Semi-empirical model of the effect of scattering on single fiber fluorescence intensity measured on a turbid medium

S. C. Kanick; D. J. Robinson; H. J. C. M. Sterenborg; A. Amelink

doi:10.1364/BOE.3.000137

1. Introduction

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 (λ_m) determine the likelihood that fluorescent emission photons propagate to the detector used in the measurement. In order to quantitatively analyze fluorescence in tissue, it is important to obtain an intrinsic fluorescence signal that is independent of the optical property effects [8–10]. This approach would yield a quantity that is proportional to the product of the concentration and quantum yield of the fluorophore within the optically sampled volume, and would be comparable between measurements of samples with different background tissue optical properties.

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 mm³. 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.

2. Methods

2.1. Monte Carlo model

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 (P(θ)) at excitation (λ_x) and emission (λ_m) wavelengths. Excitation photons were initialized by selecting a location on the fiber face, which is modeled in contact with the turbid medium at the air/medium interface z = 0, and were launched into a direction within the fiber cone of acceptance, where the acceptance angle was given as $Θ_{a} = asin (\frac{NA}{n_{medium}})$ ; both the location and the direction were sampled from uniform distributions. The index of refraction (n) of the medium and fiber were specified at 1.37 and 1.45, respectively, and were held constant between λ_x and λ_m. The numerical aperture (NA) of the fiber was set as 0.22. Reflection and refraction due to the index of refraction mismatch at the medium/fiber and the surrounding medium/air interface were calculated using the Fresnel equations and Snell’s law. This code simulated propagation of excitations photons by stochastically selecting step sizes (s_n) from an exponential distribution weighted by μ_s(λ_x), and each scattering angle was selected from P(θ)(λ_x). At discrete points along each individual step, excitation photons were stochastically checked for a fluorescence event, with the probability given by $e^{- μ_{a}^{f} s_{n}}$ , where $μ_{a}^{f}$ is the specific absorption coefficient of the fluorophore. Stochastic absorption by the fluorophore resulted in an isotropic scattering event, and propagation of the emission photon was continued at the scattering properties at λ_m. Emission photons propagating within the turbid medium that cross the medium interfacial boundary at z = 0, were checked for contact with the fiber face; those in contact and traveling at an angle within the fiber cone of acceptance were collected, the rest were terminated. Excitation photons contacting the fiber face at any angle were terminated and did not contribute to the collected fluorescence intensity. This calculation returned the fraction of the number of collected fluorescence photons and the number of excitation photons for each simulation, calculated as:

F_{S F ratio}^{M C} = \frac{TMPC}{TXPL}

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_{fib}}{μ_{s}^{'}}

; a limit that was confirmed to not influence model outputs for the range of optical properties investigated in this study. Model outputs of

F_{S F ratio}^{M C}

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(r,z) [m⁻²]), which is calculated as previously described [27], and of the photon probability density of fluorescence collected by the fiber (F_col(r,z) [m⁻³]), which represents the spatial location of origin for all collected fluorescence photons [18]. Note that these quantities involve ratio calculations and do not depend on the number of launched excitation photons. From these maps, the dimensionless escape probability density profile of emission photons $(H (r, z) = \frac{F_{col} (r, z)}{μ_{a}^{f} (r, z) Φ_{x} (r, z)})$ , which is defined as the probability of emission photon collection per fluorescence photon generated, was calculated. Note that the fluorescence generated at a location (r,z) is proportional to the product of $μ_{a}^{f} (r, z)$ and Φ_x(r,z). These 2-D spatial profiles were used to calculate effective values for the volume sampled and the excitation fluence and escape probability within the sampled volume, by properly weighting each respective quantity by the collected fluorescence that originated at the corresponding location. A scalar effective optical sampling depth (〈Z^MC〉 [m]) is calculated as the weighted average depth of the collected emission photons, given as

〈 Z^{M C} 〉 = \frac{\sum_{i = 1}^{n_{z}} z_{i} (\sum_{j = 1}^{n_{r}} F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}{\sum_{i = 1}^{n_{z}} (\sum_{j = 1}^{n_{r}} F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}

where Δa_j is the area of a voxel at position r_j and Δz is the z-dimensional length of each voxel [27]. A scalar effective excitation fluence within the optically sampled volume

(〈 Φ_{x}^{M C} 〉 [m^{- 2}])

was calculated from the weighted average of Φ_x(r,z), with the collected fluorescence photon probability density F_col(r,z) as weight factors, as

〈 Φ_{x}^{M C} 〉 = \frac{\sum_{i = 1}^{n_{z}} (\sum_{j = 1}^{n_{r}} Φ_{x} (r_{j}, z_{i}) F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}{\sum_{i = 1}^{n_{z}} (\sum_{j = 1}^{n_{r}} F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}

Similarly, a scalar for the effective escape probability within the optically sampled volume

(〈 H_{m}^{M C} 〉 [-])

was calculated from the weighted average of the escape probability density distribution H_m(r,z), with the collected fluorescence photon probability density F_col(r,z) as weight factors, as

〈 H_{m}^{M C} 〉 = \frac{\sum_{i = 1}^{n_{z}} (\sum_{j = 1}^{n_{r}} H_{m} (r_{j}, z_{i}) F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}{\sum_{i = 1}^{n_{z}} (\sum_{j = 1}^{n_{r}} F_{col} (r_{j}, z_{i}) Δ a_{j}) Δ z}

2.2. Monte Carlo simulations

MC simulations were performed over a broad range of biologically relevant [28] reduced scattering coefficient (μ′_s) values that were individually specified at λ_x and λ_m, with: μ′_s(λ_x) = [0.1,0.25,0.5,1,2,4,8] mm⁻¹ and μ′_s(λ_m) = [0.25,0.5,0.75,1.0] × μ′_s(λ_x). This series of simulations was performed at all specified μ′_s combinations using the Modified Henyey-Greenstein (MHG) PF [24] with the anisotropy specified as g₁ = 0.9 and γ, which characterizes the first two moments of the phase function and is given as $γ = \frac{1 - g_{2}}{1 - g_{1}}$ , was set as γ = 1.4.

A subset of simulations further investigated the influence of PF over a selected range of reduced scattering values, μ′_s(λ_x) = [0.5,1,2] mm⁻¹ and μ′_s(λ_m) = [0.5,1.0] × μ′_s(λ_x), using the Henyey-Greenstein (HG) PF with g₁ = [0.5,0.9] and γ = [1.5,1.9] and the MHG PF with combinations of g₁ = [0.8,0.9,0.95] and γ = [1.4,1.5,1.6,1.7,1.8,1.9].

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 $μ_{a}^{f} = 0.1 {mm}^{- 1}$ 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.

2.3. Semi-empirical model of the single fiber fluorescence intensity

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

F = (λ_{x} / λ_{m}) μ_{a}^{f} Q^{f} \int_{V} Φ_{x} (r) H_{m} (r) d^{3} r

where Φ_x(r) [J m⁻²] is the excitation fluence, H_m(r) [−] is the escape probability of emission photons,

μ_{a}^{f} [m^{- 1}]

is the absorption coefficient of the fluorophore at the excitation wavelength, and Q^f [−] is the fluorescence quantum yield. The ratio (λ_x/λ_m) accounts for the difference in photon energy between the emission and excitation wavelengths, and will be omitted in the remainder of the equations since in the Monte Carlo simulations this ratio is set to unity. The volume integral of Φ_xH_m is dependent on optical properties; however the intrinsic fluorescence, given by the product of

μ_{a}^{f} Q^{f}

is independent of optical properties and has dimensions [m⁻¹].

This study develops an approximate solution to Eq. (5) for a SFFL measurement by representing the volume integral of Φ_xH_m as the product of an effective optically sampled volume and the effective Φ_x and H_m values within that volume, thus redefining Eq. (5) as,

F_{S F} \approx μ_{a}^{f} Q^{f} 〈 V 〉 〈 Φ_{x}^{V} 〉 〈 H_{m}^{V} 〉

where 〈V〉 is the effective sampling volume, and

〈 Φ_{x}^{V} 〉

and

〈 H_{m}^{V} 〉

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

〈 V 〉 \approx A_{1} 〈 Z^{M C} 〉 d_{f}^{2}

with 〈Z^MC〉 the effective sampling depth and A₁ a proportionality factor that depends on the geometrical shape of the effective volume. Furthermore, the effective excitation fluence and escape probability within the sampled volume can be written as

〈 Φ_{x}^{V} 〉 = P_{x} 〈 Φ_{x}^{M C} 〉

〈 H_{m}^{V} 〉 = 〈 H_{m}^{M C} 〉

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

Φ_{x}^{M C}

scales with

d_{f}^{- 2}

. Substituting Eqs. (7–9) in Eq. (6), dividing by P_x, which is then replaced by TXPL yields SFFL expressed as a percentage of incident excitation photons, as

F_{S F ratio}^{M C} = \frac{T M P C}{T X P L} \approx A_{1} μ_{a}^{f} Q^{f} 〈 Z^{M C} 〉 d_{f}^{2} 〈 ϕ_{x}^{M C} 〉 〈 H_{m}^{M C} 〉

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

〈 Z^{M C} 〉 = d_{f} A_{2} {(μ_{s, avg}^{'} d_{f})}^{- A_{3}}

〈 Φ_{x}^{M C} 〉 = d_{f}^{- 2} B_{1} e^{\frac{- 1}{B_{2} (μ_{s}^{'} (λ_{x}) d_{f}) + 1}}

〈 H_{m}^{M C} 〉 = C_{1} e^{\frac{- C_{3}}{C_{2} (μ_{s}^{'} (λ_{m}) d_{f}) + 1}}

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 μ′_s,avg(λ_x,λ_m)d_f, where μ′_s,avg(λ_x,λ_m) is calculated as the average of μ′_s(λ_x) and μ′_s(λ_m). The effective relative excitation fluence

〈 Φ_{x}^{M C} 〉

was observed to scale with

d_{f}^{- 2}

as expected, and to follow an exponential expression that depended on the product μ′_s(λ_x)d_f. The effective escape probability

〈 H_{m}^{M C} 〉

was observed to follow an exponential function that dependent on the product μ′_s(λ_m)d_f. Substituting Eqs. (11–13) into Eq. (10) results in

\frac{F_{S F ratio}^{M C}}{μ_{a}^{f} Q^{f} d_{f} ν_{n}} = ζ_{1} {(μ_{s, avg}^{'} d_{f})}^{- ζ_{2}} e^{(\frac{- 1}{ζ_{2} (μ_{s}^{'} (λ_{x}) d_{f}) + 1} - \frac{ζ_{3}}{ζ_{2} (μ_{s}^{'} (λ_{m}) d_{f}) + 1})}

where [ζ₁,ζ₂,ζ₃] are fitted parameters. This represents a reduction from the parameter set specified in Eqs. (10–13). Here, ζ₁ represents the product of A₁, A₂, B₁ and C₁. 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 A₃, B₂ and C₂ were observed to have overlapping 95% confidence intervals [29], which led to the reduction of these terms to a single fitted parameter, ζ₂. 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 z = 0 (between fiber/medium and the annular air/medium interfaces). This parameter was found to be dependent on d_f, and to follow the form: $ν_{n} = \frac{1}{1 + ɛ d_{f}}$ , with ɛ = 0.17mm⁻¹. This form was identified from comparing simulations of the fiber surrounded by air with the fiber surrounded by a refractive index matching the fiber; this factor is analogous to offset factors described previously [11].

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 $μ_{a}^{f} Q^{f}$ within the sampled turbid medium. For brevity, the quantity $F_{S F}^{sim} [m]$ will be used throughout this manuscript to refer to the expression

F_{S F}^{sim} = \frac{F_{S F ratio}^{M C}}{μ_{a}^{f} Q^{f} ν_{n}}

3. Results

3.1. Influence of scattering properties and fiber diameter on $F_{S F}^{sim}$

3.1.1. Case I: μ′_s(λ_x) = μ′_s(λ_m)

MC simulations investigated the relationship between single fiber fluorescence and variations in μ′_s, initially specified as equivalent at λ_x and λ_m, and varied over the range [0.1 – 8.0] mm⁻¹. Figure 1 A and B shows $F_{S F}^{sim}$ collected by single fiber probes with d_f ∈ [0.2 – 1.0] mm. These data show a fiber-diameter specific nonlinear relationship between $F_{S F}^{sim}$ and μ′_s. Inspection of $F_{S F}^{sim}$ data sampled by the d_f = 0.2 mm fiber shows a 60% decrease in intensity as μ′_s increases across the investigated range. However, the d_f = 1.0 mm fiber shows an initial decrease in $F_{S F}^{sim}$ of 25% as μ′_s increases from 0.1 to 0.5 mm⁻¹, and $F_{S F}^{sim}$ then doubles in intensity as μ′_s increases from 0.5 to 8 mm⁻¹.

Fig. 1 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 $F_{S F}^{sim}$ vs. μ′_s. C and D shift the x-axis to dimensionless reduced scattering μ′_sd_f. E and F shift the y-axis to a dimensionless form of fluorescence, as $F_{S F ratio}^{M C} / d_{f}$ .

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Inspection of the fiber-diameter specific $F_{S F}^{sim}$ 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 μ′_sd_f, shifted the $F_{S F}^{sim}$ data on the x-axis such that the minimum $F_{S F}^{sim}$ values for each fiber specific profile aligned at the μ′_sd_f value of 0.5; the effect of this transformation is clearly shown in Figures 1 C and D. Second, expression of the ordinate as the dimensionless ratio of $F_{S F}^{sim} / d_{f}$ brought measurements from different fiber diameters onto an overlapping profile; this observed proportionality between fiber diameter and collected fluorescence is consistent with previous analysis of SFFL [20]. Figures 1 E and F show the resulting dimensionless relationship between $F_{S F}^{sim} / d_{f}$ and μ′_sd_f that is observed for measurements from all investigated fiber diameters; there exists more than a factor of 2 variation in the observed magnitude of $F_{S F}^{sim} / d_{f}$ across the investigated μ′_sd_f range. These data exhibit a distinct U-shaped profile characterized by two phases: (1) for small μ′_sd_f values (μ′_sd_f < 0.5), $F_{S F}^{sim} / d_{f}$ decreases in response to increases in μ′_sd_f, and (2) for larger μ′_sd_f values (μ′_sd_f > 0.5), $F_{S F}^{sim} / d_{f}$ increases in response to increases in μ′_sd_f. This bi-phasic behavior is consistent with previous observations of the influence of scattering on fluorescence collected at or near the source [20, 21]; the underlying mechanisms of these phases and their respective dependence on scattering parameters is described in detail in Section 4.

3.1.2. Case II: μ′_s(λ_x) ≥ μ′_s(λ_m)

The data investigated in Figure 1 are for the case μ′_s(λ_x) = μ′_s(λ_m); however, in tissue, μ′_s(λ) is understood to follow a wavelength-dependent expression (e.g. Mie or Rayleigh approximations) such that μ′_s(λ_x) > μ′_s(λ_m). MC simulations were used to investigate $F_{S F}^{sim}$ for the case of independent variation of μ′_s(λ_x) (range: [0.1 – 8.0] mm) and μ′_s(λ_m) (specified as μ′_s(λ_m) = [0.25,0.5,0.75,1.0]×μ′_s(λ_x)). Figures 2 A and B show linear and log representations of the full $F_{S F}^{sim} / d_{f}$ data set plotted vs. μ′_s(λ_x)d_f. Here, stratification of $F_{S F}^{sim} / d_{f}$ measurements at μ′_s(λ_x)d_f values are attributable to the influence of μ′_s(λ_m) on the collected intensity. These data show clear deviation of $F_{S F}^{sim} / d_{f}$ from the smooth curve displayed in Figures 1 E and F due to the independent influence of both μ′_s(λ_x) and μ′_s(λ_m) on SFFL intensity.

Fig. 2 Effect of independent variation of μ′_s(λ_x) and μ′_s(λ_m) on dimensionless single fiber fluorescence intensity, $F_{S F}^{sim} / d_{f}$ plotted vs μ′_s(λ_x)d_f ; vertical stratification is due to influence of μ′_s(λ_m) variation. Linear and log plots given on A and B, respectively.

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3.2. Influence of scattering phase function on $F_{S F}^{sim}$

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 $F_{S F}^{sim}$ , as described in Section 2.2. The $F_{S F}^{sim}$ showed minimal influence from variation among different phase functions, with < 3% variation between $F_{S F}^{sim} / d_{f}$ 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 λ_m, the simulated $F_{S F}^{sim}$ values showed no observable difference if the PF were interchanged between the wavelengths. These results demonstrate that SFFL is insensitive to the form of the PF for all investigated scattering properties and fiber diameters.

3.3. Influence of fiber NA on $F_{S F}^{sim}$

This study utilized a subset of MC simulations to investigate in the influence of fiber NA on $F_{S F}^{sim}$ , as described in Section 2.2. Simulated data showed that the effect of fiber NA on $F_{S F}^{sim}$ is well approximated by an NA² 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.

3.4. Investigation and modeling of factors underlying $F_{S F}^{sim}$ dependence on scattering properties

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(λ_x) and μ′_s(λ_m) within the sampled medium; these quantities were calculated as described in Section 2.1. Figure 3A shows a dimensionless description of effective optical sampling depth, given here as 〈Z^MC〉/d_f, plotted vs. μ′_s_,_avgd_f, with μ′_s,avg calculated as the average of μ′_s(λ_x) and μ′_s(λ_m) for each measurement. These 〈Z^MC〉/d_f data exhibit a power law that shows a decreasing relationship with increasing μ′_s,avgd_f, resulting in a 10-fold decrease over the investigated μ′_s,avgd_f range. This relationship is well-characterized by Eq. (11); fitting this equation to these data yielded estimated values for A₂ = 0.71 ± 0.01 and A₃ = 0.36 ± 0.01,and resulted in accurate estimates of 〈Z^MC〉/d_f over the full range of investigated μ′_s,avgd_f values (r = 0.996); model predictions are visualized by the solid black line on the plot.

Fig. 3 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, $〈 Φ_{x}^{M C} 〉 d_{f}^{2}$ vs. dimensionless reduced scattering at the excitation wavelength, μ′_s(λ_x)d_f. C) Escape probability of emission photons, $〈 H_{m}^{M C} 〉$ vs. dimensionless reduced scattering at the emission wavelength, μ′_s(λ_m)d_f. Fitted model estimates visualized by solid black lines.

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MC simulations also returned scalar metrics representative of effective excitation fluence and effective emission escape probability within the optically sampled volume. Figure 3B displays $〈 Φ_{x}^{M C} 〉 d_{f}^{2}$ vs. μ′_s(λ_x); these data show that $〈 Φ_{x}^{M C} 〉$ nonlinearly depends on μ′_s(λ_x)d_f, with an observed 2.5-fold increase across the investigated range. The form of the observed relationship is empirically described by Eq. (12); fitting the data to this model yielded estimated parameters of B₁ = 0.88 ± 0.01 and B₂ = 0.27 ± 0.03 and resulted in an accurate description of the simulated data (r = 0.977); fit quality is visualized by the model estimated black line. Figure 3C displays $〈 H_{m}^{M C} 〉$ vs. μ′_s(λ_m)d_f ; these data show a nonlinear dependence on μ′_s(λ_m)d_f, with a 2.9-fold increase in the likelihood of collection associated with increasing μ′_s(λ_m)d_f. This relationship is described by Eq. (13); the model fit to these data provides estimated parameter values of C₁ = 0.12 ± 0.07, C₂ = 0.10 ± 0.04, and C₃ = 2.22 ± 0.61 and returns accurate estimates of $〈 H_{m}^{M C} 〉$ (r = 0.967) over the investigated range of μ′_s(λ_m). These results indicate that the Eqs. (12) and (13) describe both the magnitude and dynamic trends of the respective dependencies of $〈 Φ_{x}^{M C} 〉$ vs. μ′_s(λ_x)d_f and $〈 H_{m}^{M C} 〉$ vs. μ′_s(λ_m)d_f.

3.5. Semi-empirical model of $F_{S F}^{sim}$

Figure 4 shows $F_{S F}^{sim} / d_{f}$ simulated by the MC model vs. estimated by the fit of Eq. (14). Here the estimated parameter values of ζ₁ = 0.0935 ± 0.003, ζ₂ = 0.31 ± 0.01, and ζ₃ = 1.61 ± 0.05 resulted in the minimum weighted residual error between simulated and model-estimated $F_{S F}^{sim}$ 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 $F_{S F}^{sim} / d_{f}$ vs. μ′_s(λ_x)d_f ; this plot visualizes the capability of the model to describe the influence of both μ′_s(λ_x) and μ′_s(λ_m) on the collected fluorescence intensity. These results indicate that Eq. (14) provides an accurate description of the SFFL intensity over a wide range of μ′_s(λ_x), μ′_s(λ_m), and d_f, and is valid for all investigated forms of the PF.

Fig. 4 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.

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Fig. 5 Dimensionless single fiber fluorescence intensity estimated by fitted model (× marks) and returned by MC simulations (○ marks). Data include independent variation of μ′_s(λ_x) and μ′_s(λ_m), and are plotted vs. μ′_s(λ_x)d_f. Linear and log plots given on A and B, respectively.

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4. Discussion

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, $F_{S F}^{sim} / d_{f}$ , 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 $F_{S F}^{sim} / d_{f}$ 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 $F_{S F}^{sim} / d_{f}$ accurately describes simulated fluorescence intensities over a wide range of biologically relevant scattering properties.

4.1. Influence of scattering properties on $F_{S F}^{sim}$

The fluorescence model, given in Eq. (14), utilizes empirical functions to represent the individual components of the SFFL measurement, including 〈Z^MC〉/d_f, $〈 Φ_{x}^{M C} 〉$ , and $〈 H_{m}^{M C} 〉$ . This approach provides insight into the mechanisms underlying the bi-phasic relationship observed between $F_{S F}^{sim} / d_{f}$ and the dimensionless reduced scattering coefficient, as visualized in Figure 1F. For μ′_sd_f < 0.5, denoted as phase (1), increases in μ′_s at either λ_x and λ_m result in a decrease in $F_{S F}^{sim} / d_{f}$ . In this scattering region, the average depth of origin for collected fluorescence (and in turn the sampling depth) follows a similar trend, while the effective excitation fluence and effective emission probability are relatively insensitive to changes in this dimensionless scattering region; these trends are visualized in Figure 3. Figure 6A shows $F_{S F}^{sim} / d_{f}$ data in phase (1) following a smooth and continuous dependence on μ′_s,avgd_f ; this scattering dependence is shared by 〈Z^MC〉/d_f. These observations suggest that the left hand side of the $F_{S F}^{sim} / d_{f}$ profile is dominated by volume effects. Here, collected fluorescent photons originate from relatively deep locations in the medium, and an increase in μ′_s,avg represents an impediment to light transport (either for excitation or emission photons), resulting in a decreased tissue volume optically sampled, and a reduced collected intensity. Conversely, for μ′_sd_f > 0.5, denoted as phase (2), increases in μ′_s at excitation or emission wavelengths result in an increase in $F_{S F}^{sim} / d_{f}$ . In this scattering region, $〈 Φ_{x}^{M C} 〉$ increases as μ′_s(λ_x) increases, as shown in Figure 3B. This observation is attributable to the fluence ’build-up’ within the turbid medium near the fiber-tip for increasing scattering [30]. Also in this scattering region, $〈 H_{m}^{M C} 〉$ increases in response to an increase in μ′_s(λ_m), as shown in Figure 3C. This phenomenon can be understood as follows. Scattering at the emission wavelength has two counteracting effects on fluorescence collection. First, fluorescent photons traveling towards the detecting fiber may be scattered away from the fibertip, decreasing the SFFL signal. This attenuating effect will be more pronounced for fluorescent photons that are emitted from relatively deep locations within the sample, while for photons originating close to the fibertip the attenuation due to scattering is expected to be small due to the small path traveled to the fibertip. Second, fluorescent photons traveling away from the detecting fiber may be backscattered towards the fibertip, increasing the SFFL signal. The balance of these counteracting effects will be depth dependent; photons originating from large depths inside the medium are expected to suffer more from attenuation due to scattering than benefit from fluorescence backscattering, while the opposite is true for photons originating close to the fibertip. Since for high scattering coefficients the effective sampling depth is relatively small, the net effect is that the benefit from fluorescence backscattering outweighs the attenuation of fluorescence due to scattering, resulting in an increase in effective escape probability with increasing scattering coefficient. $F_{S F}^{sim} / d_{f}$ data in phase (2) were observed to smoothly follow a dimensionless reduced scattering parameter dependent on the harmonic average of μ′_s(λ_x) and μ′_s(λ_m); this relationship was gained from inspection of the dependence of $〈 Φ_{x}^{M C} 〉$ and $〈 H_{m}^{M C} 〉$ on the reduced scattering coefficient. Specifically, the harmonic average reduced scattering coefficient is given as

μ_{s, h - avg}^{'} = \frac{1 + ζ_{3}}{\frac{1}{μ_{s}^{'}} + \frac{ζ_{3}}{μ_{s}^{'}}}

These observations suggest that the right hand side of the

F_{S F}^{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.

Fig. 6 Dimensionless single fiber fluorescence intensity plotted vs. dimensionless reduced scattering calculated as A) mean of μ′_s(λ_x) and μ′_s(λ_m) and B) harmonic average expression including μ′_s(λ_x) and μ′_s(λ_m), given in Eqn 16. Smooth dependence of fluorescence intensity vs. these respective scattering parameters suggests region μ′_sd_f < 0.5 is dominated by volume effects, region μ′_sd_f > 0.5 is dominated by excitation fluence and emission probability within sampled volume.

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4.2. Application of semi-empirical model of $F_{S F}^{sim}$ to extract intrinsic fluorescence in turbid media

The semi-empirical model developed in this study provides a method to return scattering-independent F_SF quantities provided that μ′_s(λ_x) and μ′_s(λ_m) are determined, e.g. from a white-light reflectance measurement. This approach is in contrast to other techniques that utilize raw reflectance to correct raw fluorescence for the influence of scattering properties. Such an approach is not appropriate for single fiber measurements, because reflectance intensities collected by single fibers (R_SF) are not only sensitive to μ′_s, but (in contrast to SFFL) are also heavily influenced by the PF [24, 31, 32]. Due to this difference in PF dependence of F_SF and R_SF, the ratio of these two quantities will also be PF dependent. The magnitude of this dependence can best be appreciated by considering R_SF measurements of two (hypothetical) turbid media with μ′_s values of 0.5 and 2.0 mm⁻¹, both with the same intrinsic fluorescence, and measured by a fiber with d_f = 1.0 mm. If the PF within the two media were varied from γ = 1.9 to γ = 1.4 (a change that would increase the likelihood of large-angle scattering events), the resulting R_SF would increase by a factor of 2.3 for μ′_s = 0.5 mm⁻¹ and a factor of 1.4 for μ′_s = 2.0 mm⁻¹ [24, 25]. For a smaller fiber of d_f = 0.2 mm, the effects are amplified to factors of 3.1 and 1.5 for each respective case. Importantly, the variation in PF would have a negligible effect on the raw F_SF ; such a difference in sensitivity to PF is attributable to the isotropic release of emission photons during propagation of fluorescent light. In contrast to R_SF, which relies on the likelihood of forward directed incident light to undergo a large-angle scattering event (defined by the PF), the isotropic release of a fluorescent photon greatly reduces the sensitivity of F_SF to PF. Therefore, for single fiber measurements (and likely other geometries which collect light close to the source fiber), a fluorescence correction algorithm that utilized a ratio of F_SF and R_SF could result in inaccurate estimation of intrinsic fluorescence by up to a factor of > 3 for small dimensionless scattering values.

The PF-specific analysis presented in this study indicates that quantitative analysis of SFFL requires determination of μ′_s(λ_x) and μ′_s(λ_m) independent of PF. This could be achieved using a multi-diameter SFR measurement, as described recently by our group [25,26]. The MDSFR approach utilizes the γ-specific R_SF vs. μ′_sd_f relationship for measurements using multiple fibers at each investigated wavelength. By specification of a background scattering model within the sampled tissue (e.g. Mie and or Rayleigh scattering) it is possible to determine μ′_s and γ across the a range of wavelengths. Moreover, this calculation can be made in the presence of absorption from tissue chromophores, requiring only specification of the basis set of absorbing constituents and their respective specific absorption coefficients. This multi-fiber approach can be executed using as few as two optical fibers with different diameters [26]; moreover, such a device can easily be developed to sample both R_SF and F_SF. The combined multi-diameter SFR and SFFL would return paired local measurements of fluorescence and tissue optical properties within the same (shallow) sampling volume. Such a technique has the potential to provide clinically useful information for tissue diagnostics and monitoring of administered therapies. The localized measurement volume would allow quantitative characterization of heterogeneities in the spatial distribution of an administered fluorescent compound; this may be advantageous compared with a volume-averaged metric gained from diffuse optical measurements. Moreover, the measurement volume can be selected at a specific area of interest (e.g. in the center of an identified malignant area, or on the border between suspicious and normal tissue). This multi-fiber approach faces challenges that must be properly assessed, including proper identification of background scattering models for determination of μ′_s(λ), the influence of μ_a on F_SF, and the influence of heterogeneities on both R_SF and F_SF ; ongoing studies are investigating these issues.

4.3. Limitations and future work

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, and did not consider background absorption effects. Absorption within the sampled medium, at both excitation and emission wavelengths, is expected to have a substantial influence on the raw fluorescence intensity collected and the volume probed during measurement. Further complicating matters, the magnitude of the absorption attenuation is expected to be heavily influenced by the paired scattering properties at excitation and emission wavelengths. An ongoing study will characterize the influence of absorption on the individual components of the SFFL model. Additionally, the MC model utilized in this study was validated by comparison with model returned outputs reported in the literature; future work will conduct experimental validation in optical phantoms.

5. Conclusions

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 μ′_s(λ_x) ∈ [0.1 – 8] mm⁻¹ and μ′_s(λ_m) ∈ [0.25 – 1] × μ′_s(λ_x), and scattering phase functions (P(θ)), with both Henyey-Greenstein and Modified Henyey-Greenstein forms with anisotropy in the range 0.5 – 0.95 and γ ∈ [1.4 – 1.9], and a wide range of fiber diameters (d_f ∈ [0.2 – 1.0] mm). The model accurately estimates F_SF given μ′_s(λ_x) and μ′_s(λ_m), and is insensitive to the anisotropy and higher order moments of the PF. Results indicate that correction for the influence of scattering on F_SF requires estimation of scattering optical properties from a paired measurement of white-light reflectance.

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Semi-empirical model of the effect of scattering on single fiber fluorescence intensity measured on a turbid medium

Abstract

1. Introduction