Persistent homology-based optical properties of microscopic turbid media for realistic light propagation analysis

Abstract

The optical properties of microscopic turbid media are critical for understanding light-tissue interactions with applications in biomedical imaging and diagnostics. However, traditional scattering coefficient-based methods are limited in their ability to capture topological heterogeneities within tissue structures, which play a crucial role in describing the relationship between microscopic tissue characteristics and their corresponding light propagation behaviors. In this study, we propose using persistent homology-based persistent images (PIs) as a descriptor and optical property of microscopic tissues. As a proof of concept, we analyzed particle-distributed turbid media with uniform and clustered particle distributions by persistent homology analysis, demonstrating that PIs can capture topological characteristics that are not discernible using traditional scattering coefficient-based methods. Light propagation simulations using the beam propagation method (BPM) demonstrated that PIs correlate with optical behaviors, such as beam centroid displacement and distortion, providing a foundation for linking microscopic topological heterogeneities to light propagation behaviors. Our results validate PIs as a meaningful and predictive optical property, bridging microscopic turbid media topology with their light propagation behaviors. This work establishes PIs as a potential optical property of microscopic tissue, capturing its topological characteristics and offering predictive insights into light propagation behaviors.

1. Introduction

In biomedical optics, the propagation of light in scattering media plays a critical role in understanding and advancing optical imaging and diagnostic techniques. Recent advancements in computational and analytical methods have significantly accelerated progress in this field, enabling more accurate simulations and deeper insights into light-tissue interactions [1]. Traditional approaches in tissue optics have relied on the radiative transport equation (RTE), which is primarily solved using Monte Carlo (MC) simulations [2]. These methods provide a robust framework for calculating macroscopic optical properties such as absorption and scattering coefficients, which are essential for describing light transport as energy. However, these conventional models face challenges when applied to microscopic scales, as macroscopic optical properties fail to describe finer microscopic details, leading to the neglect of the wave nature of light, such as diffraction and interference. At microscopic levels, structural changes in tissues manifest as subtle alterations in their optical properties, making it crucial to capture these wave phenomena accurately.

On a microscopic scale, tissues are characterized by diverse cellular structures and refractive index variations arising from subcellular components such as nuclei, mitochondria, and cytoplasm [3]. These refractive heterogeneities influence light scattering and propagation. To address these wave phenomena, the Helmholtz wave equation serves as the mathematical framework for describing light propagation in tissues. The Beam Propagation Method (BPM) emerges as a numerical approach to solve this equation, incorporating diffraction and interference effects, which are essential at microscopic scales. BPM operates under the paraxial approximation of the Helmholtz equation, making it well-suited for modeling forward-propagating light in scattering media [4,5]. Unlike MC simulations, which rely on probabilistic modeling of photon transport, BPM explicitly accounts for phase information, allowing it to capture diffraction effects. Finite-difference time-domain (FDTD) methods, on the other hand, provide a full-wave solution to Maxwell’s equations, making them capable of modeling multiple scattering effects [6–8]. Even though BPM does not inherently model multiple scattering events as accurately as MC or FDTD, it remains a computationally efficient approach for investigating microscopic refractive heterogeneities. While FDTD offers a rigorous wave-optical solution, its high computational cost makes it difficult for large-scale microscopic simulations [9]. In contrast, BPM provides a balanced trade-off by incorporating wave-optical effects while maintaining feasibility for simulating large-scale tissue models. Due to its efficiency and ability to capture diffraction and interference, BPM provides a practical approach for investigating the interplay between structural variations and light propagation at the microscopic scale. BPM was employed in the study by Glaser et al. [9], where refractive index turbulence was statistically modeled to represent spatial fluctuations. While this approach could capture the refractive heterogeneities through a fractal model using parameters like fractal dimension and variance of refractive index fluctuations, it overlooks the distinct topological features of cellular components, such as size, shape, and spatial arrangement.

The limitations of fractal model in capturing topological features, such as size, shape, and spatial arrangement of cellular structures, highlight the need for alternative analytical tools. Persistent homology, a tool from topological data analysis, emerges as a promising approach to address these limitations. Persistent homology extracts structural information directly from refractive index distributions by analyzing the spatial organization of scatterers. This technique quantifies topological features such as connected components, loops, and voids that persist across multiple scales within tissue structures [10–12]. By utilizing persistent homology, it becomes possible to capture the structural complexity and heterogeneities of microscopic tissues, addressing the limitations of existing fractal models and conventional optical descriptors, such as scattering and absorption coefficients, and introducing a new paradigm for optical characterization that is distinct from fractal-based or radiative transfer-based approaches. Moreover, persistent homology allows for the representation of topological details, such as size, shape, and spatial distribution, as quantifiable parameters. These parameters can then be integrated into light propagation simulations to link microscopic structural variability with optical behavior, offering a topology-aware approach for the study of light-tissue interactions on a microscopic scale.

In this research, we propose the concept of using persistent homology to describe and quantify the refractive heterogeneities and topological features of microscopic tissue models. To validate the proposed concept, this research employs simplified tissue models composed of refractive particles distributed in varying configurations. These models are used to examine how topological characteristics, captured by persistent images (PIs), correspond to light propagation characteristics analyzed through BPM-based simulations. By bridging the gap between microscopic structural variability and light propagation characteristics, this approach aims to position persistent homology as both a descriptor of tissue refractive heterogeneities and a potential optical property for microscopic turbid media. Ultimately, this development of PI-based optical properties seeks to advance our understanding of microscopic light-tissue interactions and contribute to the development of realistic light propagation models for biomedical applications.

2. Method

2.1. Persistent homology analysis of refractive index structure

Three dimensional refractive structures of the particle-distributed turbid media were analyzed using persistent homology, capturing their topological characteristics. Persistent diagrams (PDs) were first generated, where the birth and death of topological features, such as connected components and loops, were represented as points in a two-dimensional plane. The input data for persistent homology consists of refractive index distributions corresponding to the refractive particle-distributed turbid media. These distributions were represented as voxelized binary grids, where each voxel encodes the presence (scatterer) or absence (background medium) of refractive particles. Persistent homology was then applied to these voxelized structures using the HomCloud library [10], computing topological features based on Manhattan distance as described and illustrated in reference [13]. Figure 1 shows persistent homology analysis of turbid media with distinct topological characteristics captured by PDs. The left panels illustrate the spatial distributions of refractive particles with varying sizes (top) and different distribution patterns (bottom). The corresponding PDs on the right demonstrate how persistent homology distinguishes topological differences based on particle size and spatial organization.

Fig. 1.

Persistent homology analysis of uniform and clustered microscopic turbid media with varying particle sizes and spatial organization.

While PDs describe the topological characteristics, their variability across simulation samples, particularly in random arrangements of refractive spheres, requires a more robust representation as a tissue parameter. To address this, PIs were constructed as a vectorized representation of PDs, providing a more structured and stable representation of topological features for numerical applications such as statistical analysis, machine learning, and numerical computation [11,13]. In this study, PIs were used to compute shared topological characteristics across multiple PDs. The transformation from PDs to PIs involves a series of processing steps to convert the persistence points (b,d), where b and d represent birth and death times of topological features, into a structured grid-based format. First, each persistence point was assigned a weight based on its persistence d-b, using an arctangent-based weighting function to emphasize longer-lived topological features. These weighted persistence points were then convolved with a Gaussian kernel (standard deviation of 0.002) to generate a smooth intensity distribution. Finally, this smoothed representation was mapped onto a fixed-resolution grid, transforming the PD into a structured two-dimensional feature space, which constitutes the PI [13]. By averaging the patterns derived from multiple PDs through principal component analysis, the mean PI was obtained, encapsulating the shared topological characteristics across simulation samples.

2.2. Generation of scattering media

To create the refractive index distributions for persistent homology’s topological analysis and BPM-based light propagation simulation, two types of particle-distributed turbid media were generated: the uniformly random particle-distributed media (uniform model) and the clustered particle-distributed media (cluster model). These 2 types of models were prepared in order to observe how differences in topological characteristics impact light propagation characteristics in microscopic setting. To study the effect of microscopic variability, up to 1,000 simulation samples of both models were generated in a random manner in term of particle distribution location. Both models consist of spherical particles with a refractive index of n = 1.4, distributed within a physical space of 100×100×100 μm. Refractive index value was chosen as it falls within the general range of refractive indices commonly observed in biological organelles, such as mitochondria and nuclei [14,15]. The background refractive index was set to 1.33, corresponding to the refractive index of water. In the uniform model, spheres of varying radii are placed uniformly throughout the simulation space. The non-overlapping condition ensures that refractive particles do not intersect or overlap during the generation process. The clustered model, on the other hand, introduces heterogeneity by grouping spheres into localized clusters. The center of each cluster is defined, and the scatterers are positioned around these centers based on a Gaussian distribution with a standard deviation of 10 μm.

The density of refractive particles is systematically varied at 20,000/mm³, 40,000/mm³, 60,000/mm³, 80,000/mm³ and 100,000/mm³ to explore the effects of increasing scatterer concentration. The spatial discretization is performed on a grid with 201×201×201 voxels, where each voxel corresponds to 0.5 μm, representing a total physical size of 100×100×100 μm.

2.3. Optical propagation calculation

To investigate the effects of varying topological characteristics of the turbid media on light propagation characteristics, BPM was employed to simulate light propagation through the turbid media. BPM was selected for this study due to its efficient computational approach for modeling forward light propagation in media with spatially varying refractive index distributions, such as the turbid media analyzed in this study. Unlike methods requiring intensive computational resources, BPM balances accuracy and efficiency, capturing amplitude and phase changes of light while propagating through refractive index-varying microscopic structures [9]. This makes it suitable for analyzing beam centroid displacement and beam distortion, key optical characteristics investigated in this study. In this study, absorption is neglected to focus on the impact of microscopic refractive index structures on optical scattering characteristics. The input beam was initialized with a Gaussian profile, characterized by a beam waist of 1.5 μm (FWHM) and a wavelength of 638 nm resulting in a numerical aperture (NA) of approximately 0.96 for this BPM-based light propagation system. Light propagation was simulated through multiple turbid media, where each medium exhibited distinct topological characteristics. The beam was focused at a focal depth of μm.

2.4. Optical characteristics evaluation

To evaluate the effects of microscopic refractive index structures on light propagation, beam centroid displacement and beam distortion are quantified as a beam propagation characteristic. These metrics characterize how the light beam deviates and distorts due to scattering and refractive heterogeneities within the medium. Beam centroid displacement quantifies the deviation of the beam’s centroid from its original position as it propagates through the scattering media. The centroid displacement is computed as the difference between the beam’s center in a scattering medium and its unperturbed position in a homogeneous medium (water with refractive index 1.33). Such deviations arise from heterogeneity-induced beam steering, where refractive index gradients alter the beam’s trajectory. This phenomenon has been previously reported in studies that examine Gaussian and Bessel beam propagation in biological tissues [16]. Beam distortion measures the deformation of the beam profile, defined as deviations from an ideal Gaussian profile. In this analysis, beam distortion is assessed by comparing the irradiance distribution of the propagated beam to its ideal, unperturbed shape. Distortion occurs due to scattering and refraction from microscopic structures and can be quantified using residual differences between the perturbed beam profile and a fitted Gaussian function [16].

2.5. Scattering coefficient calculation

To investigate the relationship between topological characteristics and their scattering coefficients, the scattering coefficient μs of the turbid media was determined using two methods. The first method is through the analysis of attenuation of beam intensity I from BPM propagation based on the Beer–Lambert law. the scattering coefficient μs can be calculated from the slope of the linear equation of Beer–Lambert law as expressed by the following equation: $I/I_0=\exp (-{\mu }_{s}zf)$ . Here $I_0$ is the beam intensity in water medium at the focal depth zf. The second method is through the calculation based on Mie theory [17]. Mie theory was applied to determine the scattering coefficient using parameters such as particle size, density, and refractive index contrast of the spherical particles. In this study, particle sizes were set at 3, 4, and 6 μm, with a refractive index of 1.4, and particle densities ranged from 10,000/mm³ to 500,000/mm³.

3. Results

3.1. Persistent image and light propagation characteristics

To evaluate the performance of optical property representation, the relationships of PIs and light propagation characteristics across turbid media were analyzed. Figure 2 illustrates examples of the simulated light intensity distributions at the focal plane for the generated turbid media with different scatterer densities, showing the propagated beam profiles alongside the corresponding refractive index structures. The beam spot remained relatively intact in low-density models, while significant distortion and displacement occur in higher-density models. This progression highlights the increasing influence of microscopic scattering on light propagation as the particle density and spatial complexity increase.

Fig. 2.

Heterogeneous scattering media examples with varying scatterer concentrations (50k, 200k, 300k, 400k [1/mm³]) and their corresponding PIs and simulated light distributions at the focal plane.

In an attempt to capture the topological characteristics of each topological group using PIs, the Mean Squared Error (MSE) between a PI at a given simulation sample and a reference PI were computed. In this analysis, the reference PI was taken from N=1,000, representing the state where characteristic convergence is considered to be achieved. Since a sufficiently large number of simulation samples provides a stable representation of the topological features, characteristic convergence is defined as the point at which further increasing the number of samples does not result in substantial changes to the PI structure. In this study, 1,000 simulation samples (N=1,000) was chosen as the reference point, as it is large enough to observe the stabilization behavior of PIs and assess how MSE changes at lower simulation samples. Choosing a smaller maximum N, such as 100, could have made it difficult to capture the full extent of topological variations and their convergence trends. Figure 3(a) shows the MSE of PIs decreased rapidly as the number of simulation samples increases. At 500 simulation samples (N=500), the MSE was significantly reduced and stabilized compared to smaller N. More specifically, it was reduced to less than 1% of its initial value at N=10. This result suggests that PIs, as a descriptor of microscopic tissue structures, require a sufficient number of simulation samples (e.g., 500 in this case) to achieve robust parameterization, which we defined as the point of convergence.

Fig. 3.

(a) MSE convergence of PI across various simulation samples with referential PI at 1,000 simulation samples. (b) Standard deviation of beam centroid displacement. (c) Standard deviation of beam distortion. Standard deviations of beam characteristics were investigated for 5 simulation trials across 1,000 simulation samples.

Figures 3(b) and (c) show the convergence behavior of beam centroid displacement and beam distortion. Initially (N<500), at low simulation trial these beam characteristics exhibited high variability, reflecting the diversity of microscopic structural interactions with light. However, as the number of simulation samples increased (N>500), both metrics stabilized, and their standard deviations decreased significantly.

Together, these results demonstrate that both the topological characteristics represented by PIs and the optical metrics, including beam centroid displacement and distortion, converge and stabilize after sufficient simulation samples.

3.2. Light propagation characteristics across volume fraction

To understand how variations in topological characteristics influence light propagation, the relationship between tissue particle densities and beam propagation metrics, including centroid displacement and beam distortion were analyzed. Figures 4(a) and (b) demonstrate how changes in tissue particle density influenced beam centroid displacement and beam distortion. The boxplots illustrate the degree of variability of these metrics at N=500, a point where significant convergence has been achieved. For centroid displacement, a general trend of increasing variability was observed as particle density increases, indicating that denser media led to more pronounced and diverse scattering effects. Similarly, for beam distortion, higher particle densities resulted in larger distortion magnitudes, reflecting the increased complexity of light-tissue interactions at higher densities. Statistical outliers were not included in the boxplots. For lower-density models (e.g., 20k and 40k), distortion values occasionally extended to ranges comparable to those observed in the 60k model. However, these instances were classified as outliers and were therefore not included in the boxplots. These findings demonstrate that changes in topological characteristics (e.g., particle density in this case) enhance variability in centroid displacement and beam distortion, underscoring the topological impact on light propagation behaviors.

Fig. 4.

Beam centroid displacement (a) and beam distortion (b) as functions of particle densities.

3.3. Impact of topological particle-distribution patterns on light propagation characteristics

To investigate how turbid media with the same particle densities but different topological distribution patterns influence light propagation characteristics, 2 topological distinct models which include uniform model, and the cluster model were compared. While both models maintain the same particle density of 100,000 particle/mm³, their spatial organization differs as characterized by their PIs and beam propagation metrics. Figures 5(a, b) illustrate examples of uniform model and cluster models, respectively. The uniform model exhibited a homogeneous particle distribution, whereas the clustered model featured particles grouped into localized clusters. These topological differences of 500 simulation samples were captured by their respective PIs, shown in Fig. 5(c) and Fig. 5(d). The PI for the uniform model (Fig. 5(c)) displays a more uniform distribution of topological features, reflecting the homogeneous spatial organization of particles. In contrast, the PI for the clustered model (Fig. 5(d)) exhibited higher-intensity regions, which was attributed to faster birth-death transitions caused by particles being closely packed within clusters. These differences highlight the ability of PIs to capture and characterize the topological distinctions between uniform and clustered turbid media. To investigate how topological differences influence optical behavior, we quantified beam centroid displacement and beam distortion, as shown in Fig. 5(e) and Fig. 5(f), respectively. As shown in Fig. 5(e), both models exhibited a similar mean displacement due to having the same particle density. However, the clustered model displayed a larger variance of 23% in beam displacement values, reflecting the greater heterogeneity in light-tissue interactions caused by the localized clusters. A t-test showed no significant difference in centroid displacement between the models (p = 0.903). For the beam distortion (Fig. 5(f)), the clustered model showed a 105% significantly larger beam distortion variance compared to the uniform model. A statistically significant difference (p = 0.000263) was observed in beam distortion between the models. As shown in Fig. 5, the uniform and clustered models exhibited distinct differences in their topological and optical characteristics. The persistent images (Figs. 5(c, d)) captured these differences, with the clustered model demonstrating higher-intensity regions indicative of localized particle clustering. These topological variations directly influenced optical behavior, as seen in the beam centroid displacement (Fig. 5(e)) and beam distortion (Fig. 5(f)). The clustered and uniform model showed different topological characteristics and beam characteristics, highlighting the impact of topological characteristics on light propagation and wavefront distortion.

Fig. 5.

Turbid media with (a) uniform and (b) clustered particle distributions, along with their corresponding persistent images (c,d) respectively. (e) Beam centroid displacement and (f) beam distortion characteristics. The dashed green line in the boxplots represents the mean value of the beam characteristics.

3.4. Relationship between PI-Based classifications and light propagation characteristics

To validate the hypothesis that PI groups correspond to distinct light propagation characteristics, the relationship between PI-based classifications and beam behavior metrics, including centroid displacement and distortion were analyzed. As shown in Fig. 6, the beam centroid displacement and distortion are presented as boxplots for uniform and cluster models across different particle densities. For each particle density, results were calculated based on 3 independent simulation trials (n=1,2,3) to account for variability. Notably, the results from these independent trials showed consistency within each model, underscoring the reliability of PI-based classifications. The observed trends in displacement and distortion correspond to differences in topological characteristics associated with varying particle densities, which are reflected in the complexity of their respective PIs. Importantly, within each PI group, the beam characteristics remained consistent across independent simulation trials (n=1,2,3). These findings confirmed that similar PI groups (e.g., models with comparable topological features) produce consistent light propagation characteristics, while different PI groups yield distinct optical behaviors.

Fig. 6.

Beam centroid displacement and distortion for uniform and cluster models (500 samples) across different simulation trials (n=1,2,3), demonstrating that similar PI groups yield consistent beam characteristics.

3.5. Comparison to scattering coefficients

To quantify the relationship between scattering coefficients, volume fractions, and particle sizes, we compared the values obtained from BPM simulations with those predicted by Mie theory. Figure 7 shows the relationship between scattering coefficients and volume fractions for three particle sizes (3 μm, 4 μm, and 6 μm). The theoretical values derived from Mie theory were represented as solid lines, while the scattering coefficients calculated from BPM simulations were depicted with 95% shaded confidence intervals (CIs). The shaded confidence intervals around the BPM results highlight the structural influence on scattering coefficients, where variability arises from the differences in spatial arrangements. For all particle sizes, both theoretical and BPM-derived scattering coefficients increased with volume fraction. The BPM-derived scattering coefficients closely followed the theoretical predictions, particularly at lower volume fractions. This agreement confirms the accuracy of BPM in capturing the fundamental scattering behavior at lower volume fractions. However, at higher volume fractions, slight deviations between the two methods were observed.

Fig. 7.

Scattering coefficient calculated from BPM system of turbid media with particle size condition 3 μm, 4 μm, 6 μm in comparison with Mie theoretical values of each condition. The shaded region represents 95 percent confident interval (CI).

4. Discussion

4.1. Microscopic variability and convergence in beam and topological characteristics

Microscopic turbid media exhibit variability due to the randomness of particle distributions and the resulting diversity in their topological and optical characteristics. While this variability is reflected in the light propagation characteristics, the topological features quantified by PIs capture shared underlying topological characteristics across different media. However, due to the microscopic variability of tissue structures and their resulting beam behaviors, analyzing only a small number of simulation samples is insufficient to fully parameterize or distinguish the differences between media with varying topological characteristics (e.g., different volume fractions). To address this, sufficient simulation samples are necessary to ensure that the variability is accounted for and meaningful differences between PI groups can emerge, which we refer to as convergence. This stabilization indicates that a sufficient number of simulation samples is essential to achieve consistent beam characteristics, which are inherently tied to the convergence of topological features observed in Fig. 3(a). Together, these findings highlight the importance of sufficient simulation trials in ensuring robust parameterization of PIs as a microscopic optical property. The convergence of both PIs and beam characteristics ensures that the variability introduced by random particle distributions is captured, enabling a reliable relationship between microscopic topological characteristics and microscopic light propagation behaviors.

4.2. Persistent homology enables distinction between uniform and cluster characteristics

The findings presented in this study demonstrate the utility of PIs as a topological descriptor for linking microscopic tissue structures to microscopic and macroscopic light propagation characteristics. Specifically, PIs provide the ability to encapsulate topological heterogeneities that are not captured by conventional methods, offering a novel paradigm for studying microscopic light-tissue interactions. The results from Section 3.3 highlight a critical advantage of using persistent homology to characterize microscopic turbid media. Traditional approaches, such as scattering coefficient-based methods, would fail to differentiate between uniform and clustered particle-distributed media if their particle densities and volume fractions were identical. This is because scattering coefficients primarily capture the overall magnitude of light attenuation, neglecting the topological and spatial organization of particles within the tissue. In contrast, persistent homology-based PIs could distinguish between these two models by encoding their unique topological characteristics. Specifically, uniform model exhibit uniformly distributed topological features in the PI, while clustered models reveal localized regions of higher topological intensity, corresponding to particle aggregation. These differences were reflected in beam propagation characteristics, where the clustered model exhibited a statistically significant increase in beam distortion (p = 0.000263), while no significant difference was observed in beam displacement (p = 0.903). Under the current model conditions of 100,000 particles/mm³ and clustering configuration, the effect of topological differences on displacement appears weak, suggesting that centroid displacement is less sensitive to localized clustering compared to distortion. However, further investigations with varying particle densities and spatial distributions could provide additional insights into the role of topology in displacement behavior. This demonstrates that persistent homology enables a deeper understanding of tissue heterogeneities, providing a level of detail that conventional scattering-based metrics cannot achieve. By revealing differences between uniform and clustered models, persistent homology-based PIs emerge as a reliable descriptor for analyzing microscopic tissue structures and their optical properties.

4.3. Persistent images as optical properties of microscopic turbid media

The results from Section 3.4 provide the cornerstone for proposing PIs as an optical property of microscopic turbid media. By analyzing the relationship between PI groups and light propagation behaviors, we demonstrate that models with similar PIs exhibit consistent optical characteristics, while those with distinct PIs show significantly different behaviors. This finding is pivotal in validating the hypothesis that PIs not only serve as descriptors of topological heterogeneity but also encapsulate meaningful optical properties. Specifically, light propagation metrics, such as beam centroid displacement and distortion, correlate with the topological features captured by PIs. While PIs primarily describe the structural heterogeneities of refractive index distributions in turbid media, their relationship with light propagation characteristics suggests their potential role as a topology-based optical property. Since PIs capture topological characteristics of refractive index distributions, which fundamentally influence light propagation, they can serve as an indirect descriptor of optical behaviors that are linked with classical optical properties such as scattering and absorption coefficients. To further establish PIs as a functional optical property, future work could consider a method to generate refractive structures based on PI as input. By parameterizing tissue heterogeneities into a single PI representation, this approach could provide a reliable framework for simulating and analyzing light propagation in microscopic turbid media, with potential applications in diagnostics and tissue characterization.

In contrast to our PI-based approach, fractal analysis primarily focuses on self-similarity across scales, which can be useful for describing hierarchical organization. However, it does not inherently capture topological features of cellular components, such as size, shape, and spatial arrangement, which are essential for understanding the microscopic refractive index variations that influence light propagation behavior. Furthermore, while fractal analysis relies on multiple parameters (e.g., fractal dimension, correlation length) to describe tissue structure, these parameters often exhibit a range of possible values [9]. This lack of determinism can complicate the accurate characterization of tissue properties. In contrast, PIs consolidate the complex heterogeneities of tissue into a single deterministic descriptor, facilitating a more comprehensive and convenient representation of certain tissue characteristics.

4.4. BPM-derived and Mie-based scattering coefficients

At lower volume fractions, the scattering coefficients derived from BPM simulations show good agreement with those predicted by Mie theory. The results from BPM simulations exhibit variance across different samples, as demonstrated by 500 simulation samples. When these results are averaged, the scattering coefficient approaches the value predicted by Mie theory. Each BPM simulation sample represents a unique spatial configuration of particle distributions, capturing distinct scattering paths that light might traverse in a particle-distributed model. By averaging across these 500 simulation samples, the collective behavior of light scattering can be interpreted as analogous to the cumulative effects of multiple scattering in a macroscopic environment. Therefore, this averaging process mirrors the statistical nature of multiple scattering events in tissues, where light undergoes numerous scattering interactions before forming a stable macroscopic optical response, such as the scattering coefficient observed in this case.

However, at higher volume fractions, the results from Section 3.5 illustrate a notable discrepancy between scattering coefficients derived from the BPM and those predicted by Mie theory. While Mie theory predicts a linear relationship between scattering coefficient and volume fraction, BPM results indicate saturation at higher volume fractions.

Interestingly, this saturation aligns with the results from experimental and theoretical studies [18–20], such as the Dependent Scattering Theory (DST), which accounts for dependent scattering effects and better captures inter-particle interactions and interference dynamics. Notably, one factor contributing to the discrepancy observed at high volume fractions is that BPM does not consider backscattering effects, while Mie theory accounts for scattering in all directions, including both forward and backward scattering. This limitation of BPM may partially explain why the derived scattering coefficients diverge from Mie predictions as particle density increases, where backward scattering could become more significant due to enhanced inter-particle interactions. In contrast, the consistency observed at lower volume fractions could be explained by the minimal inter-particle interactions and interference effects at low particle densities. This allows Mie theory’s assumption of independent scattering to remain valid, leading to agreement between BPM and Mie predictions.

These findings underscore the potential of the BPM simulation for studying light-tissue interactions in scenarios where Mie theory may oversimplify the scattering dynamics. By bridging experimental observations and theoretical advancements like DST, the BPM-derived scattering coefficients offer a compelling alternative for understanding light behavior in turbid media.

4.5. Limitations and future directions

While this study employs turbid media composed of refractive particles to represent simplified tissue models similar to biological tissues in their ability to exhibit optical characteristics and demonstrate the potential of PIs as microscopic optical properties, they differ from real biological tissues in that they lack cellular-level structures and more detailed refractive index variations. Future work could extend this framework to real biological tissues by leveraging 3D refractive index maps, which could be obtained from techniques such as optical coherence tomography, quantitative phase imaging, or refractive index tomography. These refractive index maps could help establish PIs for different biological tissues (e.g., PI for skin tissue, PI for brain tissue), allowing for a more biologically representative analysis.

The findings in this study highlight the potential of PIs and BPM, although certain limitations and areas for improvement could remain. One notable limitation is that BPM does not model multiple scattering or backscattering effects. BPM is best suited for forward-propagating waves and is typically applied in weakly scattering media. However, in this study, the refractive index contrast $\left(\mathrm{\Delta }n\right)$ remains within a range where BPM assumptions hold (Δ n /n « 1), ensuring that BPM remains valid for analyzing forward-scattering interactions. Previous studies utilizing BPM for similar particle-embedded scattering media have also reported comparable $\mathrm{\Delta }n$ values, demonstrating the versatility of BPM under such conditions. For instance, Wang et al. [21] employed BPM in a holographic 3D particle imaging framework with $\mathrm{\Delta }n=0.26$ (n = 1.59 embedded in n = 1.33). Similarly, Kamilov et al. [22] applied BPM for optical tomographic image reconstruction where $\mathrm{\Delta }n=0.03$ (n = 1.548 embedded in n = 1.518). Furthermore, previous studies [23–25] have discussed that BPM is applicable to biological tissues, which exhibit similar volume fractions, scattering coefficient, and forward-scattering characteristics with the turbid media used in this work. While BPM provides an efficient computational approach for capturing key wavefront distortions and centroid displacement, it does not account for backscattering and multiple scattering interactions that may become prominent in longer range of beam propagation or in denser tissue structures. Future research could explore incorporating alternative simulation methods, such as FDTD, to provide a more comprehensive assessment of scattering dynamics in persistent homology-based tissue characterizations. Another limitation is the grid size used in this study, which was set to 0.5 μm. While sufficient for capturing general topological and optical features, this resolution may obscure finer topological details and refractive index variations. Future research should investigate the impact of higher-resolution grids to ensure that smaller, yet potentially significant, features are accurately represented.

Currently, this work focuses on the 0th dimension of persistent homology, which captures connected components in the turbid media. This choice is effective for characterizing the presence and distribution of particle clusters, directly relating to the optical properties analyzed. However, higher dimensions of persistent homology, such as the 1st dimension (loops) or 2nd dimension (voids), could reveal additional insights into the topological heterogeneities of tissues, particularly for models with intricate geometries. Incorporating these higher-dimensional features may allow for a more comprehensive understanding of the relationship between tissue structure and light behavior.

Additionally, the computational cost of persistent homology analysis remains a challenge when extending this approach to large-scale tissue datasets. For instance, for a 3D dataset of size 256×256×256, the computation time for a single sample is approximately 2.5 minutes [26]. Scaling this up to a dataset of 1,000 tissue models would require over 42 hours to generate PI representations of tissue characteristics. This computational demand must be considered when applying persistent homology-based analysis to real-world biomedical applications.

For imaging and diagnostic applications, one potential solution is the precomputed construction of a PI database for different tissue types. By precomputing the required 42 hours for each tissue type, a standardized PI dataset could be developed to serve as a reference for diagnostics and imaging applications. In diagnostics, tissue structures that deviate from the generalized patterns in the PI database could be identified as possibly "abnormal." For imaging, 3D refractive structures could be reconstructed based on a given PI input to facilitate light propagation simulations and further optical analysis.

Following the development of a precomputed PI database for different tissue types, a logical next step is the generation of turbid media informed by PIs. The results of this study suggest that PIs provide a meaningful characterization of topological features, which could serve as input parameters for creating synthetic turbid media. These models would enable further exploration of the relationship between topological features and light propagation characteristics, potentially leading to more realistic in silico tissue simulations.

To experimentally reinforce the computational framework established in this study, an initial verification step involves constructing physical turbid media using polyethylene bead suspensions and performing laser irradiation to observe beam distribution patterns, followed by an analysis of beam centroid displacement and beam distortion, as done in this simulation study. The framework will also be extended to real biological tissues to further advance the application of persistent homology in biomedical optics.

By addressing these limitations and extending the methodology, persistent homology can become an even more effective tool for characterizing the optical properties of turbid media. The integration of PIs and BPM results not only provides a deeper understanding of light propagation at the microscopic scale but also positions PIs as a potential microscopic optical property, bridging the topological heterogeneities of microscopic turbid media with their optical behaviors.

5. Conclusion

This study demonstrates the effectiveness of using persistent homology-based methods to analyze the optical properties of microscopic turbid media and their relationship with light propagation. By employing PIs to capture the topological characteristics of refractive particle-distributed turbid media with both uniform and clustered particle distributions, we have established a foundation for linking microscopic topological characteristics to microscopic optical behaviors. The findings show that PIs can distinguish topological differences between turbid media, even when they share the same volume fractions and particle densities. These topological characteristics are correlated with distinct light propagation behaviors, including beam centroid displacement and distortion. Models with similar PIs exhibit consistent light propagation characteristics, while those with distinct PIs show significant differences. This supports the hypothesis that persistent images provide meaningful and predictive descriptors of light-tissue interactions, positioning them as a potential microscopic optical property that encapsulates the topological heterogeneities of tissue structures and governs light propagation behaviors.

Funding

Japan Society for the Promotion of Science10.13039/501100001691 (23H04133); Japan Science and Technology Agency10.13039/501100002241 (JPMJFR234T).

Disclosures

The authors declare no conflicts of interest related to this article.

Data availability

Data underlying the results presented here are not publicly available but may be obtained from the authors upon reasonable request.