Tagging photons with gold nanoparticles as localized absorbers in optical measurements in turbid media

Serge Grabtchak; Kristen B. Callaghan; William M. Whelan

doi:10.1364/BOE.4.002989

1. Introduction

Advances in detection of localized inclusions in multiple-scattering biological tissues (i.e., turbid or diffusive media) are mostly driven by needs of practical applications of biomedical optics. In general, the localized inclusion can be considered as a region in turbid media with optical properties different from those of the background. Such regions may contain exogenous (i.e., indocyanine green [1–3], gold nanoparticles [4–6], quantum dots [7–9] etc.) or endogenous (i.e., hemoglobin [10, 11]) chromophores, nonscattering liquids [12, 13] (i.e., cerebrospinal fluid, water) or malignant tissues [14, 15]. Various approaches have been developed to detect objects of certain geometries in turbid media [16–26].

However, rather than being of a primary interest by themselves, localized inclusions can play a different role by acting as tools for probing spatial distributions of migrating photons in turbid media which in turn may provide researchers with a better understanding of conditions for detectability of inclusions.

Introducing a localized inclusion with optical properties different from the surrounding medium produces a perturbation that changes the original photon density distribution. In the formalism of the perturbation approach [27, 28] developed for fluence with the diffusion approximation, the distribution of photons can be reconstructed using a point localized absorber as a probe that perturbs photon density in the selected location. When a small localized absorber is placed at some position between the light source and fluence-based detector, a reduction in the resulting fluence is measured. The measurements are repeated after moving the absorber to a new position eventually covering most of the plane (or even the volume) of interest, and finally the photon-path-distribution function can be reconstructed from multiple point measurements. In essence, this process is equivalent to tagging photons with the localized absorber because only photons that pass through the region of the physical location of the absorber will be removed from a subsequent migration and detected as a reduction in the signal. Analytical expressions for the photon-path-distribution function derived in [27, 28] allowed predicting of so-called banana-shape regions where the flux is concentrated inside the turbid medium (with photon densities higher near the source and detector and lower at the midway point). This is important because it can provide clear information about optimal detector sensitivity to tumors or another localized inhomogeneity in the selected point in a turbid medium [27].

Two important aspects of the perturbation approach from Ref [27]. should be emphasized: 1) the distribution function describes the fraction of the total number of photons diffusing from the source to the detector that must travel through the selected point in space; 2) the distribution function assumes an isotropic 4π collection window for the detector that corresponds to fluence detection. If the detector has an anisotropic detection window as in the case of radiance detection [29, 30], then the photon density distribution function derived for fluence can’t be applied.

Detecting optical radiance is less understood compared to fluence for studying propagation of light in turbid media with only a few groups of researchers exploring its utility [29–39]. This is likely due to the relative complexity of radiance-based techniques with an angular-sensitive detector and rotation along its axis which imposes certain difficulties for practical use. On the other hand, the lack of a practical demand for radiance applications has produced only few theoretical studies [38, 40, 41] that would enrich experiments and stimulate further developments as was done for fluence based techniques. Indeed, the simplicity of basic fluence measurements, the equivalency between the detector and the light source (for example, the isotropic spherical diffuser can act in both capacities) and a relative straightforwardness of results interpretation have triggered a spectacular growth in applications of fluence-based techniques [12, 20, 22, 42–57]. Hence, the lack of the perturbation formalism that can be easily applied to analysis of radiance data for localized inclusions in turbid media seriously impedes further progress of the radiance-based methods.

The current work presents new experimental data and novel two-dimensional (2D) analysis of radiance detection of the localized absorptive inclusion formed by gold nanoparticles (Au NPs) in the infinite turbid medium (Intralipid-1%) geometry when the absorptive target is translated along the line connecting the light source and detector. This is the configuration that produces “banana” patterns in fluence measurements, and it was used earlier by us to study effects of voids in turbid media [37]. Radiance data were presented as spectro-angular maps of the radiance extinction ratio (RER) that were introduced previously [37, 39]. The maps were analyzed using the novel analytical expression for the relative angular photon distribution function developed in the current work within the framework of the perturbation approach as in [27] but extended to radiance.

2. Methods and materials

2.1. Set-up for radiance measurements in Intralipid-1%

Optical radiance corresponds to a variation in the angular distribution of photons measured in the selected point in turbid media. To obtain radiance data, an optical probe with a well-defined angular detection window must be rotated along its axis. A schematic of the experimental set-up is shown in Fig. 1(a). The set-up has been used in number of publications [30, 39, 58] so only a brief description is given here. Tissue illumination was provided by a tungsten halogen white light source (20 W) connected to a fiber with a 2-mm spherical diffuser at the end with a total output power ~18 mW. A 600-micron side firing fiber (Pioneer Optics) acted as a radiance detector. To prevent any secondary rays that do not undergo the total internal reflection to couple back to the fiber from other directions, a cap from a heat-shrink white polymer tube with a hole aligned to the entrance window was placed at the end of the side-firing fiber [39]. Both illuminating and detecting fibers were threaded through 15-gauge needles for mechanical stability. The side firing fiber was mounted on a computer-controlled rotation stage (Thorlabs). Radiance data were obtained by rotating the side firing fiber over a 360° range with a 2° step. The side firing fiber was connected to a computer-controlled USB 4000 spectrometer (Ocean Optics) that collected spectra at every angular step. Depending on a number of averages, it took between 4 and 8 minutes to acquire a complete single angular profile.

Fig. 1 a) Experimental set-up for radiance measurements of gold target in a Intralipid-1%, b) conceptual top-view diagram illustrating the principle of distance dependent experiments.

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The sample containing the capillary filled with the colloidal gold was controlled in the X,Y-plane by two translation stages. Au NPs based inclusion was prepared by filling two types of capillaries with the colloidal gold: ~3.5-mm (~0.8 mL, ~6 cm immersion length) and 2-mm diameter (~0.1 mL, ~6 cm immersion length) thin-wall quartz capillary tubes. The colloids (BBI Solutions) were formed by spherical 5-nm Au NPs with the plasmon resonance at 520 nm. Two different stock concentrations were used in various experiments: 3.95x10¹³ and 7x10¹⁴ particles/mL (referred as 20x in the text).

The Lucite box with blackened 18-cm walls was filled with Intralipid-1% suspension and accommodated both detecting and illuminating fibers as well as the capillary tube with the colloidal gold. Intralipid-1% suspension was prepared by volume dilution of Intralipid-20% stock (Sigma Aldrich Canada). The size of the box ensured an approximation of infinite medium geometry.

For distance dependent experiments, the source-detector separation was kept constant at 30 mm, and the target was translated with 5 mm (2 mm in some experiments) increments from 5 mm (3 mm in some experiments) to 25 mm away from the detector keeping it in line between the source and detector (i.e., at 0° angle) as shown in Fig. 1(b). In all experiments, for every target position a set of radiance profiles was acquired by rotating the detecting fiber over a 360° range with a 2° step. The reference measurement was taken in the Intralipid solution for the corresponding source-detector separation prior to inserting the capillary with Au NPs.

Optical properties of Intralipid-1% were extensively studied by us earlier [58], and published data for the effective attenuation coefficient (μ_eff(λ)) and the diffusion constant (D(λ)) were used in simulations in the current work. Of a particular importance for the current work are the spectral dependencies of μ_eff(λ) for Intralipid-1% and the absorption coefficient (μ_a(λ)) for 5-nm Au colloidal solutions that are shown in Fig. 2(a), 2(b).

Fig. 2 a) Effective attenuation coefficient of Intralipid-1% and absorption coefficient of 5-nm Au NPs colloidal solutions with a) 3.95x10¹³ particles/mL and b) 7x10¹⁴ particles/mL (marked as 20x) concentrations.

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Radiance data were presented in a form of spectro-angular maps that define relative photon distribution in the angular domain. Spectro-angular mapping approach relies on obtaining two data sets, one for the phantom without the inclusion and the other for the phantom with the inclusion. Individual spectra (i.e., radiance profiles) collected at various radiance detector angles in 0°-360° range in the phantom without inclusion were assembled into “the Intralipid matrix”, I_Intralipid with columns corresponding to wavelengths and rows to angles. Placing the Au target at a certain distance from the detector along a selected angle and repeating the measurements yielded “the Intralipid+Au matrix”, I_{Intralipid + Au}. An element by element ratio of two matrices, I_Intralipid/I_{Intralipid + Au} minimized contributions from Intralipid, white light source and illuminating/detecting fibers, highlighting the target signal in a particular background. The ratio is defined as the Radiance Extinction Ratio (RER) and can be presented via surface or contour plots demonstrating intensity and spectral variations vs. angle [37, 39].

2.2. Perturbation modeling for radiance

Diffusion approximation (DA) is a simplification which allows one to obtain an analytical solution of the radiative transfer equation (RTE) and thus, analytical expressions for fluence and radiance. This simplification can be achieved when absorption is much less than scattering (which is true for most biological tissues) and when measurements are performed at distances from the light source that are much bigger than the reciprocal reduced scattering coefficient. A clear physical meaning and the ease of manipulation with fluence or radiance under DA makes it attractive for testing theoretical predictions against experimentally observed effects.

Consider the isotropic dc light source placed at r_S and the radiance detector located at r_D in the infinite homogenous medium that exhibits absorption and multiple scattering effects and can be treated within the framework of the diffusion theory. Since radiance measurements are always performed interstitially inside tissues of interest, infinite medium geometry corresponds to practical applications of radiance. Radiance measured by the detector can be described by a familiar expression [59]:

\begin{array}{l} I_{0} (| r_{S} - r_{D} |, θ, λ) = \frac{S_{0}}{4^{2} π^{2} D (λ)} [1 + 3 (\frac{D (λ)}{| r_{S} - r_{D} |} + μ_{e f f} (λ) D (λ)) \cos θ] \\ \cdot \frac{\exp (- μ_{e f f} (λ) | r_{S} - r_{D} |)}{| r_{S} - r_{D} |} \end{array}

where S₀ is the source power,

| r_{S} - r_{D} |

is the source-detector separation, θ is the angle between the direction of propagation and scattering, λ is the wavelength,

D (λ) = 1 / 3 ({μ^{'}}_{s} (λ) + μ_{a} (λ))

is the diffusion coefficient, μ_eff(λ) is the effective attenuation coefficient of the background turbid medium.

Once the inclusion is formed at position r by an absorbing sphere of radius a, the new value of radiance affected by the perturbation, $I_{p e r t} (| r_{S} - r_{D} |, θ, λ)$ will be measured. The radiance extinction ratio (RER) introduced earlier can be constructed from two values as RER = $I_{0} (| r_{S} - r_{D} |, θ, λ)$ / $I_{p e r t} (| r_{S} - r_{D} |, θ, λ)$ . This ratio singles out properties of the inclusion making the absolute radiance measurements unnecessary [37]. The ratio also emphasizes attenuation of light in the phantom with the inclusion such that values of RER>1correspond to the increased extinction relative to the bare phantom while values of RER<1indicate that addition of the inclusion reduces the overall extinction and results in a higher photon density. When the inclusion is added to the phantom it replaces the equivalent volume of the background medium, and the differential optical properties of two volumes will play an important role in determining the directional changes in RER values. In principle, different perturbation metrics can be introduced from RER as { $I_{0} (| r_{S} - r_{D} |, θ, λ)$ - $I_{p e r t} (| r_{S} - r_{D} |, θ, λ)$ }/ $I_{0} (| r_{S} - r_{D} |, θ, λ)$ ≡ΔI/ $I_{0}$ but as will be shown later, it bears essentially the same information as RER.

Following the diffusion approximation, radiance can be approximated by first two terms (isotropic fluence and anisotropic flux) of a series expansion in spherical harmonics [60]:

I_{p e r t} (| r_{S} - r_{D} |, θ, λ) = \frac{1}{4 π} Φ_{p e r t} (| r_{S} - r_{D} |, λ) + \frac{3}{4 π} D (λ) \nabla Φ_{p e r t} (| r_{S} - r_{D} |, λ) \cdot \hat{s}

where

Φ_{p e r t} (| r_{S} - r_{D} |, λ)

is the fluence measured in the presence of the perturbation,

\hat{s}

is the unit vector pointing to the direction of the perturbed flux propagation. One can see that knowledge of the perturbed fluence allows determining perturbed radiance.

In order to obtain the expression for fluence in the presence of the inclusion, it is instructive to invoke the perturbation approach developed for fluence for the infinite medium geometry in the steady-state condition for a perfectly absorbing sphere [27, 28]. Following this approach, the perturbed fluence can be presented as a sum of the fluence measured in the absence of the perturbation and the fluence that is due to photons that come from the source r_S and pass through the region occupied by the inclusion at r, because only these photons will be removed by absorption of the inclusion:

Φ_{p e r t} (| r_{S} - r_{D} |, λ) = Φ_{0} (| r_{S} - r_{D} |, λ) + Φ_{1} (| r_{S} - r_{D} |, r, λ)

The corresponding expressions for both terms from Eq. (3) are written below (the first is known from the diffusion theory and the second was derived in [27]):

Φ_{0} (| r_{S} - r_{D} |, λ) = \frac{S_{0}}{4 π D (λ)} \frac{\exp (- μ_{e f f} (λ) | r_{S} - r_{D} |)}{| r_{S} - r_{D} |}

Φ_{1} (| r_{S} - r_{D} |, r, λ) = - k \frac{a S_{0}}{4 π D (λ)} \frac{\exp (- μ_{e f f} (λ) | r - r_{S} |)}{| r - r_{S} |} \frac{\exp (- μ_{e f f} (λ) (| r_{D} - r | - a))}{| r_{D} - r |}

where a is the radius of the absorbing sphere and k is the coefficient introduced by us when defining the boundary condition on the surface of the absorbing sphere and is set to 1 for the perfect absorber such that Eq. (5) matches the corresponding expression from Ref [27].

Note, that the positions of the source and detector are fixed while the inclusion can be moved which makes r a spatial variable. Substituting Eq. (3) into Eq. (2) results in the following expression:

\begin{array}{l} I_{p e r t} (| r_{S} - r_{D} |, θ, λ) = \frac{1}{4 π} Φ_{0} (| r_{S} - r_{D} |, λ) + \frac{1}{4 π} Φ_{1} (| r_{S} - r_{D} |, r, λ) \\ + \frac{3}{4 π} D (λ) \nabla Φ_{0} (| r_{S} - r_{D} |, λ) \cdot \hat{s} + \frac{3}{4 π} D (λ) \nabla Φ_{1} (| r_{S} - r_{D} |, r, λ) \cdot \hat{s} \end{array}

Inspection of Eq. (6) indicates that since the third term does not depend on the variable r, it will be zero. The next step is performing necessary differentiation in Eq. (6) and setting the coordinate system. In the coordinate system the detector lies at the origin, the source is placed at the distance d from the detector along X-axis and the inclusion is moved from the detector to the source along X-axis. Then, r transforms to x and the following expressions can be written for the perturbed and unperturbed radiance:

\begin{array}{l} I_{p e r t} (d, θ, λ) = \frac{S_{0}}{4^{2} π^{2} D (λ)} [\frac{\exp (- μ_{e f f} (λ) d)}{d} - a k \frac{\exp (- μ_{e f f} (λ) (d - a))}{x (d - x)} \\ - 3 a k D (λ) \frac{(d - 2 x) \exp (- μ_{e f f} (λ) (d - a))}{{(d - x)}^{2} x^{2}} \cos θ] \end{array}

I_{0} (d, θ, λ) = \frac{S_{0}}{4^{2} π^{2} D (λ)} [1 + 3 (\frac{D (λ)}{d} + μ_{e f f} (λ) D (λ)) \cos θ] \frac{\exp (- μ_{e f f} (λ) d)}{d}

It should be noted that the selected geometry has 2D symmetry such that the absorbing sphere (or cylinder) is reduced to a circle and rotation of the detector and translation of the target are performed in the same plane. The expression for RER will take the final form as following:

\begin{array}{l} RER = \frac{I_{0} (d, θ, λ)}{I_{p e r t} (d, θ, λ)} \\ \equiv \frac{1 + 3 (\frac{D (λ)}{d} + μ_{e f f} (λ) D (λ)) \cos θ}{1 - \frac{a d k}{x (d - x)} \exp (- μ_{e f f} (λ) (d - a)) - \frac{3 a D (λ) (d - 2 x) d k}{{(d - x)}^{2} x^{2}} \exp (- μ_{e f f} (λ) (d - a)) \cos θ} \end{array}

As follows from Eq. (9), modeling RER requires the knowledge of the measurement geometry, size of the inclusion and optical properties of the background medium.

While the perfect absorber can be a useful concept, realistic absorbers exhibit finite absorption values. For realistic absorbers, the fluence on the surface of the absorber does not vanish completely but is reduced to some non-zero value so that k≠1. To account for the finite absorption of the inclusion, we used a simple approximation based on the Beer-Lambert law of optical attenuation. The absorbing circle of radius a (for 2D case) was assigned an absorption coefficient μ_a(λ). The circle was subject to uniform illumination from all directions. The fluence at the surface is determined by the transmission through the inclusion’s medium with a 2a optical path that has replaced the similar shape of the turbid medium and is also immersed into the turbid medium characterized by μ_eff(λ). Then, fluence on the surface of the absorber was approximated to be reduced by $k = (1 - \exp [- (μ_{a} (λ) - μ_{e f f} (λ)) 2 a])$ . This new boundary condition was incorporated via k into the derivation (i.e., Eqs. (5), (7), (9)) so that Eq. (9) can be used for both perfect and finite absorbers given by k values.

3. Results and discussion

Presenting data via RER is extremely useful for isolating signatures of the localized inclusion in spectral and angular domains in radiance measurements. To illustrate this, Fig. 3 shows spectro-angular contour plots of raw radiance data for the Intralipid phantom (Fig. 3(a)) and for the phantom with the 3.5 mm Au NPs inclusion (3.95x10¹³ particles/mL) (Fig. 3(b)) that was placed between the source and detector (i.e., at 0° angle) at a distance of 5 mm from the detector (with source-detector separation at 30 mm). In the color coding used in the contour plots, red corresponds to a high signal with a maximal optical transmission while blue highlights areas with lowered transmission. For both plots, the maximal transmission is observed along 0° angle that is consistent with preferential forward scattering in Intralipid solutions [58]. Spectral responses of both plots include contributions from wavelength-dependent white light source, illuminating and detecting fibers. (Optical signatures of the Intralipid solution can be isolated by normalizing to measurements in water [58].) Close examination of two plots suggests that other than a red-shift of the maximal transmission for Fig. 3(b), nothing indicates the presence at 0° of the Au NPs inclusion with certain spectral characteristics.

Fig. 3 a) Spectro-angular contour plot of raw radiance data for Intralipid-1% ; b) spectro-angular contour plot of raw radiance for Intralipid-1% with 3.5 mm inclusion with Au NPs located at 0° and 5mm from the detector. Color bars have units of counts per second. Source-detector separation 30 mm.

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However, the ratio of the two plots expressed as RER in Fig. 4(a) clearly isolates the signature of the inclusion. Red color corresponds to the regions of highest attenuation that occurs in the vicinity of 520 nm along 0° angle. Figure 4(b) presents the same data as the surface plot that emphasizes the spectral response which is characteristic for 5-nm Au NPs in water (compare with Fig. 2(a)). Figure 4(a) also shows that RER undergoes changes with wavelength such that for λ>680 nm RER<1. The contour line corresponding to RER = 1 is presented in white. Reduction in RER values indicates that photon density has increased for certain wavelengths relative to Intralipid solution after inserting the Au NPs inclusion. (Some asymmetry seen in RER = 1 contour line was caused by a small angular misalignment in the position of the inclusion). Thus, away from the plasmon resonance, the Au NPs inclusion not only can lose its strong absorptive properties but, in fact, may even become more transparent than the background turbid medium it has replaced.

Fig. 4 a) Spectro-angular contour plot of the radiance extinction ratio (RER) the Au NPs inclusion. b) Spectro-angular surface plot of the radiance extinction ratio (RER) of the Au NPs inclusion. Inclusion: 3.5 mm capillary with 5nm Au NPs at 0° at 5 mm from the detector. Source-detector separation 30 mm.

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Continuing to move the Au NPs inclusion with 5 mm step toward the light source yields four more spectro-angular contour plots (target at 10 mm, 15mm, 20 mm and 25 mm away from the detector) presented in Figs. 5(a), 5(b), 5(c), 5(d)). A characteristic absorption peak around 520 nm is observed in all plots. A noticeable angular broadening manifested in the increased horizontal ellipticity of contour lines while moving the inclusion away from the detector is another characteristic feature. From the intensity scale, one can see that RER tends to achieve higher values when the inclusion is closer to the source or to the detector while in the middle of the distance RER is minimal. These observations are quantified in Fig. 6 which shows the variation of maximal values of RER (observed along 0°) and Full-Width-Half-Maximum (FWHM) of the angular profile at 520 nm with distance.

Fig. 5 Spectro-angular contour plots of the radiance extinction ratio (RER) the Au NPs inclusion located at a) 10 mm, b) 15 mm, c) 20 mm, d) 25 mm from the detector at 0°. Inclusion: 3.5 mm capillary with 5nm Au NPs. Source-detector separation 30 mm.

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Fig. 6 Variation of maximal RER values observed along 0° and full-width-half-maximum (FWHM) measured at 520 nm with distance extracted from Figs. 4(a), 5(a)-5(d).

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Because it is extremely difficult to present the variation of RER in the spectral, spatial and angular domains at the same time, we concentrated on two most interesting and important representations. Since Au NPs are used mostly because of their optical properties at the plasmon resonance, the first plot in Fig. 7(a) represents the distance variation in the relative angular photon density distribution probed by the Au NPs inclusion at the wavelength of the plasmon resonance, 520 nm. To create this plot, five individual angular profiles of RER at 520 nm measured at 5, 10, 15, 20, and 25 mm (from Figs. 4(a), 5(a)-5(d)) were combined into the spatio-angular contour plot as in Fig. 7(a). Due to the nature of interaction between the inclusion and photons, highest values of RER correspond to highest photon densities such that the plot can be considered as a negative image of the photon density in the turbid medium. Since the radius of contour lines corresponds to angular distribution of photons, one can see that photons are better localized in the vicinity of the detector rather than of the light source. Indeed, the radius of contour lines gradually increases while moving the target from the detector to light source. It becomes more obvious when comparing the profiles measured at 5 and 25 mm because these two positions are symmetric relative to the midpoint (i.e., both have 5 mm separation from the detector or from the source). Variation of RER values along 0° angle exhibits a minimum midway between the source and the detector while showing rising toward the source and detector. Applying this knowledge to detection of inclusions means that the closer the inclusion is to the detector, the better the angular detectability (or angular localization). When data from Fig. 7(a) were represented (not shown) via the spatio-angular contour plot using different metrics, ΔI/I₀ (that was described in the modeling section) the resulting plot had the appearance very similar to the one in Fig. 7(a) with the maximal value at ~0.63. Since both plots contained essentially the same information and demonstrated similar features, we proceeded with RER-based analysis.

Fig. 7 Experiment for 3.5 mm target with 3.95x10¹³ Au NPs/mL: a) spatio-angular contour plot of RER measured at 520 nm, b) spectro-spatial contour plot of RER measured along 0°. Plots created from data in Figs. 4(a), 5(a)-5(d).

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The second way of data representation is shown in Fig. 7(b) where the variation of RER along 0° angle in the 450-900 nm spectral range with distance is displayed as the spectro-spatial contour plot. One can notice that almost all wavelengths exhibit the pattern observed for 520 nm in Fig. 6 along 0° i.e., rising toward the detector and source with the minimum half-way between the detector and source. The shape of contour lines in 450-500 nm spectral range somewhat resembles the absorption spectrum of Au NPs with the peak at 520 nm. The contour line with RER = 1 separates the spectral range of the increased extinction (<680 nm) from the one of the decreased extinction (>680 nm). In accordance with RER definition, RER<1indicates that Intralipid solution with the added inclusion becomes more transparent than the Intralipid alone.

Results of modeling using Eq. (9) for the perfect absorber of 3.5 mm diameter in the Intralipid-1% solution are presented in Fig. 8 which shows the spatio-angular contour plot of RER at 520 nm. Simulations were always performed with the same incremental step for the detector-target separation as in the experiment. This figure can be compared directly with Fig. 7(a). Figure 8 shows qualitatively the same behavior as the Fig. 7(a): better angular localization of photons near the detector, much broader angular distributions of photons closer to the light source and well-defined minimum in RER at the midpoint between the source and the detector. Quantitative comparison indicates that simulated values are about 30% lower than the experimental ones. Also, the simulated plot shows that for the inclusion located closer to the light source the contour lines start bending upward which is opposite to what was measured in the experiment. This plot demonstrates the relative angular distribution of photons at the selected wavelength in a very characteristic asymmetric “hourglass” pattern and can be considered analogous to known banana-type patterns for fluence.

Fig. 8 Simulated spatio-angular contour plot of RER at 520 nm using Eq. (9) for 3.5 mm diameter perfect absorber.

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By definition, the absorption coefficient of the perfect absorber is much larger than the extinction of the surrounding medium. The concept of the perfect absorber might be useful when comparing optical properties at the selected wavelength, however extending the concept to a wider spectral range would result in the uniform absorption spectrum of the black body. Since the absorption coefficient of the perfect absorber is not present explicitly in Eq. (9), simulations can’t generate the plot with spectrally resolved features similar to the one in Fig. 7(b).

Performing simulations using Eq. (9) for the realistic absorber resulted in two plots shown in Figs. 9(a), 9(b): spatio-angular and spectro-spatial contour plots. Comparing Fig. 9(a) with Fig. 7(a) one can see that in spite of the increased difference in absolute values of RER between the two plots, there is still a satisfactory qualitative correspondence. The increased gap between the experimental and simulated values should not come as a surprise. The perfect absorber has already introduced values that were lower by ~30% so that any further decrease in the absorption will only increase the discrepancy.

Fig. 9 Simulation of a realistic absorber (3.5 mm diameter with 3.95x10¹³ Au NPs/mL): a) spatio-angular contour plot of RER at 520 nm using Eq. (9), b) spectro-spatial contour plot of RER at 0° using Eq. (9)

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Detailed comparison of Fig. 9(b) with Fig. 7(b) demonstrates a very good degree of correspondence between the two plots. (Note that the plot in Fig. 9(b) appears more discrete because fewer wavelengths were used in the simulation than in the experiment.) With the difference in absolute values still present, the contour line patterns remain similar. Both plots demonstrate contour lines clearly delineating the plasmon peak around 520 nm at both ends of the detector-target separation and a very distinct evolution of contour lines up to ~700 nm. The contour line with RER = 1 is marked in white in both contour plots. The origin of this line becomes clear after inspecting Fig. 2(a) that contains the wavelength-dependent absorption coefficient of Au NPs in water and the effective attenuation coefficient of Intralipid-1%. Below ~650 nm absorption of Au NPs dominates over the attenuation of Intralipid-1%, however for wavelength >650 nm this reverses. Once this relation was incorporated into the boundary condition, the crossover also appears in the simulated contour plot. It is interesting that while the crossover in Fig. 2(a) occurs at ~650 nm, for the Au NPs inclusion placed in the turbid medium the wavelength at which the crossover is observed changes with distance. Not only modeling and experiment produce almost the same shape for the corresponding contour line but the values of crossover wavelengths are close (~690 nm and ~700 nm for 5mm detector-target separation for the experiment and theory, correspondingly). Some deviations between the model and the experiment occur for λ>700nm where simulation shows a slightly different spectral dependence which is also manifested in different distance dependent patterns.

In order to get closer to perfect absorber conditions, colloidal solution with a 20x increased concentration of Au NPs (7x10¹⁴ particles/mL) was used in subsequent experiments. To improve the spatial resolution in mapping of the relative angular photon distribution function, the size of the capillary tube was reduced from 3.5 mm to 2 mm, and the incremental step in the translation of the target was set to 2 mm covering the range from 3 to 25 mm away from the detector. The corresponding optical properties of the 20x Au NPs solution and Intralipid-1% are shown in Fig. 2(b). Figure 10 presents experimental data in two adopted formats: spatio-angular plot for 530 nm (Fig. 10(a)) and spectro-spatial plot for 0° angle (Fig. 10(b)). (The offset seen in Fig. 10(a) in 0° position is due to the target’s misalignment.) Increased extinction values (up to ~4) indicate that this combination of the concentration and tube size produced much stronger perturbation than the set of parameters used previously.

Fig. 10 Experiment for 2.0 mm target with 7x10¹⁴ Au NPs/mL: a) spatio-angular contour plot of RER measured at 530 nm, b) spectro-spatial contour plot of RER measured along 0°.

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The main tendencies however remain the same: a) the extinction is maximal near the detector, goes through the minimum midway between the source and detector and rises again near the source to values lower than those near the detector; b) angular localization of photons is better when the target is closer to the detector such that the relative angular photon distribution function resembles the shape of an asymmetric hourglass. Figure 10(b) shows values of RER>1 for the entire spectral range of interest (450-900 nm). This behavior is consistent with Fig. 2(b) that indicates that the absorption coefficient of Au NPs is larger than the effective attenuation coefficient of Intralipid-1% in the entire spectral range and no crossing occurs. Another feature that is seen in both in Fig. 10(b) and Fig. 2(b) is some red shift and spectral broadening of the plasmon resonance. Even though at low Au NP concentrations the resonance is well defined at 520 nm (as can be seen in Fig. 2(a)), higher concentrations result in the reduced interparticle spacing that is manifested in spectral broadening of the plasmon resonance and red-shifting to ~530 nm. Similar effects occur during cluster formation and aggregation of Au NPs [61]. These phenomena highlight the importance of the actual concentration of Au NPs in studied turbid media when the local concentration may significantly exceed the initial stock concentration of Au NPs solution thus affecting spectral signatures of the detected Au NPs inclusions.

Results of simulations corresponding to these measurements are shown in Fig. 11(a), 11(b). Figure 11(a) displays the spatio-angular plot for 530 nm for the realistic absorber. Simulations for the perfect absorber produced a virtually indistinguishable plot indicating that optical properties of Au NPs near the plasmon resonance with 20x concentration approximate well those of the perfect absorber. Comparing Fig. 11(a) with Fig. 10(a) shows a reasonable degree of qualitative agreement between the experiment and the theory: patterns of contour lines are very similar as well as the degree of angular localization of photons near the detector and source with the increased degree of deviation closer to the light source. Absolute values of theoretical RER are about twice as low as the measured values. As in the case of the previous sample, the flatness of contour lines in the vicinity of the light source observed experimentally indicates a stronger degree photon delocalization than is seen in simulations.

Fig. 11 Simulation of the realistic absorber (2.0 mm diameter with 7x10¹⁴ Au NPs/mL (i.e.,20x)): a) spatio-angular contour plot of RER at 530 nm using Eq. (9), b) spectro-spatial contour plot of RER at 0° using Eq. (9)

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Comparing Fig. 11(b) and Fig. 10(b) indicates that modeling correctly predicts RER>1 in the entire spectral range, shows strong localization of photons near the detector, minimum photon density midway between the source and the detector and lower values of RER in the vicinity of the source. It is interesting to note a remarkable degree of similarity of contour lines in 450-650 nm spectral range. It appears that very strong absorption and relatively broad plasmon resonance of 20x Au NPs solution smear the plasmon signature in contour lines by producing almost uniform band in 450-600 nm range both in the experiment and modeling. Other than the expected difference in absolute values of RER, modeling produces a different spectral behavior in 700-900 nm range which is also reflected in the distance dependence. A close examination of all modeling results for both samples indicates that this is common feature most likely brought by the boundary condition used in modeling.

Translating the target in-line between the source and fluence detector and performing the perturbation analysis [27, 28], produces a curve that is completely symmetric relative the midpoint and shows the minimal photon density midway between the source and detector. Extending target translation to the entire X,Y-plane in the perturbation approach for fluence produces the so called banana pattern. The banana pattern is also symmetric relative the midpoint so that it has the identical widths measured at the same distance away from the source and detector [28]. It is brought by the fact that the expression for photon density distribution function for fluence is completely symmetric [27] and replacing the source with the detector results in the same pattern. Inspection of Eq. (9) that can be considered analogous to the photon density distribution function but for radiance indicates the absence of such symmetry.

Qualitatively the difference between fluence and radiance in the perturbation analysis can be explained as follows. When the inclusion is closer to the fluence detector, almost all photons that pass through the region occupied by the inclusion are detected by the detector but this number is relatively low compare with the total number of photons emitted by the light source. For the inclusion closer to the light source, it acts like a filter such that almost all photons that originate from the source also pass though the inclusion’s location. However, due to properties of the turbid medium only a small fraction of these photons reach the detector. Hence, the fractions in both cases are the same. Introducing angular sensitivity for the detector (as in the case of radiance) changes the picture described for fluence. Multiple scattering reduces the number of photons reaching the finite angular detection window from the specific direction for radiance (as opposed to unlimited 4π for fluence) after travelling long distances when the inclusion is closer to the light source. Thus, it produces lower values for RER as oppose to the case when the inclusion is closer to the detector. The same multiple scattering also continuously broadens photon angular distribution with distance which results in the loss of directionality. Hence, angular profiles measured closer to the light source will exhibit such broadening effects as seen in the experiment.

The observed discrepancies between the experiment and simulations can be attributed to several factors. Even though radiance detection was performed in-plane in our experiments, the source of the perturbation was a non-uniformly illuminated cylinder containing Au NPs. Thus, it is expected that absolute RER values obtained for the cylinder will deviate from those for the thin disk as in the 2D model. To account for the actual shape of the inclusion the model needs to be extended to a 3D geometry. Another factor is related to applicability of diffusion approximation for describing radiance in the angular domain [40, 58, 59]. It has been shown theoretically that the deviations for radiance, obtained under diffusion approximation from more accurate Monte-Carlo modeling, can be observed for distances from the source equal up to five reciprocal reduced scattering coefficients [59]. In fact, the deviation between radiance under diffusion approximation and more accurate analytical solution of RTE was shown to exist for Intralipid-1% up to 10 mm away from the source [38]. Since diffusion approximation was used twice in calculating RER, it is responsible for reported differences for contour lines in vicinity of the light source between simulated and experimental spatio-angular plots. Validity of diffusion approximation can be also undermined by very high absorption values of the localized inclusion which may exceed locally the scattering coefficient of Intralipid.

It is remarkable that the simple model used in the current work was able to confirm most essential features observed in the experiments. It not only validates the experimental approach but provides important details about the relative angular photon distribution function which plays a critical role in interpreting radiance data for localized inclusions. In order to obtain a complete quantitative agreement between theory and experiment, perturbation approach should be incorporated into more sophisticated and accurate treatments of radiance [40, 62–64]. By improving accuracy, such models would provide the basis for extracting the concentration of nanoparticles in the localized inclusion and the distance from the inclusion to the detector from experimental data. The universality of the model also allows extending it to analysis of the inclusion positioned at any distance and angle relative to the detector-source direction thus generalizing model predictions and parameter extractions.

Radiance is an interstitial spectroscopic technique with some spatial resolution capabilities. Once fully developed, it can identify an inclusion spectroscopically, localize it in the angular domain by specifying the angular and radial coordinates and quantify it by providing the absorption coefficient or concentration. It all can be achieved with a single detector-source pair by rotating the detector over 360° range. A single fluence-based detector-source pair lacks spatial sensitivity and can’t detect an inclusion. As a result, configurations with arrays of sources and fluence detectors on the boundaries of regions of interest are employed for this purpose. Such technique, diffuse optical tomography (DOT) is a truly imaging technique that uses sophisticated back-projection algorithms to reconstruct the image. To constrain the inverse problem and improve the resolution of the image, DOT is usually paired with ultrasound or MRI techniques. In addition, highly non-linear reconstruction algorithms are required when there is a large discrepancy between the refractive index of the inclusion and the surrounding medium (as in the case of Au NPs in biological tissues). Thus, all techniques have their merits and limitations. Radiance-based approach can’t substitute DOT in applications where true imaging is needed. We consider radiance as a complementary technique that can be used interstitially for diagnostic purposes (for example, in prostate) when its capabilities (identify, localize and quantify) and simple inexpensive hardware satisfy medical needs.

4. Conclusion

We present new experimental results on detecting signatures of Au NPs based localized inclusions in Intralipid-1% phantom when the inclusion was translated along the line connecting the detector and light source. The inclusion acts as a perturbation and allows tagging photons that passed through the inclusion’s location recreating their distribution in the turbid medium. It permits drawing conclusions about detectability of the inclusion anywhere within 30-mm source-detector separation.

Using diffusion approximation and extending the perturbation approach to radiance, we develop the novel analytical expression for the radiance extinction ratio (RER) that has a meaning of the relative photon angular distribution function and is equivalent to the photon path density distribution function for fluence. The derived expression for radiance is capable of capturing qualitatively most essential features and trends observed in the experiments: strongly asymmetric distribution of photons in the angular domain between the source and detector with highest density of photons near the detector and the minimum midway between the source and the detector, better angular confinement of photons in vicinity of the detector and continuous deterioration of the angular resolution once the inclusion is moved away from the detector. This is distinctly different from the photon distribution seen by the fluence detector indicating that the anisotropic detector paired with a single light source is capable of providing spatial sensitivity as opposed to the isotropic detector. Angular detectability is always better for the inclusion located closer to the anisotropic detector but not the light source.

We show that differential optical properties of the inclusion and the background medium are manifested in measured spectra leading to very distinct transitions at RER = 1 when optical attenuations match.

The observed discrepancies between the model and experiment call for more accurate modeling approaches that can further aid in extracting such important parameters as the concentration of nanoparticles in the localized inclusion and the distance between the inclusion and detector.

Acknowledgments

The authors acknowledge financial support from Natural Sciences and Engineering Research Council, the Canadian Institutes of Health Research, Atlantic Canada Opportunity Agency, and Undergraduate Student Research Award from Natural Sciences and Engineering Research Council (to K.B.C). The work was also partially funded by the Canadian Research Chair Program to W.M.W.

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Tagging photons with gold nanoparticles as localized absorbers in optical measurements in turbid media

Abstract

1. Introduction

2. Methods and materials

2.1. Set-up for radiance measurements in Intralipid-1%

2.2. Perturbation modeling for radiance

3. Results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (11)

Equations (9)

Biomedical Optics Express