2.1.

Simulation of Light Transport and Fluorescence in Blood Tube Model

2.1.1.

Description of simulation

Light transport in an infinitely wide blood slab (Fig. 1 ) with different blood hematocrit levels (Hct) and containing fluorescent ICG in varying concentrations $\left(C_{\mathit{ICG}}\right)$ was simulated using the Monte Carlo method.^{14, 15} Fluorescence was mimicked following the approach of Hyde ¹⁶ Briefly, the probability of survival of an excitation photon at each interaction was assumed to be $p=({\mu }_s+{\mu }_{a,\mathit{ICG}})∕({\mu }_s+{\mu }_{a,\mathit{ICG}}+{\mu }_a)$ , where ${\mu }_s$ and ${\mu }_a$ represent the scattering and absorption coefficients of the blood medium, and ${\mu }_{a,\mathit{ICG}}$ is the absorption coefficient of ICG. The “weight” of the excitation photon was reduced by a factor $p$ at each interaction event. Furthermore, at each interaction, the excitation photon was either scattered in a new direction with probability $p_s={\mu }_s∕({\mu }_s+{\mu }_{a,\mathit{ICG}})$ or transformed into a fluorescent photon with probability $p_f={\mu }_{a,\mathit{ICG}}∕({\mu }_s+{\mu }_{a,\mathit{ICG}})$ . The photon direction after a scattering event was chosen from a probability distribution described by a Henyey-Greenstein phase function defined by the mean cosine of scattering angle (g).¹⁵ When an excitation photon was transformed into a fluorescent photon, the new photon direction was chosen at random, assuming isotropy of the fluorescence. Fluorescent photons were then tracked with the traditional Monte Carlo scheme, which at each interaction reduced the photon weight by $\left({\mu }_s\right)∕({\mu }_s+{\mu }_{a,\mathit{ICG}}+{\mu }_a)$ . A unit quantum yield for the fluorescence was used, which was of no consequence for the analysis of the change of the fluorescence signals with $C_{\mathit{ICG}}$ and Hct.

Fig. 1

Blood medium for Monte Carlo simulation. Photons incident on the blood slab at the origin of the axes were tracked as they scattered in the medium. The probability for a photon to become fluorescent increased as the ICG concentration increased. Fluorescent and excitation photons that escaped out of the slab were tabulated based on the distance between the exit point and the axis of cylindrical symmetry of the model represented by the direction of illumination.

A pencil beam of excitation light reached the propagation medium, at the origin of the axes in the direction normal to the plane of the slab (Fig. 1). Reflected light and back-fluorescence emerged from the blood medium on the side illuminated by the excitation beam, while transmitted light and forward-fluorescence emerged after crossing the blood slab. The contribution of photons exiting the blood slab was quantified in bins 0.1 mm wide defined by the distance between the locus of emergence and the origin of the axes, taking into account the cylindrical symmetry of the model. All photons exiting within the width of the bin were tabulated, irrespective of the angle between the photon direction and the direction normal to the blood slab. Photons at the excitation wavelength (wavelength = 784 nm) and fluorescent photons (wavelength = 830 nm) were tabulated separately. The tally represented the radially resolved reflected light, transmitted light, back-fluorescence, and forward-fluorescence signals. These estimates were divided by the number of photons injected and by the area of the annulus with radius equal to the distance $\left(d\right)$ between the center of the bin and the locus of incidence of the excitation beam to determine the throughput, defined as the probability per unit area, for each light signal.¹⁵ The throughputs were noted $R\left(d\right)$ (reflected light), $T\left(d\right)$ (transmitted light), $\mathit{BF}\left(d\right)$ (back-fluorescence), and $\mathit{FF}\left(d\right)$ (forward-fluorescence). Each simulation followed the trajectories of $1.5\times {10}^6$ photons injected in the medium. The thickness of the blood slab was set at 4.5 mm, a common diameter for hemodialysis tubing. The simulations neglected the influence of the tubing material on the optical signals because such a tubing is near-transparent at the visible and near-infrared wavelengths of interest (between 750 and 850 nm).

2.2.

Empiric Validation

3.1.

Simulations: Variations of Signal Intensities as a Function of Lateral Distance and Hematocrit

The back-fluorescence and reflected light throughputs $\mathit{BF}\left(d\right)$ and $R\left(d\right)$ decreased 35-fold and 15-fold, respectively, as a function of distance $d$ over the first 0.5 mm measured from the locus of illumination [Figs. 2a and 2b ]. The throughputs decreased more gradually and approximately exponentially (linear trend on the semilogarithmic plots of Fig. 2) starting from $d\sim 1.5\mathrm{mm}$ . Increasing the hematocrit resulted in a diminution of $\mathit{BF}\left(d\right)$ at all distances $d$ [Fig. 2a]. In contrast, increasing the hematocrit increased $R\left(d\right)$ for $d<1.5\mathrm{mm}$ but decreased $R\left(d\right)$ for $d>1.5\mathrm{mm}$ [Fig. 2b].

Fig. 2

Probability of emergence per unit area (throughput) as a function of distance between the direction of illumination and the locus of emergence for back-fluorescence (a), reflected light (b), forward-fluorescence (c), and transmitted light (d) when the blood ICG concentration is ${10}^{-4}\mathrm{g}∕\mathrm{L}$ . The spread between the traces obtained for different hematocrit levels near the illumination axis (distance <1 mm) is much smaller for the back-fluorescence and reflected light throughputs when compared to the forward-fluorescence and transmitted light throughputs. Near the illumination axis, changing the hematocrit had a smaller effect on the signals measured on the same side of the slab as the illumination beam but changed substantially for the signals measured on the opposite side.

Contrasting with the decay of $\mathit{BF}\left(d\right)$ and $R\left(d\right)$ observed for small values of $d$ , the forward-fluorescence and transmitted light throughputs $\mathit{FF}\left(d\right)$ and $T\left(d\right)$ were approximately unchanged for $d<1\mathrm{mm}$ [Figs. 2c and 2d]. For $d>1\mathrm{mm}$ , the throughputs $\mathit{FF}\left(d\right)$ and $T\left(d\right)$ decreased exponentially with $d$ . Increasing the hematocrit Hct reduced $\mathit{FF}\left(d\right)$ and $T\left(d\right)$ in an approximately exponential fashion. Importantly, $\mathit{FF}\left(d\right)$ measured near the illumination axis was much more sensitive to hematocrit changes than $\mathit{BF}\left(d\right)$ .

3.2.

Simulations: Variations of Integrated Throughtputs as a Function of ICG Concentration and Hematocrit

The integrated back-fluorescence throughputs ${\overline{\mathit{BF}}}_{0.3}$ and ${\overline{\mathit{BF}}}_{2.0}$ increased with $C_{\mathit{ICG}}$ [Figs. 3a and 4a ]. The increase was near linear for $C_{\mathit{ICG}}<{10}^{-3}\mathrm{g}∕\mathrm{L}$ . The slope $\Delta \overline{\mathit{BF}}∕\Delta C_{\mathit{ICG}}$ diminished for $C_{\mathit{ICG}}>{10}^{-3}\mathrm{g}∕\mathrm{L}$ , minimally for ${\overline{\mathit{BF}}}_{0.3}$ , but markedly for ${\overline{\mathit{BF}}}_{2.0}$ . Throughput ${\overline{\mathit{BF}}}_{2.0}$ was more sensitive to Hct changes and decreased by ∼50%, whereas ${\overline{\mathit{BF}}}_{0.3}$ decreased by ∼30% when Hct increased in agreement with the trends noted on the back-fluorescence profile as a function of lateral distance $d$ [Fig. 2a].

Fig. 3

Simulated back-fluorescence (a), reflected light (b), forward-fluorescence (c), and transmitted light throughputs (d) measured near the illumination axis as a function of blood ICG concentration $\left(C_{\mathit{ICG}}\right)$ for different hematocrits (◻: 24%; ◼: 30%; ○: 36%; ●: 42%; △: 48%). The throughputs were integrated over a 0.5-mm-wide annulus whose inner radius was 0.3 mm for the back-fluorescence and reflected light and 0.0 mm for the forward-fluorescence and transmitted light.

Fig. 4

Simulated back-fluorescence (a), reflected light (b), forward-fluorescence (c), and transmitted light throughputs (d) measured away from the illumination axis as a function of blood ICG concentration $\left(C_{\mathit{ICG}}\right)$ for different hematocrits (◻: 24%; ◼: 30%; ○: 36%; ●: 42%; △: 48%). The throughputs were integrated over a 0.5-mm-wide annulus whose inner radius was 2 mm. Note that the back-fluorescence intensity is ∼20 times less intense than that measured near the illumination axis [Fig. 3a], while the forward-fluorescence intensity is reduced only by a factor 3.5 [Fig. 3]. The back-fluorescence response becomes nonlinear for ICG concentrations $>{10}^{-3}\mathrm{g}∕\mathrm{L}$ .

The integrated reflected light throughputs ${\overline{R}}_{0.3}$ [Fig. 3b] and ${\overline{R}}_{2.0}$ [Fig. 4b] decreased when $C_{\mathit{ICG}}$ increased. Reflectance throughput ${\overline{R}}_{0.3}$ increased in near-equal steps when Hct increased for all ICG concentrations, In contrast, reflectance throughput ${\overline{R}}_{2.0}$ decreased when Hct increased, with little change being observed for Hct ⩽ 36% and more substantial decreases for higher hematocrits. The summation distance (2.0 to 2.5 mm) for calculating ${\overline{R}}_{2.0}$ was near the distance range where the curves $R\left(d\right)$ calculated for different values of Hct intersect [Fig. 2b], giving rise to this complex trend.

Integrated forward-fluorescence throughputs ${\overline{\mathit{FF}}}_{0.0}$ and ${\overline{\mathit{FF}}}_{2.0}$ increased near linearly as a function of $C_{\mathit{ICG}}$ only for $C_{\mathit{ICG}}⩽{10}^{-3}\mathrm{g}∕\mathrm{L}$ [Figs. 3c and 4c]. Increasing the hematocrit reduced both forward-fluorescence throughputs in a near-exponential fashion as noted on the forward-fluorescence profile as a function of $d$ [Fig. 2c].

Integrated transmitted light throughputs ${\overline{T}}_{0.0}$ and ${\overline{T}}_{2.0}$ decreased as a function of $C_{\mathit{ICG}}$ [Figs. 3d and 4d]. The slope of the decrease was less when the hematocrit was larger. Increasing the hematocrit reduced the transmittance throughputs for all ICG concentrations in a near-exponential fashion.

3.3.

Simulations: Evaluation of ICG Calibration Equations

Table 2 summarizes the optimal values of the scaling factors and exponents for estimating $C_{\mathit{ICG}}$ from the optical measurements based on the model equations 1, 2, 3. [Eq. 4] could not describe the variations of $C_{\mathit{ICG}}$ satisfactorily.] We noted that for all values of $C_{\mathit{ICG}}$ , increasing Hct decreased the back-fluorescence throughput and the transmitted light throughput. For Eq. 1, the negative value of exponent α amounted to multiplying the decreasing back-fluorescence signal by a correction factor that increased with Hct [ $A\cdot {\overline{T}}_{0.0}^{\alpha }$ or $A\cdot {\overline{T}}_{2.0}^{\alpha }$ ), resulting in a near-linear dependence of $C_{\mathit{ICG}}$ on ${\overline{\mathit{BF}}}_{0.3}$ or ${\overline{\mathit{BF}}}_{2.0}$ . A similar observation accounted for the negative value of exponent β in Eq. 3. In contrast, increasing Hct increased the reflected light throughput ${\overline{R}}_{0.3}$ . The factor $(D\cdot {\overline{R}}_{0.3}^{\delta })$ with a positive value of δ in Eq. 3 increased and compensated for the diminishing value of ${\overline{\mathit{BF}}}_{0.3}$ . Overall, Eq. 1, with the back-fluorescence and transmitted light throughputs measured near the illumination point, and Eq. 2, with the forward-fluorescence and transmitted light throughputs measured at a distance from the illumination axis, resulted in the smallest residual errors.

Table 2

Mean value ± standard error for the estimated coefficients (scaling factors A , B , D ; exponents α, β, δ) of the model calibration equations as estimated from the simulations or derived experimentally. In the simulations, parameters ( BFi , FFi , Ri , and Ti ) represented the back-fluorescence, forward-fluorescence, reflected light, and transmitted light throughputs (probability of photon emergence per unit area) measured near the axis of illumination (i=0.0,0.3) or at a 2-mm distance (i=2.0) . Parameters BFexp , FFexp , Rexp , and Texp designated the corresponding experimental signals. The quality of the fit was evaluated from the mean squared residual error and the average relative residual error between the data and the value predicted by the model: |(CICG,predicted−CICG)∕CICG| . The experimental measurements were repeated on different days with slightly different equipment settings. The experimental results were combined to fit the data to the model equations and determine a single best-fit exponent while allowing for different scaling factors for the two data sets. Note that Eq. 4 could not describe the variations of CICG satisfactorily when used to approximate the data (simulated or experimental) with the model equation.

3.4.

Empiric In Vitro Measurements of Optical Signals

Variations of the experimental optical signals with blood ICG concentration and with blood hematocrit reflected the trends predicted from the simulations. Back-fluorescence increased near-linearly with $C_{\mathit{ICG}}$ , with a slightly reduced slope for the higher values $C_{\mathit{ICG}}$ [Fig. 5a ]. For the same value of $C_{\mathit{ICG}}$ , the back-fluorescence intensity was more intense when Hct was reduced. The reflected light intensity decreased slightly and gradually when $C_{\mathit{ICG}}$ increased [Fig. 5b]. Reflected light intensity decreased when hematocrit decreased, but the decreasing trend was not as gradual as that observed in the simulations [Fig. 3b]. Small position shifts in replacing the blood tube in its holder after each rinse could have affected more acutely the reflected light signal than the other measurements since the reflected light was measured over the narrowest surface area very near the illumination fiber.

Fig. 5

Back-fluorescence (a), reflected light (b), forward-fluorescence (c), and transmitted light intensities (d) measured experimentally in blood as a function of blood ICG concentration $\left(C_{\mathit{ICG}}\right)$ for different hematocrits (◼: 28%; ○: 35%; ●: 42%). The trends observed for each signal as ICG concentration and hematocrit changed are in agreement with the simulation predictions. The absolute intensities cannot be compared because the collection optics and the level of amplification were different for back-fluorescence, reflected light, forward-fluorescence, and transmitted light.

Forward-fluorescence [Fig. 5c] increased with $C_{\mathit{ICG}}$ , presenting larger variations with Hct when compared to back-fluorescence, in agreement with the simulation results. Last, transmitted light intensity was strongly dependent on Hct and decreased gradually with $C_{\mathit{ICG}}$ [Fig. 5d].

Experimental observations from two separate blood loop experiments were combined to estimate the parameters of Eqs. 1, 2, 3 applied to the experimental measurements. The exponents for model equations 1, 2, 3 (α = −0.26, β = −0.99, δ = 0.99) were similar to their counterparts determined from the analysis of the simulation results (Table 2). Notably, parameter α, derived from experimental measurements obtained from a distance of 0.8 mm from the illumination center, was between −0.18 and −0.40, the values obtained in the simulations by integration over the 0.3 to 0.8 mm and 2.0 to 2.5 mm ranges of distance from the illumination beam, respectively. Equation 3 was as accurate as Eq. 2 and only slightly less accurate than Eq. 1 with respect to estimating the circulating ICG concentration.

The pump flow rate and volume of blood in the circuitry derived from the measured optical signals and Eqs. 1 to 3 (Table 3 ) differed by less than 10% from the empiric pump rate and blood volume in the circuitry. Thus, the model calibration equations could be used to convert the transient optical changes associated with ICG injections to useful first-pass ICG concentration curves within the framework of indicator-dilution theory.

Table 3

Empiric volumetric flow rates set by the blood pump and reservoir blood volume compared with the flow rate and blood volume predicted from the optical signals by application of the calibration model equations in Table 2 to the in vitro measurements of fluorescence and diffusely scattered excitation light across the blood tube. The empiric volume flow rate was determined by the setting of the Masterflex pump and verified with the “bucket and stopwatch” method by timing the emptying of the circuitry in a graduated cylinder. The empiric volume was that used to fill the priming reservoir of the blood loop with a graduated cylinder. After conversion of the optical measurements to circulating ICG concentrations, the concentration curves were analyzed using the equations from indicator dilution theory (Refs. 5, 6) to calculate blood flow and reservoir volume.

4.1.

Effect of $C_{\mathit{ICG}}$ on Integrated Throughputs

In the simulations, the integrated back-fluorescence and forward-fluorescence throughputs increased near linearly with $C_{\mathit{ICG}}$ only for $C_{\mathit{ICG}}<{10}^{-3}\mathrm{g}∕\mathrm{L}$ . Several groups reported^{12, 21} that when the ICG concentration is elevated, its fluorescence emission is partially absorbed by dye molecules in the environment (inner filter effect), resulting in a decreasing slope for the fluorescence intensity versus $C_{\mathit{ICG}}$ relationship. Fluorescent light detected at a large distance from the locus of illumination traveled farther and deeper in the blood medium and was more susceptible to being absorbed by ICG when $C_{\mathit{ICG}}$ was elevated. Thus, the inner filter effect was more noticeable for the forward-fluorescence throughputs detected after traversal of the blood medium when compared to the back-fluorescence throughputs detected on the side of the illumination. For the back-fluorescence throughputs, ${\overline{\mathit{BF}}}_{2.0}$ was more sensitive to the inner filter effect when compared to ${\overline{\mathit{BF}}}_{0.3}$ . The experimentally measured back-fluorescence and forward-fluorescence intensities largely reflected the inner filter effect and the fact that forward-fluorescence was more sensitive to this effect than back-fluorescence. In addition to the inner filter effect, which was reproduced in the simulations, aggregation of ICG monomers into polymer chains has been reported, which could have reduced the emission yield for high ICG concentrations in the experimental measurements.²⁰

The transmitted light and reflected light throughputs decreased when $C_{\mathit{ICG}}$ increased, in agreement with increased absorption of the light traveling through the blood medium by more numerous ICG molecules. Reflected light throughput ${\overline{R}}_{0.3}$ represented the contribution of photons, which, on average, traveled the least in the medium, and was therefore least dependent on $C_{\mathit{ICG}}$ .

4.2.

Consequences for the Design of an ICG Fluorescence Probe for Hemodialysis Applications

In the simulations, all fluorescence throughputs were dependent on Hct, which affected the absorption and scattering properties of the propagation medium. Back-fluorescence throughput ${\overline{\mathit{BF}}}_{0.3}$ estimated near the point of entry of the illumination beam was least affected by hematocrit changes. As noted earlier, ${\overline{\mathit{BF}}}_{0.3}$ was the fluorescent throughput least dependent on the inner filter effect, and the most intense of those studied in the simulations. These aggregate results suggest that monitoring the back-fluorescence intensity near the illumination point is the most favorable configuration for a measurement probe used to derive circulating ICG concentration based on optical fluorescence measurements.

The throughputs estimated from the simulations corresponded to the signal intensity per unit area detected over an annular window centered on the illumination axis, equivalent to a slit detector, positioned radially with respect to the direction of illumination of the medium. These throughputs must be multiplied by the area of the annuli and integrated along the radial direction to compare the signals measured with circular detectors of different apertures. The calculations show that 48% of all back-fluorescence photons emerge within an aperture of 0.5 mm and 64% within an aperture of 1 mm when the hematocrit is 36%. In contrast, only 6% of all forward-fluorescence photons emerge within an aperture of 0.5 mm and 17% within an aperture of 1 mm for the same hematocrit. Similarly, 33% of all reflected photons exit the medium within 0.5 mm of the illumination axis, while that is true for only 8% of all transmitted photons. Thus, a small-diameter circular detector or fiber optic bundle positioned to measure the back-fluorescence or reflected light intensity captures a large fraction of the available signal. A much larger aperture is necessary to capture a substantial fraction of the available forward-fluorescence or transmitted light. A measurement system designed to capture forward-fluorescence would be ineffective unless it had a large aperture. Such a configuration, however, is limited by the finite diameter of the dialysis tube (∼5 mm) and would make the measurement more sensitive to the variability of the interface between the detector and the curvature of the blood tube.

Correcting the back-fluorescence signal for variations associated with the hematocrit may be done using the transmitted light [Eq. 1] or the reflected light [Eq. 3]. The transmitted light approach produced slightly more accurate estimates of the ICG concentration both in the simulations and in the experimental tests. Practically, using the reflected light signal for the correction may be more convenient with respect to the construction of the probe since the fiber optic bundles used to measure the two signals could be adjacent rather than facing each other on opposite sides of the blood tube. Measuring the reflected light anywhere between 0 and 1 mm from the illumination axis yields a factor that increases by 60% when the hematocrit increases from 24% to 48% and could be used to account for hematocrit changes.

4.3.

Relation to Previous Work

Several authors have examined the effect of the medium optical properties on the detected fluorescence emission intensity with the goal of estimating the concentration of a fluorophore in a turbid medium independently of the medium’s optical properties.^{22, 23, 24} In particular, Weersink and colleagues²² demonstrated that the ratio (back-fluorescence/reflected light) measured at specific distances from the illumination point source was linearly related to fluorophore concentration over a wide range of tissue optical properties. These results were equivalent to those derived in our study for the ratio of forwardfluorescence/transmitted light (coefficient β ∼ −1), but they differed from those we obtained for the (back-fluorescence/reflected light) ratio (coefficient δ ∼ 0.8 to 1). The distances from the illumination point at which the optical measurements were obtained were different in our study and in the work of Weersink ²² Furthermore, our study considered a propagation medium equivalent to blood and therefore more absorbent $({\mu }_a\sim 0.2\text{to}0.4{\mathrm{mm}}^{-1})$ than the tissue phantoms examined in the earlier work $({\mu }_a\sim 0.001\text{to}0.1{\mathrm{mm}}^{-1})$ . It is noteworthy that previous work mostly envisaged fluorescence measurements from the surface of optically thick tissues and organs. Therefore, the use of transmitted excitation light and forward-fluorescence emission was not considered. Our study shows that such quantities would provide useful information if measured from optically thin tissue that can be transilluminated, such as the earlobe or the fingertips.

4.4.

Limitations

Mention should be made of several limitations of the study. The propagation medium for the simulations was assumed to be semi-infinite, while the experiment were performed with blood contained in a cylindrical plastic tube. The semi-infinite blood slab was chosen for the simulations to characterize the interactions between blood hematocrit, ICG concentration, and optical measurements of fluorescence, transmitted light, and reflected light intensities, independently of a particular shape for the experimental measurement chamber. As indicated in discussing the design of a fluorescence probe, the integrated back-fluorescence and reflected light signals are concentrated near the point of entry of the illumination beam and therefore are less sensitive to the shape of the medium when compared to the forward-fluorescence and transmitted light signals. The simulations assumed an infinitely thin pencil of light whereas the experiments used a 100-μm optic fiber to carry the excitation light to the blood medium. Last, the simulations did not consider absorption, scattering, or refraction in the blood tube. Most plastic materials used for dialysis tubing are near-transparent at visible and near-infrared wavelengths, and their refractive index (∼1.35 to 1.55) is close to the refractive index of blood.¹⁸

Simulations and experimental results were obtained for blood ICG concentrations $C_{\mathit{ICG}}<2.5\times {10}^{-3}\mathrm{g}∕\mathrm{L}$ . It is known that ICG fluorescence increases linearly only within a limited range of concentrations $(<1\text{to}2\times {10}^{-3}\mathrm{g}∕\mathrm{L})$ . Aggregation of ICG monomers into polymer chains reduces the emission yield for higher concentrations.^{12, 20} It is possible that the proposed model equations would not extend much beyond the range of concentrations examined in the study.

The ICG calibration equations tested on the simulation results considered only combinations of fluorescent quantities and incident light quantities that were both obtained proximal to the axis of illumination or distal from the axis of illumination. Other combinations that mixed proximal and distal measurements (i.e., ${\mathit{BF}}_{0.3}\cdot T_{2.0}$ ) were not tested, as these combinations would be less practical to implement.

Blood ICG concentration $C_{\mathit{ICG}}$ estimated using the model equations reproduced accurately the ICG concentration used as a parameter in the Monte Carlo simulations [Fig. 6a ]. Concentration $C_{\mathit{ICG}}$ predicted from the experimental data using the model equations was more spread out around the line of identity [Fig. 6b]. The variability of the prediction was not related to the hematocrit, suggesting that the effect of Hct on the predicted $C_{\mathit{ICG}}$ had been accounted for by the model. Rather, the spread of the predicted $C_{\mathit{ICG}}$ more noticeable for the higher ICG concentrations likely reflected experimental errors and the fact that the minimization approach used to estimate the parameters of the model was carried out on the sum of relative differences between the model predictions and $C_{\mathit{ICG}}$ . This approach was chosen to increase the contribution of the low values of $C_{\mathit{ICG}}$ in the selection of the model parameters. In the intended application of estimating cardiac output and circulating blood volume using fluorescence dilution methods,⁶ concentrations < 20% of the maximum circulating $C_{\mathit{ICG}}$ $(\sim 1\text{to}2\times {10}^{-3}\mathrm{g}∕\mathrm{L})$ are the most important to estimate accurately due to the pointy shape of the dilution traces.

Fig. 6

(a) Blood ICG concentration $\left(C_{\mathit{ICG}}\right)$ predicted with Eq. 1 applied to back-fluorescence and transmitted light throughputs ( ${\overline{\mathit{BF}}}_{0.3}$ and ${\overline{\mathrm{T}}}_{0.0}$ ) as a function of actual $C_{\mathit{ICG}}$ concentration used in Monte Carlo simulation. Symbols: concentration predicted from simulated data; solid line: line of identity. (b) Blood ICG concentration $\left(C_{\mathit{ICG}}\right)$ predicted with Eq. 1 applied to back-fluorescence and transmitted light intensities ( ${\overline{\mathit{BF}}}_{\exp }$ and ${\overline{T}}_{\exp }$ ) measured experimentally. Symbols: concentration predicted from experimental data obtained for different values of the hematocrit (Hct); solid line: line of identity. The larger dispersion around the line of identity in the concentrations predicted from the experimental data reflects the larger average relative residual error for the experimental data (8.6% versus 2.8%; Table 2).

Last, the optical coefficients of the blood were chosen to correspond to blood hemoglobin 100% saturated in oxygen. The excitation (784 nm) and emission wavelengths (830 nm) used in the simulations correspond to the peak of absorption and fluorescence emission of ICG. These wavelengths are near the isobestic point of hemoglobin (805 nm) such that the absorption characteristics of hemoglobin vary only by a small amount with ${\mathrm{O}}_2$ saturation for wavelengths between 784 and 830 nm (Ref. 17). Thus, the simulation results should remain largely valid even for blood that is not fully saturated in oxygen.