1.

Introduction

Most optical-physiological diagnosis methods are based on the insertion of light, with known parameters to a tested tissue, followed by the measurement of the re-emitted light. When investigating light–tissue interactions, human tissue is usually dealt with as a semi-infinite surface. Hence, the light reflected from the tissue (volume reflection) is commonly investigated.^1–6 Very few methods test the transmitted light^7–10 or ballistic photons.¹¹ In this work, we suggest looking at the full profile of scattered light (transmitted and reflected) from a circle of tissue, such as a fingertip joint, ear lobe, or pinched tissue, at all possible exit angles. This full profile will reveal new behaviors of photon migration in tissue.

The human tissue is one of the most complex optical mediums since it is a turbid one, which means it is not homogeneous. Hence the optical properties, such as the absorption coefficient ${\mu }_a$, the scattering coefficient ${\mu }_s$, and the anisotropy factor $g$, are unknown and vary in different areas and physiological states. Furthermore, during respiration the blood vessels, which are the main cause for light absorption, vary in size.

Since the blood vessel distribution in the tissue is not homogenous, several researchers^12–15 raised the question whether the attenuation due to absorption in the blood also depends on the distribution of the blood vessels in the tissue. Liu et al.¹⁰ used an epoxy phantom to demonstrate that the inner part of larger blood vessels is less effective in light absorption because it has lower exposure to light (“shielding effect”). By using computer simulation, Jacques¹³ found that the reflectance from a semi-infinite tissue increases with a direct correlation to the vessel size, even though blood volume was kept constant.

Measurements of light–tissue interaction involve either direct methods which need no theoretical models of light propagation, or indirect methods. Direct methods include attenuation measurements in calibration solutions¹⁶ or integrating spheres for the measurement of the total light transmission through a blood sample and of the total reflected light.^7–9

Indirect methods are based on mathematical simulation models of photon migration. Photon migration simulations are commonly described by the diffusion theory.¹⁷ However, in the close region near the light source, diffusion theory does not accurately describe the light distribution.¹⁸ The Monte Carlo (MC) simulation method assumes that all photons begin as ballistic photons, and the change in their direction is due to the scattering coefficient of the medium.^{1,11,13,19,20}

In this work, we calculated via MC simulation the distribution of collected photons after being scattered and absorbed in a two-dimensional (2-D) model of a circular cross section of tissue containing blood vessels. The results were compared to three-dimensional (3-D) where the transmission profile was calculated. A near infrared (NIR) wavelength of 850 nm was chosen due to its high penetration depth and minimal scattering.²¹ The numerical simulation was repeated ten times to eliminate vessel location dependency. The transmission and reflectance profiles via blood vessel diameter was extracted from our results in order to compare them to traditional known results.

2.

Materials and Methods

2.1.

Model of Tissue with Blood Vessels

The 2-D 10-mm diameter circular cross section of “tissue” in which circles of “blood vessels” were introduced is shown in Fig. 1. To understand the influence of the vessel diameter, several simulations were held where a different vessel diameter was chosen. In order to fulfill the constant blood volume (denoted $K$) condition, the number of vessels $n$ having a diameter $D$ was chosen using

Eq. (1)

$$K=n\pi {(D/2)}^2,$$

so the total volume of the blood is constant for all blood vessel diameter groups.

Fig. 1

Scheme of simulated 2-D model. Blood vessels (small red circles) were randomly placed in a homogenic tissue (black circle). A beam of photons with a waist of $w_0$ was introduced at one end (green arrows) in a direction of 0 deg central angle and collected at all central angles. Reflected photons (violet arrows) are defined as photons exiting the tissue in a central angle higher than 135 deg. Transmitted photons (blue arrows) are defined as photons exiting from the tissue on the opposite side of the tissue in a central angle lower than 45 deg.

The location of these $n$ vessels was chosen randomly while verifying that there are no overlaps.

A 3-D model of a cylinder of tissue with absorbent blood vessels randomly placed in the $xz$ plane and uniform in the $y$ direction were created with the same method and properties as described in the 2-D model (Fig. 2). Ten repetitions of the simulation, where the number of vessels and their diameter was the same but their location was changed, were held to determine the variations in transmission due to the random vessel location. The illumination was set to be a square of $1.5\times 1.5{\mathrm{mm}}^2$, and the transmitted photons were collected from a 1 mm thick slice (gray area in Fig. 2) along the cylinder surface. Although the cylindrical geometry description is well known in the literature,²² to the best of our knowledge this is the first time its full scattering profile for 360 deg over the sample has been simulated.

Fig. 2

The cross-section of simulated 3-D model of tissue with blood vessels.

3.

Results and Discussion

The simulation described in Sec. 2 was held for several vessel diameters while maintaining a constant blood volume. The mean fraction of photons that exited the tissue at several central angles ($\theta$ in Fig. 1) from 0 deg to 180 deg is presented in Fig. 3 for each vessel diameter. The mean of 10 different vessel distributions was calculated. As can be seen from Fig. 3, for central angles lower than 135 deg the photon transmission decreases for lower vessel diameters, while for central angles higher than 135 deg the photon reflection decreases for higher vessel diameters. Thus, $\theta =135\mathrm{deg}$ is an isobaric point to vessel diameter.

Fig. 3

The distribution of collected photons. The percent of photons exiting from the tissue in different central angles for various vessel diameters. For central angles lower than 135 deg the photon transmission increases for higher vessel diameters; for central angles higher than 135 deg the photon reflection decreases for higher vessel diameters.

In order to facilitate the presentation, reflected photons are defined as those that exit the tissue near the entrance point at a central angle of $|\theta |>135\mathrm{deg}$. Forward transmitted photons are defined as those that exit the tissue at a central angle of $|\theta |<45\mathrm{deg}$. Figure 4(a) and 4(b) presents the average fractions of the reflected and forward transmitted photons, respectively, for several vessel diameters while maintaining a constant blood volume. The standard deviation (STD) due to vessel placement is presented in both figures. These results match the “shielding effect”¹⁰ since for larger blood vessels the total effective absorption and scattering are lower. As a result, the attenuation $n$ is lower, hence the transmission is higher and the reflectance is lower as seen in Fig. 4(a) and 4(b). Note that the transmission curve in Fig. 4(b) matches the results presented by Jacques¹³ for reflection. The changes in the scattering values near 850 nm wavelength are negligible and the changes in the absorption are still dominant.³ It follows that less absorption will lead to more transmission and less reflectance as demonstrated.

Fig. 4

The fraction of (a and c) reflected and (b) forward transmitted photons as a function of blood vessel diameter while keeping a constant blood volume, obtained by computerized simulation for (a and b) 2-D and (c) 3-D.

Due to the fact that real tissues are 3-D and not 2-D, the 3-D simulation described in Sec. 2 was held, and the fraction of the reflection as a function of blood vessel diameter was presented [Fig. 4(c)], while keeping a constant blood volume. Despite the differences in the slopes and the variables, the 3-D curves resemble those received in the 2-D simulation.

However, the exact position of the isobaric point as well as the profile of transmitted photons strongly depends on the optical properties as well as the geometry of the tissue. As seen in Table 2, even if only the $D_t$ changes while the number of vessels ($n$) is kept constant, the behavior of the transmitted and reflected photons switches and the isobaric point moves to a higher angle ($\sim 160\mathrm{deg}$) for $D_t=15\mathrm{mm}$ and to a lower angle ($\sim 30\mathrm{deg}$) for $D_t=5\mathrm{mm}$.

The main result in the 2-D simulation is that for a central angle lower than the isobaric point the photon transmission is lower for vessels of lower diameter while for a central angle higher than the isobaric point the photon reflection is smaller for vessels of higher diameter. The latter result is in contrast to that obtained by Jacques,¹³ who used a semi-infinite model of the tissue and a wavelength of 585 nm. While repeating Jacques’ conditions we were able to reproduce his results. This outcome reinforces the correctness of our model. The reasons that we found that the photon reflection is smaller for vessels of higher diameter are the difference in: (1) the wavelengths (585 nm at Jacques and 850 nm in this research); and following this (2) the absorption constant ($168{\mathrm{cm}}^{-1}$ at Jacques) is 36 times higher than the absorption constant for the wavelength chosen in our paper²¹ which has a value of $4.67{\mathrm{cm}}^{-1}$. Furthermore, (3) an 850 nm wavelength provides higher penetration depth and (4) lower tissue scattering.³ However, both the scattering profile as well as the value of the isobaric point strongly depend on optical properties (like the absorption and scattering constants) and the exact geometry of the case (as we shown for different $D_t$).

4.

Conclusion

The main result is that there is a central angle that below which the photon transmission decreased for lower vessel diameters while above this angle the opposite occurred. The isobaric point (found at $\theta =135\mathrm{deg}$ under the presented studies conditions) could be used as a point of measurement for optical characterization purposes. Furthermore, the findings of our work dealing with the changes in reflection and transmission for different blood vessels diameter with the same volume could be used for different applications such as pulse oximetry in the ear lobe as well as in pinched tissue or in the fingertip joint. The changes in the scattering profile due to variations in blood vessel diameter are more substantial than the change in optical properties in the NIR region, and hence may be used to improve existing pulse oximetry methods. The current pulse oximetry technique derives oxygen saturation in arterial blood from light transmission pulses (photoplethysmography) signals in two wavelengths. The current technique requires calibration and the calibration process saves the need for separation between absorption and scattering effects, and also the need for exact knowledge of the light extinction coefficients. By using our new computer simulation for the evaluation of the transmission and reflection of NIR photons after being scattered and absorbed, the accurate values of extinction coefficients can be extracted and will improve the pulse oximetry technique as well as other NIR methods for the assessment of oxygen saturation.