Open Access
Add to BibSonomy
Abstract
Acoustic light waveguides use a technology that employs acoustic waves to create pressure-dependent refractive index distribution and guide light deep into biological tissue similar to an optical fiber. The method by which acoustic optical waveguides increase light transmission in biological tissues occurring inside scattering-dominated medium has not been clarified. To understand the phenomena occurring inside the scattering-dominated medium, we performed Monte Carlo simulations of photon transport in acoustic optical waveguides. The findings indicate that the larger the change in the refractive index in the scattering-dominated media, the greater the effect of photon confinement. In addition, as the refractive index gradient was increased, the near-field internal fluence was found to be greatly enhanced. The transition depth, which indicates the region where the internal fluence is enhanced by the refractive index gradient, was determined as a function of the radius at which the refractive index change is given.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The benefits of in vivo optical imaging of biological tissues comprise high spatial and temporal resolution, the capacity to choose the molecule to be excited, a minimally invasive nature, and the availability of light sources and optical equipment. Nevertheless, the light absorption and scattering properties of biological tissues make imaging deep into the tissue a difficult problem [1]. Light at wavelengths shorter than that of red light is strongly absorbed by hemoglobin in the blood, whereas light with wavelengths longer than that of infrared light is absorbed mostly by water, which makes up more than 70% of the human body. The impact of absorption in biological tissues is small between these wavelengths, i.e., from visible light to near-infrared light (wavelength: 0.5–1.2 µm) [2], making it ideal for biological tissue imaging. Scattering, on the other hand, is present at all wavelengths, and the effect of scattering in biological tissue at wavelengths from visible to near-infrared light is two orders of magnitude stronger than absorption [3]. Therefore, overcoming the problem posed by scattering in biological tissues is necessary for deep imaging of biological tissues.
The imaging limit of a typical optical microscope is approximately 100 µm by means of the mean free path (MFP) of a photon [4]. Various methods have been devised to overcome scattering in biological tissues and to image regions deeper than that. Chemically altering the refractive index of the tissue of the removed organ to make it transparent for light to pass through is one imaging technique, but it is quite invasive [5,6]. However, in vivo imaging approaches such as confocal microscopy and multiphoton excitation microscopy, permit imaging beyond the scattering MFP by filtering out scattered photons up to the depth of the photon diffusion limit. For imaging deeper than that, invasive techniques such as microprisms and gradient-index (GRIN) lenses that are implanted in biological tissue are used [7,8]. As a non-invasive method, ultrasound-assisted imaging techniques are also used for the imaging [9–13]. Photoacoustic tomography is an ultrasound technique that utilizes optical waves to create acoustic waves and then uses the ultrasound information such as pressure wave to conduct imaging [9]. However, because the brain is covered by the skull, this method is difficult to use. Another approach employs ultrasound to modify the input wavefront by shifting the frequency of the light wave to converge the lights at the target position [10]. This procedure enables focusing of light at depths beyond the optical diffusion limit with ultrasonic resolution (${\sim} 30\; \mathrm{\mu m}$), but has the problem of low light intensity or the requirement for a high-precision wavefront control technique.
Acoustic light waveguides are another approach that uses acoustic waves to create a pressure-dependent refractive index distribution to guide light deep into biological tissues, similar to optical fibers [11–13]. The acoustic optical waveguide is valuable method as a deep imaging for biological tissues as it is non-invasive and does not need advanced control approaches such as wavefront engineering methods. This procedure is based on the property that light is bent by the gradient of the refractive index, and the larger the gradient in the refractive index, the greater the bending of the light. In non-scattering media, the usefulness and mechanism of utilizing optical waveguides to enhance light transmission have been demonstrated by experiments and numerical simulations employing geometrical optics techniques [14,15]. However, for light to travel in the desired direction the refractive index gradient must be greater, which requires a larger pressure gradient in the acoustic wave. Since a significant pressure gradient might damage the biological tissue, the pressure gradient that can be applied in the imaging of biological tissue is limited. Experimental studies of acoustic optical waveguides have reported light transmission at relatively small ultrasonic peak pressures, on the order of 100 kPa to 10 MPa, can improve light transmission without damaging tissues, even in tissues where scattering is dominant, such as biological tissues [11–13]. The mechanism by which light can reach the depth of the biological tissue even at moderately low peak pressures due to acoustic optical waveguides has not been clarified.
Monte Carlo simulations of light propagation in scattering and absorbing media are widely used to understand the interaction between light and biological tissues [e.g., 16–18]. Therefore, the purpose of this study is to clarify how the pressure change (in this case, the refractive index change) can be improve the transmission of light in acoustic optical waveguide using Monte Carlo simulations of photon transport in a scattering medium with a refractive gradient. In order to quantitatively evaluate the optical properties of the scattering medium, Intralipid diluent, which is a phantom of biological tissue, was employed here, and its optical properties were evaluated using Mie scattering theory and employed in Monte Carlo simulation. In the acoustic optical waveguide experiments [11–13], the longitudinal pressure distribution induced by acoustic waves can be regarded as approximately uniform. Therefore, in the present simulation, a quadratic refractive index profile with cylindrical symmetry in the radial direction was used to reproduce the refractive index distribution in acoustic optical waveguide. From the results of Monte Carlo simulations, it was found that the larger the refractive index change, which is the difference between the peak and minimum refractive indices, the greater the confinement effect of photons and the higher is the light transmission, even in a scattering-dominated medium. Furthermore, it was found that increasing the gradient of the refractive index greatly enhances the internal fluence in the region close to the photon incident plane, and here the transition depth, which is a measure of the depth at which the internal fluence is enhanced, was clarified as a function of the radius at which the refractive index change is given. In addition, it was suggested that not only the bending of light caused by the refractive index gradient but also the reflection of photons at the refractive index boundary is important as a mechanism for enhancement of light transmission in acoustic optical waveguide. In Section 2, we present the condition of numerical simulation. Section 3 present our numerical results and section 4 present their discussion. The conclusions of our study are presented in the final section.
2. Methods
2.1 Optical properties
The scattering of light by biological tissue depends on the microscopic structure of the tissue. Biological tissues are made up of micron-sized cells that include organelles such as nuclei, cytoplasm, and mitochondria. Since scattering is caused by light striking these cells or organelles, their size is crucial in determining the scattering properties. Since cell organelles are only a few micrometres in size, the scale of a scatterer is about the same as the wavelength of visible to near-infrared light. Therefore, Mie scattering theory [19] is applied to evaluate the scattering properties such as scattering coefficient ${\mu _s}$ and anisotropy parameter g.
In Mie scattering theory, the scattering coefficient is given by
where ${\rho _s}$ is a number density of scattering particles, ${Q_s}$ is a scattering efficiency and A is a particle cross-sectional area.$\; {Q_s}$ is a function of size parameter $x = 2\pi a/\lambda $ and relative refractive index $m = {n_p}/{n_{med}}$, where a is a particle radius, $\lambda $ is the wavelength of light and ${n_p}$ and ${n_{med}}$ are refractive indices of particle and medium. The scattering phase function and anisotropy parameter are also evaluated by the Mie scattering theory.In this study, a 1% dilution of Intralipid, a phantom of biological tissue, was employed as a model of biological tissue where scattering is dominant. Taking into consideration the measured particle size distribution of scatterer particles in Intralipid [20], the scattering coefficient and anisotropy parameter were evaluated from Mie scattering theory. The scattering coefficient, anisotropy parameter utilized in the Monte Carlo calculations are ${\mu _s} = 41.9{\; \textrm{c}}{\textrm{m}^{ - 1}}$ and $g = 0.684$, respectively, assuming a light wavelength of 0.6 µm. The absorption coefficient was set to ${\mu _a} = 0.2{\; \textrm{c}}{\textrm{m}^{ - 1}}$, which is roughly two orders of magnitude smaller than the scattering coefficient.
2.2 Monte Carlo method
We utilize the Monte Carlo approach to recreate the behavior of light in a scattering medium [21]. Light is considered as a collection of photons (photon packet) in the Monte Carlo simulation, and the behavior of each photon packet is computed using absorption and scattering coefficients of biological tissue. The distance l traveled by a photon packet before it interacts with biological tissue is given by the following equation utilizing a scattering coefficient ${\mu _s}$, an absorption coefficient ${\mu _a}$ and a uniform random number $\xi (0 < \xi < 1)$,
After the distance l, the photon packet is assumed to be absorbed and scattered at the same time. The absorption of a photon packet is described as a weight of the packet, which the weight is initially set at one. After the absorption event, the weight is reduced by following equation,
where ${W^{old}}$ and ${W^{new}}$ are the weight of the photon packet before and after the event. For scattering, the polar angle $\theta $ and azimuthal angle $\phi $, which represent the change in direction of travel of the scattered photons, depend on the optical properties of the tissue and the random value sampling. The angle $\theta $ is evaluated using the Henyey–Greenstein phase function given by where g is the anisotropy parameter. The angle $\phi $ is given by where $\eta $ is a uniform random number $\eta (0 < \eta < 1)$. Finally, the computation of a photon packet is terminated when the weight becomes sufficiently small or the packet escapes from the computational domain.To store the results of the Monte Carlo simulation, the medium was divided into grids. We set up a two-dimensional grid in the r and z directions. The grid separation was set to be $\Delta r = 5\; \mathrm{\mu m}$ and $\Delta z = 20\; \mathrm{\mu m}$ in the r and z directions, respectively.
2.3 Graded refractive index model
The simulation model is finite in the longitudinal direction (1 cm) and infinite in the radial extent direction, and one billion photon packets with a pencil beam of infinitesimal size are incident at the origin in the z-direction. We referred experimental methods of acoustic optical waveguides [11–13], the distribution model of the refractive index was built up as a cylindrical model in which the refractive index changes in the radial direction around the axis of photon incidence (z-axis), as shown in Fig. 1. The cylindrical volume encompassed within ${r_n}$ as shown in Fig. 1(b) is the region where the refractive index is altered. The refractive index has a maximum value of $\; {n_{max}}$ at the center and a minimum value of ${n_{min}}$ at ${r_n}$. In the outside of ${r_n}$, the refractive index is set to be constant at ${n_{min}}$. The most common radial refractive index profile in a GRIN fiber is a parabolic profile [12], therefore, we set the refractive index profile as a quadratic function within the range ${r_n}$. This region in the radial direction was divided into multiple sections, and the refractive index was adjusted to vary in steps as indicated in Fig. 1(b). Meanwhile, the refractive index profile in the z direction is constant. Such a refractive index profile can be achieved by forming a ring of photoacoustic driving pulses and focusing them into the medium around the observation target using a combination of axicons and lenses [22]. The carbon ink solution consisting of small carbon colloids, which are photon absorbers, is placed around the target. The photoacoustic driving pulses ablate these carbon colloids along the propagation direction of the pulses and generate cylindrical nonlinear acoustic waves around the target [13].
Fig. 1. Simulation model with refractive index gradient. (a) Infinite plate with a thickness of 1 cm and a photon packet is incident from the origin in the z direction (red arrow). The refractive index varies cylindrically symmetric in the radial direction r around z-axis. (b) Graded refractive index model. ${r_n}$ is the radius till which the refractive index is varied. The change is approximated by a quadratic function (blue curve), which is divided into some parts in the radial direction. The refractive index is ${n_{max}}$ at the center and ${n_{min}}$ outside the radius ${r_n}$. (c) Fresnel reflection and refraction model. The direction vectors of photon travel by reflection ${\boldsymbol r}$ and refraction ${\boldsymbol t}$ by incident direction vector u and normal vector of the refractive index surface ${{\boldsymbol n}^r}$.
From our experimental visualization of the acoustic wave in water by the shadowgraph technique [13], we set the radius till which the refractive index is varied to ${r_n} = 70\; \mathrm{\mu m}$ and the number of divisions to 20, which is small enough to neglect the effect of the number of divisions on the results. Furthermore, the maximum and minimum refractive indices estimated from the pressure distribution of the acoustic wave are ${n_{min}} = 1.334\; \; \textrm{and}\; {n_{max}} = 1.336,\; $ and the pressure disparity between them is approximately 10 MPa [13]. To analyze the effect of the refractive index change $\mathrm{\Delta }n = {n_{max}} - {n_{min}}$, ${n_{min}} = 1.334$ was kept constant and $\mathrm{\Delta }n$ was varied up to 0.7. Assuming a linear relationship between the pressure difference and the change in refractive index [23], the pressure difference at $\mathrm{\Delta }n = 0.7$ corresponds to 3.5 GPa. To further investigation the effect of the radius ${r_n}$ of the refractive index change, ${r_n}$ was varied 10, 70, 350, 1000, 3000, 7000 $\; \mathrm{\mu m\;\ }$ and $\infty $. The number of divisions of the radius of the refractive index change was made so small that the effect of the number of divisions on the results was negligible, as was the case with ${r_n} = 70\; \mathrm{\mu m}$.
The model of reflection and refraction, explained by the Fresnel equation, was adopted to handle photons hitting the refractive index boundary. The reflectance at the refractive index boundary is given by
To validate our Monte Carlo program, we simulated the total diffuse reflectance and total diffuse transmittance for photons incident perpendicularly on an infinite plate, as well as the internal fluence of the model with surfaces of different refractive indices, referring to the simulation of Wang et al [21]. The results of both simulations are in good agreement with the results of Wang et al [21], with a relative error of less than 0.05%.
3. Results
3.1. Effect of change in refractive index difference Δn
Figure 2 shows the effect of the difference between the maximum and minimum refractive indices, $\Delta n$, on the internal fluence per unit cross-sectional area within ${r_n} = \; 70\mathrm{\;\ \mu m}$. Comparing the condition of Δn = 0.002 which is the refractive index difference estimated from experimental results [13] and the condition where the refractive index does not change, there was no difference in the relationship between depth and internal fluence in the present simulation. Despite the enhancement in light transmission in the optical waveguide experiments [11–13], this simulation result for Δn = 0.002 did not show the effect. When the change in refractive index Δn was greater than Δn = 0.02, the internal fluence at each depth increased. This can be thought of as a result of the photon confinement effect induced by the refractive index change (refractive index gradient). To increase the internal fluence by one order of magnitude at the depth of 0.2 cm, where the effect of the change in refractive index was the largest, the change in the refractive index should be about Δn = 0.7.
Fig. 2. Effect of difference between the maximum and minimum refractive indices, Δn, on the internal fluence of the medium per unit cross-sectional area within ${r_n} = \; 70\mathrm{\;\ \mu m}$.
The slope of the line representing the photon decay is unaffected by Δn in the deep region (depth > 0.6 cm) and remains constant. This may be because few photons reach the deep area, and the behavior is isotropic diffusion. The difference in refractive index between the top and bottom surfaces of the medium and the outside (air: n = 1.0) increases, as Δn increases in this depth region, enhancing the reflectivity of photons at that surface and the amount of photons remaining inside, leading to a larger internal fluence. The internal fluence near the incident surface (depth < 0.02 cm), on the other hand, decreases as Δn increases. This is due to the effect of specular reflection at the photon incidence, which has small effect on the fluence. The slope of the decay of the internal fluence varies with Δn in the region between the surface and the deep region. The larger the Δn, its slope decreases which means light is transmitted to deeper regions. This can be seen from the results of the fluence distribution per unit cross-sectional area inside the medium (for without refractive index change, Δn = 0.4 and 0.7), as shown in Fig. 3. The fluence takes logarithm to the base 10 of the fluence ${\phi _{rz}}({{r_i},{z_i}} )\; [{\textrm{c}{\textrm{m}^{ - 2}}} ]$ of each grid defined by
Fig. 3. 2D fluence distribution within the medium takes logarithm to the base 10 of the fluence ${\phi _{rz}}$ of each grid, from left to right, for $\Delta n\; $= 0, 0.4 and 0.7. The white lines indicate the radius of the refractive index change ${r_n}$.
3.2. Effect of the refractive index gradient of refractive index variation
Figure 4 shows the effect of the radius ${r_n}$ at which the refractive index changes on the internal fluence of the medium per unit cross-sectional area within ${r_n}$. The refractive index change is set to Δn = 0.2, which is significant enough to affect the internal fluence by the refractive index gradient.
Fig. 4. Effect of radius ${r_n}$ at which the refractive index changes on the internal fluence of the medium per unit cross-sectional area within ${r_n}$. The refractive index change set to Δn = 0.2. ${r_{inf}}\; $ is the result of ${r_n} = \infty $.
In deep areas (depth > 0.5 cm), the internal fluence reduces with increasing $\; {r_n}$, and the result for ${r_n} = \infty \; ({{r_{inf}}} )$ are consistent with the result of diffusion theory [24] given by
where $z\; $ is the depth and $\; {\varphi _0}$ is the incident irradiance, which is one here. k is a scalar that depends on the amount of backscattered reflection [25]. ${\mu _{eff}}$ is the effective attenuation coefficient given byThe internal fluence gradually increases as the radius of the refractive index changes, ${r_n}$, is reduced from ${r_{inf}}$. This increase in internal fluence is saturated at ${r_n} = 1000\; \mathrm{\mu m}$, so even if ${r_n}$ is further reduced, the internal fluence does not increase, and the equation of that line is given by
where ${k_{far}} = 82.9{\; }[{\textrm{c}{\textrm{m}^{ - 2}}} ]$ and ${\mu _{far}} = 4.98{\; }[{\textrm{c}{\textrm{m}^{ - 1}}} ]$. The increase in the internal fluence in the region of large ${r_n}$ is considered to be a result of the effect of bending of light toward the greater refractive index and the confinement of photons by reflection at the refractive index boundary, induced by the refractive index change. However, when ${r_n}$ is small (${r_n} < 1000\; \mathrm{\mu m})$, most of the photons in the deep area are out of the region of refractive index change ${r_n}$, and few photons reach the deep region, so the internal fluence is considered to decay at a constant rate regardless of the radius ${r_n}$.When the radius of the refractive index change ${r_n}$ is small (${r_n}\; $< 1000 µm), the internal fluence increases significantly in the shallow region (depth < 0.5 cm). The decay rate of the internal fluence, ${\mu _{near}}$, is clearly different from the decay rate of diffusion theory, ${\mu _{eff}}$, and each line indicating the internal fluence is given by
where ${k_{near}}\; $ and ${\mu _{near}}{\; }$ are functions of the radius of refractive index change ${r_n}\; $ and given by and This implies that the smaller ${r_n}$ is, the larger the refractive index gradient is, and thus the photons are confined within the radius ${r_n}$ and are delivered deeper into the medium. This situation is also apparent from the fluence distribution ${\phi _{rz}}({{r_i},{z_i}} )\; [{\textrm{c}{\textrm{m}^{ - 2}}} ]$ within the medium, as shown in Fig. 5. The smaller ${r_n}$ is, the greater the confinement effect of photons within ${r_n}$ and the light transmission is enhanced.Fig. 5. 2D fluence distribution within the medium takes logarithm to the base 10 of the fluence ${\phi _{rz}}$ $\; $ of each grid. The blue vertical lines indicate the radius of the refractive index change ${r_n}$. The refractive index change set to Δn = 0.2. The unit of ${r_n}\; $ is $\mathrm{\mu m}$.
In acoustic optical waveguide, knowing the depth to which the light transmittance increases is extremely important finding because it provides an important clue to the pressure gradient to be applied in the experiment. The depth ${d_{trans}}\; $ can be estimated as the depth of transition from the near field fluence ${\varphi _{near}}\; $ to the far-field fluence $\; {\varphi _{far}}$. By substituting Eq. (13) into (14), the depth is given as a function of the radius of the refractive index change ${r_n}$, by
Figure 6 shows the relation between the radius of the refractive index change ${r_n}$ and the transition depth ${d_{trans}}.$ In this study (Δn = 0.2, corresponding to the pressure difference of approximately 1.0 GPa), the transition depth is saturated around ${r_n} = 100\; \mathrm{\mu m}$, and the influence of increasing light transmission in the near field disappears at that transition depth ${d_{trans}} = 0.3\; \textrm{cm}$. To significantly improve the light transmission, it is essential to maximize the effect in the near field, which is achieved by taking ${r_n}$ small (${r_n} < 100\; \mathrm{\mu m}$).
Fig. 6. Relation between radius of refractive index change ${r_n}$ and transition depth ${d_{trans}}$. The refractive index change set to Δn = 0.2.
4. Discussion
The optical properties of Intralipid diluent with a volume concentration of 1% were evaluated using Mie scattering theory in agreement with the experimental results of Intralipid diluent with a volume concentration of 1% [26,27]. Furthermore, the results are in agreement (10–20 cm−1) with the reduced scattering coefficient of human and animal skin measured in vitro and ex vivo [28]. Thus, the diluted solution may be regarded as a phantom that simulates biological tissue in terms of reduced scattering coefficient. However, because the anisotropy parameter of actual biological tissue is about g = 0.9–0.95 [29], it cannot be considered a phantom that replicates biological tissue in terms of the anisotropy parameter. The results of this study relate to a more severe case with a lower fraction of forward scattering than the actual biological tissue because the anisotropy parameter is a measure of the forward scattering.
In this study, Mie scattering theory was employed to evaluate the optical coefficients. The scattering condition changes from independent scattering to multiple scattering as the volume concentration of the scattered particles in Intralipid diluent increases. The relationship between the volume concentration of Intralipid diluent and the scattering coefficient or reduced scattering coefficient is related to the volume concentration of Intralipid diluent and varies with the wavelength of light, but the linear correlation is maintained in the range of 1%–2% volume concentration of Intralipid diluent [30–32]. Therefore, it is reasonable to apply Mie scattering theory to the volume concentration of 1% Intralipid diluent employed in this study, assuming independent scattering.
The fluence within the medium increases as the refractive index change Δn increases. In the present study (${r_n} = 70\; \mathrm{\mu m}$), a substantial refractive index change Δn = 0.7 is required to enhance the fluence inside the medium by an order of magnitude. Given that the refractive index change Δn is proportional to the pressure due to acoustic waves [23], this corresponds to a large pressure difference of approximately 3.5 GPa. For a refractive index change Δn = 0.002 corresponding to a peak pressure of about 10 MPa, the present Monte Carlo simulations show no confinement effect of photons in the scattering medium, which is quite different from the experimental results [11–13]. In this simulation, the thickness of the sample was assumed to be 10 mm and the volume concentration of the intralipid was assumed to be 1.0%. On the other hand, in the experiment in [13], the thickness of the sample is varied from 2 to 10 mm and the volume concentration of the intralipid is varied from 0.09% to 0.9%. As the concentration and thickness increase, the transmittance decreases exponentially, but a transmittance improvement of about one order of magnitude is maintained under all conditions. In addition, quadratic refractive index profile in the radial direction with a constant depth direction as the effect of the acoustic wave were assumed in the simulation, while the refractive index profiles in the experiments were due to the pressure gradient generated by transient ultrasound waves [11], ultrasound waves by multi-segment cylindrical transducer array [12] and laser-induced nonlinear acoustic waves [13]. Therefore, the difference between the Monte Carlo simulation and the experimental results may be in these difference conditions.
In general, the refractive index gradient caused by acoustic waves has been considered to be the main factor in the improvement of light transmission by optical waveguides. However, according to our simulation results, the refractive index gradient due to acoustic waves alone cannot explain the improvement in light transmission by optical waveguides, and there must be other key factors. From a geometrical optics point of view, the relationship between the pressure difference and the angle of incidence at which light is bent in the depth direction by the change in refractive index in a medium without scatterers is considered to be that the angle of incidence of light should be within 0.5–2.5 degrees from the depth direction when the pressure difference is about 100 kPa to 10 MPa. In biological tissues where scattering is dominant, even if the incident direction of light is coincident with the depth direction, the direction of light travel is likely to deviate significantly from the depth direction due to scattering. Therefore, the results of Monte Carlo simulations suggest that the effect of light bending in the depth direction due to refractive index distribution, which has been cited as a factor in the improvement of light transmission by acoustic optical waveguides, would be small.
When the incident direction of light is coincident with the depth direction, and the direction of light changes due to scattering in biological tissue, the light must be extremely strong forward scattering in order to travel in the depth direction. The anisotropy parameter increases as the size parameter increases in Mie scattering theory. The sudden change in acoustic pressure induced by ultrasound irradiation might generate bubbles to form in the focal region (cavitation). As a result, these bubbles become scatterers, which may increase the apparent size of the scatterers and increase the anisotropy parameter [33]. The thermal effect is another potential effect of ultrasonic irradiation. When the temperature of the medium is increased by ultrasound irradiation, the volume expands and the scattering coefficient decreases [34,35], or the fluidity increases, and the scattering coefficient decreases [36] in the case of scatterers formed by lipids.
The confinement effect of an optical waveguide is thought to be due to the refractive index gradient, which bends photons toward the larger refractive index [12,14]. In non-scattering medium, relatively small pressure gradients, like those that induce a refractive index change Δn = 0.002, can only bend photons adequately for angular changes of a few degrees in the direction of travel. In general, scattering causes a greater shift in the direction of photon propagation (scattering angle) in a scattering-dominated medium such as biological tissue than in a non-scattering medium. Figure 7 shows the probability distributions of the number of scattering and the position of the photons after scattering were evaluated by Monte Carlo simulation. As a result, we can prove that for Δn = 0.002, there is little effect of the refractive index change on the photon confinement, and many photons are ejected from within the radius of the refractive index change after the first scattering. However, for Δn = 0.2, after the first scattering, the photons are moderately confined to the radius of the refractive index change, but after a few scatterings, many of them are ejected from the region. Therefore, within the radius of refractive index change, the number of photons that are bent by the refractive index gradient decreases. The larger the refractive index change, the greater the reflectivity at the refractive index boundary, and thus the more likely photons will be reflected at that boundary and confined within the radius of the refractive index change. In a highly scattering medium such as biological tissue, the effect of light confinement may be due to the reflection of photons at the refractive index boundary rather than the bending of photons by the refractive index gradient.
Fig. 7. Probability distributions of the location of a photon after it has been scattered once, twice and five times. The probability takes logarithm of the probability of the photon's position with base 10. The left figures are results for $\Delta n\; $= 0.002 and the right figures are results for $\Delta n\; $= 0.2. The white lines indicate the radius of the refractive index change ${r_n} = 70\; \mathrm{\mu m}$.
5. Conclusions
To elucidate the phenomena occurring within the medium in biological tissues where scattering is dominant, Monte Carlo simulations of photon transport in acoustic optical waveguides were performed. The findings suggest that for a radius of refractive index change of ${r_n} = 70\; \mathrm{\mu m}$, which is similar to the experiment, the larger the refractive index change Δn, which is the difference between the peak and minimum refractive indices, the greater the confinement effect of photons and improved the light transmission, even in a scattering-dominated medium. However, in contrast to the experimental results [11–13], a large refractive index change of Δn = 0.7 (equivalent to a pressure difference of approximately 3.5 GPa) was needed to enhance the internal fluence by one order of magnitude. Furthermore, decreasing the radius of the refractive index change ${r_n}$, which is equivalent to increasing the refractive index gradient (which is experimentally equivalent to increasing the pressure gradient), can greatly improve the internal fluence near the field of photon incident plane. Here the transition depth, which is a measure of the depth at which the internal fluence is enhanced, was defined as a function of the radius at which the refractive index change is given. At Δn = 0.2 (corresponding to the pressure difference of approximately 1.0 GPa), the effect of the near field saturates at a depth of 0.3 cm, implying that ${r_n}\; $ should be smaller than 100 $\; \mathrm{\mu m}$ to increase the photon confinement effect. In acoustic optical waveguide in scattering dominated medium, the effect of the refractive index gradient on the photon trajectory bending is small, suggesting that the reflection of photons at the refractive index boundary is also important for the confinement effect of photons to increase the light transmission.
Funding
Japan Agency for Medical Research and Development (21dm0207076h0003).
Acknowledgments
The authors would like to thank A. Ishijima, S. Wunderl, Z. Xu (Univ. Tokyo, Japan) for supporting with the analysis.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, and J. Wyatt, “Estimation of optical pathlength through tissue from direct time of flight measurement,” Phys. Med. Biol. 33(12), 1433–1442 (1988). [CrossRef]
2. R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nat. Med. 9(1), 123–128 (2003). [CrossRef]
3. Y. Yamada and S. Okawa, “Diffuse Optical Tomography: Present Status and Its Future,” Opt. Rev. 21(3), 185–205 (2014). [CrossRef]
4. V. Ntziachristos, “Going deeper than microscopy: the optical imaging frontier in biology,” Nat. Methods. 7(8), 603–614 (2010). [CrossRef]
5. D. S. Richardson and J. W. Lichtman, “Clarifying tissue clearing,” Cell 162(2), 246–257 (2015). [CrossRef]
6. E. A. Susaki, K. Tainaka, D. Perrin, H. Yukinaga, A. Kuno, and H. R. Ueda, “Advanced CUBIC protocols for whole-brain and whole-body clearing and imaging,” Nat. Protoc. 10(11), 1709–1727 (2015). [CrossRef]
7. G. I. S. Ancataldo, L. U. S. Ilvestri, A. N. N. A. L. Etizia, A. L. M. Ascaro, and L. E. S. Acconi, “Advanced fluorescence microscopy for in vivo imaging of neuronal activity,” Optica 6(6), 758–765 (2019). [CrossRef]
8. T. H. Chia and M. J. Levene, “Microprisms for in vivo multilayer cortical imaging,” J. Neurophysiol. 102(2), 1310–1314 (2009). [CrossRef]
9. L. V. Wang and J. Yao, “A practical guide to photoacoustic tomography in the life sciences,” J,” Nat. Methods. 13(8), 627–638 (2016). [CrossRef]
10. H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3(12), 1–9 (2017). [CrossRef]
11. M. N. Cherkashin, C. Brenner, G. Schmitz, and M. R. Hofmann, “Transient Light Waveguides Deep Into Scattering Media by Transversal Ultrasound,” Biophot Cong: Biomed. Opt. 2020, JTh2A.15 (2020). [CrossRef]
12. M. Chamanzar, M. G. Scopelliti, J. Bloch, N. Do, M. Huh, D. Seo, J. Iafrati, V. S. Sohal, M.-R. Alam, and M. M. Maharbiz, “Ultrasonic sculpting of virtual optical waveguides in tissue,” Nat. Commun. 10(1), 92 (2019). [CrossRef]
13. A. Ishijima, U. Yagyu, K. Kitamura, A. Tsukamoto, I. Sakuma, and K. Nakagawa, “Nonlinear photoacoustic waves for light guiding to deep tissue sites,” Opt. Lett. 44(12), 3006–3009 (2019). [CrossRef]
14. M. G. Scopelliti, H. Huang, A. Pediredla, S. Narasimhan, I. Gkioulekas, and M. Chamanzar, “Overcoming the tradeoff between confinement and focal distance using virtual ultrasonic optical waveguides,” Opt. Express 28(25), 37459–37473 (2020). [CrossRef]
15. M. G. Scopelliti and M. Chamanzar, “Ultrasonically sculpted virtual relay lens for in situ microimaging,” Light: Sci Appl 8(1), 65 (2019). [CrossRef]
16. L. V. Wang, R. E. Nordquist, and W. R. Chen, “Optimal beam size for light delivery to absorption-enhanced tumors buried in biological tissues and effect of multiple-beam delivery: a Monte Carlo study,” Appl. Opt. 36(31), 8286–8291 (1997). [CrossRef]
17. T. J. Pfefer, J. K. Barton, E. K. Chan, M. G. Ducros, B. S. Sorg, T. E. Milner, J. S. Nelson, and A. J. Welch, “A three-dimensional modular adaptable grid numerical model for light propagation during laser irradiation of skin tissue,” IEEE J. Select. Topics Quantum Electron. 2(4), 934–942 (1996). [CrossRef]
18. S. V. Patwardhan, A. P. Dhawan, and P. A. Relue, “Monte Carlo simulation of light-tissue interaction: three-dimensional simulation for trans-illumination-based imaging of skin lesions,” IEEE Trans. Biomed. Eng. 52(7), 1227–1236 (2005). [CrossRef]
19. Bohren and Huffman, “Absorption and scattering of light by small particles,” Wiley: New York, (1983).
20. R. Michels, R. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16(8), 5907–5925 (2008). [CrossRef]
21. L. Wang, S. L. Jacques, and L. Zheng, “CML - Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef]
22. D. Veysset, A. A. Maznev, T. Pezeril, S. Kooi, and K. A. Nelson, “Interferometric analysis of laser-driven cylindrically focusing shock waves in a thin liquid layer,” Sci. Rep. 6(1), 24 (2016). [CrossRef]
23. T. A. Pitts and J. F. Greenleaf, “Three-dimensional optical measurement of instantaneous pressure Three-dimensional optical measurement of instantaneous,” J. Acoust. Soc. Am. 108(6), 2873–2883 (2000). [CrossRef]
24. B. C. Wilson and S. L. Jacques, “Optical reflectance and transmittance of tissues: Principles and applications,” IEEE J,” Quantum Electron. 26(12), 2186–2199 (1990). [CrossRef]
25. S. L. Jacques, “Simple Theory, Measurements, And Rules Of Thumb For Dosimetry During Photodynamic Therapy,” Proc. SPIE 1065, 100–108 (1989). [CrossRef]
26. H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid- 10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef]
27. S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. C. van Gemert, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers Med. Sci. 12(5), 510–519 (1992). [CrossRef]
28. A. N. Bashkatov, E. A. Genina, and V. V. Tuchin, “Optical properties of skin, subcutaneous, and muscle tissues: A review,” J. Innov. Opt. Health Sci. 04(01), 9–38 (2011). [CrossRef]
29. S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013). [CrossRef]
30. P. Di Ninni, F. Martelli, and G. Zaccanti, “Effect of dependent scattering on the optical properties of Intralipid tissue phantoms,” Biomed. Opt. Express 2(8), 2265–2278 (2011). [CrossRef]
31. G. Zaccanti, S. Del Bianco, and F. Martelli, “Measurements of optical properties of high-density media,” Appl. Opt. 42(19), 4023–4030 (2003). [CrossRef]
32. A. Giusto, R. Saija, M. A. Iatì, P. Denti, F. Borghese, and O. I. Sindoni, “Optical properties of high-density dispersions of particles: application to intralipid solutions,” Appl. Opt. 42(21), 4375–4380 (2003). [CrossRef]
33. H. Kim and J. H. Chang, “Increased light penetration due to ultrasound-induced air bubbles in optical scattering media,” Sci. Rep. 7(1), 16105 (2017). [CrossRef]
34. V. A. McGlone, P. Martinsen, R. Künnemeyer, B. Jordan, and B. Cletus, “Measuring optical temperature coefficients of Intralipid®,” Phys. Med. Biol. 52(9), 2367–2378 (2007). [CrossRef]
35. B. Cletus, R. Künnemeyer, P. Martinsen, and V.A. McGlone, “Temperature-dependent optical properties of Intralipid® measured with frequency-domain photon- migration spectroscopy,” J. Biomed. Opt. 15(1), 017003 (2010). [CrossRef]
36. J. Laufer, R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Effect of temperature on the optical properties of ex vivo human dermis and subdermis,” Phys. Med. Biol. 43(9), 2479–2489 (1998). [CrossRef]
