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

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## Abstract

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## 1. Introduction

#### 1.1. Applications and Optical Spectra in Biological Samples

_{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.

#### 1.2. Basic Principles of Microscopy and Optical Properties of Tissues

_{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].

#### 1.3. Measuring Tissue’s Optical Properties

## 2. Theoretical Concepts for Optical Properties in Biological Studies

#### 2.1. Scattering

_{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

_{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

#### 2.1.3. Scattering Coefficient

_{S}), which is tissue- and wavelength-dependent and is related to the size of the particles (tissue molecules). If σ

_{S}(m

^{2}) is the scattering cross-section, then:

_{S}(m

^{2}) 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:

_{S}is the density of the scattering particles per volume (cm

^{−3}).

#### 2.1.4. Reduced Scattering Coefficient

^{−1}). The reduced scattering coefficient is defined as:

^{−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:

#### 2.1.5. Scattering and Anisotropy

_{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:

#### 2.2. Refraction

_{A}is the absorption coefficient.

#### 2.3. Absorption

#### 2.3.1. Definition and Mathematical Terms

_{A}(cm

^{−1}) is described as:

_{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

^{−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

_{H}

_{2O}= 0.65, the absorption coefficient of water at 970 nm is μ

_{A}= f

_{H}

_{2O}μ

_{A,H}

_{2O}= 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

_{0}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:

_{260}

_{nm}is the extinction coefficient of the medium. From Equation (18), we can derive that the concentration of a nucleotide is:

#### 2.3.4. Hemoglobin and Blood

#### 2.3.5. Melanosomes, Melanocytes and Melanin

#### 2.3.6. Chlorophyll

#### 2.3.7. Adipose Tissue

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

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

#### 3.1. The Monte Carlo Simulation: A Short History

#### 3.2. Numerical Solution of the Monte Carlo Method

_{square}= A

^{2}and the circle surface is E

_{circle}= πR

^{2}. Taking the ratio of the two surfaces we get:

^{®}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.

_{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].

_{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:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Diagrammatical summary of the approaches toward the determination of biological samples’ optical properties.

**Figure 2.**Diagrammatical representation of the scattering variances described above. The different excitation processes are presented, i.e., that of Rayleigh scattering, Stokes–Raman scattering and Anti-Stokes–Raman scattering (adapted and reproduced from https://en.wikipedia.org/wiki/Raman_scattering, accessed 6 November 2021, under the CC BY-SA 3.0 license).

**Figure 3.**Diagrammatical representation of the refraction of light. A beam of light passes through a medium with refraction index n

_{1}, to a new medium with refraction index n

_{2}. The angle of incidence at point (0,0) is θ

_{1}, while the angle of refraction is θ

_{2}(adapted and reproduced from https://en.wikipedia.org/wiki/Refraction, accessed 6 May 2021).

**Figure 4.**Simplified representation of Beer–Lambert theorem principle and measurement of nucleotides with respect to their absorption—incoming light (hv

_{1}) with intensity I

_{0}, travels through a path (L), which goes through a solution with concentration (C) and extinction coefficient (ε), and exits the solution (hv

_{2}) with intensity I

_{1}.

**Figure 5.**The absorption spectra of the five nucleotides adenine (A), thymine (T), guanine (G), cytosine (C) and uracil (U).

**Figure 6.**The molecular structures, both in 2D (

**left**) and 3D (

**right**), of an RNA chain consisting of adenosine (A), guanosine (G), uridine (U) and cytidine (C).

**Figure 7.**Two in-house evaluations of RNA absorbance with respect to the wavelength (

**A**) and absorbance with respect to RNA concentration (

**B**).

**Figure 8.**The absorption spectra of chlorophyll a extracted with diethyl ether, as reproduced by PhotoChem CAD (

**A**) and in-house experimental spectrophotometric data (unpublished data) (

**B**).

**Figure 10.**One of the first applications of the Monte Carlo method was the calculation of pi. Two shapes, a square and a circle, are filled with random dots and superimposed. It is proven that the ratio of the two surfaces multiplied by 4 approximates the number pi.

**Figure 11.**The simplified algorithm of a Monte Carlo simulation for a photon passing through a biological medium (adopted and reproduced from Zhu and Liu (2013), Zhu, C.; Liu, Q. Review of Monte Carlo mod-eling of light transport in tissues. J. Biomed. Opt.

**2013**, 18, 50902, DOI:10.1117/1.JBO.18.5.050902 [108], under the Creative Commons CC BY 4.0 license https://creativecommons.org/licenses/by/4.0/legalcode).

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

Coefficient | Symbol | Units |
---|---|---|

Absorption | μ_{A} | cm^{−1} |

Scattering | μ_{S} | 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} |

**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}) | f_{Ray} | b_{Mic} | 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] |

**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 | µ_{a} (1/mm) | µ_{s} (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 |

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Lambrou, G.I.; Tagka, A.; Kotoulas, A.; Chatziioannou, A.; Matsopoulos, G.K.
Physical and Methodological Perspectives on the Optical Properties of Biological Samples: A Review. *Photonics* **2021**, *8*, 540.
https://doi.org/10.3390/photonics8120540

**AMA Style**

Lambrou GI, Tagka A, Kotoulas A, Chatziioannou A, Matsopoulos GK.
Physical and Methodological Perspectives on the Optical Properties of Biological Samples: A Review. *Photonics*. 2021; 8(12):540.
https://doi.org/10.3390/photonics8120540

**Chicago/Turabian Style**

Lambrou, George I., Anna Tagka, Athanasios Kotoulas, Argyro Chatziioannou, and George K. Matsopoulos.
2021. "Physical and Methodological Perspectives on the Optical Properties of Biological Samples: A Review" *Photonics* 8, no. 12: 540.
https://doi.org/10.3390/photonics8120540