2.1.

Experimental Setup

A schematic of the experimental setup for FDF measurements is shown in Fig. 1. The sample was illuminated from underneath with a broad collimated beam of light. The fiber-optic probe was placed over the sample and gradually advanced via a micromanipulator to measure the FDF $\psi (z)$.

Fig. 1

Schematic of the experimental setup used for light attenuation measurments (side view).

The apparatus consisted of an aluminum stand (not shown) with a sample tray, the three-axis micromanipulator with attached fiber-optic probe, and the illumination system. The stand was painted flat black so as not to reflect light. The micromanipulator, bolted to the top of the stand, allowed the probe to move along $x$, $y$, and $z$ axes as well as to rotate up to 90 deg around the horizontal axis. The fine adjustment allows the probe to be advanced into the sample in steps as small as $10\mu \mathrm{m}$.

The light sources (Shanghai Dream Lasers) were a 660 nm, 600 mW laser (model SDL-LM-660-600T) and a 532 nm, 1000 mW laser (model SDL-532-1000T). To remove the nonuniformity in the 660-nm laser’s beam profile, the beam was focused with a 45-mm double convex (DCX) lens ($F=250\mathrm{mm}$) onto a holographic diffuser plate. The outgoing beam was collimated using an 83-mm diameter DCX lens with $F=173\mathrm{mm}$ (Anchor Optics AX74813) and reflected upward by a $108\mathrm{mm}\times 145\mathrm{mm}\times 3\mathrm{mm}$ mirror (Anchor Optics AX27536) to illuminate the tissue sample.

We used a custom-built optical probe (optrode) using the FG200LCC optical fiber (Thorlabs). The NA of the fiber and its diameter were 0.22 and $200\mu \mathrm{m}$, respectively (see Appendix A for more details).

2.2.

Measuring Forward-Directed Flux

Measuring the FDF required calibrating the position of the probe, such that the micromanipulator would read 0 when the tip of the needle touched the glass. Next, the probe was raised and the sample (heart tissue, homogenized heart tissue, or phantom) was placed under the probe. The needle was then advanced in small steps into the sample, starting on the surface and continuing until the probe tip touched the glass bottom ($z=0$). To keep the change in intensity more uniform over the entire thickness of the medium, the step size ($\mathrm{\Delta }z$) varied with $z$, so that the change in power between steps did not exceed 10%. The initial value of $\mathrm{\Delta }z$ was chosen between 100 and $200\mu \mathrm{m}$. When change in power between steps exceeded 10%, the step size was cut in half. As a result, $\mathrm{\Delta }z$ was reduced to 50, 20, and $10\mu \mathrm{m}$ at depths of $z=1\mathrm{mm}$, 200 and $100\mu \mathrm{m}$, respectively.

2.3.

Tissue Samples

All tissue experiments were performed in accordance with National Institutes of Health Guidelines in the use of laboratory animals and approved by the SUNY Upstate Medical University Animal Care and Use Committee. Female Yorkshire pigs (26 to 32 kg) undergoing 48-hr sepsis studies were anesthetized using a continuous infusion of ketamine/xylazine. Our choice of the specimens was determined in part by the availability of freshly harvested hearts from another study, which enabled us to achieve our objectives without sacrificing additional animals. Upon completion of the study, pigs were euthanized and the hearts were removed. The right ventricle was dissected from each heart and laid flat on a piece of glass with the epicardial side facing the light source and the endocardial side facing the fiber-optic probe. A small incision was made in the endocardial side with a scalpel at the fiber-optic probe insertion site to prevent a dimple from forming on the surface as the flat tipped and comparatively large-diameter needle went in.

In some of the experiments, the measurements of FDF were carried out in homogenized tissue. The heart ventricles were separated from the valves, veins, arteries, atria, auricle, and fat. The 100 to 200 cc of isolated muscle tissues were then homogenized for 2 to 3 min into a fine paste by the Omni Macro ES digital programmable homogenizer by Omni International. The homogenized tissue was then placed in the sample tray and spread to an even thickness of 5 to 10 mm. The measurement protocol for these samples was the same as that used for the intact heart tissue.

2.4.

Phantoms

We used a mixture of India ink, as the absorber, with IL 20% emulsion (Sigma Aldrich), as the scattering media. The experiments were conducted for IL concentration of 8% being close to the scattering properties of cardiac tissue at 660 nm, see discussion below.

2.5.

Monte Carlo Light Transport Simulations

To simulate measurements of FDF, $\psi$, at different depths, $\psi (z)$, using optical fibers of varying NA (i.e., varying maximum acceptance angle, $\theta$, where $\mathrm{NA}=n\sin (\theta )$ and $n=\text{index of refraction of the tissue}$), we performed MC simulations using the open-source software packages Monte Carlo modeling of photon transport in multi-layered tissues (MCML)³⁷ and CONV.³⁸

Using MCML, we obtained the tissue response to an infinitely narrow light beam. The “total transmittance” MCML routine was modified to allow simulation of the FDF at any given depth $z$ inside the tissue. This was achieved by allowing user-defined placement of the $z$-plane, where total transmittance was calculated, which, in the original routine, was fixed at the tissue’s rear surface. The output of the modified program produced the FDF $\psi (z,r,\alpha )$ as function of the distance, $r$, from the beam axis, and the angle of photon incidence $\alpha$ with respect to a line normal to a given $z$-plane (grid spacing, $\mathrm{\Delta }\alpha =1\mathrm{deg}$; $\alpha =0\mathrm{deg}$ to 90 deg).

The function $\psi (z,r,\alpha )$ was then convolved using the CONV package to simulate broad-field illumination. We simulated a circular, flat incident beam of radius $R=1.5\mathrm{cm}$ with total incident energy of 1.0 J. The probe was located in the center of the beam ($r=0$). Increasing $R$ above the set value did not affect the convolution results. This suggests that the condition of uniform broad-field illumination was satisfied. To obtain responses for probes with different NAs, the function $\psi (z,r=0,\alpha )$ was convolved with the function representing the angular sensitivity of the respective probe.

The following parameters were used as inputs to the program: (1) the photon absorption coefficient, ${\mu }_{\mathrm{a}}$; (2) the photon scattering coefficient, ${\mu }_{\mathrm{s}}$; (3) the scattering anisotropy factor, $g$; (4) the refractive index of both the tissue ($n_{\text{tissue}}$) and surrounding ambient air medium ($n_{\text{air}}=1.0$); and (5) the thickness of the tissue, $d$. The numerical values of the parameters were chosen based on the literature data obtained for cardiac tissues in different species for wavelengths ranging from 650 to 700 nm (see Table 5, Appendix C). The absorption coefficient was held constant at ${\mu }_{\mathrm{a}}=0.1{\mathrm{mm}}^{-1}$ for all simulations. This value is approximately the median of the reported values (0.018 to $0.150{\mathrm{mm}}^{-1}$). The values of ${\mu }_{\mathrm{s}}$ were set at 5, 10, and $20{\mathrm{mm}}^{-1}$, which cover the range of values reported in the literature ($6.27{\mathrm{mm}}^{-1}$ in Ref. 31, $10.4{\mathrm{mm}}^{-1}$ in Ref. 32, and $21.8{\mathrm{mm}}^{-1}$ in Ref. 28). The refractive index in the tissue was set at 1.4 following Ref. 39. (Respective measurements were carried in bovine muscle illuminated with 632.8-nm light.)

We explored how the FDF, $\psi (z)$, as function of depth ($z=0$ to 5 mm), was affected by the anisotropy coefficient, $g$ (from 0.10 to 0.99), and fiber-optic probe maximum acceptance angle, $\alpha$ (from 1 deg to 36 deg). The tissue thickness, $d$, was set at a value of 5.0 mm. For each MCML simulation, a total of 100,000,000 photon packets were launched.

3.1.

Measuring $\psi (z)$ in Heart Tissue

Representative samples of forward flux measurements in the intact pig right ventricular wall at 660 and 532 nm are shown in Fig. 2(a). The normalized FDF is plotted on a log scale. The data points for 530 nm lie well below those for 660 nm, which reflects the higher rate of attenuation of green light as compared with red light. However, both curves have an important similarity. Each has a steeper slope at smaller $z$ and a lower one at larger $z$ suggesting that the curves represent a sum of two decaying exponents.

Fig. 2

(a) FDF plots at 660 nm (red) and 532 nm (green) in pig right ventricular wall. Red and green solid lines show the respective two-exponent fits. (b) Relative error of the fits.

Confirmation that the data in Fig. 2(a) are indeed in accordance with the sum of two-exponential decay functions as presented in Eq. (1) are obtained by using the FIT function of MATLAB,

Similar measurements were carried out in different locations, in five different hearts. The recording sites were chosen primarily across the apical section of the right ventricle, which lacks fat patches and large blood vessels. An example of the measurement sites we chose is shown in Fig. 3, which depict the endocardial and epicardial surfaces of the right ventricle in one of the hearts and all the measurement sites. The number of measurements conducted on each heart was 20 (10 with red light and 10 with green light).

Fig. 3

Picture of (a) endocardial and (b) epicardial sides of pig right ventricular wall in one of the hearts. The red-solid circles represent measurement sites for 660-nm light, and the green-dashed circles represent measurement sites for 532-nm light. (c) Plot of average FDF data across all recording sites for 660 and 532-nm light. (d) Averaged fit error.

Figure 3 shows the averaged FDF data and fit curves in one of the preparations. It is notable that while the scatter of the light intensity at any given depth is quite significant, the accuracy of the individual fits [see Fig. 3(d)] remains relatively high. This reflects the fact that while the optical properties of the tissue vary from one location to another, $\psi (z)$ remains as a two-exponential function. The average fit parameters $k_1$, ${\delta }_1$, and ${\delta }_2$ for different hearts and their standard deviations for 532- and 660-nm light are shown in Table 1 for five hearts each with 10 measurements for red and green light, for a total of 100 measurements.

Table 1

Optical characteristics of pig ventricular myocardium at 660 and 532 nm. Five hearts were examined and 10 measurements for red and green light each were taken for a total of 100 measurements.

Due to signal normalization to $\psi (0)$, the parameter $k_2$ is calculated as $k_2=1-k_1$. The attenuation length for ballistic photons ${\delta }_1$ at 530 nm is almost two times smaller than at 660 nm (0.13 versus 0.24 mm). The difference is even greater for ${\delta }_2$ (0.58 versus 2.1 mm). The observed differences are the manifestation of reduced light absorption and scattering at longer wavelengths.

The interpretation of the long spatial constant is rather straightforward. It represents a spatial decay of fluence rate in the diffuse regime, with ${\delta }_2$ being the attenuation length commonly symbolized as $\delta$ in the literature. Indeed, our experimentally measured values of ${\delta }_2$ are within the range of values reported in Refs. 28, 31, 32, and 40 for cardiac tissue (1.4 to 3.6 mm). The origin of the fast exponent is less obvious. The numerical value of ${\delta }_1$ is not reported in the literature. However, according to our experiments, its inverse (${\delta }_1^{-1}$) is between 5 and $8{\mathrm{mm}}^{-1}$, which is not too far off from the values that are reported in Refs. 28, 31, and 32 for the scattering coefficient in biological tissues (6.27 to $21.8{\mathrm{mm}}^{-1}$). This suggests that ${\delta }_1$ may represent the decay of ballistic photons in the tissue. It may also reflect a heterogeneity of the optical properties of the myocardial surface layer or both.

To assess the potential contribution of myocardial heterogeneity on $\psi (z)$, we carried out the following experiment. After measuring $\psi (z)$ in intact heart tissue, the tissue was homogenized, and the experiments were repeated. Figure 4 compares $\psi (z)$ before and after homogenization. If the first exponent was the result of optical heterogeneity, after homogenization, we would have observed a significant reduction, or the complete disappearance, of the fast exponent. However, this was not the case.

Fig. 4

Comparison of $\psi (z)$ measured in the same heart, once as intact tissue and once after being homogenized, illuminated with 660-nm light. The data were collected with the optrode made from the 25-gauge needle and the FG200LCC optical fiber ($\mathrm{NA}=0.22$).

One can see that after homogenization, the first exponent is largely preserved, which suggests that it is not the result of tissue heterogeneity. The average values of ${\delta }_1$ and ${\delta }_2$ obtained in intact and in homogenized tissues ($N=100$ and 5, respectively) at 660 nm are compared in Table 2. The differences are insignificant, suggesting that the process of homogenization does not affect the decay constants.

Table 2

Optical characteristics of intact and homogenized pig ventricular myocardium at 660 nm.

3.2.

Monte Carlo Simulations

To determine a potential link between the fast exponent and the rate of decay of ballistic photons, we carried out MC simulations of FDF measurements reproducing our experimental protocol. The results are presented in Fig. 5.

Fig. 5

(a) MC simulations of the FDF, $\psi (z)$, for fiber-optic probes with large ($\mathrm{NA}=0.22$) and small ($\mathrm{NA}=0.024$) NAs ($\lambda =660\mathrm{nm}$, ${\mu }_{\mathrm{s}}=5{\mathrm{mm}}^{-1}$, ${\mu }_{\mathrm{a}}=0.1{\mathrm{mm}}^{-1}$, $g=0.80$) and (b) the error of curve fit is shown on the right.

Figure 5(a) shows simulated $\psi (z)$ for two probes with different NAs. One had an $\mathrm{NA}=0.22$, matching the probe used in our experiments, whereas the second probe had a much smaller NA ($\mathrm{NA}=0.024$) and consequently should accept fewer diffuse photons. The parameters (see figure legend) were chosen to be close to those reported in the literature for pig myocardium.³¹ As in the tissue experiments, both plots could be well approximated with the sum of two exponents [see fit error plots in Fig. 5(b)].

The slope of the slow exponent ${\delta }_2$ is nearly identical in both cases (for $\mathrm{NA}=0.024$ and for $\mathrm{NA}=0.22$). As expected, ${\delta }_2$ corresponds to attenuation length reflecting the decay of diffuse photons

We compared the ${\delta }_2$ predicted by Eq. (2) with the respective value obtained from fitting the FDF, $\psi (z)$. The value of ${\mu }_{\mathrm{s}}^{\prime }$ was calculated using the following equation:

The slopes of the fast exponents were also similar. Despite a 10-fold difference in the NA ($\mathrm{NA}=0.22$ and $\mathrm{NA}=0.024$), the values of ${\delta }_1$ were 0.24 and 0.20, respectively. The only significant difference between the two cases was in the distribution of the weights between the fast and the slow exponents, which could be anticipated. Indeed, the weight of the slow exponent $k_2$ obtained using the probe with larger NA should be greater because it accepts a larger portion of diffuse photons.

Notably, the shorter ${\delta }_1$ obtained from the fit was very close to the rate of decay of ballistic photons ${\delta }_{\mathrm{b}}=0.196\mathrm{mm}$ for the given values of ${\mu }_{\mathrm{a}}$ ($0.1{\mathrm{mm}}^{-1}$) and ${\mu }_{\mathrm{s}}$ ($5{\mathrm{mm}}^{-1}$)

This result suggests that for a given set of optical characteristics, the decay rate is not significantly influenced by diffuse photons despite relatively large NAs of the probe. This was unexpected considering that the NA for measuring ${\delta }_{\mathrm{b}}$ is usually ${10}^{-3}$ smaller than that of NA used in our experiments.

To determine how well this observation holds at different parameters, we carried out a series of MC simulations by varying the scattering coefficient ${\mu }_{\mathrm{s}}$, the anisotropy coefficient $g$, and the NA of the probe NA.

The simulation results are shown in Fig. 6. Panel (a) shows the dependence of ${\delta }_1^{-1}$ on $g$ at three different values of ${\mu }_{\mathrm{s}}$ for a probe with $\mathrm{NA}=0.22$, which we used in our experiments. Panel (b) shows the same data after normalization by ${\mu }_{\mathrm{s}}$. Panel (c) shows the effect of NA on ${\delta }_1^{-1}$ at different $g$ for ${\mu }_{\mathrm{s}}=10{\mathrm{mm}}^{-1}$.

Fig. 6

Data from the MC simulations. (a) The dependence of ${\delta }_1^{-1}$ on anisotropy $g$ for ${\mu }_{\mathrm{s}}=5$, 10, and $20{\mathrm{mm}}^{-1}$. (b) The data from panel A replotted with ${\delta }_1^{-1}$ normalized by dividing by ${\mu }_{\mathrm{s}}$. (c) The dependence of ${\delta }^{-1}$ on anisotropy $g$ for five different values of the optical probe maximum acceptance angle $\alpha$ (here, ${\mu }_{\mathrm{s}}=10{\mathrm{mm}}^{-1}$). In panels A and B, the NA of the probe was $\mathrm{NA}=0.22$ ($\alpha \approx 10\mathrm{deg}$). The absorption coefficient was ${\mu }_a=0.1{\mathrm{mm}}^{-1}$ for all plots.

Our findings can be summarized as follows:

The accuracy of the fit can be appreciated from Fig. 6, panel (b), which shows plots of Eq. (5) for ${\delta }_1^{-1}/{\mu }_{\mathrm{s}}$ versus the anisotropy coefficient $g$, superimposed on the normalized MC simulation data. Please note that all the above is valid for ${\mu }_{\mathrm{s}}\gg {\mu }_{\mathrm{a}}$, the condition that is usually satisfied in the cardiac and other biological tissues.⁴¹

3.3.

Using Forward Directed Flux Measurements for Estimating ${\mu }_{\mathrm{s}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and ${\mu }_{\mathrm{a}}$

In cardiac tissues, the reported values of $g$ fall in the range between 0.78³¹ and 0.96.²⁸ In pig myocardium, at 700 nm, it is reported to be around $g=0.91$.³² Using this value of $g$ as well as the experimentally derived values of ${\delta }_1$ and ${\delta }_2$, we estimated the key optical parameters of the tissue, such as ${\mu }_{\mathrm{s}}$, ${\mu }_{\mathrm{s}}^{\prime }$, and ${\mu }_{\mathrm{a}}$ and compared them with the respective values reported in the literature. The value of ${\mu }_{\mathrm{s}}$ was obtained from Eq. (5) by substituting the numerical value of $g$ and the average value of ${\delta }_1$ obtained in our experiments. Then, we calculated the reduced scattering coefficient ${\mu }_{\mathrm{s}}^{\prime }$ from Eq. (3) using the earlier derived value of ${\mu }_{\mathrm{s}}$. Finally, we plugged ${\mu }_{\mathrm{s}}^{\prime }$ into Eq. (2) and solved for ${\mu }_{\mathrm{a}}$. The respective results are shown in Table 3.

Table 3

Optical parameters of pig myocardial tissue at 660 nm.

One can see that our estimates of all three parameters are consistent with the data reported in the literature for pig myocardium obtained using completely different techniques. The difference between our data and respective values obtained in Ref. 32 are $<8\%$ for ${\mu }_{\mathrm{s}}$ and ${\mu }_{\mathrm{s}}^{\prime }$, and $<15\%$ for ${\mu }_{\mathrm{a}}$, which can be the result of myocardial heterogeneity and difference in wavelength (Data from the literature was obtained at 700 nm, and ours was obtained at 660 nm).

3.4.

Using Forward Directed Flux Measurements for Estimating ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, $g$ and ${\mu }_{\mathrm{s}}$ in Intralipid-Based Tissue Phantoms

As we will show below, the values of ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, $g$, and ${\mu }_{\mathrm{s}}$ can be obtained from FDF even without prior knowledge of $g$. All we need is to repeat the measurements of $\psi (z)$ after adding a known amount of absorber. By comparing the changes in ${\delta }_2$ after adding an absorber, one can determine the values of ${\mu }_{\mathrm{s}}^{\prime }$ and the absorption coefficient ${\mu }_{\mathrm{a}}$. This method is known in the literature as the added absorber method.⁴² Indeed, adding an absorber with a known absorption coefficient $\mathrm{\Delta }{\mu }_{\mathrm{a}}$ changes the attenuation length ${\delta }_2$ to a new value ${\delta }_{2\mathrm{\Delta }}$ that can be expressed using the following formula:

Figure 7 shows sample plots of $\psi (z)$ in 4% IL phantom with and without an absorber (India ink) at 660 nm (panel A) and 532 nm (panel C). The data fit average error was within 1% to 2% [panels (b) and (d)]. The addition of an absorber has a negligible effect on ${\delta }_1$ because $\mathrm{\Delta }{\mu }_{\mathrm{a}}$ is very small. In contrast, the values of ${\delta }_2$ undergo a significant change, which is roughly proportional to the square root of ${\mu }_{\mathrm{a}}$ [see Eq. (6)].

Fig. 7

Forward-directed flux measurements in phantoms at 660 and 532 nm. The data was collected with the optrode made from the 25G needle and the FG200LCC optical fiber (0.22 NA). (a) shows the FDF measurements for 660 nm with the corresponding error of the fit in panel (b), (c) shows the FDF measurements for 532nm light with the corresponding error of fit in panel (d).

Table 4 shows the values of ${\delta }_1$ and ${\delta }_2$ as well as estimated values of ${\mu }_{\mathrm{s}}^{\prime }$, ${\mu }_{\mathrm{a}}$, $g$ and ${\mu }_{\mathrm{s}}$ for different phantoms at 660 and 532 nm. Notably, the values of $g$ obtained in our experiments are within 8% of the values reported in the literature (0.64 for red light and 0.73 for green light) for this particular type of IL.⁴³ In addition, they reproduce the increase in $g$ due to the reduction of the illumination wavelength from 660 to 532 nm.

Table 4

Optical characteristics of IL phantoms at 660 and 532 nm derived from the FDF measurements.

To compare our values of ${\mu }_{\mathrm{s}}$ with those reported in the literature, we also estimated values of ${\mu }_{\mathrm{s}}$ for 20% emulsion IL under the assumption that ${\mu }_{\mathrm{s}}$ is proportional to IL concentration. Using the scattering coefficient for undiluted IL given in the literature (${\mu }_{\mathrm{s}}=70{\mathrm{mm}}^{-1}$ for red light and $123{\mathrm{mm}}^{-1}$ for green light), we can expect ${\mu }_{\mathrm{s}}=5.6{\mathrm{mm}}^{-1}$ for red light and ${\mu }_{\mathrm{s}}=9.84{\mathrm{mm}}^{-1}$ for green light at 8% concentration. The estimated values are close to the measured values in Table 4.