An extended analytical approach for diffuse optical imaging

Abstract

In this work, we introduce an analytical method to solve the diffusion equation in a cylindrical geometry. This method is based on an integral approach to derive the Green’s function for specific boundary conditions. Using our approach, we obtain comprehensive analytical solutions with the Robin boundary condition for diffuse optical imaging in both two and three dimensions. The solutions are expressed in terms of the optical properties of tissue and the amplitude and position of the light source. Our method not only works well inside the tissue but provides very accurate results near the tissue boundaries as well. The results obtained by our method are first compared with those obtained by a conventional analytical method then validated using numerical simulations. Our new analytical method allows not only implementation of any boundary condition for a specific problem but also fast simulation of light propagation making it very suitable for iterative image reconstruction algorithms.

An accelerated photo-magnetic imaging reconstruction algorithm based on an analytical forward solution and a fast Jacobian assembly method

“…In our previous work [45], we obtained a comprehensive analytical solution for the diffusion equation based on an integral method for the Robin boundary condition. This solution is obtained by deriving a particular Green’s function based on the integral method.…”

“…The solution of the homogeneous diffusion equation in 2D cylindrical polar coordinates for a Dirac delta like point source was previously presented [45]. Implementing this photon density solution in equation (2), and using the separation of variables method lead to the final expression describing the laser induced temperature variations [51].…”

“…Both diffusion [33–45] and Pennes’ bio-heat equations [46–50] have been extensively solved analytically on regular geometries such as cylindrical, spherical and infinite or semi infinite slabs. These methods directly provide continuous spatiotemporal distribution of temperature inside the studied medium [51].…”

Real-time photo-magnetic imaging

“…Technically, utilizing the properties of the Dirac delta function and this boundary condition leads to an extensive photon density expression that will be used in the source term of Eq. (2) [17]. After implementing the heat source term, the separation of variables method is used to obtain the final temperature expression at any position (r,θ) and at any time t [13]:…”

Ex vivo validation of photo-magnetic imaging

“…Considering the optical absorption of the medium, the photon’s energy is converted to heat [11] and used as a source term in the bio-heat equation to generate spatiotemporal temperature maps [12]. This system of combined equations is usually solved using either the finite element method (FEM) [6,7] or analytical methods [13–16]. As a final step, these simulated temperature maps are compared to the measured ones obtained by MRT to solve the PMI inverse problem and obtain high-resolution optical absorption maps.…”