Physical and Methodological Perspectives on the Optical Properties of Biological Samples: A Review

Abstract

The optical properties of biological systems can be measured by imaging and microscopy methodologies. The use of X-rays, γ-radiation and electron microscopy provides information about the contents and functions of the systems. The need to develop imaging methods and analyses to measure these optical properties is increasing. On the other hand, biological samples are easily penetrated by a high-energy input, which has revolutionized the field of tissue optical properties and has now reached a point where light can be applied in the diagnosis and treatment of diseases. To this end, developing methodologies would allow the in-depth study of optical properties of tissues. In the present work, we review the literature focusing on optical properties of biological systems and tissues. We have reviewed the literature for related articles on biological samples’ optical properties. We have reported on the theoretical concepts and the applications of Monte Carlo simulations in the studies of optical properties of biological samples. Optical properties of biological samples are of paramount importance for the understanding of biological samples as well as for their applications in disease diagnosis and therapy.

1. Introduction

1.1. Applications and Optical Spectra in Biological Samples

Since the development of the first microscope by the pioneer Zacharias Jansen (16th century), Robert Hooke (17th century) and later by Anton van Leeuwenhoek (18th century), it has become evident that biological entities have optical properties. The field of imaging methodologies now focuses on visualizing and understanding biomolecules, such as proteins. For the biology of the 21st century, high-throughput methodologies provide immense potential for the investigation of biological systems, yet the data produced exceed our capacity of understanding them.

The fact that biological phenomena are mostly stochastic, or more simply of a “contemplative” nature, has always been a main problem. For example, we study the presence of a protein (X_t) in a cell type under a given condition (e.g., the effect of a growth factor over time) by detecting the protein levels or concentrations (X) at a given time (t). The question we can pose is whether we can precisely predict protein levels at time t + 1 (i.e., X_t+1). The answer is no, due to uncertainty, while we can calculate the probability for the possible values of X_t+1. Therefore, the stochastic method is an important tool for understanding biological phenomena.

From the biological perspective, in both basic and applied research, one of the methods to find markers (biomarkers) is microscopy. Biomarker discovery is of great importance for the diagnosis and prognosis of many diseases, as for example in human tumors. A very interesting example for the use of microscopy and the optical investigation of tumors comes from melanoma, which is one of the most aggressive tumor types known. The spectral investigation of melanomas is even more critical, since they are found on skin, and the early differential diagnosis could be facilitated by the use of optical methods (i.e., electromagnetic radiation in the spectrum of light) [1,2,3,4,5,6,7,8].

Moreover, part of the current research focuses on finding either indicators related to the “predisposition” of a chemotherapy-resistant tumor [9,10,11], or specific biomarkers for each tumor as therapeutic targets. Finding early biomarkers for the diagnosis [12] or even the prognosis of neoplastic diseases is of great interest [13,14,15]. The search for biomarkers has not been successful because the system of a eukaryotic cell is too complex to identify or group such biomarkers. On the other hand, the finding of biomarkers mainly concerns the identification of different molecules, such as genes or proteins, and thus leaves a very large field for the use of photons for disease diagnosis and even treatment. Although light can be used directly on tissues for microscopic investigation, recent reports have suggested electromagnetic radiation in the visual spectrum as a potential therapeutic agent assisted by non-chemical agents in breast or ovarian tumors [16,17,18,19,20,21,22,23,24]. These are examples wherein photons (i.e., electromagnetic radiation) could be used for diagnostic or therapeutic purposes. The main idea, which has been already stated, is that tissues have different properties depending on their homeostatic position [1,2,3,4,5,6,7,8]. In other words, it is expected that the optical properties of a normal cell are distinct from a cancer cell, and cells from different tissues will have different optical properties.

The most modern approach, especially for neoplasms, is “patient-specific treatment” or “patient-tailored treatment”. It is hypothesized that the diversity among patients in terms of both diagnosis and treatment is large, and therefore, both diagnosis and treatment should focus on each patient individually, bypassing the hitherto known model of holistic approaches to treatment [25]. It makes sense to discuss the aspect of molecular markers or biomarkers when a biopsy is possible. However, what happens in cases when biopsy is unavailable? A valuable tool would be the use of electromagnetic radiation in the optical spectrum. Such an example comes from central nervous system (CNS) tumors, which are the biggest challenges, both in terms of diagnosis and treatment. The first important difference is that sampling alone is much more difficult in these tumors as compared to others, which makes them hard to study. Meanwhile, CNS biology all together, along with the developmental biology of living organisms, are, by themselves, one of the greatest challenges in modern research. Examples of this complexity relate to the understanding of CNS tumor biology and the differentiation mechanisms during development. Several molecular biomarkers have been identified for CNS tumors, such as Epidermal Growth Factor Receptor (EGFR), MYC proto-oncogene-bHLH transcription factor (MYC), receptor tyrosine-protein kinase erbB-2 (HER2), etc. However, the difficulty of obtaining a biopsy from CNS tumors, combined with the fact that a biopsy takes place only after diagnosis but not during different stages of the disease, makes CNS tumors one of the greatest research challenges. Any intervention in the CNS involves the risk of collateral damage depending on positive or negative therapeutic effects. Therefore, the administration of treatments or interventions using electromagnetic radiation in the optical spectrum would be an excellent tool. To date, several attempts have been made to apply photodynamic therapy in CNS neoplasms, but merely as an adjunctive method to chemical/molecular therapies [26,27,28,29,30,31].

From the methodological point of view, recent advances have helped us to unravel the genomic or proteomic profile for each type and subtype of disease/biological phenomenon (i.e., tumor), but biological phenomena are actually dynamic. Only a percentage of biological samples (i.e., patients) become distinct. For example, a group of genes is found to be overexpressed in some types of CNS tumors but not others, which instinctively gives us a first indication that observed profiles (either expressional or genomic) are not the cause of pathogenesis or the origin of a type of tissue. A systematic error was the assumption that the observed factor measured during sampling was related to the true condition of a biological entity. Biological systems are dynamic, with their state changing with time and environmental conditions. Furthermore, biological systems owe part of their dynamic nature to the exchange of energy and mass with their environment, which makes them a more complicated system to comprehend. Thus, examination of a biological system at a specific time point would give us information for the specific time point only, while the system would completely change in the very next step of its time evolution. It is practically impossible to study a biological system with immensely large degrees of freedom.

In another domain of science, the use of optical spectra has been employed for the monitoring of bacterial and viral properties and biology. Such a phenomenon is anti-microbial resistance (AMR), which is also directly linked to the aforementioned terms of “patient-specific treatment” and “patient-tailored treatment”. AMR is an acquired property of bacteria against antibiotics. The phenomenon has increased significantly in recent years, due to the extensive use of antibiotics [32]. A recent report from the World Health Organization (WHO) indicated that AMR is expected to become one of the leading causes of death worldwide in the following decades. Towards that end, a recent study reported the development of a new method for detecting antibiotics resistance, exploiting the optical properties of bacteria [32]. The method utilizes “optical tweezers” based on photonic crystal cavities for the trapping of individual bacteria. As the bacteria are trapped, it is possible to add antibiotics and study the subsequent changes in the bacteria’s optical properties. In particular, the observed physical property concerns the resonance wavelength and transmission of the bacterium under investigation [32]. In another recent study, a very interesting method has been developed for the detection of live and dead bacteria. In particular, this study has exploited the optical properties of silver nanoparticles, which have the ability to adhere differently on live and dead bacteria [33]. This difference was measured using surface-enhanced Raman scattering, which was able to discriminate between the two bacterial states [33]. Furthermore, a new method has been developed for the detection and investigation of biofilms—a bacterial mechanism that contributes to the AMR. Biofilms are “syntrophic consortia of microorganisms in which cells stick to each other and often also to a surface” [34,35]. These adherent cells become embedded within a slimy extracellular matrix that is composed of extracellular polymeric substances (EPSs) [35]. The cells within the biofilm produce the EPS components, which are typically a polymeric conglomeration of extracellular polysaccharides, proteins, lipids and DNA [34,35]. Because they have a three-dimensional structure and represent a community lifestyle for microorganisms, they have been metaphorically described as “cities for microbes” [34,35]. Biofilms are a very solid resistance mechanism against antibiotics (and other drugs, since not only bacteria form biofilms), since they do not allow the access of a drug to the entirety of the pathogens. Towards that end, a novel method has been developed in order to study the formation of biofilms. The proposed method included the design of an “optoelectronic device based on an on-chip dual array of interdigitated micro- and nanoelectrodes that combines together optical and electrical techniques to monitor the growth of a biofilm and to analyze the effect of antibiotics on the bacteria” [36]. The proposed approach included the exploitation of the optical as well as electrical properties of bacterial biofilms. The addition of the electrical approach allowed the concurrent study of the developing phases of the bacterial biofilm, and the impedance changes provided insight to the evaluation of the biofilm’s growth and maturation [36]. A similar report presented a “biosensing” system, which is reported to be able to detect bacterial cells in low-volume fluid samples [37]. In this method, similarly to the previous one, the scientists have combined a “dielectrophoretic cell-collecting technique with evanescent-field sensing” [37].

Similarly to bacterial properties, the investigation of viral properties and biology is of paramount importance. In particular, this topic has regained attention due to the recent SARS-CoV-2 pandemic. Viruses are difficult to study and their investigation requires a large range of methods, which are both time- and resources-consuming. A method proposed as an alternative to alleviate these difficulties requires the use of optical and electrical properties measurements. Recent studies have indicated the utilization of trapping nanoparticles able to detect the optical, electric and dielectric properties of small molecules, includes viruses [38]. Moreover, if this methodology is enhanced with the simultaneous trapping of numerous molecules, through the use of an array, then a multitude of biological particles can be studied synchronously [39]. Thus, it becomes evident that the use of optical properties as a means of study is not only limited to the study of eukaryotic cells and tissue, but can also apply to the study of prokaryotes and viruses.

In conclusion, light (as defined above) is non-invasive and can be effectively used at different time points under certain circumstances. To use light as a diagnostic, therapeutic and exploratory tool, we first need to understand its permeability properties in biological samples.

1.2. Basic Principles of Microscopy and Optical Properties of Tissues

The basic optical properties of biological samples can be described by a set of physical parameters. These are presented as coefficients, which include absorption (μ_A), scattering (μ_Σ), anisotropy (γ), reflection (μ_R) and the scattering function p(θ,ψ), where θ is the polar and ψ is the azimuthal scattering angle. All the coefficients are presented in units of cm⁻¹ and the scattering function p(θ,ψ) in units of sr⁻¹ [40]. These coefficients with variations were obtained by several studies depending on the wavelength and the type of tissue used [40,41,42,43]. Although extensive studies were carried out on the optical properties of tissues, there is no absolute match between individual tissue measurements and in vivo measurements. In particular, since biological tissues are dynamical systems, a measurement of the optical property of tissues will manifest different results from a measurement at another time, since the tissues’ composition, such as water, lipids, proteins, and transcriptome, would have changed over time. At the same time, changes in tissue’s optical properties also differ individually. The same tissue between two subjects of the same species could hold different optical coefficients [44].

The probability of scattering can be negligible when the section of tissue/sample is very thin. The scattering function can be used to analyze images obtained from transmission microscopy and confocal microscopy. If the biological samples are thicker, random scattering occurs, with factor ψ ignored. In this case γ = cosθ holds. The optical properties of the tissues are summarized in

Table 1. The principal optical coefficients of tissues (adapted from Jacques SL (2013) [40]).

Coefficient	Symbol	Units
Absorption	μA	cm−1
Scattering	μS	cm−1
Scattering function	p(θ,ψ)	sr−1
Anisotropy	γ = cos(θ)	Arbitrary
Real refractive index	n′	Arbitrary
Reduced scattering	${\mu }_S^{\prime }={\mu }_S(1-g)$	cm−1

.

1.3. Measuring Tissue’s Optical Properties

The most generalized categorization of tissue optical properties measurements can be defined by ex vivo, in vivo and in vitro measurements. The first concerns the estimation of excised, or isolated, tissues (usually thin-dissected tissue probes), the second concerns “live” measurements on tissues, on a whole organism, and the third concerns the evaluation of optical properties under in vitro conditions, such as for example cell or tissue cultures. The in vivo case can be separated into two further subcategories, namely, invasive measurements and non-invasive measurements. Another proposed categorization has been reported to be based on the method of optical properties’ determination, that is, “direct” and “indirect” approaches [45]. “Direct” measurements include those experiments that determine optical properties independently of any modeling approach for light propagation. On the other hand, “indirect” approaches concern the determination of optical properties through the solution of an “inverse problem”. In other words, the optical properties are computed through a light propagation model. The aforementioned schema is summarized in Figure 1. The “indirect” measurements are further divided into photometric and photothermal methods.

Several methods have been developed over the years for the creation of precise models. The need for accurate predictions for characterizing the optical properties of biological samples has become more imperative since light offers an invaluable tool in the investigation of tissue properties. Thus, methods developed include the “adding-doubling” [46] and “inverse adding-doubling” [47], the “diffusion approximation” [48] and the Monte Carlo methodologies. To date, the majority of the developed methodologies have been designed to estimate/calculate the main parameters, which include anisotropy, the scattering coefficient and the absorption coefficient.

2. Theoretical Concepts for Optical Properties in Biological Studies

2.1. Scattering

Scattering describes the direction change of an electromagnetic wave, which can be divided into two main categories: elastic (Rayleigh scattering) and inelastic (Stokes–Raman and Anti-Stokes–Raman) scattering. In elastic scattering, there is no change in energy, where λ_in = λ_out (λ: wavelength). In the case of inelastic scattering, energy is absorbed from the electromagnetic wave directed towards the medium, where λ_in < λ_out.

2.1.1. Rayleigh Scattering

The Mie and Rayleigh theory lies behind the scattering phenomena. The Mie theorem describes the scattering of ideal particles as they pass through a medium [49]. A good approach to the scattering effect is to simulate it with a mixture of ideal particles of different sizes as they penetrate a tissue. Many reports have studied the scattering of light in tissues through the autocorrelation function (autocorrelation functions are used in signal processing; they make up a set of tools for the detection of patterns in noisy signals, and their main principle relies on the correlation of a signal on itself, but with a temporal difference. In other words, it is the correlation of the signal at time t_i with respect to the same signal at time t_i+1), where patterns are detected regarding the change of scattering coefficients in a tissue [50]. To simplify, tissues are known to be heterogeneous bodies, where optical factors change spatially. Different values will be acquired by measuring the changes of the scattering coefficient at different levels of a tissue, which can be auto-correlated with itself to find patterns. Ideally, the best-case scenario would be detecting a periodic or quasi-periodic pattern in a signal, from which we could predict the pattern. Another method for studying the optical properties of a medium comes from the Wiener–Khinchin theorem. The theorem of Wiener–Khinchin states that a wide-sense-stationary random process is spectrally composed, which can be resolved from the variable’s frequencies distribution or the power spectrum of the process. When the optical function′s coefficients are self-correlated, the optical properties present a spectral composition given by the distribution of frequencies [51,52]. Rayleigh energy theorem is also known as Parseval’s theorem [53]. This theorem reports that the energy of a signal in the time-domain can be expressed in terms of the average energy in its frequency components. In other words, Rayleigh’s theorem refers to the scattering of light by tissues, considering particles smaller than one-tenth the wavelength of the light. In contrast, Mie’s theorem refers to particles that are larger and longer in wavelength than the light, and approximates the phenomenon to spheres of both small and large sizes.

2.1.2. Stokes–Raman and Anti-Stokes–Raman Scattering

A simultaneous exchange of energy and exchange in direction is called Raman scattering. An example of this phenomenon is when a molecule gains energy when photons from a laser in the visible spectrum are shifted to lower energy. This is also known as normal Stokes–Raman scattering [54]. The absorption of a photon by a molecule excites the electron to an imaginary state where, in the case of Raman scattering, the electron falls to a higher energy level, while in Anti-Stokes–Raman scattering this energy level is lower, as compared to the initial ground-state. This phenomenon can be described in terms of wavenumbers (a wavenumber is the spatial frequency of a wave). Thus, if ${\tilde{v}}_o$ is the wavenumber of the photon source and ${\tilde{v}}_e$ is the wavenumber of the electron transition, then the Stokes–Raman scattering is equal to

$${\tilde{v}}_o-{\tilde{v}}_e({\mathrm{nm}}^{-1})$$

(1)

while the Anti-Stokes–Raman scattering is equal to

$${\tilde{v}}_o+{\tilde{v}}_e({\mathrm{nm}}^{-1})$$

(2)

Raman scattering has found application in numerous experimental procedures, to which we will refer further on in a dedicated section.

In Figure 2 we summarize the properties of the scattering phenomena.

2.1.3. Scattering Coefficient

The scattering coefficient is a measure of the ability of particles to scatter photons. When an electromagnetic wave (mostly light, which we will refer to) passes through a medium with particles of size n, the wave is scattered. Each material has a specific scatter coefficient (μ_S), which is tissue- and wavelength-dependent and is related to the size of the particles (tissue molecules). If σ_S (m²) is the scattering cross-section, then:

$${\sigma }_S=Q{A}_S({\mathrm{m}}^2)$$

(3)

where A_S (m²) is the cross section of the tissue’s part that scatters light and Q is the fractional scattering efficiency. Furthermore, the scattering coefficient (μ_s cm⁻¹) is related to the particle (tissue’s molecules) density and cross-section, as:

$${\mu }_S={\rho }_S{\sigma }_S({\mathrm{cm}}^{-1})$$

(4)

where ρ_S is the density of the scattering particles per volume (cm⁻³).

In Rayleigh scattering, it is hypothesized that the scattering is associated with the interaction between photons and the electron cloud of an atom, while in Mie scattering, the particle is spherical and bigger than the photon′s wavelength, so the general solution is derived following the same method as in scattering from a dielectric sphere. Mie scattering is not the same in all directions, and is thus called anisotropic. Anisotropic scattering shows an uneven distribution in the scatter directions. The distribution changes with particle sizes as well as the angular intensity depending on the dielectric constants of the medium.

2.1.4. Reduced Scattering Coefficient

Another scattering coefficient is the reduced scattering coefficient (${\mu }_S^{\prime }$ cm⁻¹). The reduced scattering coefficient is defined as:

$${\mu }_S^{\prime }=\alpha {\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-b}({\mathrm{cm}}^{-1})$$

(5)

where λ is the wavelength, α is the value of ${\mu }_S^{\prime }$ at 500 nm and b is the scattering power. Basically, the reduced scattering coefficient is a measure that incorporates the scattering coefficient and the anisotropy (γ):

$${\mu }_S^{\prime }={\mu }_S(1-\gamma )({\mathrm{cm}}^{-1})$$

(6)

where γ is the anisotropy with units of cm⁻¹. This coefficient describes the path of a photon undergoing a random walk of step size $\frac{1}{{\mu }_S^{\prime }}$ (cm), where each step involves isotropic scattering. As a matter of fact, Equation (4) introduces the connection between the reduced scattering coefficient and anisotropy. Such a description is equivalent to a photon moving with many small steps $\frac{1}{{\mu }_S^{\prime }}$, with only a partial deflection angle θ if there are many scattering events before an absorption event (i.e., ${\mu }_A<<{\mu }_S^{\prime }$). This situation of scattering-dominated light transport is called the diffusion regime, and ${\mu }_S^{\prime }$ is useful when visible and near-infrared light propagates through biological tissues. In its expanded form, the reduced scattering coefficient can be written as:

$${\mu }_S^{\prime }(\lambda )={\alpha }^{\prime }\left(f_{Ray}{\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-4}+(1-f_{Ray}){\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-b_{Mie}}\right)({\mathrm{cm}}^{-1}\mathrm{nm})$$

(7)

This coefficient has been extensively studied and measured in various tissues. Examples of such measurements include the evaluation of the reduced scattering coefficient in brain, bone, fibrous, adipose and breast tissues [50,55,56,57,58,59,60,61,62,63,64,65]. Results of these measurements are summarized in an extensive review by Jacque (2013), and are presented in

Table 2. Factors that affect scattering coefficient of tissues ($a={\mu }_{S,500\mathrm{nm}}^{\prime }$, such as ${\mu }_S^{\prime }(\lambda )=\alpha {\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-b}$ and ${\mu }_S^{\prime }(\lambda )={\mu }_{S,500\mathrm{nm}}^{\prime }{f}_{Rayleigh}{\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-b}+f_{Mei}{\left(\frac{\lambda }{500\mathrm{nm}}\right)}^{-b}$) [50,55,56,57,58,59,60,61,62,63,64,65] (adapted from Jacques (2013), Jacques, S.L. Optical properties of biological tissues: A review. Phys. Med. Biol. 2013, 58, R37–R61. Doi 10.1088/0031-9155/58/11/R37 [40]. Published after the license © Institute of Physics and Engineering in Medicine. Reproduced by permission of IOP Publishing. All rights reserved).

#	a (cm−1)	b	a′ (cm−1)	fRay	bMic	Ref.	Tissue
Skin
1	48.9	1.548	45.6	0.22	1.184	Skin	Anderson et al., 1982 [66]
2	47.8	2.453	42.9	0.76	0.351	Skin	Jacques 1996 [67]
3	37.2	1.390	42.6	0.40	0.919	Skin	Simpson et al., 1998 [68]
4	60.1	1.722	58.3	0.31	0.991	Skin	Saidi et al., 1995 [61]
5	29.7	0.705	36.4	0.48	0.220	Skin	Bashkatov et al., 2011 [69]
6	45.3	1.292	43.6	0.41	0.562	Dermis	Salomatina et al., 2006 [62]
7	68.7	1.161	66.7	0.29	0.689	Epidermis	Salomatina et al., 2006 [62]
8	30.6	1.100	na	na	na	Skin	Alexandrakis et al., 2005 [70]
Brain
9	40.8	3.089	40.8	0.00	3.088	Brain	Sandell and Zhu 2011 [42]
10	10.9	0.334	13.3	0.36	0.000	Cortex (frontal lobe)	Bevilacqua et al., 2000 [55]
11	11.6	0.601	15.7	0.53	0.000	Cortex (temporal lobe)	Bevilacqua et al., 2000 [55]
12	20.0	1.629	29.1	0.81	0.000	Astrocytoma of optic nerve	Bevilacqua et al., 2000 [55]
13	25.9	1.156	25.9	0.00	1.156	Normal optic nerve	Bevilacqua et al., 2000 [55]
14	21.5	1.629	31.0	0.82	0.000	Cerebellar white matter	Bevilacqua et al., 2000 [55]
15	41.8	3.254	41.8	0.00	3.254	Medulloblastoma	Bevilacqua et al., 2000 [55]
16	21.4	1.200	21.4	0.00	1.200	Brain	Yi and Backman 2012 [50]
Breast
17	31.8	2.741	31.8	0.00	2.741	Breast	Sandell and Zhu 2011 [42]
18	11.5	0.775	15.2	0.58	0.000	Breast	Sandell and Zhu 2011 [42]
19	24.8	1.544	24.8	0.00	1.544	Breast	Sandell and Zhu 2011 [42]
20	20.1	1.054	20.2	0.18	0.638	Breast	Sandell and Zhu 2011 [42]
21	14.6	0.410	18.1	0.41	0.000	Breast	Spinelli et al., 2004 [71]
22	12.5	0.837	17.4	0.60	0.076	Breast, premenopausal	Cerussi et al., 2001 [56]
23	8.3	0.617	11.2	0.54	0.009	Breast, postmenopausal	Cerussi et al., 2001 [56]
24	10.5	0.464	10.5	0.00	0.473	Breast	Durduran et al., 2002 [57]
Bone
25	9.5	0.141	9.7	0.04	0.116	Skull	Bevilacqua et al., 2000 [55]
26	20.9	0.537	20.9	0.00	0.537	Skull	Firbank et al., 1993 [58]
27	38.4	1.470	na	na	na	Bone	Alexandrakis et al., 2005 [70]
Other soft tissues
28	9.0	0.617	11.5	0.61	0.000	Liver	Parsa et al., 1989 [60]
29	13.0	0.926	13.0	0.00	0.926	Muscle	Tromberg 1996 [67]
30	12.2	1.448	13.0	0.44	0.731	Fibroadenoma breast	Peters et al., 1990 [72]
31	18.8	1.620	18.8	0.00	1.620	Mucous tissue	Bashkatov et al., 2011 [69]
32	28.1	1.507	27.7	0.23	1.165	SCC	Salomatina et al., 2006 [62]
33	42.8	1.563	42.5	0.10	1.433	Infiltrative BCC	Salomatina et al., 2006 [62]
34	31.9	1.371	31.5	0.15	1.157	Nodular BCC	Salomatina et al., 2006 [62]
35	16.5	1.240	na	na	na	Bowel	Alexandrakis et al., 2005 [70]
36	14.6	1.430	na	na	na	Heart wall	Alexandrakis et al., 2005 [70]
37	35.1	1.510	na	na	na	Kidneys	Alexandrakis et al., 2005 [70]
38	9.2	1.050	na	na	na	Liver and spleen	Alexandrakis et al., 2005 [70]
39	25.4	0.530	na	na	na	Lung	Alexandrakis et al., 2005 [70]
40	9.8	2.820	na	na	na	Muscle	Alexandrakis et al., 2005 [70]
41	19.1	0.970	na	na	na	Stomach wall	Alexandrakis et al., 2005 [70]
42	22.0	0.660	na	na	na	Whole blood	Alexandrakis et al., 2005 [70]
43	16.5	1.640	16.5	0.00	1.640	Liver	Yi and Backman 2012 [50]
44	8.1	0.980	8.1	0.00	0.980	Lung	Yi and Backman 2012 [50]
45	8.3	1.260	8.3	0.00	1.260	Heart	Yi and Backman 2012 [50]
Other fibrous tissues
46	33.6	1.712	37.3	0.72	0.000	Tumor	Sandell and Zhu 2011 [42]
47	30.1	1.549	30.1	0.02	1.521	Prostate	Newman and Jacques 1991 [73]
48	27.2	1.768	29.7	0.61	0.585	Glandular breast	Peters et al., 1990 [72]
49	24.1	1.618	25.8	0.49	0.784	Fibrocystic breast	Peters et al., 1990 [72]
50	20.7	1.487	22.8	0.60	0.327	Carcinoma breast	Peters et al., 1990 [72]
Fatty tissue
51	13.7	0.385	14.7	0.16	0.250	Subcutaneous fat	Simpson et al., 1998 [68]
52	10.6	0.520	11.2	0.29	0.089	Adipose breast	Peters et al., 1990 [72]
53	15.4	0.680	15.4	0.00	0.680	Subcutaneous adipose	Bashkatov et al., 2011 [69]
54	35.2	0.988	34.2	0.26	0.567	Subcutaneous fat	Salomatina et al., 2006 [62]
55	21.6	0.930	21.1	0.17	0.651	Subcutaneous adipocytes	Salomatina et al., 2006 [62]
56	14.1	0.530	na	na	na	Adipose	Alexandrakis et al., 2005 [70]

.

2.1.5. Scattering and Anisotropy

Measuring the scattering coefficient is a difficult task. One way is to estimate the transmission of light (T) through a medium, such as:

$${\mu }_S=-\frac{\ln (T_c)}{L}({\mathrm{cm}}^{-1})$$

(8)

where T_C is the transmission of light without any scattering and L is the length of the path. The problem with this measurement is that the step of transmission must not exceed the step of the random walk (1/μ_S), which is estimated at approximately 100 nm. Although there are several technical restrictions when measuring the anisotropy factor, it has been estimated in various tissues [74,75]. For example, a single photon passing through a medium scatters and deflects with angle θ compared to the initial path. It can be proven that the scattering function p(θ) is a probability of scattering:

$${\displaystyle \underset{0}{\overset{\pi }{\int }}p(\theta )2\pi \sin (\theta )d\theta =1}$$

(9)

where the shape of the scatter function depends on the medium (this would be the tissue in our case). Isotropic scattering shows a constant p(θ):

$$p(\theta )=\frac{1}{4\pi }(°)({\mathrm{sr}}^{-1})$$

(10)

Thus, the anisotropy factor (γ) can be defined as:

$$\gamma =\langle \cos (\theta )\rangle ={\displaystyle \underset{0}{\overset{\pi }{\int }}p(\theta )2\pi \cos (\theta )2\pi \sin (\theta )d\theta }(\mathrm{arbitrary}\mathrm{units}(\mathrm{a}.\mathrm{u}.))$$

(11)

where γ is the average over cos(θ) of scattered photons in all directions. Therefore, scattering is isotropic when γ = 0, light scatters forward when γ = 1, and light totally scatters reversely when γ = −1. However, in almost all tissues, the aforementioned values of γ deviate from those encountered. In real tissues, it is −1 < γ < 1. In real-life measurements, tissues follow the Henyey–Greenstein function, which is described as:

$$p(\theta )=\frac{1}{4\pi }\frac{1-{\gamma }^2}{4\pi {\left(1+{\gamma }^2-2\gamma \cos (\theta )\right)}^{\frac{3}{2}}}(s{r}^{-1})$$

(12)

Interestingly, this function was derived from a study on scattering in interstellar clouds.

2.2. Refraction

Refraction is the phenomenon by which wave particles (and in particular, light) change their trajectories and velocity after entering a medium (Figure 3).

The incidence angle is related to the refraction angle by Snell’s theorem:

$$n_1\sin {\theta }_1=n_2\sin {\theta }_2$$

(13)

where n is the refraction index and θ is the respective angle. The refraction index is a complex coefficient ($n\in C$), which can be described as:

$$n=n^{\prime }+n^{″}(\mathrm{a}.\mathrm{u}.)$$

(14)

where n′ is the real part representing the deposition of energy, and therefore affects the velocity of light when it passes through a medium; n″ is the imaginary part describing the energy scattering, and it defines the absorption coefficient as:

$${\mu }_A=\frac{4\pi n^{″}}{\lambda }({\mathrm{cm}}^{-1})$$

(15)

where μ_A is the absorption coefficient.

2.3. Absorption

2.3.1. Definition and Mathematical Terms

The absorption coefficient is probably the most interesting and most studied factor in tissue’s optical properties. This is because the properties of a biological material are closely connected to its absorption coefficient. Additionally, absorption is linked to the diagnostic and therapeutic properties of light in tissues. The absorption coefficient μ_A (cm⁻¹) is described as:

$${\mu }_A=\frac{1}{{\rm T}}\frac{\partial {\rm T}}{\partial L}({\mathrm{cm}}^{-1})$$

(16)

where T is the transmitted light after a path of length L. The ratio $\frac{\partial {\rm T}}{T}$ represents the exponential reduction of transmitted light with increasing L ($\partial L$). It is proven that:

$$T=e^{-{\mu }_{A}L}={10}^{-\epsilon CL}=e^{\frac{-4\pi n^{″}L}{\lambda }}\iff T=e^{\frac{-4\pi n^{″}L}{\lambda }}\left(\mathrm{arbitrary}\mathrm{units}\right(\mathrm{a}.\mathrm{u}.\left)\right)$$

(17)

where C is the concentration of a molecule or substance (mol/lt) and ε is the extinction coefficient of the material.

In optical theory, light transmission T is defined as:

$$T=e^{\frac{-4\pi n^{″}L}{\lambda }}(\mathrm{a}.\mathrm{u}.)$$

(18)

where n is the complex number of the refraction index. Then, we can derive:

$${\mu }_A=\ln (10){\displaystyle \sum _i{C}_i{\epsilon }_i}\left({\mathrm{cm}}^{-1}\right)$$

(19)

In other words, the absorption coefficient is equal to the sum of the products of the concentration of different molecules (C_i) multiplied by the extinction coefficient of each molecule (ε_i). These equations have been used for the characterization of biological samples based on their absorption properties.

2.3.2. Water

One of the most interesting examples is water. The diffraction coefficient of water at 970 nm has been calculated to be n = 3.47 × 10⁻⁶. The absorption coefficient for water at 970 nm has been calculated to be μ_A = 4πn″/λ = 0.45 cm⁻¹. If the percentage of water in a tissue is f_H2O = 0.65, the absorption coefficient of water at 970 nm is μ_A = f_H2Oμ_A,H2O = 0.65 × 0.45 = 0.29 cm⁻¹. It seems that the role of water in tissue absorption properties is not negligible, and it should be taken into account [40,41]. In particular, in vivo, water (as well as blood in humans and mammals) dominates the absorption spectrum of tissues with respect to light or other electromagnetic radiations [76,77,78].

2.3.3. Nucleotides

The properties of nucleotide solutions comprise a main topic in molecular biology. In an in vitro system, nucleotides are easily detected and measured, as the absorption properties have been extensively studied. Nucleic acids, either single-stranded (RNA) (which is actually no longer the case, since RNA is not considered anymore to exist in “strands”, and is known to take three-dimensional conformations as well as circular conformations) or double-stranded (DNA), are known to absorb light at 260 nm and 280 nm, respectively. Nucleic acid absorption follows the theorem of Beer–Lambert. In particular, the absorption of light at 260 nm is calculated as:

$$A_{260\mathrm{nm}}=\log \left(\frac{I_0}{I}\right)\left(\mathrm{Au}\right)$$

(20)

where I₀ is the intensity of the incident light and I is the intensity of the emitted light. The Beer–Lambert theorem connects the absorption to the concentration:

$$A_{260\mathrm{nm}}={\epsilon }_{260\mathrm{nm}}CL\left(\mathrm{Au}\right)$$

(21)

where C is the nucleotide concentration under investigation, L is the length of the path and ε_260nm is the extinction coefficient of the medium. From Equation (18), we can derive that the concentration of a nucleotide is:

$$C=\frac{A_{260\mathrm{nm}}}{{\epsilon }_{260\mathrm{nm}}L}(\mathrm{moles}/\mathrm{lt})$$

(22)

The principle of this measurement (in vitro) is illustrated in Figure 4.

Determining the concentration of nucleotides in a solution is important to a variety of experimental procedures in biology. For example, RNA plays an essential role in the regulatory paths of biological entities, but is considered to be one of the more difficult molecules to keep intact (due to the abundant presence of RNases). Measuring RNA concentration after extraction from biological samples can be accomplished by spectrophotometry. In Figure 5, the absorbance spectra of the five nucleotides are presented (adenine (A), cytosine (C), thymine (T), guanine (G) and uracil (U) (Figure 6)), reproduced with the help of the PhotoChem CAD software [79,80,81]. Moreover, Figure 7 presents in-house spectrophotometric measurements of total RNA (unpublished data).

2.3.4. Hemoglobin and Blood

One of the most extensively studied molecules is hemoglobin [82,83,84]. Hemoglobin is abundant and easily studied since it is in the circulatory system of every mammal (even lower organisms have hemoglobin, which transfers oxygen to their tissues). As a consequence, the optical properties of blood have also been studied for diagnosis. Blood poses a problem, since it can be considered as a complex, heterogeneous solution whose constitution changes with time. The diagnostic value of blood is of paramount importance, since it can secondarily reflect the pathogenesis of other organs. For example, one can derive measurements from the evaluation of neurotransmitters in blood, which is difficult to perform directly in the brain.

2.3.5. Melanosomes, Melanocytes and Melanin

Melanosomes are the organelles that are responsible for the absorption of UV light. They consist of special organelles, since they constitute the main defense against UV radiation. Melanosomes are found in melanocytes and the pigment that facilitates UV absorption of melanin. Several studies investigated melanocytes and their absorption spectra, and showed that melanocytes absorb light mostly at short wavelengths and reduce absorption at longer wavelengths [85]. The exact mechanisms of melanogenesis are still unknown, and are a subject of intensive investigation [86]. The reason is that melanoma, one of the most aggressive tumors known, is derived from melanocytes. Recent works have highlighted that the regulation of melanocyte and UV protection are mainly controlled by the OPN3 gene, which is calcium-dependent and is activated through the CAMKII and CREB transcription factors [87].

Light has been extensively used as a tool in the study and diagnosis of melanoma. There are several available techniques and methods, and the non-invasive ones, such as coherence tomography, digital dermoscopy and reflectance confocal microscopy, are especially influential. Reflectance confocal microscopy is an optical imaging technique that uses a laser diode as a source of coherent monochromatic light, which penetrates the tissue and illuminates a single point [87,88].

2.3.6. Chlorophyll

As in the case of mammalian eukaryotic cells, melanosomes are those organelles that absorb light, while mesosomes in bacteria and plant chloroplasts contain chlorophyll. Chlorophylls are the main reason why we perceive leaves as green, since they absorb light in the blue and red spectra and reflect green [89]. Two types of chlorophyll exist in the photosystems of green plants: chlorophyll a and b. Photosynthesis, the major function of chloroplasts and chlorophyll, is a unique procedure in nature, wherein energy is produced directly from light, and is thus one of the first evolutionary processes developed in the early stages of life’s emergence. The exact mechanism of photosynthesis in chloroplasts and chlorophyll is still unknown. The study of light absorbance of chlorophylls has shown that these molecules absorb light with maxima at 460 nm and 650 nm. In Figure 8, the absorbance spectra of chlorophyll a are depicted as reproduced by the PhotoChem CAD software [79,80,81] (Figure 8A), as well as from in-house experimental data (unpublished data) (Figure 8B), where chlorophyll a (Figure 9) was extracted using methanol and diethyl ether.

2.3.7. Adipose Tissue

In contrast to melanocytes, adipose tissues possess diverse absorbance spectra, yet all have been found to converge at 930 nm [90]. In different tissues, for example, water, blood and adipose tissue, the measurements could be separated significantly at wavelengths between 400 nm and 700 nm, yet all converge at wavelengths larger than 900 nm [90]. It is noteworthy that many different biological tissues show similar absorption spectra, which are difficult to discriminate, at least at the visible and hyper-visible spectra [40].

2.4. Optical Properties in Non-Spherical Particles: The Case of Real Life

All the aforementioned theoretical concepts related to light propagation through a medium make a significant assumption: particles are homogeneous. On the other hand, real life examples are full of cases where the scattering and propagation of light takes place through non-spherical, non-ideal and non-isotropic media. For example, the remote sensing of the Earth’s and planetary atmospheres, which relies largely on analysis of the parameters of radiation scattered by aerosols, clouds, and precipitation [91]. Thus, many natural and artificial particles have “nonspherical overall shapes or lack a spherically symmetric internal structure” [91]. The first challenges were posed by astrophysics, where the analysis of propagated light through anisotropic media was dominant. Further on, the same problem appeared in the biological sciences, since no biological system can be considered homogeneous.

The problem posed by non-homogeneous shapes (and particles) can be solved numerically, yet several proposals have been made for scattering through non-spherical shapes, yet it is proposed that these properties can be measured through experimentation or computation. Several methods have also been proposed for the computation of optical properties in “real-life” shapes, some of which include the Separation of Variables Method (SVM) [92], the Finite Element Method (FEM) [93,94], the Finite Difference Time Domain Method (FDTD) [95,96], the Point Matching Method (PMM) [97], the Integral Equations Method (IEM) [98,99], the Discrete Dipole Approximation (DDA) [100,101], the Fredholm Integral Equation Method (FIE) [102,103,104], the T-Matrix Approach [105] and the Superposition Method for Compounded Spheres and Spheroids (SMCSS) [91]. However, no matter the modeling method, there is a common denominator in all of them: the space and time derivatives of the electric and magnetic fields are approximated using finite difference schemes with space and time “discretizations”. These “discretizations” are selected in such ways that minimize computation errors and “ensure numerical stability” of the algorithm [91]. In other words, instead of taking the complete shape through which light scatters, they consider small infinitesimal approximations.

3. Methods for Studying the Optical Properties of Tissues: The Case of Monte Carlo Simulations

3.1. The Monte Carlo Simulation: A Short History

Monte Carlo (the term was coined in the Los Alamos National Laboratory, in the late 1940s) methodology is a stochastic method, which has been developed for the solution of problems that were not able to be solved otherwise. In other words, this method attempts to solve deterministic problems by probability. The term was borrowed from the city of Monte Carlo, due to the infamous casino. The first usage of the method was in the 18th century by Georges-Louis Leclerc, Comte de Buffon, who formulated a simple and interesting question: in a wooden floor, where the wooden parts leave small gaps between them, a needle is left to fall from height h; what is the probability that the needle will either stay on the wooden board or fall between the narrow gaps [106]? Later on, Enrico Fermi experimented with the Monte Carlo method by studying neutron beam transmission. A few years later, Stanislav Ulam and Nicholas Metropolis worked in the “Manhattan” project and formulated the latest version of the Monte Carlo method. After the work of Ulam and Metropolis, John von Neumann realized the significance of the method and developed an algorithm for Monte Carlo simulations using the EVAC computer.

3.2. Numerical Solution of the Monte Carlo Method

This method consists of the most significant tools in the “Manhattan” project, despite the low processing power at the time. The Monte Carlo method is based on the massive collection of data and their subsequent analysis. This is due to the fact that the method requires the use of pseudo-random numbers in order to calculate the expected results. One of the first applications of the method was the estimation of pi (π). A square with side A and a circle with radius (R), where R = A/2, are drawn. Both shapes are filled with an immensely large number of dots, and then they are superimposed. The square surface is E_square = A² and the circle surface is E_circle = πR². Taking the ratio of the two surfaces we get:

$$\frac{E_{circle}}{E_{square}}=\frac{A^2}{\pi R^2}\left(\mathrm{dimensionless}\right)$$

(23)

However, since both shapes are filled with a large amount of dots, their number approximates the surface of each shape. By counting the number of dots, we can calculate the ratio, which will give us the approximation of π when multiplied by 4. We have performed a small experiment using the Microsoft Excel^® software and 10,000 dots (Figure 10). We were able to approximate π as 3.1408 (the Monte Carlo simulation, in some cases, can be very time-demanding; in the present case, the calculation of π required 1.2 s in an i7-8core, 24 GB RAM desktop computer). The Monte Carlo method can be applied to a series of disciplines, including microscopy.

One of the important variables in the Monte Carlo simulation in tissues is the thickness (d) of the tissue. Ideally, a tissue should have a depth equal to the photon′s step (that would be 1/μ_S). Accordingly, tissue’s geometry is of great importance [107]. The final step is to determinate the optical coefficients μ_A, μ_S, γ, and μ_R, where they are all the function of f(x,y,z,λ). These coefficients are calculated by the application of an algorithm, which is presented in Figure 11 [108].

To evaluate the optical coefficients, the first step is the determination of the starting point, where the photon first appears. This can be defined as (x,y,z)_i=1 = (0,0,0) and therefore, (a_x, a_y, a_z)_i=1 = (0,0,1). Then, the step of the photon can be defined as:

$$S=\frac{-\ln (1-RND)}{{\mu }_t}\left(\mathrm{m}\right)$$

(24)

where RND is a random number generator. In addition, the trajectory of the photon can be described as:

$$\left\{\begin{array}{l}{x}_i=x_{i-1}+a_{x,i-1}{S}_{i-1}\\ y_i=y_{i-1}+a_{y,i-1}{S}_{i-1}\\ z_i=z_{i-1}+a_{z,i-1}{S}_{i-1}\end{array}\right\}.$$

(25)

The probability of scattering and/or absorption is described as:

$$P_{absorbance,scatter}=\frac{{\mu }_A}{{\mu }_A+{\mu }_S}(\%)$$

(26)

In the case of photon scattering, the new direction of the photon must be estimated, and this is defined by:

$$\cos \theta =\left(\begin{array}{l}\frac{1}{2\gamma }+\left(1+{\gamma }^2-{\left(\frac{1-{\gamma }^2}{1-\gamma +2\gamma \cdot RND}\right)}^2\right),\gamma \ne 0\\ 1-2\cdot RND,\gamma =0\\ \phi =2\pi \cdot RND\end{array}\right).(°)$$

(27)

This model has been applied to mouse tissues by Ren et al. (2010) [107]. This research group calculated the optical coefficients via light transmission. Their results are summarized in the following table (

Table 3. The calculation of optical coefficients in a mouse simulation (adapted and reproduced from Ren et al. (2010) [107] with permission from ©The Optical Society).

Tissue	µa (1/mm)	µs (1/mm)	g	n	Number of Triangle Meshes
Adipose	0.005045	20.4545	0.94	1.35	2000–100,000
Skeleton	0.08138	26.0896	0.9	1.50	25,000
Heart	0.078594	6.7104	0.85	1.42	3500
Lung	0.262956	36.818	0.94	1.38	6000
Kidney	0.088107	16.846	0.86	1.45	2500
Stomach	0.015044	18.4973	0.92	1.40	4000

).

The Monte Carlo approximation is an excellent tool for the study of the optical properties of biological samples. The challenge for the study of biological systems comes from their heterogeneity and dynamic evolution. One of the basic requirements for a Monte Carlo simulation is that the properties of the tissue do not change over time. This is not the case for in vivo systems, and it is a fact that complicates the prediction of tissue properties. It is very likely that such approaches will help us to further understand the physiology of biological systems. The ultimate goal of such research would be to understand not only the physiology at a given time, but also the dynamic changes that a tissue undergoes, as well as the ability to predict those changes. It could also provide excellent evidence for the transition of a tissue or cellular system from a healthy physiological status to pathophysiology. Approximately 30–40 years ago, such simulations were very difficult to perform, since computation capacity was not available in abundance (even 20 years ago), and even if available, it was extremely expensive and inaccessible by small research groups.

4. Conclusions

Since the invention of microscopy, it has become apparent that biological samples have optical properties. These properties begin from the fact that biological samples absorb and emit specific wavelengths of light, and the absorption and emission are functions of their homeostasis at a certain time point. The optical properties of biological samples play important roles in both diagnosis and therapy. The ability of light to penetrate through a tissue and to deposit or absorb energy is a key factor to be used as a diagnostic or therapeutic tool. Therefore, the first step in designing devices or technologies in this direction is to understand the properties of biological samples by determining their optical properties. Then, by applying light to the samples, we can predict the distribution of light, and the distribution of the emission or absorption of energy, based on the properties of biological samples.