Direct inversion scheme of time-domain fluorescence diffuse optical tomography by asymptotic analysis of peak time

TL;DR

The paper addresses locating a fluorescent point target inside a scattering medium using time-domain FDOT measurements by exploiting the peak time of the temporal response. It develops asymptotic expansions for the approximate peak time in regimes of small and large fluorescence lifetime $\ell$, establishing explicit $t$–to–$\lambda$ relations that translate peak timing into a sphere radius $r=\sqrt{vD\,\lambda^2-\frac{|x_d-x_s|^2}{4}}$ and, from three source-detector pairs, a tetrahedral geometry to pinpoint the target. Key contributions include closed-form asymptotic formulas for $t^{p}_0$ and $t^{p}_\infty$, a direct inversion for $\lambda$ from $t_{peak}$, and a practical, non-iterative 3-SD reconstruction scheme with numerical validation. The approach enables fast and robust FDOT target localization and provides a foundation for extending to non-point targets and curved measurement surfaces via Green's function and parabolic-scaling analyses.

Abstract

This paper proposes a direct inversion scheme for fluorescence diffuse optical tomography (FDOT) to reconstruct the location of a point target using the measured peak time of the temporal response functions. A sphere is defined for the target, with its radius determined by the peak time, indicating that the target lies on the sphere. By constructing a tetrahedron with edges determined by the radii, we identify the location of the target as the vertex of the tetrahedron. Asymptotically, we derive the relationship between the radius of the sphere and the peak time. Several numerical tests are implemented to demonstrate the accuracy and performance of the asymptotic relationship and the inversion scheme.

Figures (4)

• Figure 3.1: Peak time $t_{peak}$ and the asymptotic $t_{peak}^s$. For (b) and (c), we set $\ell = 100 \; {\rm ps}$.
• Figure 3.2: Peak time $t_{peak}$ and the approximate peak time $t_{peak}^{l_0}$. For (b) and (c), we set $\ell = 1000 \; {\rm ps}$.
• Figure 3.3: Peak time $t_{peak}$ and the approximate peak time $t_{peak}^{l}$. For (b) and (c), we set $\ell = 1000 \; {\rm ps}$.
• Figure 4.1: Tetrahedron with four vertices $O,\;A,\;B$ and $C$. Point $A$ is the center point of the initial S-D pair. Points $B$ and $C$ are the center points of the second and third S-D pairs, respectively. Point $C$ is the target location.

Theorems & Definitions (12)

• Lemma 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5
• Definition 2.6
• Theorem 2.7
• proof