Approximate peak time to time-domain fluorescence diffuse optical tomography for nonzero fluorescence lifetime

Abstract

This paper concerns an inverse problem for fluorescence diffuse optical tomography (FDOT) reconstructing locations of multiple point targets from the measured temporal response functions. The targets are multiple fluorescent point objects with a nonzero fluorescence lifetime at unknown locations. Peak time, when the temporal response function of the fluorescence reaches its maximum, is a robust parameter of the temporal response function in FDOT because it is most less suffered by the artifacts, such as noise, and is easily determined by experiments. We derive an approximate peak time equation based on asymptotic analysis in an explicit way in the case of nonzero fluorescence lifetime when there are single and multiple point targets. The performance of the approximation is numerically verified. Then, we develop a bisection algorithm to reconstruct the location of a single point target from the algorithm proposed in [4] for the case of zero fluorescence lifetime. Moreover, we propose a boundary-scan algorithm for the reconstruction of locations of multiple point targets. Finally, several numerical experiments are implemented to show the efficiency and robustness of the addressed algorithms.

Figures (5)

• Figure 3.1: Peak time, approximate peak time, and relative error for different physical parameters
• Figure 3.2: Peak time, approximate peak time, and relative error for different S-D pairs
• Figure 4.1: (a) peak time, (b) approximate peak time, and (c) relative error for S-D pairs defined by \ref{['SD8_index']}
• Figure 5.1: Noise-free peak times measured by different S-D pairs \ref{['SD_index']} in Example \ref{['exa_mult01']}: (a) 3D contour plot of peak times for $m,\,n=0,\,1,\,\cdots,\,20$, (b) peak times for $m=0,\,1,\,\cdots,\,20,\,n=4,\,5,\,6,\,16,\,17,\,18$
• Figure 5.2: Peak times, noisy peak times ($\hat{\delta}=0.1\%$) and smoothed noisy peak times for $m=0,\,1,\,2,\,\cdots,\,20$ and $n=5,\,17$

Theorems & Definitions (20)

• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Remark 2.5
• Theorem 3.1
• proof