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**|**Biomed Opt Express**|**v.3(1); 2012 January 1**|**PMC3255331

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Biomed Opt Express. 2012 January 1; 3(1): 137–152.

Published online 2011 December 14. doi: 10.1364/BOE.3.000137

PMCID: PMC3255331

Center for Optical Diagnostics and Therapy, Department of Radiation Oncology, Erasmus Medical Center, PO Box 2040, 3000 CA Rotterdam, The Netherlands

Received 2011 September 29; Revised 2011 November 11; Accepted 2011 November 11.

Copyright © 2011 Optical Society of America

This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License, which permits download and redistribution, provided that the original work is properly cited. This license restricts the article from being modified or used commercially.

This article has been cited by other articles in PMC.

Quantitative determination of fluorophore content from fluorescence measurements in turbid media, such as tissue, is complicated by the influence of scattering properties on the collected signal. This study utilizes a Monte Carlo model to characterize the relationship between the fluorescence intensity collected by a single fiber optic probe (*F _{SF}*) and the scattering properties. Simulations investigate a wide range of biologically relevant scattering properties specified independently at excitation (

Detection and quantitation of fluorescence is important for many biomedical and clinical applications. The optical detection of fluorescent endogenous compounds [1] such as collagen and NADH, or exogenous compounds that include labelled markers, can be used for diagnostic purposes [2, 3]. The measurement of therapeutic compounds, such as photosensitizers used in photodynamic therapy [4, 5], may provide insight into the pharmacokinetic distribution and pharmacodynamic activity in tissues of interest and may play a role in monitoring administered therapies [6]. However, quantitation of fluorescence in tissue *in vivo* is complicated by the influence of the tissue optical properties on the collected fluorescence signal [7]. Absorption by chromophores within the tissue causes attenuation that is (non-linearly) proportional to the absorption coefficient at the excitation and emission wavelengths. Scattering within tissue is known to have a complicated effect on fluorescence measurements: the properties at the excitation wavelength (*λ _{x}*) affect the delivered excitation light profile and the properties at the emission wavelength (

Previously developed methods to extract intrinsic fluorescence spectra involve the acquisition of a paired measurement of fluorescence and white-light reflectance, where the latter is used to inform a correction of the influence of optical properties on fluorescence. This general approach has been extensively investigated for multi-fiber fluorescence probes, with separate source(s) and detectors [8–16]. These probes collect multiply scattered, or diffuse, light and sample volumes of tissue on the orders of several mm^{3}. An alternative approach for fluorescence measurements is to use small fiber optic probes that utilize a single optical fiber to both deliver excitation light and collect emitted fluorescence [5, 17–21]; such a measurement results in a localized sampling volume, with the majority of the collected signal originating very close to the probe face [18]. Single fiber fluorescence (SFFL) measurements collect photons that have undergone few scattering events, and in turn, have a very small light propagation path, making the collected intensity less sensitive to tissue chromophores and scattering properties than diffuse measurements. The influence of scattering on collected SFFL intensity has been previously investigated and was observed to be nonlinear and fiber-diameter specific [20]. Furthermore, the SFFL intensity was observed to be insensitive to variations in the scattering phase function (PF) [19]. The underlying mechanism of these factors was not fully elucidated. These and other previous studies accounted for the influence of scattering on SFFL by characterizing ranges of fiber diameters and optical property combinations where the SFFL signal was insensitive to optical properties [17, 19, 20]. While this approach may be useful for specific applications with well-known ranges of optical properties, it does not return a quantitative description of tissue fluorescence that is independent of optical properties, and therefore, does not provide a reliable comparison of measurements performed on different tissue locations or with different fiber diameters.

To the best of the authors’ knowledge, there is currently no analytical or empirical description of the influence of scattering properties on the fluorescence intensity sampled by a single fiber. The present study investigates the detailed mechanisms associated with the influence of scattering properties on the SFFL intensity measured in a turbid medium, and develops a mathematical model to correct for these influences. This represents a first step towards a full correction of collected SFFL intensities for the influence of optical properties (*i.e.* both scattering and absorption). Monte Carlo (MC) simulations are used to investigate SFFL measurement of a wide range of scattering properties that are independently specified at excitation and emission wavelengths; simulations also included a wide range of fiber diameters. Simulated data are used to identify and characterize a semi-empirical model that expresses SFFL intensity as a function of a dimensionless scattering property (given as the product of scattering coefficient and fiber diameter). The resulting model is applicable to all investigated fiber diameters and provides insight into the physics underlying the SFFL measurement.

The Monte Carlo (MC) code utilized in this study is a customized version of the MCML program [27] that is modified to emulate single fiber fluorescence measurements of a homogeneous turbid medium. The code allows independent specification of both the scattering coefficient (*μ _{s}*) and scattering phase function (

$${F}_{SF\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{ratio}}^{MC}=\frac{\mathit{TMPC}}{\mathit{TXPL}}$$

(1)

where TXPL is the total number of excitation photons launched and TMPC the total number of emission photons collected. Excitation and emission photons propagating within the medium far from the fiber face do not contribute to the collected fluorescence intensity and were terminated at a hemispherical limit from the fiber face of $10\frac{{d}_{\mathit{fib}}}{{\mu}_{s}^{\prime}}$; a limit that was confirmed to not influence model outputs for the range of optical properties investigated in this study. Model outputs of ${F}_{SF\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{ratio}}^{MC}$ were validated by comparison with previously published fluorescence intensities over a range of background optical properties [18].

During photon propagation, the photon positions were tracked in a discrete voxel grid to yield individual 2D(r,z) probability density profiles for all incident excitation photons, for all fluorescence emission photons, and a separate profile for all collected fluorescence photons. Specifically, the code generated 2D maps of the relative excitation light fluence (Φ* _{x}*(

$$\langle {Z}^{MC}\rangle =\frac{\sum _{i=1}^{{n}_{z}}{z}_{i}\left(\sum _{j=1}^{{n}_{r}}{F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}{\sum _{i=1}^{{n}_{z}}\left(\sum _{j=1}^{{n}_{r}}{F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}$$

(2)

where Δ*a _{j}* is the area of a voxel at position

$$\langle {\Phi}_{x}^{MC}\rangle =\frac{\sum _{i=1}^{{n}_{z}}\left(\sum _{j=1}^{{n}_{r}}{\Phi}_{x}\left({r}_{j},{z}_{i}\right){F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}{\sum _{i=1}^{{n}_{z}}\left(\sum _{j=1}^{{n}_{r}}{F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}$$

(3)

Similarly, a scalar for the effective escape probability within the optically sampled volume
$\left(\langle {H}_{m}^{MC}\rangle [-]\right)$ was calculated from the weighted average of the escape probability density distribution *H _{m}*(

$$\langle {H}_{m}^{MC}\rangle =\frac{\sum _{i=1}^{{n}_{z}}\left(\sum _{j=1}^{{n}_{r}}{H}_{m}\left({r}_{j},{z}_{i}\right){F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}{\sum _{i=1}^{{n}_{z}}\left(\sum _{j=1}^{{n}_{r}}{F}_{\mathit{col}}\left({r}_{j},{z}_{i}\right)\Delta {a}_{j}\right)\Delta z}$$

(4)

MC simulations were performed over a broad range of biologically relevant [28] reduced scattering coefficient (*μ*′* _{s}*) values that were individually specified at

A subset of simulations further investigated the influence of PF over a selected range of reduced scattering values, *μ*′* _{s}*(

Additionally, simulations investigated variations in *NA* from the baseline value of 0.22 over the range [0.1 – 0.4]. This subset of simulations was performed using the same scattering properties as the subset of simulations used to investigate the influence of PF.

Simulations of each possible combination of scattering properties were performed for a range of fiber diameters, with *d _{f}* = [0.2,0.4,0.6,1.0] mm. The absorption of the fluorophore was given as
${\mu}_{a}^{f}=0.1{\text{mm}}^{-1}$ in all simulations; this study did not consider absorption due to background chromophores. In total, the data presented in this study include 616 MC simulations, each launching at least 20 million photons.

The fluorescence signal F (in units of Joules [J]) collected by a fiber optic probe is given by the integral [12]

$$F=\left({\lambda}_{x}/{\lambda}_{m}\right){\mu}_{a}^{f}{Q}^{f}{\int}_{V}{\Phi}_{x}\left(\mathbf{r}\right){H}_{m}\left(\mathbf{r}\right){d}^{3}r$$

(5)

where Φ* _{x}*(

This study develops an approximate solution to Eq. (5) for a SFFL measurement by representing the volume integral of Φ* _{x}H_{m}* as the product of an effective optically sampled volume and the effective Φ

$${F}_{SF}\approx {\mu}_{a}^{f}{Q}^{f}\langle V\rangle \langle {\Phi}_{x}^{V}\rangle \langle {H}_{m}^{V}\rangle $$

(6)

where *V* is the effective sampling volume, and
$\langle {\Phi}_{x}^{V}\rangle $ and
$\langle {H}_{m}^{V}\rangle $ are the effective excitation fluence and effective escape probability within the sampled volume, respectively. These quantities can be related to the scalar outputs from the MC simulations defined in Section 2.1 by approximating the effective sampling volume as

$$\langle V\rangle \approx {A}_{1}\langle {Z}^{MC}\rangle {d}_{f}^{2}$$

(7)

with *Z ^{MC}* the effective sampling depth and

$$\langle {\Phi}_{x}^{V}\rangle ={P}_{x}\langle {\Phi}_{x}^{MC}\rangle $$

(8)

$$\langle {H}_{m}^{V}\rangle =\langle {H}_{m}^{MC}\rangle $$

(9)

where *P _{x}* is the total power output from the fiber, which in these simulations is proportional to the number of launched photons TXPL. Eq. (8) properly accounts for differences in the incident excitation intensity emitted from the fiber face for different numbers of launched photons TXPL, but does not correct for differences in incident excitation light intensity due to differences in fiber diameter. Since the incident excitation intensity is inversely proportional to the fiber area, it is expected that
${\Phi}_{x}^{MC}$ scales with
${d}_{f}^{-2}$. Substituting Eqs. (7–9) in Eq. (6), dividing by

$${F}_{SF\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{ratio}}^{MC}=\frac{TMPC}{TXPL}\approx {A}_{1}{\mu}_{a}^{f}{Q}^{f}\langle {Z}^{MC}\rangle {d}_{f}^{2}\langle {\varphi}_{x}^{MC}\rangle \langle {H}_{m}^{MC}\rangle $$

(10)

As described in Section 2.1, the MC simulations used in this study were used to return information about how SFFL intensity and the effective terms presented in Eq. (10) are influenced by scattering properties at the excitation and emission wavelengths. Inspection of the simulated data led to the identification of candidate empirical expressions to describe each quantity; from these a set of equations was selected on the basis of fit quality and model simplicity, and is given as

$$\langle {Z}^{MC}\rangle ={d}_{f}{A}_{2}{\left({\mu}_{s,\mathit{avg}}^{\prime}{d}_{f}\right)}^{-{A}_{3}}$$

(11)

$$\langle {\Phi}_{x}^{MC}\rangle ={d}_{f}^{-2}{B}_{1}{e}^{\frac{-1}{{B}_{2}\left({\mu}_{s}^{\prime}\left({\lambda}_{x}\right){d}_{f}\right)+1}}$$

(12)

$$\langle {H}_{m}^{MC}\rangle ={C}_{1}{e}^{\frac{-{C}_{3}}{{C}_{2}\left({\mu}_{s}^{\prime}\left({\lambda}_{m}\right){d}_{f}\right)+1}}$$

(13)

where [*A*_{1,2,3}, *B*_{1,2}, *C*_{1,2,3}] in Eqs. (10–13) are fitted parameters. The effective sampling depth *Z ^{MC}* was observed to follow an exponential decay with respect to the product of

$$\frac{{F}_{SF\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathit{ratio}}^{MC}}{{\mu}_{a}^{f}{Q}^{f}{d}_{f}{\nu}_{n}}={\zeta}_{1}{\left({\mu}_{s,\mathit{avg}}^{\prime}{d}_{f}\right)}^{-{\zeta}_{2}}{e}^{\left(\frac{-1}{{\zeta}_{2}\left({\mu}_{s}^{\prime}\left({\lambda}_{x}\right){d}_{f}\right)+1}-\frac{{\zeta}_{3}}{{\zeta}_{2}\left({\mu}_{s}^{\prime}\left({\lambda}_{m}\right){d}_{f}\right)+1}\right)}$$

(14)

where [*ζ*_{1},*ζ*_{2},*ζ*_{3}] are fitted parameters. This represents a reduction from the parameter set specified in Eqs. (10–13). Here, *ζ*_{1} represents the product of *A*_{1}, *A*_{2}, *B*_{1} and *C*_{1}. Fitted parameters were estimated using a Levenberg-Marquardt algorithm coded into a Matlab script (version 2009a, MathWorks). Confidence intervals of the estimated parameters were calculated from the square root of the diagonal of the covariance matrix [29]. During the model fit analysis, the estimated values for *A*_{3}, *B*_{2} and *C*_{2} were observed to have overlapping 95% confidence intervals [29], which led to the reduction of these terms to a single fitted parameter, *ζ*_{2}. This substitution did not result in a significant increase in model residual error.

Continuing the description of the terms in Eq. (14), *ν _{n}* represents the influence of the index of refraction mismatch at

Equation (14) represents a fiber diameter dependent expression that relates fluorescence collected by a single fiber with diameter *d _{f}* that has been distorted by scattering at excitation and emission wavelengths, to the intrinsic fluorescence
${\mu}_{a}^{f}{Q}^{f}$ within the sampled turbid medium. For brevity, the quantity
${F}_{SF}^{\mathit{sim}}\left[\text{m}\right]$ will be used throughout this manuscript to refer to the expression

$${F}_{SF}^{\mathit{sim}}=\frac{{F}_{SF\hspace{0.17em}\mathit{ratio}}^{MC}}{{\mu}_{a}^{f}{Q}^{f}{\nu}_{n}}$$

(15)

MC simulations investigated the relationship between single fiber fluorescence and variations in *μ*′* _{s}*, initially specified as equivalent at

Effect of reduced scattering coefficient (equivalent at *λ*_{x}, *λ*_{m}) on single fiber fluorescence intensity. Linear and log scales of the data are presented in the following panel pairings: A and B show collected
${F}_{SF}^{\mathit{sim}}$ vs. *μ*′ **...**

Inspection of the fiber-diameter specific
${F}_{SF}^{\mathit{sim}}$ vs. *μ*′* _{s}* profiles led to the identification of two dimensionless transformations that are important for interpretation of the data. First, transformation of the abscissa to dimensionless reduced scattering, given as the product

The data investigated in Figure 1 are for the case *μ*′* _{s}*(

In tissue the exact form and wavelength-dependence of the PF is not well characterized. This study utilized a subset of MC simulations to investigate in detail the influence of PF on
${F}_{SF}^{\mathit{sim}}$, as described in Section 2.2. The
${F}_{SF}^{\mathit{sim}}$ showed minimal influence from variation among different phase functions, with < 3% variation between
${F}_{SF}^{\mathit{sim}}/{d}_{f}$ values returned from the 19 simulated PFs at each of the dimensionless reduced scattering values (data not shown). For simulations specifying different PFs at *λ _{x}* and

This study utilized a subset of MC simulations to investigate in the influence of fiber NA on
${F}_{SF}^{\mathit{sim}}$, as described in Section 2.2. Simulated data showed that the effect of fiber NA on
${F}_{SF}^{\mathit{sim}}$ is well approximated by an NA^{2} proportionality, with < 5% mean residual error between estimates of *F _{SF}* measured by fibers of NA= [0.22] and NA= [0.1,0.4] in the investigated scattering range (data not shown), with increasing deviations associated with decreasing dimensionless reduced scattering values.

MC simulations were used to investigate the dependence of optical sampling depth, excitation fluence, and emission escape probability within the sampled volume on *μ*′* _{s}*(

A) Dimensionless sampling depth *Z*^{MC}/*d*_{f} vs. the product of average of reduced scattering coefficients at excitation and emission wavelengths, *μ*′_{s,avg} and *d*_{f}. B) Excitation fluence within the sampled volume,
$\langle $ **...**

MC simulations also returned scalar metrics representative of effective excitation fluence and effective emission escape probability within the optically sampled volume. Figure 3B displays
$\langle {\Phi}_{x}^{MC}\rangle {d}_{f}^{2}$ vs. *μ*′* _{s}*(

Figure 4 shows
${F}_{SF}^{\mathit{sim}}/{d}_{f}$ simulated by the MC model vs. estimated by the fit of Eq. (14). Here the estimated parameter values of *ζ*_{1} = 0.0935 ± 0.003, *ζ*_{2} = 0.31 ± 0.01, and *ζ*_{3} = 1.61 ± 0.05 resulted in the minimum weighted residual error between simulated and model-estimated
${F}_{SF}^{\mathit{sim}}$ values. The model estimates were strongly correlated with simulated outputs, with the quality of the fit given by the Pearson correlation coefficient of *r* = 0.991 and displayed by the proximity of the data points to the plotted line of unity. The mean absolute residual between simulated and model estimated values is < 3% and all data points have a mean residual error that is < 10% of the simulated value. Figures 5A and B show simulated and model estimated
${F}_{SF}^{\mathit{sim}}/{d}_{f}$ vs. *μ*′* _{s}*(

Dimensionless single fiber fluorescence intensity estimated by fitted model vs. MC simulated values. Data include variations of *μ*′_{s}(*λ*_{x}) and *μ*′_{s}(*λ*_{m}). Line of unity included for comparative purposes.

This study utilizes a Monte Carlo model to characterize the relationship between the fluorescence intensity collected by a single fiber (*F _{SF}*) and the scattering properties within an optically sampled turbid medium. Simulated data were used to identify a relationship between dimensionless fluorescence intensity,
${F}_{SF}^{\mathit{sim}}/{d}_{f}$, and dimensionless reduced scattering. We found that the collected fluorescence does not scale exclusively with dimensionless reduced scattering at the excitation wavelength, nor with dimensionless reduced scattering at the emission wavelength; rather it shows a more-complicated dependence on the reduced scattering coefficients at both wavelengths. These data were used to develop a semi-empirical model that expresses
${F}_{SF}^{\mathit{sim}}/{d}_{f}$ as the product of an effective sampling volume, and the effective excitation fluence and the effective escape probability within the effective sampling volume. The influence of scattering properties on each of these components was identified and mathematically described using simulation outputs. The semi-empirical model of
${F}_{SF}^{\mathit{sim}}/{d}_{f}$ accurately describes simulated fluorescence intensities over a wide range of biologically relevant scattering properties.

The fluorescence model, given in Eq. (14), utilizes empirical functions to represent the individual components of the SFFL measurement, including *Z ^{MC}*/

$${\mu}_{s,h-\mathit{avg}}^{\prime}=\frac{1+{\zeta}_{3}}{\frac{1}{{\mu}_{s}^{\prime}}+\frac{{\zeta}_{3}}{{\mu}_{s}^{\prime}}}$$

(16)

These observations suggest that the right hand side of the ${F}_{SF}^{\mathit{sim}}/{d}_{f}$ profile (phase (2)) is dominated by a combination of excitation fluence build-up and increased fluorescence escape probability close to the fibertip for increasing reduced scattering coefficients at excitation and emission wavelengths, respectively. These observations are consistent with mechanisms that were previously proposed, but not explicitly investigated, in studies of localized [21] or single fiber measurements [20] of fluorescence.

The semi-empirical model developed in this study provides a method to return scattering-independent *F _{SF}* quantities provided that

The PF-specific analysis presented in this study indicates that quantitative analysis of SFFL requires determination of *μ*′* _{s}*(

In order to appropriately utilize the semi-empirical model of SFFL presented in this study, it is important to consider the assumptions and approximations utilized in its development. The mathematical modeling approach utilized in this study represents the collected fluorescence intensity in terms of the product of three factors contributing to fluorescence that were extracted from Monte Carlo models outputs; these relationships are presented in the transition from Equation 6 to 10. A critical assumption of this modeling approach is that the effective scalar values for these components are representative of the more complicated 2-D maps of these properties. The empirical models of each of the components expresses a high quality of fit, providing evidence that this assumption is reasonable. Another important point of this study is the specific investigation of a single optical fiber in contact with a turbid medium; the exact form of the expressions governing light transport have been defined for this geometry. While the approach to modeling SFFL utilized here is extensible to modifications in measurement geometry, it is important to note that changes to the geometry will result in changes to the excitation and emission light distributions, and will require assessment of the appropriateness and accuracy of the specified model structures. Such modifications include interstitial placement of the fiber optic in the sampled medium, or placement of the fiber optic into a probe face surrounded by epoxy, metal, or other optical fibers; ongoing work is investigating these influences. Another important consideration is that this study characterized the scattering dependence of *F _{SF}*

In summary, the current study utilized MC simulations to investigate the influence of scattering properties on fluorescence intensity collected by a single fiber probe. Simulated data were used to identify an underlying dimensionless relationship between fluorescence intensity and dimensionless reduced scattering. Results indicate that the mathematical model of *F _{SF}* is valid over a wide range of reduced scattering coefficients, in the range

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