and Cauchy data.