Inverse Problems in Diffuse Optical Tomography Applications

Abstract

Optical computed tomography, known as Diffuse Optical Tomography (DOT), uses near-infrared light to quantitatively investigate in vivo and without ionizing radiations the spatial distribution of optical properties in biological tissues. While the potentiality of this technology as a screening and diagnostic tool is very promising, significant issues have hindered in the past its development. These were mainly related to: (1) choice of light sources capable of deep penetration in the tissue; (2) complexity of the physics of light crossing biological tissues. Focusing on this latter issue, one must observe that in the tissue light undergoes to strong scattering, so that emerging photons are the result of diffusive (random) paths, which are not easily traceable back. This, in turn, implies an inherent difficulty in reconstructing the distribution of the optical coefficients that gave rise to those paths, the so-called DOT inverse problem. As a matter of fact, from the mathematical viewpoint, this problem displays severe ill-conditioning and an appropriate numerical treatment of regularization is needed in order to avoid aberrant image reconstruction. The present material provides a somewhat “gentle introduction” to the solution of the inverse problem arising from DOT screening applications. We detail one of the possible procedures to address this problem and we propose a framework to obtain a computational solution along with numerical tests to validate the implementation. Much attention is paid to obtain a fast computation time (less than a few minutes), which is an essential requirement to produce an imaging instrument suitable for mass screening.

Appendix: Green’s Function Solution of the DE Model

The DE model and its successive manipulations in the Rytov procedure yield relation that represent instances of the inhomogeneous modified Helmholtz equation of the form:

$$\displaystyle \begin{aligned}{}[\varDelta-\alpha^2]\phi(r)=f(r). \end{aligned} $$

(1.27)

We are interested in finding an analytic solution to this partial differential equation. It is clear that this will be possible only on simple geometries. In particular, we will refer to the solution on a infinite or semi–infinite domain and we will “pretend” that it can be used as it is for our finite domain. With this aim, for a linear operator $\mathcal {L}$, we introduce the Green’s function G = G(r − r ^′), such that

$$\displaystyle \begin{aligned}\mathcal{L}G(r-r^\prime)=\delta(r-r^\prime) \end{aligned}$$

where δ is the Dirac delta centered in r ^′. It holds the following (see e.g., [7])

Lemma 1.1

Given the partial differential equation $\mathcal {L}\phi (r) = f(r)$, r ∈ Ω, if G(r − r ^′) is the Green’s function with respect to the linear partial differential operator $\mathcal {L}$, then a solution to the PDE is given by the convolution between the source term f and the Green’s function

$$\displaystyle \begin{aligned} \phi(r)=\int_\varOmega G(r-r^\prime) f(r^\prime)\, dr^\prime {} \end{aligned} $$

(1.28)

The Green’s function for our operator at hand $\mathcal {L}=(\varDelta -\alpha ^2)$ has the following expression for n = 2 or 3 (see e.g. [7])

$$\displaystyle \begin{aligned} G(r-r^\prime) = \left\{ \begin{array}{lcl} -\displaystyle \frac{1}{2\pi}K_0(\alpha||r-r^\prime||) & \qquad n=2, \\ {} -\displaystyle \frac{1}{4\pi}\frac{\mathrm{e}^{-\alpha||r-r^\prime||}}{||r-r^\prime||} & \qquad n=3, \end{array} \right. {} \end{aligned} $$

(1.29)

where K ₀ is the modified Bessel function of the second kind of order zero (in Matlab® you can compute it with the command besselk). Observe that the Green’s functions in (1.29) refer to an infinite domain and only satisfy the radiation condition |G|→ 0 for |r|→∞. As a result, the corresponding solution obtained from the convolution procedure does not satisfy the proper boundary conditions on the finite domain Ω. In order to partially correct this fact, you can use the so–called dipole approximation which allows to enforce null solution over a plane, which is a convenient condition in our case for the part of the breast laying on the solid plate. To fix ideas, let us think this plane to correspond to z = 0. When a point source is placed at a small depth ℓ into the sample, an equivalent opposite “sink” is placed at − ℓ. The following dipole solution is obtained for a source located in x ^′ = (0, 0, ℓ)

$$\displaystyle \begin{aligned}G_D(x, x^\prime) =G(x, (0,0,\ell)-G(x, (0,0,-\ell) ). \end{aligned}$$

This expression guarantees that ϕ = 0 on the plane.