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**|**HHS Author Manuscripts**|**PMC3544305

Opt Lett. Author manuscript; available in PMC 2013 January 14.

Published in final edited form as:

Opt Lett. 2012 November 15; 37(22): 4783–4785.

PMCID: PMC3544305

NIHMSID: NIHMS430433

Anand T. N. Kumar, Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, Massachusetts 02129, USA;

Anand T. N. Kumar: ude.dravrah.hgm.rmn@ramukna

We show that a multiexponential model for time-resolved fluorescence allows the use of an absorption-perturbation Monte Carlo (MC) approach based on stored photon path histories. This enables the rapid fitting of fluorescence yield, lifetimes, and background tissue absorptions in complex heterogeneous media within a few seconds, without the need for temporal convolutions or MC recalculation of photon path lengths. We validate this method using simulations with both a slab and a heterogeneous model of the mouse head.

Several methods have been developed for the efficient Monte Carlo (MC) computation of light transport with intrinsic [1–3] and fluorescence [4–7] contrast in turbid tissue. For fluorescence, the most efficient approach is the adjoint fluorescence MC (aFMC, or reverse-emission MC [4]), which is based on convolving the absorption and emission Green’s function distributions with the fluorescence decay. In addition to a temporal double convolution, aFMC requires recomputation of the distributions for changes in tissue absorptions and a fine spatial and temporal binning for accuracy [4]. These aspects make aFMC computationally intensive for tomographic applications. Here we present an alternative fluorescence MC approach using a multiexponential model for time-resolved diffuse fluorescence (TRF) [8], which incorporates fluorescence lifetimes into Green’s functions implicitly through a reduced absorption. This allows a direct MC calculation of the entire TRF using perturbation MC [1,3,7], using photon path histories in Beer–Lambert factors. The method avoids the convolutions and spatial binning errors inherent in aFMC, while allowing rapid recalculation for changes in fluorophore or tissue absorption.

Consider a diffuse medium, with background absorption and scattering {
${\mu}_{a}^{x},{\mu}_{s}^{x}$} at excitation (*λ _{x}*) and {
${\mu}_{a}^{e},{\mu}_{s}^{e}$} at the emission (

$${U}_{F}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)=\sum _{n}{A}_{n}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t){e}^{-{\mathrm{\Gamma}}_{n}t},$$

(1)

with the time-dependent decay amplitudes given by

$${A}_{n}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)={\int}_{0}^{t}{\text{d}t}^{\prime}\left[\int {\text{d}}^{3}{rW}_{n}^{x,e}({\mathbf{r}}_{s},{\mathbf{r}}_{d},\mathbf{r},{t}^{\prime}){\eta}_{n}(\mathbf{r})\right].$$

(2)

Here
${W}_{n}^{x,e}={G}_{n}^{x}({\mathbf{r}}_{s},\mathbf{r},t){G}_{n}^{e}(\mathbf{r},{\mathbf{r}}_{d},t)$ is a convolution of the Green’s functions
${G}_{n}^{x,e}$, which are just the Green’s functions *G ^{x}*,

A direct application of the path history approach to evaluate
${G}_{n}^{x}$ and
${G}_{n}^{e}$ in Eq. (2) would, however, be computationally intensive since it requires the storage of path histories from each source/detector to every voxel within the medium. Alternately, we first recognize that the quantity in the square brackets in Eq. (2) is simply the Born approximation term for the time-resolved (TR) photon fluence for an absorption perturbation equivalent to *η _{n}*(

Consider the imaging medium as divided into subregions [based for, e.g., on anatomically segmented images from magnetic resonance imaging (MRI) or computed tomography (CT)] indexed by “*j*.” The regionwise background and fluorophore absorptions are denoted as
${\mu}_{a}^{mj}$ and
${\mu}_{fn}^{xj}$. If
${L}_{k}^{j}$ denotes the path length of the *k*th photon in the *j*th region as computed by the MC simulation using the mean scattering coefficient,
${\mu}_{s}^{m}$, Eq. (2) can be approximated as (using Beer–Lambert law to calculate Φ)

$${A}_{n}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)={q}_{n}\sum _{k}^{N(t)}\underset{{e}^{-\left({\mu}_{a}^{mj}-\frac{{\mathrm{\Gamma}}_{n}}{v}\right){L}_{k}^{j}}\left[1-{e}^{-{\mu}_{fn}^{xj}{L}_{k}^{j}}\right]}{\overset{}{j=1J}},$$

(3)

where ‘*k*’ runs over the *N*(t) photons arriving between times 0 and *t* at detector **r*** _{d}*. Note that the summation over

We tested the accuracy of the history-FMC (hFMC) method [Eqs. (1) and (3)] by comparing it with the aFMC method. For the aFMC, Green’s functions from each **r*** _{s}* and

The accuracy of the entire TRF as predicted by hFMC was quantified as the rms error:
$E={[(1/N){\mathrm{\sum}}_{k}{({U}_{k}^{\text{his}}-{U}_{k}^{\text{adj}})}^{2}/{U}_{k}^{\text{adj}}]}^{1/2}$, which was less than 5% across the entire range of detectors and optical properties studied. Figure 2 shows a representative TRF for a single source–detector pair for the slab case, computed using the aFMC (solid blue) and hFMC (red circles). Also shown is the prediction of hFMC without the use of the reduced absorption in Eq. (3) (dotted black curve), which results in *E >* 15% and a lifetime error of *>*20% (Fig. 1, dashed-dotted curve), with the mismatch in the early time points of the TRF particularly noteworthy (Fig. 2). These results indicate the nontrivial role played by the reduced absorption in accurately predicting entire TRF. Equation (3) is thus also applicable for early photon tomography [11].

We also tested the accuracy of hFMC for a heterogeneous, anisotropic medium with complex boundaries and multiple fluorophores. We used a publicly available segmented mouse atlas [12] [Fig. 3(a)], retaining only the head region (29 mm × 17 mm × 17 mm, 0.3 mm voxels) and assigned optical properties {
${\mu}_{a}^{x},{\mu}_{s}^{x},g$} ranging from {0.001 mm^{−1}, 1 mm^{−1}, 0} (eyes) to {0.02 mm^{−1}, 12.5 mm^{−1}, 0.9} (brain). For each tissue segment,
${\mu}_{a}^{e}=0.8{\mu}_{a}^{x}$, and
${\mu}_{s}^{e}=0.8{\mu}_{s}^{x}$. Four brain regions were assigned fluorophores with distinct lifetimes: cerebellum (0.6 ns), cerebrum (0.8 ns), striatum (1 ns) and rest of the brain (1.5 ns), all with
${\mu}_{f}^{x}={10}^{-4}\phantom{\rule{0.16667em}{0ex}}{\text{mm}}^{-1}$. Figure 3(b) shows representative TRFs computed using hFMC and aFMC (10^{8} photons used for both) for a single source (*S*) and two detectors (D1 and D2) located on the mouse head. The two methods showed excellent agreement (*E <* 3%) across a range of detectors on the surface [open circles in Fig. 3(a)]. While aFMC required more than 3 h/source or detector using two quad-core Xeon 5472 CPUs, hFMC took less than 1 s for recalculation of the full TRF for any set of background absorption, fluorophore yield, and lifetimes.

(Color online) (a) Heterogeneous absorption map of a digitized mouse, with four brain regions assigned fluorophores with distinct lifetimes as indicated. (b) Representative TRF curves predicted by aFMC (solid blue) and hFMC (red dots) for a source *S*(x) **...**

The simulations presented here illustrate the feasibility of the hFMC method for rapid fitting of fluorescence yield and lifetime in heterogeneous turbid media. The increasing availability of high-resolution CT or MRI anatomical maps in conjunction with optical tomography makes hFMC particularly relevant for multimodality imaging. The hFMC can allow a rapid and accurate estimation of fluorescence in anatomical segments (e.g., fluorophore-labeled plaques or cancer cells within organs) rather than recovering full distributions, which are limited by the ill posedness of diffuse optical tomography. In addition, the approach can naturally take advantage of lifetime multiplexing [10] when lifetime sensitive probes are available, while also being applicable to other TR techniques such as early photon tomography [11]. In combination with hardware-accelerated MC implementations [13], the hFMC approach could be generalized to include fitting of tissue scattering, thereby making real-time recovery of *in vivo* fluorescence yield and lifetimes a possibility.

This work was supported by the National Institutes of Health grant R01 EB015325.

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