Monte Carlo (MC) simulation of photon transport has been frequently employed to estimate the light distribution within tissues [1, 2]. Realistic tissue models containing layered structures and vessels have helped to improve the understanding of general light-tissue interactions [1, 3, 4]. This has been particularly useful in optimizing laser therapies, such as port-wine stain treatment [5-8]. More light is delivered to a blood vessel as the radius increases, and when the vessel is closer to the surface [7]. The MC method has also been used in predicting redistribution of light energy caused by changes in tissue parameters, such as absorption (μ_a) and scattering (μ_s) coefficients [9-11].

The scattering coefficient of blood can change due to various physiological conditions, including glucose and lipid levels [12-15]. Noninvasive measurement of light distribution changes on the surface may provide information related to such changes in blood composition, without the need for blood extraction. This may lead to more accurate, yet still noninvasive, diagnostic devices. For example, the ratio of surface reflectance measured at two different locations from a common source has been proposed as a potential parameter to monitor scattering changes within a tissue [15]. Since large vessels such as arteries and veins are superficially distributed across human skin, it is important to understand the influence of the vessels on light distribution. However, there has been little investigation regarding the variation of surface light distribution according to scattering changes in a large blood vessel of varying geometry.

We ran a series of MC simulations for various vessel radii and depths while changing the scattering coefficient of blood, and observed the change in surface fluence rates. Once we understood the effects of vessel geometry, we tested the feasibility of using multiple source-detector pairs, featuring a near and a far detector with a common source, near a large vessel, for noninvasive measurement of scattering changes of the blood. The goal of this study is to assess the feasibility of sensitive noninvasive measurement of blood-scattering changes from the surface, using favorable blood-vessel geometry and detected light changes for multiple source-detector pairs. We aim for these simulations to improve the accuracy of noninvasive optical devices for measuring blood-scattering changes.

The MC method can model photon paths by randomizing the photon step size and scattering angle as the photon experiences scattering and absorption events while passing through a medium. Thorough descriptions of the MC method are found elsewhere [4, 16].

For this study, MC simulations were run using a mesh-based MC program [17]. This code provides an efficient and easy-to-use implementation of MC simulations through a MATLAB interface and GPU-based processing capabilities [3]. Most importantly, a mesh-based MC program allows for more flexibility in modeling vessel geometry. Various blood-vessel geometries were modeled in a 100 mm × 100 mm × 100 mm domain with approximately 30,000 nodes. Each simulation was run with 1 million photons and a point source of illumination. The number of photons was chosen to maintain reasonable simulation time for the various simulations, while also providing reliable results for further analysis. The topics of noise and number of photons used in the simulations will be discussed further in the Discussion section. A single run of the MC simulation took approximately 30 to 40 minutes to complete, and the post-processing analysis was done using a custom-built script for MATLAB (MATLAB 6.1, The MathWorks Inc., Natick, MA, 2000).

Table 1 shows the optical properties of the epidermis/dermis of the skin, and of blood, at 800 nm that were previously determined [12, 18-20]. The wavelength was chosen because it is in the near-infrared (NIR) range, which ensures large tissue penetration and minimizes the compounding effects of blood oxygenation levels [21].

[TABLE 1.] Summary of optical properties of tissues used in the simulations, with the individual references listed

Summary of optical properties of tissues used in the simulations, with the individual references listed

A three-dimensional mesh was created for each MC simulation with a different blood-vessel geometry. For the first set of simulations, the radii of blood vessels varied (1, 2, and 5 mm) running parallel to the skin’s surface. Blood-vessel depths varied from 2 to 10 mm below the surface, in increments of 2 mm. These depths were chosen based on physiologically relevant blood-vessel depths in the body [22]. Figure 1 provides an example of the rendered mesh for MC simulation and variation in blood-vessel geometry.

[FIG. 1.] Example of rendered tissue model for MC simulation, with blood-vessel variation in terms of depth and radius. The various colored tetrahedrons in the mesh represent different optical properties.

Calculation of the surface fluence rate in a homogenous skin-tissue model provided a baseline for comparing the effects of other blood-vessel geometries. Figure 2(a) shows the radial distribution of the surface fluence rates (in logarithmic scale) in the homogenous tissue model, for different μ_s. Only three values of μ_s are shown, for the sake of clarity. As anticipated, the majority of the light-tissue interactions occurred near the source. There was a faster decay of surface fluence rates farther from the source as μ_s increased. Immediately near the source, fluence rates increased as scattering increased (Fig. 2(b)). Slight asymmetry, especially immediately near the source, can be attributed to mesh resolution and the number of photons, which is a common issue for stochastic simulations such as Monte Carlo simulations.

[FIG. 2.] (A) Logarithm of the surface fluence rate for the homogenous tissue model (epidermis/dermis only, no vessel), for μs of 150, 200, and 250 cm-1. (B) Log of the surface fluence rate near the source.

As mentioned, the ratio of detected signals from near and far detectors (as depicted in Fig. 1) is sensitive to scattering changes within a tissue [15]. For the homogenous tissue model, Fig. 3 shows the logarithmic relationship of the ratio of near and far signals as μ_s changes from 150 to 250 cm^-1, in increments of 10 cm^-1. For simplicity, the near detector was fixed at 10 mm, similar to a previous study [15]. For all positions of the far detector, the ratio of the two signals produced a linear relationship in response to the μ_s changes. A far detector at 15 mm was less sensitive to μ_s changes, since the distance between the near and far detectors was not large enough. Signals from source-detector distances longer than 20 mm may suffer from noise problems, as hinted in the actual experiment [15]. Based on this observation, a general analysis of the effects of blood-vessel geometry was performed using a near detector at 10 mm and a far detector at 20 mm from the source. A more in-depth analysis of detector position will be discussed in later sections.

[FIG. 3.] Log of the ratio of surface fluence rates at distances of 10 mm :( 15, 20, 25, 30) mm per μs change (150 cm-1 to 250 cm-1), for the homogenous tissue model.

For each blood-vessel depth and radius, simulations were run with μ_s changes of blood near the value listed in Table 1, ranging from 400 cm^-1 to 490 cm^-1. The vessel ran directly through the middle of the x-y plane of the skin-tissue model. The point source was located directly on the surface, in the middle of the x-y plane (coordinates (50 mm, 50 mm), as in Fig. 1).

Thirty simulations were run, for three radii (1, 2, and 5 mm) and ten values of blood μ_s. The log of the ratio of surface fluence rates was taken for the 10 and 20 mm distances along the blood vessel for each of the radii (Fig. 4(a)), to explore the ratio changes depending on μ_s. The fitted slopes for radii of 1, 2, and 5 mm were -3.72 × 10^-4, 6.48 × 10^-4, and 1.64 × 10^-3 respectively. As the vessel size increased, the ratio of surface fluence rates between the near and far distances increased.

Similarly, 50 more simulations at 5 depths and 10 values of μ_s were conducted, for depths of 2, 4, 6, 8, and 10 mm. Here the blood-vessel radius was fixed to 2 mm, to constrain the study to more physiologically relevant parameters [23]. Regression analysis was performed using similar source-detector positions. The results are shown in Fig. 4(b). The signal-ratio changes were most sensitive to the scattering changes when the vessel depth was fixed at 4 mm. Deeper embedding of vessels had less effect on the overall measurement.

[FIG. 4.] Log of the ratio of surface fluence rates for distances of 10 to 20 mm along the vessel, for different blood-vessel (A) radii and (B) depths.

Differences in surface fluence rates for a tissue model with a blood vessel, compared to the homogenous tissue model, were quantified and analyzed. The difference map for various blood-vessel radii and depths, using a blood μ_s of 400 cm^-1, is presented in Fig. 5. For the radius simulations, the blood vessel was placed directly underneath the source (e.g. a 2-mm-radius vessel was placed at a depth of 2 mm). For the depth simulations, the radius was fixed at 2 mm. For all cases, the difference was greatest along the blood vessel, as expected. In general, the difference increased as the radius increased (Fig. 5(a)). The peak difference of surface fluence rate occurred at a blood-vessel depth of 4 mm. Minimal difference of surface fluence rates compared to those for the homogenous tissue model was evident for blood vessel depths of 8 and 10 mm (Fig. 5(b)).

[FIG. 5.] Differences in surface fluence rate from the homogenous tissue model, for each (A) radius simulation and (B) depth simulation, for a blood μs of 400 cm-1. For the radius simulations, the blood vessel was placed directly underneath the source (e.g. a 2-mm-radius vessel was placed at a depth of 2 mm). For the depth simulations, the radius was fixed at 2 mm. The red dot indicates the source’s location.

Using the vessel geometry with a radius of 2 mm and a depth of 4 mm, which corresponds closely to the human lower arm/wrist region, a series of MC simulations was performed for a range of blood-scattering changes between 400 cm^-1 and 490 cm^-1. In this study, 7% of blood volume fraction was incorporated into the overall tissue domain, in addition to the blood-scattering changes in the vessel, to provide a more realistic scenario of tissue-scattering changes [24]. For each μ_s, we binned the surface fluence rate from an 8 mm × 2 mm area at distances of 5, 10, 20, 25, and 30 mm from the source (Fig. 6). The detector was positioned along the blood vessel, which we had found to be the position that was most affected by blood-scattering changes (Fig. 5). Then we calculated the ratio between near detectors at 5, 10, and 15 mm and far detectors at 10, 15, 20, 25, and 30 mm. The ratios were normalized and linear-fitted to demonstrate the sensitivity of the two-detector approach (Fig. 7). Additionally, the slope of the linear-fit line and root-mean-square error (RMSE) between fitted and observed surface fluence rates were calculated, to quantify the effects of various near- and far-detector combinations (Fig. 8).

[FIG. 6.] Example of the analysis performed to quantify the effect of detector distance in the two-detector approach. The fluence rate is shown in logarithmic scale, for clarity. Surface fluence rate was binned from an 8 mm × 2 mm area at distances of 5, 10, 15, 20, 25, and 30 mm away from the source along the blood vessel.

As shown in Fig. 3, a near detector at 10 mm and a far detector at 15, 20, 25, or 30 mm could detect scattering changes. For these simulations, the slope of the linear-fit line could be increased by moving the near detector closer to the source, from 10 to 5 mm (Figs. 7 and 8). Conversely, the slope decreased as the near detector was moved farther from the source, from 10 to 15 mm. These results indicate that high contrast in the intensity of detected signal between the near and far detectors is needed for high sensitivity to blood-scattering changes.

The calculated RMSE for the fitted ratios revealed a limit to the distance at which the far detector can be placed (Fig. 8). When the near detector was placed at 5 mm, the RMSE continued to decrease until the far detector was placed at 20 mm; for distances greater than this, a large increase in RMSE was observed. For far detectors of 25 and 30 mm, many outliers in the surface fluence rate were observed, which affected the linear fit of the ratio to μ_s changes (Fig. 7). Thus, although high contrast between near and far detectors increases the sensitivity of changes in μ_s, increased noise levels for farther detectors affects the reliability of the results.

[FIG. 7.] Ratio of near- and far-detector surface fluence rates compared to μs changes and the corresponding linear fit, for a near detector at a distance of (A) 5, (B) 10, and (C) 15 mm from the source. From left to right, the far detector was placed 10, 15, 20, 25, and 30 mm away from the source.

[FIG. 8.] Calculated linear-fit slope (left axis, black bar) and RMSE (right axis, outlined bar) for a near detector at a distance of (A) 5, (B) 10, and (C) 15 mm. From left to right, the far detector was placed 10, 15, 20, 25, and 30 away from the source.

This study has presented the trends that could be used to improve noninvasive measurement of blood-scattering changes, including targeting of favorable blood-vessel geometry, and a two-source-detector-pairs approach using both near and far detectors. However, vessel radius and depth not only depend largely on the location of the vessel in the body, but also can vary widely from person to person [25]. These variations make it difficult to select an appropriate location for noninvasive measurement of blood properties. The vessel depth along the arm’s length can range from 1 mm (distal) to a maximum of approximately 10 mm (proximal) [22, 26], but the radius can vary. In general, blood vessels distal to the human body are closer to the surface and have a smaller radius [27]. Our results indicate that the blood vessel depth and radii that produce the most sensitive change in surface fluence rate, according to blood-scattering changes, are similar to those located in the lower arm/wrist region [23].

Our results also indicate the need for further studies. The compounding effects of multiple vessels on the measurement needs to be clarified. Also, the exact range of scattering variation of the blood for a typical person is unknown, and depends on many external factors [28]. A separate measurement may help to understand the related physiology better. We believe that using a more realistic range of scattering changes according to certain physiological conditions (i.e. glucose levels), will allow us to focus on specific problems.

This work specifically looks at situations solely involving scattering changes of blood, similar to previous works regarding optical coefficients of blood [12, 13, 15]. Other interactions of light with blood, including absorption, may be of interest for future studies, but adding another variable to our simulations would increase the number and overall complexity of the simulations.

The surface fluence rate is not what is acquired using a detector in contact with the skin’s surface; rather, it is proportional to the gradient of the fluence rate, which is related to the optical flux or diffuse reflectance [29]. However, the results provide information that will be useful for light-energy distribution, in general.

The excessive simulation noise that occurs due to mesh resolution and the number of photons prevents definitive conclusions concerning our two-detector method. That said, trends were observed by increasing the contrast in detected signal between the near and far detectors. Simulation noise is particularly an issue when a three-dimensional MC model is considered. However, having a source-detector separation longer than several centimeters usually makes it difficult to overcome the low signal-to-noise ratio [30, 31]. Therefore, this limitation also exists in actual measurements.

The current study was conducted using a point source of illumination. With advances in the mesh-based MC program that allow for wide-field illumination [32], different source types would allow for more realistic scenarios, and might also increase the sensitivity to blood-vessel scattering changes. Increased resolution, size of the mesh domain, and number of photons in the simulations could also give more accurate results, especially near the boundaries.

Observing changes in surface fluence rate according to blood-scattering changes is key for in vivo scattering measurements of the bloodstream. The radii simulations show that the change in surface fluence rate according to blood-scattering change increases with a larger radius. Blood vessels placed at a depth of 4 mm had the most effect in surface fluence rate per μ_s change. The geometry was similar to blood-vessel size in the human lower arm/wrist. By targeting such favorable geometry, we can produce the greatest surface change for blood scattering changes.

We also present a method for sensitive detection of blood-scattering changes from the surface using a two-detector method. This method depends on high contrast in detected signal between the near and far detectors. Higher contrast can be achieved by moving the near detector closer the source. When the far detector’s distance increased, large outliers were more common. Therefore, there is a limit to the distance of the far detector, due to increased noise.