Application of the Laplace transform in time-domain optical spectroscopy and imaging

Abstract

In this paper, we consider the Laplace transform (LT) for solving different time-dependent photon migration problems occurring in the biomedical optics field. It is shown that the LT exhibits important advantages in view of accuracy, efficiency, and numerical stability compared to the classical approach that uses the Fourier transform (FT) to obtain time-dependent quan-tities from data in the frequency domain. For typical applications in tissue spectroscopy or imaging, a speed-up of up to several orders of magnitude can be accomplished by applying the LT for both numerical or analytical solution approaches. Modeling of light propagation in scattering media, such as biological tissue, in the mesoscopic and macroscopic scales is commonly performed using the radiative transport equation (RTE) or its approximation, the diffusion equation (DE). Analytical solutions of these equations in the time domain are restricted to relative simple geometries, 1–3 whereas for a series of applications, efficient analytical solutions are available in the frequency domain. 4,5 Usually, the FT is applied to obtain the time-domain solutions from the corresponding solutions in the frequency domain. Similarly, in the case of numerical sol-utions of these equations, calculations in the frequency domain are more efficient than in the time domain. Again, commonly, the FT is applied to obtain solutions in the time domains. In this paper, we show, using exemplary analytical solutions of the DE, that the application of the LT is considerably more efficient and accurate compared to the use of the FT for obtaining time-domain solutions. We present comparisons for the fluence in an infinitely extended medium, for the reflectance from a two-layered medium, and for the fluorescence in an in-finitely extended medium. Analytical solutions of the time-dependent DE are provided either by the separation of variables method or the integral trans-forms, such as the FT, which is defined as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 6 3 ; 1 9 9 fðtÞ ¼ 1 2π Z ∞ −∞ FðωÞe iωt dω; ω ∈ R: (1) However, a serious problem when using the FT [Eq. (1)] arises when FðωÞ exhibits a slow algebraic decay for increasing angu-lar modulation frequency ω. As a result, one has to deal with a highly oscillatory integrand, which is known to be difficult to integrate numerically. For example, we consider E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 3 2 6 ; 7 5 2