Color Science and Its Application in Dentistry

Abstract

This first chapter presents the basic principles of color science and their application to dentistry. It includes color coordinates, color differences, whiteness indexes, and the most relevant optical properties such as transmittance, absorption, scattering, fluorescence, translucency, and opacity. Measuring and evaluating methods are described and relevant literature and standards are reported. Therefore, this chapter is essential to understand color science and optical properties, assisting any dental professional in performing and interpreting color science studies, improving the practice of esthetic dentistry by applying such knowledge to design and manufacture restorative material systems and, finally, to restore the missing dental structures.

1.1 Color Measurements and Whiteness Indexes

A common ultimate goal of color measurement or shade specification in dentistry is the reproduction of important appearance characteristics of oral structures by prosthetic materials. Within the dental clinical setting, whenever an indirect restoration is planned for an area that is readily observed and the restoration would be easily assessed for harmony to adjacent existing natural structure, it would be ideal to quantify valid color information fast and reliably using the patient’s existing natural structure, which has characteristics throughout the visible spectrum. Such information is used to facilitate the complex color scenario that comprise the restoration, which can have similar colors under various illumination conditions, within at least acceptable limits but, more preferably, within limits of perceived color difference.

Color measurements in dentistry often involve nonhomogeneous layers of both natural and prosthetic materials with varying inherent color and translucency. In addition, teeth often present interesting texture and curved surface. By contrast, color and reflectance standards are flat and opaque, mainly due to the complicating effects of both translucency and non-planar surfaces on color measurement.

1.1.1 CIE Standard Observers

The color of an object depends on its spectral reflectance, the spectral power distribution of the light source, and, finally, the observer. When performing a color measurement it is important to standardize all the observation/measurement conditions in order to obtain reproducible and trustful results that can be comparable with previous studies.

1.1.1.1 CIE 1931 Standard Colorimetric Observer

The CIE 1931 standard colorimetric observer is defined for a 2° foveal field of observation (2° field corresponds to an area of the macula lutea that has a practically constant density of photoreceptors) and a dark surround.

The CIE 1931 2° observer was derived from results of two investigations on color matching of test stimuli with mixtures of three RGB primary stimuli, conducted by WD Wright and J Guild. The results of these studies led to the first $\overline{r}\left(\lambda \right),\overline{g}\left(\lambda \right),\overline{b}\left(\lambda \right)$ color-matching functions (CMF), which characterize the CIE 1931 2° standard observer. However, these first matching functions presented negative lobes, which referred to the fact that in some parts of the spectrum a match can be obtained only if one of the matching stimuli is added to the test stimulus.

The negative lobes present in the CMFs made calculations difficult and, therefore, in 1931, the International Commission on Illumination decided to transform from the real RGB primaries to a set of imaginary primaries X Y Z, with the difference that, in this latter case, the new CMFs that characterize the observer do not have negative lobes. In addition, an equienergy stimulus was characterized by equal tristimulus values (X = Y = Z), one of the tristimulus provide photometric quantities and the volume of the tetrad on set by the new primaries should be as small as possible.

Therefore, X Y Z tristimuli values are calculated as the integral over the visible spectrum (380 nm–780 nm):

$$ X=k\underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right)\overline{x}\left(\lambda \right)\mathrm{d}\lambda $$

$$ Y=k\underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right)\overline{y}\left(\lambda \right)\mathrm{d}\lambda $$

$$ Z=k\underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right)\overline{z}\left(\lambda \right)\mathrm{d}\lambda $$

where k is a constant; ϕ(λ) is the color stimulus function of the light seen by the observer and $\overline{x}\left(\lambda \right),\overline{y}\left(\lambda \right),\overline{z}\left(\lambda \right)$ are the CMF of the CIE 1931 standard colorimetric observer.

However, according to the CIE recommendations, the X Y Z tristimuli values can be calculated as numerical summation at wavelength intervals (Δλ) by applying the following equations:

$$ X=k\sum \limits_k{\phi}_{\lambda}\left(\lambda \right)\overline{x}\left(\lambda \right)\varDelta \lambda $$

$$ Y=k\sum \limits_k{\phi}_{\lambda}\left(\lambda \right)\overline{y}\left(\lambda \right)\varDelta \lambda $$

$$ Z=k\sum \limits_k{\phi}_{\lambda}\left(\lambda \right)\overline{z}\left(\lambda \right)\varDelta \lambda $$

In the case of a non-self-luminous object, the spectral reflection of the surface is described by the spectral reflectance factor R(λ) and the spectral transmission is described by the spectral transmittance factor T(λ). Taking into account these considerations, the relative color stimulus function ϕ(λ) for reflecting or transmitting objects is given by:

$$ \phi \left(\lambda \right)=R\left(\lambda \right)\cdot S\left(\lambda \right)\ \mathrm{or}\ \phi \left(\lambda \right)=T\left(\lambda \right)\cdot S\left(\lambda .\right) $$

where R(λ) is the spectral reflectance factor; T(λ) is the spectral transmittance factor of the object (preferably evaluated for one of the standard geometric conditions recommended by CIE), and S(λ) is the relative spectral power distribution of the illuminant (preferably, one of CIE standard illuminants).

In this case, the constant k is chosen so that the tristimulus value Y = 100 for objects that have R(λ) = 1 or T(λ) = 1 at all wavelengths:

$$ k=\frac{100}{\sum_{\lambda }S\left(\lambda \right)\cdot \overline{y}\left(\lambda \right)d\left(\lambda \right)} $$

The color-matching functions of the CIE 1931 standard colorimetric observer are given as values from 360 nm to 830 nm at 1 nm intervals [1] (Fig. 1.1). This observer should be used if the field subtended by the sample is between 1° and 4° at the eye of the observer. In technical applications, this observer is often addressed as 2° standard colorimetric observer.

Fig. 1.1

$\overline{x}\left(\lambda \right),\overline{y}\left(\lambda \right),\overline{z}\left(\lambda \right)$ color-matching functions (CMF) of the CIE 1931 standard colorimetric observer (continuous line) and the ${\overline{x}}_{10}\left(\lambda \right),{\overline{y}}_{10}\left(\lambda \right),{\overline{z}}_{10}\left(\lambda \right)$ color-matching functions of the [2] standard colorimetric observer (dotted line)

1.1.1.2 CIE [2] Standard Colorimetric Observer

As specified before, the CIE 1931 2° standard observer is recommended only for small stimuli, which is 1°–4° at the eye of the observer. However, the description of a larger stimulus that falls on a larger area of the macula lutea, or it is seen partially parafoveally, is required. In this sense, the International Commission on Illumination standardized a large-field colorimetric system [2] based on the visual observations conducted on a 10° visual field.

The adoption of a 10° colorimetric observer was based on the works of Stiles and Burch in 1959, which used different sets of monochromatic primaries, and the CMFs were obtained directly from the observations, and no appeal to heterochromatic brightness measurements or to any luminous efficiency function was required.

The color-matching functions of the 10° system are distinguished from the 2° system by a 10 subscript. The color-matching functions of the [2] standard colorimetric observer are given as values from 360 nm to 830 nm at 1 nm intervals [1] (Fig. 1.1).

The tristimulus values are calculated similar to the case of the 2° standard observer:

$$ X={k}_{10}\cdot \underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right){\overline{x}}_{10}\left(\lambda \right)\mathrm{d}\lambda $$

$$ Y={k}_{10}\cdot \underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right){\overline{y}}_{10}\left(\lambda \right)\mathrm{d}\lambda $$

$$ Z={k}_{10}\cdot \underset{380\mathrm{nm}}{\overset{780\mathrm{nm}}{\int }}{\phi}_{\lambda}\left(\lambda \right){\overline{z}}_{10}\left(\lambda \right)\mathrm{d}\lambda $$

and the k₁₀ constant for non-self-luminous objects is defined by the following equation:

$$ {k}_{10}=\frac{100}{\sum_{\lambda }S\left(\lambda \right)\cdot {\overline{y}}_{10}\left(\lambda \right)d\left(\lambda \right)} $$

Apart from the recommendations regarding the field sustained by the sample, there are additional factors that have to be taken into account when using the [2] standard colorimetric observer. The precision of the system is greater than that of the CIE 1931 trichromatic system since it has been determined with a higher number of observers but the larger stimulus area rod intrusion had to be considered as well. Further, in the case of using the 10° observer without any rod-correction, the luminance levels have to be high enough. While in the case of a 2° field one can calculate with photopic adaptation down to about 10 cd/m², this is not the case for the larger field size. In this sense, CIE [1] recommends the following: “The large-field color matching data as defined by the [2] standard colorimetric observer are intended to apply to matches where the luminance and the relative spectral power distributions of the matched stimuli are such that no participation of the rod receptors of the visual mechanism is to be expected.” This condition of observation is important as “rod intrusion” may upset the predictions of the standard observer. For daylight, possible participation of rod vision in color matches is likely to diminish progressively above, approximately, 10 cd/m² and be entirely absent at about 200 cd/m².

1.1.2 CIE Color Space

The color of a stimulus can be specified by a triplet of tristimulus values (X, Y, Z). In order to provide a convenient two-dimensional representation of the color, chromaticity diagrams were developed. In the transformation from tristimulus values to chromaticity coordinates, a normalization that removes luminance information is performed. This transformation is defined by:

$$ x=\frac{X}{X+Y+Z} $$

$$ y=\frac{Y}{X+Y+Z} $$

$$ z=\frac{Z}{X+Y+Z} $$

Chromaticity coordinates attempt to represent a three-dimensional phenomenon with just two variables. To fully specify a colored stimulus, one of the tristimulus values must be reported in addition to two of the chromaticity coordinates. Usually, the Y tristimulus value is reported since it represents the luminance information. Therefore, this color space is often called the CIE Yxy space.

The CIE Yxy space is well suited to describe color stimuli. However, the practical use of colorimetry very often requires information about whether two samples will be indistinguishable by visual observation or not. MacAdam showed that the chromaticity difference that corresponds to a just noticeable color difference will be different in different areas of the xy chromaticity diagram, and also, at one point in the diagram, equal chromaticity differences in distinct directions represent visual color differences of distinct magnitudes [3]. Many attempts were made to transform the xy diagram in such a form that the MacAdam ellipses become circles, but it should be mentioned that there is still no perfect transformation available.

Nowadays, the use of the chromaticity diagrams has become largely obsolete, and CIELAB and CIELUV color spaces are the most frequently used. One of the main characteristics of these two color spaces is that they extend tristimulus colorimetry to three-dimensional spaces with dimensions that approximately correlate with the perceived lightness, chroma, and hue of a colored stimulus. This is achieved by incorporating features to account for chromatic adaptation and nonlinear visual responses. The main aim in the development of these spaces was to provide uniform practices for the measurement of color differences, something that cannot be done reliably in tristimulus or chromaticity spaces. The International Commission on Illumination recommended, in 1976, the use of both spaces, since there was no clear evidence to support one over the other at that time. However, over time, CIELAB has been greater implemented in color applications than the CIELUV color space. This is also true for dental color studies.

1.1.2.1 CIE 1976 (L^∗a^∗b^∗) Color Space—CIELAB

The CIE 1976 (L^∗a^∗b^∗) color space, or simply CIELAB, is defined by the following equations [1]:

$$ {L}^{\ast }=116\cdot f\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)-16 $$

$$ {a}^{\ast }=500\cdot \left[f\left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)-f\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)\right] $$

$$ {b}^{\ast }=200\cdot \left[f\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)-f\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)\right] $$

where

$$ f\left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)={\left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)}^{1/3}\ \mathrm{if}\ \left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)>{\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

$$ f\left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)=\left(\raisebox{1ex}{$841$}\!\left/ \!\raisebox{-1ex}{$108$}\right.\right)\cdot \left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)+\raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\ \mathrm{if}\ \left(\raisebox{1ex}{$X$}\!\left/ \!\raisebox{-1ex}{${X}_n$}\right.\right)\le {\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

and

$$ f\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)={\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)}^{1/3}\ \mathrm{if}\ \left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)>{\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

$$ f\left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)=\left(\raisebox{1ex}{$841$}\!\left/ \!\raisebox{-1ex}{$108$}\right.\right)\cdot \left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)+\raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\ \mathrm{if}\ \left(\raisebox{1ex}{$Y$}\!\left/ \!\raisebox{-1ex}{${Y}_n$}\right.\right)\le {\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

and

$$ f\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)={\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)}^{1/3}\ \mathrm{if}\ \left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)>{\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

$$ f\left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)=\left(\raisebox{1ex}{$841$}\!\left/ \!\raisebox{-1ex}{$108$}\right.\right)\cdot \left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)+\raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\ \mathrm{if}\ \left(\raisebox{1ex}{$Z$}\!\left/ \!\raisebox{-1ex}{${Z}_n$}\right.\right)\le {\left(\raisebox{1ex}{$24$}\!\left/ \!\raisebox{-1ex}{$116$}\right.\right)}^3 $$

In these equations, X, Y, and Z are the tristimulus values of the sample and the X_n, Y_n, and Z_n are the tristimulus values of the reference white. These signals are combined into three response dimensions corresponding to the light–dark, red–green, and yellow–blue responses of the opponent theory of color vision. Finally, appropriate multiplicative constants are incorporated into the equations to provide the required uniform perceptual spacing and proper relationship between the three dimensions.

The CIE L^∗ coordinate is a correlate to perceive lightness ranging from 0, for black, to 100, for a diffuse white (L^∗ can sometimes exceed 100.0 for stimuli such as specular highlights in images). The CIE a^∗ and CIE b^∗ coordinates correlate approximately with red–green and yellow–blue chroma perceptions. They take on both negative and positive values. Both a^∗ and b^∗ have values of 0 for achromatic stimuli (white, gray, black). Their maximum values are limited by the physical properties of materials, rather than the equations themselves. The CIELAB L^∗, a^∗, and b^∗ dimensions are combined as Cartesian coordinates to form a three-dimensional color space (Fig. 1.2).

Fig. 1.2

Schematic representation of the CIELAB color space

In some applications the correlates of the perceived attributes of lightness, chroma, and hue are of more practical interest. In this case, the CIELAB color space is represented using three cylindrical coordinates (lightness—L^∗; chroma—${C}_{ab,}^{\ast }$ and hue angle—h_ab) (Fig. 1.2). The CIE L^∗ coordinate is calculated as described before, while chroma and hue angle are calculated as follows:

$$ {C}_{ab}^{\ast }=\sqrt{\left({a^{\ast}}^2+{b^{\ast}}^2\right)} $$

$$ {h}_{ab}=\arctan \left(\raisebox{1ex}{${b}^{\ast }$}\!\left/ \!\raisebox{-1ex}{${a}^{\ast }$}\right.\right) $$

1.1.3 Color-Difference Formulas

In the CIELAB color space, the color difference between two object color stimuli of the same size and shape, viewed in identical white to middle-gray surroundings, by an observer photopically adapted to a field of chromaticity not too different from that of average daylight, is quantified as the Euclidean distance between the points representing them in the space. This difference is expressed in terms of the CIELAB $\varDelta {E}_{ab}^{\ast }$color-difference formula, and calculated as:

$$ \varDelta {E}_{ab}^{\ast }=\sqrt{\varDelta {L^{\ast}}^2+\varDelta {a^{\ast}}^2+\varDelta {b^{\ast}}^2} $$

The CIELAB $\varDelta {E}_{ab}^{\ast }$ color-difference formula is widely adopted and it was used in multiple fields and applications. It represented a great step forward toward harmonization of color-difference evaluation and also color description in the technical world.

The CIELAB color space was designed with the goal of having color differences being perceptually uniform throughout the space, but this goal was not strictly achieved. Soon after its implementation, the CIELAB $\varDelta {E}_{ab}^{\ast }$ color-difference formula was reported to present inhomogeneities. To improve the uniformity of color-difference measurements, modifications of the CIELAB $\varDelta {E}_{ab}^{\ast }$ equation have been made based upon various empirical data, but in almost all cases a poor correlation between the dataset of visual judgments and the Euclidean distances was found, hence raising the need of equation optimization [4].

Keeping CIELAB coordinate system as a start, several changes were applied to components of the color-difference formula by adapting weighting functions to the three component differences. The Color Measurement Committee (CMC) of the Society of Dyers and Colorists proposed a new color-difference formula denominated CMC(l:c), which became an ISO standard for textile applications in 1995 [5]. Other CIELAB-based formula, named BFD(l:c), which is similar in structure to the CMC(l:c) formula, was proposed by Luo and Rigg, in 1987 [6]. Later, the CIE Technical Committee tried to find an optimization of the CIELAB color-difference formula mainly based on new experiments under well-controlled reference conditions. The resulting recommendation followed the general form of the CMC(l:c) formula and it was called CIE94 color-difference formula ($\varDelta {E}_{94}^{\ast }$) [7]. Luo et al. in 2001 [8], proposed a new color-difference metric named CIEDE2000 color difference, which became a CIE recommendation for color-difference computation in 2004 [1]. The CIEDE2000 formula stands as the last, in a long series, of developments improving the CIELAB formula, and outperforms the older CMC(l:c), BFD(l:c), and CIE94 formulas [5].

CIEDE2000 total color-difference formula [8] corrects for the non-uniformity of the CIELAB color space for small color differences under reference conditions. Improvements to the calculation of total color difference for industrial color-difference evaluation were made through corrections for effects of lightness dependence, chroma dependence, hue dependence, and hue–chroma interaction on perceived color difference. The scaling along the a^∗ axis is modified to correct for a non-uniformity observed with gray colors. The resulting recommendation is as follows [1]:

$$ \varDelta {E}_{00}=\sqrt{{\left(\frac{\varDelta {L}^{\prime }}{K_L{S}_L}\right)}^2+{\left(\frac{\varDelta {C}^{\prime }}{K_C{S}_C}\right)}^2+{\left(\frac{\varDelta {H}^{\prime }}{K_H{S}_H}\right)}^2+{R}_T\left(\frac{\varDelta {C}^{\prime }}{K_C{S}_C}\right)\left(\frac{\varDelta {H}^{\prime }}{K_H{S}_H}\right)} $$

$$ {L}^{\prime }={L}^{\ast } $$

$$ {a}^{\prime }={a}^{\ast}\left(1+G\right) $$

$$ {b}^{\prime }={b}^{\ast } $$

$$ G=0.5\left(1-\sqrt{\frac{{{\overline{C}}_{ab}^{\ast}}^7}{{{\overline{C}}_{ab}^{\ast}}^7+{25}^7}}\right) $$

The weighting functions S_L, S_C, S_H adjust the total color difference for variation in perceived magnitude with variation in the location of the color-difference pair in L′, a′, and b′ coordinates.

$$ {S}_L=1+\frac{0.015{\left({\overline{L}}^{\prime }-50\right)}^2}{\sqrt{20+{\left({\overline{L}}^{\prime }-50\right)}^2}} $$

$$ {S}_C=1+0.045{\overline{C}}^{\prime } $$

$$ {S}_H=1+0.015{\overline{C}}^{\prime }T $$

$$ T=1-0.17\cos \left({\overline{h}}^{\prime }-30\right)+0.24\cos \left(2{\overline{h}}^{\prime}\right)+0.32\cos \left(3{\overline{h}}^{\prime }+6\right)-0.20\cos \left(4{\overline{h}}^{\prime }-63\right) $$

Visual color-difference perception data show an interaction between chroma difference and hue difference in the blue region that is observed as a tilt of the major axis of a color-difference ellipsoid from the direction of constant hue angle. To account for this effect, a rotation function is applied to weighted hue and chroma differences:

$$ {R}_T=-\sin \left(2\varDelta \varTheta \right){R}_C $$

$$ \varDelta \varTheta =30\exp \left\{-{\left[\raisebox{1ex}{$\left({\overline{h}}^{\prime }-275\right)$}\!\left/ \!\raisebox{-1ex}{$25$}\right.\right]}^2\right\} $$

$$ {R}_C=2\sqrt{\frac{{{\overline{C}}^{\prime}}^7}{{{\overline{C}}^{\prime}}^7+{25}^7}} $$

The parametric factors K_L, K_C, K_H, are correction terms for variation in experimental conditions. Under reference conditions they are all set to 1. The reference conditions are defined by the International Commission on Illumination as [1]:

• Illumination: source simulating the spectral relative irradiance of CIE standard illuminant D65;

• Illuminance: 1000 lx;

• Observer: normal color vision;

• Background field: uniform, neutral gray with L^∗ = 50;

• Viewing mode: object;

• Sample size: greater than four degrees subtended visual angle;

• Sample separation: minimum sample separation achieved by placing the sample pair in direct edge contact;

• Sample color-difference magnitude: 0–5 CIELAB units;

• Sample structure: homogeneous color without visually apparent pattern or non-uniformity.

Few years after the CIEDE2000 (ΔE₀₀) color-difference formula was published, comparisons between CIELAB and CIEDE2000 formulas were already available in dental literature [9, 10]. In addition, some studies showed significant correlations between ΔE^∗_ab and ΔE₀₀ values of resin composites [9,10,12], metal-ceramic and all-ceramic restorations [13], and natural tooth color space [14, 15]. The majority of reported correlations showed the values obtained from these formulas are proportional, but the two color-difference formulas cannot be used interchangeably to evaluate the color differences in dentistry.

A recent study [14, 15] determined the relationship between the results provided by classic ΔE^∗_ab and CIEDE2000 formulas in the natural tooth color space using the Bland and Altman approach. The results obtained showed that in the natural tooth color space, the scale factor between both formulas values changes from 0.46 to 0.90, suggesting it is difficult an accurate scale factor between both values. Furthermore, the ΔE₀₀/ΔE^∗_ab ratio increases with the increase in ΔL^∗ and the decrease in Δb^∗. Additionally, CIEDE2000 formula reflected the color differences perceived by the human eye better than the CIELAB formula [13,14,15,17]. It was reported that CIEDE2000 (2:1:1) formula showed the best estimate to visual perception when compared to CIELAB and CIEDE2000 (1:1:1) formulas [16, 17]. CIELAB and CIEDE2000 (2:1:1) formulas were also used to evaluate the influence of gender on visual shade matching. Only CIEDE2000 (2:1:1) formula showed a statistical difference between genders [18].

1.1.4 Whiteness Indexes

In spectral terms, a white material is the one that has a constant and high (near to 100%) reflectance across the entire visible wavelength range. Whiteness is generally considered to be a one-dimensional perception defined by Ganz as an attribute of color of high luminous reflectance and low purity situated in a relatively narrow region of the color space along dominant wavelengths of 570 nm and 470 nm, approximately. In terms of CIELAB color space, this type of spectral behavior is translated into a very high lightness (L^∗) and very low (ideally zero) chroma (C^∗_ab).

For a three-dimensional color space, three color coordinates are necessary for a complete identification of any white. However, a one-dimensional color index can be more efficient for identifying the properties of white materials. Since colors perceived as white are in a three-dimensional color space, most observers are able to arrange white samples in one-dimensional order according to whiteness.

In the literature, numerous whiteness indices have been proposed for various industrial needs. Yet, only those relevant to dental applications are considered in this text. One of them is the CIE whiteness index (WIC), which was proposed by CIE in 1986 for the neutral hue preference [19].

$$ \mathrm{WIC}=Y+800\left({x}_n-x\right)+1700\left({y}_n-y\right) $$

where x, y are chromaticity coordinates defined as the ratio of individual tristimulus value and the sum of all three tristimulus values. x_n and y_n are chromaticity coordinates of the perfect white for the chosen standard observer (2° or 10°), and always under illuminant D65. The WIC formula gives relative, but not absolute, evaluations of whiteness. The higher the value of WIC, the greater the whiteness of the object.

Another whiteness index is based on the Euclidean distance of test color from the perfect white diffuser in the CIELAB color space [20].

$$ {W}^{\ast }={\left(100-{L}^{\ast}\right)}^2+{a^{\ast}}^2+{b^{\ast}}^2 $$

A modified version of CIE whiteness index (WIO) was proposed specifically for quantifying tooth whiteness based on visual perception of the Vita 3D Shade Guide [21]:

$$ \mathrm{WIO}=Y+1075.012\left({x}_n-x\right)+145.516\left({y}_n-y\right) $$

It was found that the WIO index outperformed other whiteness and yellowness indices for tooth whiteness assessment and was as reliable as the average human observer. In a study of colorimetric analysis of shade guides, it was demonstrated that the WIO gave the best fit with the instrumental color measurements.

A new CIELAB-based whiteness index for dentistry, WI_D, was proposed based on correlations with visual perception of shade guide tabs and dental materials [22].

$$ {\mathrm{WI}}_{\mathrm{D}}=0.511\ {L}^{\ast }-2.324{a}^{\ast }-1.100{b}^{\ast } $$

WI_D showed an improved correlation to the associated visual perception data compared to all other CIELAB and CIE 1931 XYZ-based whiteness or yellowness indexes tested under laboratory and clinical conditions. The results [22] showed that only WIO index was comparable to WI_D index. Further evaluations were done, including validation experiments under laboratory and typical clinical conditions, showing that the WI_D index outperformed previous indices, being the most adequate CIELAB-based index to evaluate whiteness in dentistry. Considering the popularity of CIELAB in dentistry and that color-measuring devices used in dental practice use this color space for color specification, the proposed whiteness index represents a significant step forward on measuring and evaluating whiteness in dentistry. Therefore, changes (∆WI_D) on tooth whitening treatments can be calculated as [23, 24]:

$$ \varDelta {\mathrm{WI}}_{\mathrm{D}}={\mathrm{WI}}_{\mathrm{D}}\left(\mathrm{treament}\right)-{\mathrm{WI}}_{\mathrm{D}}\left(\mathrm{baseline}\right) $$

1.2 Optical Properties and Measuring Methods

1.2.1 Optical Properties

Scattering and absorption are the main processes light suffers when traveling throughout a tissue or biomaterial.

1.2.1.1 Scattering

The scattering of the optical radiation in tissues or biomaterials occurs due to variations in the refractive index at microscopic level (cell membranes, cellular organelles, etc.). Similar to absorption, a scattering coefficient (μ_s) can be defined as:

$$ I={I}_0{e}^{-{\mu}_sx} $$

where I is the undispersed component of the light beam after crossing a non-absorbing sample with a thickness x. The inverse amount of the scattering coefficient (1/μ_s) is named scattering penetration and it is equal to the average optical path that a photon passes between two consecutive dispersions. It is also possible to define the optical thickness of a sample as the product between the dispersion penetration and the sample thickness (μ_sx—dimensionless measure).

When an incident photon with a direction described by the unit vector ${\overrightarrow{e}}_s$ suffers a scattering, the probability that its new direction of displacement is described by another unitary vector ${{\overrightarrow{e}}_s}^{\prime }$ is given by the normalized phase function $f\left({\overrightarrow{e}}_s\cdot {{\overrightarrow{e}}_s}^{\prime}\right)$. For tissues, it can be considered that the distribution probability is a function that depends only on the angle formed by the direction of the incident photon and the scattered photon (θ). Thus, this probability can be expressed in a very convenient way as a function of the cosine of the dispersion angle (${\overrightarrow{e}}_s\cdot {{\overrightarrow{e}}_s}^{\prime }= cos\theta$):

$$ f\left({\overrightarrow{e}}_s\cdot {{\overrightarrow{e}}_s}^{\prime}\right)=f\left(\cos \theta \right) $$

Therefore, the anisotropy of a material or tissue can be characterized by using the mean value of the cosine of the scattering angle. The parameter used is called anisotropy factor (factor):

$$ g=\underset{-1}{\overset{1}{\int }}\cos \theta \left[f\left(\cos \theta \right)\theta \right]d\cos \theta $$

1.2.1.2 Absorption

The absorption coefficient (μ_a) is defined as:

$$ dI={\mu}_a Idx $$

where dI is the differential variation of the intensity of a collimated luminous beam that travels an infinitesimal dx path in a homogeneous medium having the absorption coefficient μ_a. The inverse amount of the absorption coefficient (1/μ_a) is named absorption penetration and is equal to the average optical path that a photon passes between two consecutive absorptions. If the medium has a thickness (x), then:

$$ I={I}_0{e}^{-{\mu}_ax} $$

1.2.1.3 Transmittance

The transmittance (T) is defined as the ratio between the intensity of the light beam transmitted by the analyzed material (I) and the intensity of the incident light beam (I₀):

$$ T=\frac{I}{I_0} $$

Reflection and transmission greatly influence the chromatic appearance, and scattering and absorption are the main optical phenomena that affect the way optical radiation propagates through the material. Even if the effect of both phenomena is important, scattering is considered to have a greater influence on the propagation of optical radiation, since even for very thin samples (<1 mm) there is a high probability that photons will suffer multiple dispersions before crossing the sample as a result of the atomic interaction with the elements of the analyzed material. The relative probability to occur these interaction processes depends on the wavelength of the incident radiation. Therefore, when the incident radiation is polychromatic (consisting of several wavelengths), these properties are to be reported spectrally (depending on wavelength).

1.2.1.4 Radiative Transport Equation

Propagation of optical radiation through tissues (as natural tooth) could be described using fundamental electromagnetic theory. In this case, the tissue should be considered as a medium with a position-dependent permittivity so that field variations can be described using Maxwell’s equations. This approach is not possible due to the high complexity of the tissue structure as well as to the lack of precise knowledge of its permittivity. The problem can be simplified by ignoring phenomena related to wave propagation, such as polarization and interference, as well as phenomena related to microscopic particles, such as inelastic collisions.

Thus, according to this theory, the radiance $L\left(r,\overrightarrow{s}\right)$ of a light beam in the position r propagating in the direction described by the unit vector $\overrightarrow{s}$is diminished due to absorption and scattering and amplified by radiation propagating in the direction of ${\overrightarrow{s}}^{\acute{\mkern6mu}}$ but scattered in the direction of $\overrightarrow{s}$. The equation describing this phenomenon is as follows:

$$ \overrightarrow{s}\cdot \nabla L\left(r,\overrightarrow{s}\right)=-\left({\mu}_a+{\mu}_s\right)L\left(r,\overrightarrow{s}\right)+{\mu}_s\underset{4\pi }{\int }p\left(\overrightarrow{s},{\overrightarrow{s}}^{\prime}\right)L\left(r,{\overrightarrow{s}}^{\prime}\right)d{\omega}^{\prime } $$

where μ_a is the asorption coefficient, μ_s is the scattering coefficient, dω^′ is the differential solid angle in ${\overrightarrow{s}}^{\prime }$ direction, and $p\left(\overrightarrow{s},{\overrightarrow{s}}^{\prime}\right)$is the phase function.

The phase function describes the angular distribution for a single scattering and generally depends on the angle formed between $\overrightarrow{s}$ and ${\overrightarrow{s}}^{\prime }$. In general, the phase function is not known, being characterized by the anisotropy factor (g):

$$ g=\underset{4\pi }{\int }p\left(\overrightarrow{s},{\overrightarrow{s}}^{\prime}\right)\left(\overrightarrow{s}\cdot {\overrightarrow{s}}^{\prime}\right)d{\omega}^{\prime } $$

Solving of the radiative transport equation has a high degree of difficulty, which has led over time to propose different approximations for the radiance and/or phase function. Thus, the type of approximation that can be applied to calculate the distribution of a light beam through the biological tissues depends on the type of irradiation (diffuse or collimated) and the presence of a variation of the refractive index.

1.2.1.5 Translucency and Opacity

Translucency and opacity are related to the ability of a material to transmit light. The hiding power or opacity of materials has been described by its contrast ratio (CR). The opacity of a dental material was first described by Paffenbarger and Judd, in 1937 [25], and refined in 1979 for “esthetic dental filling materials” [26]. Subsequently, direct composite materials were described by CR based on luminous reflectance. The popular CR was defined as the ratio of the luminous reflectance of a translucent material on a black backing to the luminous reflectance of the same material on a white backing. Thus, it is important to note that luminous reflectance is the Y tristimulus value in reflectance, as defined by CIE.

$$ CR=\frac{Y_B}{Y_W} $$

The thickness required to produce a certain CR can be calculated when the relationship between thickness and CR is known [26]. This method offers the possibility of establishing a critical CR for a given application and then solving for the thickness required to obtain this critical value of translucency.

The translucency parameter (TP) has been used to assess the translucency of dental materials and it is defined as the CIELAB color difference for a material, at a particular thickness, on optical contact with ideal black and white backings [27]. As previously mentioned, CIE currently recommends the use of CIEDE2000 color-difference formula to improve correction between perceived and computed color differences; however, the majority of translucency studies in dental literature still uses TP associated to CIELAB color-difference formula (∆E^∗_ab). Thus, in this case, TP values are determined by calculating the color difference between readings from the same specimen over the black (B) and the white (W) backgrounds using CIELAB color-difference formula (TP_ab).

$$ {\mathrm{TP}}_{ab}={\left[{\left({L}_B^{\ast }-{L}_W^{\ast}\right)}^2+{\left({a}_B^{\ast }-{a}_W^{\ast}\right)}^2+{\left({b}_B^{\ast }-{b}_W^{\ast}\right)}^2\right]}^{1/2} $$

where the subscripts “B” and “W” refer to color coordinates over the black and the white backgrounds, respectively.

Following CIE current recommendations, the translucency parameter can be calculated using CIEDE2000 color-difference formula (TP₀₀) [28]:

$$ {\mathrm{TP}}_{00}={\left[{\left(\frac{L_B^{\prime }-{L}_W^{\prime }}{K_L{S}_L}\right)}^2+{\left(\frac{C_B^{\prime }-{C}_W^{\prime }}{K_C{S}_C}\right)}^2+{\left(\frac{H_B^{\prime }-{H}_W^{\prime }}{K_H{S}_H}\right)}^2+{R}_T\left(\frac{C_B^{\prime }-{C}_W^{\prime }}{K_C{S}_C}\right)\left(\frac{H_B^{\prime }-{H}_W^{\prime }}{K_H{S}_H}\right)\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$

where the subscripts “B” and “W” refer to lightness (L′), chroma (C′), and hue (H′) of the specimens over the black and the white backgrounds, respectively.

1.2.1.6 Opalescence and Fluorescence

Opalescence is one of the most interesting color features of teeth. This is characterized by bluish and amber shades that appear with greater or lesser intensity in the translucent incisal edge of anterior teeth. Its presence in dental reconstructions brings natural beauty to them. Most dental ceramic and resin-based composite systems allow the dental professional to simulate natural dental opalescence to adapt restorations to their natural environment.

Opalescence is caused by the scattering of the visible spectrum of light, giving the material a bluish appearance in reflected color, and an orange/brown appearance in transmitted color, because shorter light wavelengths are more scattered than longer wavelengths. Thus, blue light is scattered throughout the translucent tooth enamel and it is reflected back as a bluish-white light, which is clearly visible as the incisal halo. Structurally, tooth enamel hydroxyapatite crystals (0.16 μm long and 0.02–0.04 μm wide) act as scattering particles causing the opalescence, brightening the tooth and rendering depth and vitality to the structure. As opalescence is more evident in the incisal third of dental crowns, many clinicians, incorrectly, assume that could be specific areas of opalescence in tooth enamel. This, however, is caused by counter opalescence, a phenomenon in which light penetrates an opalescent object and it is reflected from within. Such opalescence is responsible for the orange or orange-pink coloration, commonly seen in the mamelons and incisal edge of the anterior teeth [29].

The most popular index to evaluate opalescence of tooth tissues and esthetic restorative materials is the OP-RT, which is calculated from the difference in yellow-blue (Δb^∗) and red-green (Δa^∗) coordinates between the reflected and transmitted colors measured by spectrophotometers [30].

$$ \mathrm{OP}-\mathrm{RT}={\left[{\left({\mathrm{CIE}}_{a_R}^{\ast }-{\mathrm{CIE}}_{a_T}^{\ast}\right)}^2+{\left({\mathrm{CIE}}_{b_R}^{\ast }-{\mathrm{CIE}}_{b_T}^{\ast}\right)}^2\right]}^{1/2} $$

where the subscripts R and T indicate the reflected and transmitted mode, respectively.

OP-RT values of human and bovine tooth enamel have been reported [30, 31]. As required, enamel color was measured in reflectance and transmittance modes. Two spectrophotometers were used to determine their configuration influence on the OP-RT values of bovine enamel, and one spectrophotometer was used for human enamel samples because of their reduced dimensions. Sample thickness varied between 0.7 and 1.1 mm for bovine enamel and from 0.9 to 1.3 mm for human enamel. Mean OP-RT value of bovine enamel varied between 10.6 and 19.0 based on the spectrophotometers configuration. A mean OP-RT value of 22.9 was found for human enamel. No significant correlation was found between sample thickness and OP-RT values for both bovine and human enamel.

OP-RT values of restorative materials (ceramics and resin-based composite) were reported as well [29]. Mean OP-RT values of cured direct resin-based composites have a large variation (5.7–23.7) and are usually significant different before (~25.6) and after (~12.4) light activation (polymerization). Those values are much higher than the ones for unfilled resin (3.5). In addition, mean OP-RT values of ceramic systems vary between 1.6 and 7.1. Thus, considering that most of these OP-RT values are lower than the value of tooth enamel, manufacturers should design dental restorative systems to adjust the composition of filler and matrix phases to facilitate the reproduction of natural teeth opalescence.

Fluorescence is defined as the absorption of UV light (100–400 nm invisible light) by an object and its spontaneous emission in longer wavelengths (430–450 nm visible light). Thus, dental practitioners should consider fluorescent emission of human teeth as part of a successful esthetic rehabilitation.

Bluish-white fluorescence of human teeth is resultant from a broad emission band with a diffuse peak at 410–420 nm when subjected to near UV excitation of ∼340 nm. In human teeth, fluorescence mostly occurs in dentine because of its greater amount of organic content [32]. As to the endogenous fluorophores in enamel and dentine, studies using UV excitation indicated that light is emitted from the organic matrix embedded in the inorganic calcium apatite matrix [33]. In addition, three distinct fluorescence peaks (350–360 nm, 405–410 nm, and 440–450 nm) were found in enamel and dentine. Another study reported blue fluorescence from dentine at 440 ± 10 nm, with a peak width of about 100 nm [34]. Yet, as chroma of dentine increases, fluorescence decreases. Thus, dentine presents a more intense and a greater index of fluorescence than enamel. Not surprising, fluorescence at the gingival area is significantly greater than at the incisal area. However, there is no significant difference in fluorescence between male and female teeth, nor between upper and lower teeth, nor even between tooth types (incisors, canines, premolars, and molars) taken from the same individual [32].

The fluorescence of human teeth and dental materials can be determined with a color-measuring spectrophotometer or spectroradiometer. Spectral reflectance can be measured over a white standard tile using a standard illuminant D65. A UV filter is inserted or removed to exclude or include the UV component of the illumination. From the spectral reflectance values, the subtraction spectrum is calculated by the inclusion and exclusion of the UV component, and the color difference by the UV component is defined as the fluorescence parameter ($\varDelta {E}_{ab}^{\ast }$ − FL):

$$ \varDelta {E}_{ab}^{\ast }-\mathrm{FL}={\left[{\left({L}_i^{\ast }-{L}_e^{\ast}\right)}^2+{\left({a}_i^{\ast }-{a}_e^{\ast}\right)}^2+{\left({b}_i^{\ast }-L{b}_e^{\ast}\right)}^2\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} $$

where subscripts i and e indicated UV included and excluded, respectively

Fluorescence properties of resin-based restorative systems vary depending on material, shade, and polymerization. The mean fluorescence parameter value for a resin-based composite was reported to be 2.5 before polymerization and 0.7 after polymerization [35].

1.2.2 Methods of Measuring Optical Properties

The optical properties of biological tissues or biomaterials are described in terms of the absorption coefficient, μ_a, the scattering coefficient, μ_s, the scattering function p(θ, ψ), where θ is the deflection angle of scatter and ψ is the azimuthal angle of scatter, and the real refractive index of the media. The p(θ, ψ) is appropriate when discussing only a single or few scattering events, such as during confocal reflectance microscopy, which includes optical coherence tomography. In thicker biomaterials and biological tissues where multiple scattering occurs and the orientations of scattering structures are randomly oriented, the ψ dependence of scattering is averaged and hence ignored, and the multiple scattering averages θ, such that and average parameter, g = cosθ, called the anisotropy of scatter, characterizes tissue scattering in terms of the relative forward versus backward direction of scatter.

To determine the optical properties of a biomaterial or a tissue is the initial step toward properly designing devices, interpreting diagnostic measurements or planning therapeutic protocols. Secondly, the optical properties are used in a light transport model to predict the light distribution and energy deposition. Thus, it is essential to use adequate methods to determine the optical parameters of tissues and biomaterials and they can be divided into two large groups, direct and indirect methods.

Direct methods include those based on some fundamental concepts and rules such as Bouguer–Beer–Lambert law, the single-scattering phase function for thin samples, or the effective light penetration depth for slabs. The parameters measured are the collimated light transmission, Tc, and the angular dependence of the scattered light intensity, I(θ), for thin samples or the fluence rate distribution inside a slab. These methods are advantageous in that they use very simple analytic expressions for data processing and reconstruction of optical parameters. Their disadvantages are related to the necessity to strictly fulfill experimental conditions dictated by the selected model, single scattering in thin samples, exclusion of the effects of light polarization, etc. In the case of slabs with multiple scattering, the recording detector must be placed far from both the light source and the medium boundaries.

Indirect methods obtain the solution of the inverse scattering problem using a theoretical model of light propagation in a theoretical model of light propagation in a medium. They are divided into iterative and non-iterative models. The former uses equations in which the optical properties are defined through parameters directly related to the quantities being evaluated. The latter are based on the two flux Kubelka–Munk (K-M) model and multi-flux models. In indirect iterative methods, the optical properties are implicitly defined through measured parameters. Quantities determining the optical properties of a scattering medium are listed until the estimated and measured values for reflectance and transmittance coincide with the desired accuracy. These methods are difficult, but the optical models currently in use maybe even more complicated than those underlying non-iterative methods, such as the diffusion theory, inverse adding-doubling (IAD), and inverse Monte Carlo (IMC) methods.

A schematic representation of methods to measure the optical properties of tissues is shown in Fig. 1.3.

Fig. 1.3

Schematic representation of (a) non-iterative method and (b) iterative method

1.2.2.1 Kubelka–Munk Theory

In 1948, Kubelka and Munk described a simplified mathematical model of the optical radiation interaction with the translucent isotropic materials. Taking into account that the main dental structures (dentine and enamel) and dental materials designed to replace them are translucent, the applicability of this theory in dentistry is obvious.

Based on the double-flux theory of a light beam crossing a translucent material, Kubelka and Munk mathematically processed the reflection and transmission of a sample (both homogeneous and inhomogeneous) placed and measured on different colored backgrounds [36]. They expressed the scattering (S) and absorption (K) of a material where both parameters depend on the wavelength, considering the influence of sample thickness and background.

According to the Kubelka–Munk theory the reflectance (R) and transmittance (T) are described by the following expressions:

$$ R=\frac{\sin h(Sbd)}{a\ \cos h(Sbd)+b\ \sin h(Sbd)} $$

$$ T=\frac{b}{a\ \cos h(Sbd)+b\ \sin h(Sbd)} $$

where S and K represent the scattering and absorption Kubelka-Munk coefficients, and a and b are the optical Kubelka–Munk constants. The advantage of this theory lies in the fact that both scattering and absorption coefficients can be expressed according to the reflectance and the transmittance of the material:

$$ S=\frac{1}{bd}\ln \left[\frac{1-R\left(a-b\right)}{T}\right] $$

$$ K=\left(a-1\right)S $$

where

$$ a=\frac{1+{R}^2-{T}^2}{2R} $$

$$ b=\sqrt{a^2-1} $$

If no transmittance measurements are available, the Kubelka–Munk a and b optical constants can be calculated from reflectance measurements of a sample over a black background and a white background:

$$ a=\frac{\begin{array}{l}R(B)-R(W)-{R}_B+{R}_W-R(B)\cdot R(W)\cdot {R}_B+R(B)\cdot R(W)\cdot {R}_W\\ {}+R(B)\cdot {R}_B\cdot {R}_W-R(W)\cdot {R}_B\cdot {R}_W\end{array}}{2\cdot \left(R(B)\cdot {R}_W-R(W)\cdot {R}_B\right)} $$

$$ b=\sqrt{a^2-1} $$

where R_B represents the black background reflectance, R_W the white background reflectance, R(B) is the sample reflectance over a black background, and R(W) is the sample reflectance over a white background [37].

Reflectivity (RI—reflectance of a sample of infinite thickness) can also be defined as:

$$ RI=a-b $$

Thus, the Kubelka–Munk scattering (S) and absorption (K) coefficients are described by the following equations:

$$ S\left({\mathrm{mm}}^{-1}\right)=\frac{1}{b\cdot X}\mathrm{arctgh}\left(\frac{1-a\cdot \left(R(B)+{R}_B^{\mathrm{eff}}\right)+R(B)\cdot {R}_B^{\mathrm{eff}}}{b\cdot \left(R(B)-{R}_B^{\mathrm{eff}}\right)}\right) $$

$$ \mathrm{K}\left({\mathrm{mm}}^{-1}\right)=S\left(a-1\right) $$

where X is the thickness of the measured sample, R(B) is the reflectance of the sample over the black background, and ${R}_B^{\mathrm{eff}}$ is the effective reflectance of the black background when it is in contact with the measured sample. ${R}_B^{\mathrm{eff}}$ is calculated according to suggestions proposed by Mikhail et al. [38]:

$$ {R}_B^{\mathrm{eff}}=\frac{R_B}{\left(1-{k}_1\right)\cdot \left(1-{k}_2\right)+{R}_B\cdot {k}_2} $$

where k₁ = 0.039 and k₂ = 0.540

The Kubelka–Munk transmittance (T) can be calculated as:

$$ T=\frac{b}{a\cdot \sinh (bSX)+b\cdot \cosh (bSX)} $$

Both, the simplicity in implementing the Kubelka–Munk theory and the fact that it allows the determination of an analytical solution, are the main reasons why this theory is used in many areas, including dentistry. However, it should be noted that the Kubelka–Munk theory does not provide an explicit relationship between its coefficients (S, K) and the optical coefficients (μ_s, μ_a). The correlation between these parameters has been extensively studied by several authors [36,37,41]. Nowadays, these relationships are as follows:

$$ K=2{\mu}_a $$

and

$$ S=\frac{3}{4}{\mu_s}^{\prime }-x{\mu}_a $$

Thennadil [42] demonstrated that the contribution of the term that includes the absorption coefficient (μ_a) to the Kubelka–Munk scattering coefficient (S) leads to impossible values from a physical standpoint, and, therefore, this relationship is reduced to

$$ S=\frac{3}{4}{\mu_s}^{\prime } $$

It has been concluded that the use of this formula for the estimation of the optical properties of materials produces an approximate error of 3%.

1.2.2.2 Inverse Adding-Doubling Method

The principle of the adding method, as first proposed by van de Hulst [43], is illustrated in Fig. 1.4. For a very thin slab, one can write the radiance at the two surfaces from knowledge of the phase function, since the multiple scattering is negligible. If an identical slab is added, the radiance of the final double-thick slab can be calculated by considering the successive scattering back and forth between the component layers. Computation for thicker slabs can be carried out by adding other thin layers or, more efficiently, by doubling the total thickness with each iteration (Fig. 1.4).

Fig. 1.4

Schematic representation of the adding-doubling approximation method. I incident light, R refraction, T transmission

When one-dimensional geometry is a reasonable representation, the adding-doubling method provides an accurate solution of transport equation for any phase function. It allows modeling of anisotropically scattering, internally reflecting, and arbitrarily thick layered media with relatively fast computations. The adding-doubling approximation is one simple method and it has been successfully used to determine the optical properties of turbid media, such as biological tissues.

Prahl et al. [44] proposed a method for determining the optical properties of turbid media based on the adding-doubling approximation, called inverse adding-doubling (IAD): inverse implies a reversal of the usual process of calculating reflection and transmission from optical properties, and adding-doubling indicates the method used to resolve the radiative transport equation. The adding-doubling method is sufficiently fast that iterated solutions are possible on current microcomputers and sufficiently flexible that anisotropic scattering and internal reflection at boundaries may be included.

Inverse adding-doubling (IAD) method involves direct measurements of reflectance and transmittance of the samples and a Monte Carlo simulation to determine the scattering and absorption coefficients. Reflection and transmission measurements, usually made with an integrating sphere, are converted to the optical properties of the sample (scattering and absorption coefficients) using the computer program named “iad,” developed by Scott Prahl (https://omlc.org/software/iad/manual.pdf). This program has been extensively tested and validated for accuracy and precision and it has been widely used to determine the optical characteristics of different biological tissues. The general idea is that measurements of reflection and transmission are fed into the program to extract the intrinsic optical properties of the sample to be studied. The program does it by repeatedly guessing the optical properties and comparing the expected observables with those that have been made.

To solve the radiative transport equation, the iad program must be supplied with the experimental values of total diffuse reflectance (MR) and transmittance (MT) together with the values of the scattering anisotropy factor (g) and the refraction index (n) of the sample. The program guesses a set of optical properties (μ_a and μ′_s) and then calculates values for reflectance (MR) and transmittance (MT). This process is repeated until the calculated and measured values of reflectance (MR) and transmittance (MT) are within a specified tolerance (for the sum of both relative differences, the tolerance default value is 0.01%).

The Monte Carlo simulation is a statistical method that calculates the trajectories of a great number of photons and, as a result, presents the reflectance and transmission of a sample for a given set of optical parameters. The use of this simulation minimizes systematic errors, considers the scattering phase function, and also takes into account the measuring geometry. A limiting consideration is that the accuracy of calculated quantities increases only with the square root of the number of photon histories, making the Monte Carlo particle simulation a computationally costly method. Nevertheless, as computing power is progressively becoming cheaper, this technique is being more widely applied, for example, in tissue optics. The inverse Monte Carlo simulation was included in the iad program to achieve an accurate evaluation of the sample’s optical properties. The measurements of the total diffuse reflectance (MR) and transmittance (MT) used in the IAD method to determine the optical absorption coefficient(μ_a) and reduced scattering coefficient (μ′_s) are modeled by the Monte Carlo simulation technique, which uses as stochastic simulation of light interaction with biological media.

A schematic of the experimental setup for measuring the total diffuse reflectance and total diffuse transmittance is shown in Fig. 1.5.

Fig. 1.5

Schematic representation of the experimental setup. (a–c) Configuration for reflection measurements and (d–f) configuration for transmission measurements

A polarizer is used to reduce and direct the intensity of the incident laser beam in order to prevent undesirable diffuse reflections. A mirror system to divert the laser beam was added to fix the sample after the diffuse reflection measurements, so that the diffuse transmission measurements could be performed in the same spot of the sample.

The total diffuse reflectance M_R, in terms of percentage, is calculated using:

$$ {M}_R={r}_{\mathrm{std}}\frac{R\left({r}_s^{\mathrm{direct}},{r}_s\right)-R\left(0,0\right)}{R\left({r}_{\mathrm{std}},{r}_{\mathrm{std}}\right)-R\left(0,0\right)} $$

where r_std is the reflection of the reference standard.

Total diffuse transmittance is measured under the same setup conditions (single-integrating-sphere setup using collimated light). Transmittance measurements ($T\left({t}_s^{\mathrm{direct}},{r}_s\right)$) were referenced to 100% with the lasers illuminating the open and empty port (T(0, 0)) and a dark measurement with an open port without illumination from the lasers (T_dark) (Fig. 1.5). The total diffuse transmittance, in terms of percentage, was calculated using

$$ {M}_T=\frac{T\left({t}_s^{\mathrm{direct}},{r}_s\right)-{T}_{\mathrm{dark}}}{T\left(0,0\right)-{T}_{\mathrm{dark}}} $$

One critical parameter in both reflection and transmission measurements is the sphere wall reflectance. In order to determine this parameter, two measurements are needed (Fig. 1.6). The integrating sphere rotated with respect to the incident beam so that light was directly incident upon the sphere wall between the sample port and the baffle. The reflectance sphere wall (r_w) was calculated according to the following equation:

$$ \frac{1}{r_w}={a}_w+{a}_d{r}_d\left(1-{a}_e\right)+{a}_s{r}_{\mathrm{std}}\left(1-{a}_s\right)\frac{R_{\mathrm{std}}^{\mathrm{diff}}}{R_{\mathrm{std}}^{\mathrm{diff}}-{R}_0^{\mathrm{diff}}} $$

where a_w is the fractional sphere wall area, a_d is the fractional detector area, a_s is the fractional sample area, a_e is the fractional entrance port area, r_d is the detector reflectance, and r_std is the reflectance of the reflectance standard.

Fig. 1.6

Schematic representation of the experimental setup used for the reference sphere calibration measurements. (a) measurements without the standard and (b) measurements with the standard

1.2.3 Application of Color Science and Optical Properties to Dental Structures and Dental Materials

1.2.3.1 Color of Tooth

From the point of view of a human observer, the appearance of a tooth is a complicated psycho-physiological process, which is influenced by different factors, such as the spectral power distribution of the illuminant, the sensitivity of the observer and the spectral characteristic of the tooth, mostly reflectance and transmittance, which are mainly determined by its absorption, reflection, and transmission properties. As with any other translucent sample, when light reaches the surface of a tooth, five phenomena associated with the interaction of radiant energy with tooth may occur: specular reflection at the tooth surface, diffuse reflectance, direct transmission through the tooth, and absorption and scattering of light within the different tooth structures (Fig. 1.7).

Fig. 1.7

Five phenomena (regular transmission, specular reflection, diffuse reflection, absorption, and scattering) may occur in the interaction of incident radiation and tooth

Tooth color is influenced by a combination of intrinsic color and the presence of extrinsic stains that may form on the tooth surface. Light scattering and absorption within enamel and dentine give rise to the intrinsic color of the teeth and since enamel is relatively translucent, the properties of dentine can play a major role in determining the overall tooth color.

The range and distribution of tooth color has been described in a number of studies [45] where factors such as location, gender, age, and ethnicity have been investigated. In general, the maxillary anterior teeth are slightly more yellow than mandibular anterior teeth, and the maxillary central incisors have greater value than the lateral incisors and canines [46, 47]. Although some studies have shown no differences in tooth color between males and females, few studies have reported differences in tooth color between genders with females showing lighter and less yellow incisors than males [46, 45,46,50]. In a study with Chinese urban population (N = 405), L^∗ values were significantly higher (about 1.7 units) and b^∗ values were significantly lower (about 0.9 units) for females [51]. Similarly, in a Spanish population (N = 1361) the females showed higher in L^∗ values (2.53 units) and lower b^∗ values (3.11 units) compared to males [49, 50].

In the midst of a vast array of genetically determined tooth colors, all teeth darken over the course of time. Indeed, many studies have showed the color of teeth become darker and more yellow with aging. In a group of 180 USA adults and teenagers whose maxillary central incisors were measured with a spectrophotometer, it was shown that for each year of life, the average tooth color decreased in L^∗ value by 0.22 units and increased in b^∗ value by 0.10 units [52]. Larger changes have been measured in a Spanish population using a spectrophotometer where tooth color decreased in L^∗ value by 0.6 units per year, b^∗ values increased by 0.56 units per year, and a^∗ values increased by 0.26 units per year [49, 50]. In addition, it has been shown that with increase in age, the mean value of b^∗ increases faster in males than females [52].

Studies investigating the relationship of skin color to tooth shade and color are conflicting. Most studies show no relationship, but few studies reported an inverse relationship where people with medium- and dark-skin tones were more likely to have the highest value in tooth color in comparison to people with lighter skin tones regardless of their age or gender [53]. In contrast, others studies investigated the relationship of the color of maxillary incisors and facial skin across four different ethnic groups and found that L^∗ value of tooth color had a positive correlation with L^∗ value of skin color for subjects from Saudi Arabia, India, and East Asia [54]. In addition, it was found a negative correlation between the L^∗ values of tooth and skin color for subjects of African origin. Such controversy is probably associated to differences in sample size and ethnic origin of population, as well as to the methods used to measure skin and tooth color. When considering the broad range of tooth colors measured and reported in the literature from many different study populations, it is evident that ethnicity does not pre-dispose an individual to a particular tooth color. Considering Saudi Arabian, Indian, African, and East Asian groups, the reported mean (standard deviation) values for L^∗ were 79.26 (3.16), 80.59 (3.23), 78.95 (4.33), and 78.33 (2.10), respectively; for a^∗ were 0.853 (0.99), 0.546 (0.95), 1.132 (1.51), and 0.955 (0.56), respectively; and for b^∗ were 21.05 (3.84), 19.57 (4.01), 19.30 (4.95), and 17.52 (2.75), respectively [54]. Despite the statistical difference found among groups for each color parameter, the clinical relevance of the data indicates that the spread of tooth colors from one ethnicity will greatly overlap with those of another ethnicity, indicating that all ethnicities share a large proportion of tooth colors.

Considering tooth whiteness, the WIC formula has been used in some early dental studies to assess whiteness of porcelain teeth, human teeth, and VITA Shade Guide tables [55, 56], and the W^∗ formula has been used in a few tooth bleaching studies to track whiteness changes after treatments. It was suggested that for a whiteness index to be valid, it must be used with the type of materials for which it was intended. Thus, it was reported that tooth colors appear to be outside the valid color range for using the WIC formula [21].

The index WIO has been used in several in vitro and in vivo tooth whitening studies, including tooth bleaching studies with hydrogen peroxide products and whitening studies with toothpastes containing optical whitening technologies [57, 58]. In vitro studies demonstrated that toothpastes containing blue covarine gave a statistically significant reduction in tooth yellowness and both, in vitro and clinical trials, reported improvement in tooth whiteness immediately after brushing. In addition, the higher concentration blue covarine toothpaste gave statistically significant greater tooth whitening benefits than the lower concentration blue covarine toothpaste [59].

1.2.3.2 Optical Properties of Dental Structures and Dental Materials

Measuring and reporting optical properties of natural teeth and dental materials are of particular importance, since their interaction is the basis for an esthetic match between the restorative material and the adjacent or remaining dental structure. Therefore, reported values for the various dental structures should be used in the development and manufacture of esthetic dental restorative materials.

The optical properties of esthetic restorative materials (resin composites, ceramics, etc.) and dental structures (enamel and dentin) have been extensively investigated. We can easily find reports on the anisotropy factor, absorption, scattering coefficients, and transmittance for different dental materials and dental structures. Relevant data from some of these studies are summarized below.

1.2.3.2.1 The Anisotropy Factor (g)

It characterizes the phase function and it has been experimentally evaluated for different dental tissues (enamel and dentin) and dental materials (hybrids, resin-based composites, and ceramics) for two different thicknesses (0.5 mm and 1 mm) [60]. The anisotropy factor was calculated from the average cosine of the scattering angle, and four-dimensional measurements were made for four wavelengths of the visible spectrum: 457.9 nm, 488.0 nm, 514.5 nm, and 632.8 nm. The results demonstrate that the thicker samples have a low scattering profile compared to the thinner samples, regardless of the material. It was also observed that dental materials have an anisotropy factor (g) comparable to tissues they aim to replace (Table 1.1).

Table 1.1 Anisotropy factor (g) values for different dental materials and structures

1.2.3.2.2 Absorption Coefficient (μ_a) and Scattering Coeffficient (μ_s)

In other study [60], the absorption (μ_a) and scattering (μ_s) coefficients, as well as the reduced scattering coefficient (μ_s^′) of three types of dental materials (hybrid composite, nano-composite, and zirconia) were evaluated for four different wavelengths. The determination of the coefficients was performed using the iterative, inverse adding-doubling method that combines transmission and reflection measurements with an experimental setup containing an integrating sphere and a multi-frequency laser (Fig. 1.5). According to the results of this study, both the absorption (μ_a) and scattering (μ_s) coefficients exhibit a similar spectral behavior for the two composite materials but different from that obtained for the ceramic material (Table 1.2). It was concluded that these differences may be due to the anisotropy factor (g), which exhibits different values for ceramic compared to resin-based composites.

Table 1.2 Values of the absorption (μ_a), scattering (μ_s), and reduced scattering (μ_s^′) coefficients for different dental materials

1.2.3.2.3 Absorption (K) and Scattering (S) Coefficients, Transmittance (T), and Reflectivity (RI) According to the Kubelka–Munk Theory

The correct application of Kubelka–Munk theory (K-M theory) for the analysis of dental structures and dental materials was demonstrated by Ragain and Johnston [61]. They showed that Kubelka–Munk theory can reliably estimate the diffuse reflection values of samples by comparing them with the diffuse reflectance values experimentally measured with a spectrophotometer. Since then, the Kubelka–Munk theory has been used to estimate the optical properties of various dental restorative materials. In one of such studies [62], the K-M theory was used to evaluate the scattering, absorption, and transmittance of human and bovine dentines, and two ceramic systems. The spectral behavior of the scattering coefficient (S) for both dentines (human and bovine) was found similar in all wavelengths. Although the ceramic systems and human dentine showed similar spectral behavior of the scattering coefficient (S), their values were statistically different (Fig. 1.8). Significant differences were found among the study groups for the absorption coefficient (K). However, in all cases, K values were greater for short wavelengths and the values were decreasing as the wavelength increases (Fig. 1.9). The spectral behavior of transmittance (T) showed a constant increase of values with increasing wavelengths, displaying the highest values for the longest wavelengths (Fig. 1.10). The spectral values obtained for human and bovine dentines were similar. Although the zirconia-based ceramic systems showed similar spectral behavior to human dentine, the values were statistically different. Both ceramic systems showed lower transmittance values compared to human and bovine dentines.

Fig. 1.8

Spectral distribution of Kubelka-Munk scattering coefficient (S) for human dentine (HD) and two zirconia-based ceramic systems (ZC and LV)

Fig. 1.9

Spectral distribution of Kubelka-Munk absorption coefficient (K) for human dentine (HD) and two zirconia-based ceramic systems (ZC and LV)

Fig. 1.10

Spectral distribution of Kubelka–Munk transmittance (T) for human dentine (HD) and two zirconia-based ceramic systems (ZC and LV)

In addition, the Kubelka-Munk theory was used to estimate the optical properties of a range of shades from resin-based composites with wide applicability in clinical practice [16, 17]. It was found that the scattering coefficient (S) has a maximum value at the 450 nm wavelength. Thus, lower values of S were found for longer wavelengths. Translucent shades exhibit different spectral behavior compared to other shades (Fig. 1.11). The spectral behavior of the absorption coefficient (K) is marked by a decrease in K values as the wavelength increases, thus the smallest values of K are found for the longest wavelengths. As for other dental materials, the transmittance (T) increased with the wavelength and, as expected, greater T values were found for translucent materials.

Fig. 1.11

Spectral distribution of (a) the scattering coefficient (S), (b) the absorption coefficient (K), and (c) the transmittance (T) for different shades of the same resin-based composite material

Last but not least, the Kubelka–Munk theory was used to evaluate the optical properties of 2-mm thick dentine sections from three teeth: incisor, canine, and molar [63]. Similar spectral behavior of scattering (S), transmittance (T), and reflectivity (RI) was found for the three types of dentine, despite significant different values. However, no significant difference was found for the absorption coefficient (K) among the three types of dentine (incisor, canine, and molar). Dentine from canine teeth showed the highest scattering and reflectivity values (Fig. 1.12), while dentine from molar teeth showed the highest absorption and transmittance values (Fig. 1.13). Therefore, the optical properties of human dentine are influenced by the type of tooth dentine.

Fig. 1.12

Spectral distribution of scattering (S) and reflectivity (RI) for dentine from (a) incisor, (b) canine, and (c) molar teeth. Mean value spectrum for each graph is shown in red

Fig. 1.13

Spectral distribution of absorption coefficient (K) and transmittance (T) for dentine from (a) incisor, (b) canine, and (c) molar teeth. Mean value spectrum for each graph is shown in red

Many studies evaluated the translucency of natural tooth structure and several conditions of dental materials using the translucency parameter (TP), contrast ratio (CR), and transmission (%T) [64]. Polymers from conventional and industrial polymerization showed a wide variation in translucency using wavelength-specific transmittance determinations. Direct and indirect dental composite materials, that is, resin-based composites cured by different polymerization processes, also showed a change in translucency (TP values). Changes due to depth of cure were also investigated using both TP and wavelength-specific transmittance measurements. Polymerization kinetics was evaluated using wavelength-specific transmittance measurements. The influence of bisphenol in the resin matrix composition was demonstrated using both spectral transmittance measurements. In addition, the effect of the sample thickness has been extensively evaluated for dental materials using TP, CR, and %T. Other conditions such as staining, thermocycling, and filler volume fraction were also evaluated using transmittance and TP determinations. TP associated with wavelength-specific CR was used to show changes in translucency with wavelength. TP determinations were also used to show that the translucency of translucent composites is different from that of tooth enamel. Studies on translucency of dental ceramics demonstrated differences in CR among shades of zirconia-based ceramics. Differences due to sintering conditions of these ceramics were shown using total transmission and diffuse transmittance measurements, and the effect of veneering was further demonstrated using total luminous transmittance. The curing of underlying cement was the major topic of interest in determining the translucency of these ceramics using wavelength-specific transmittance. Both CR and transmittance determinations were used to determine that CR is useful to transmission differences below 50% transmission for a wide variety of ceramic materials. TP, CR, and wavelength-specific transmittance determinations were used to show differences among shades of dental ceramic materials. TP determinations of ceramic materials were solely used to determine significant effects of the illuminant, surface texture, and surface-finishing technique. Using ceramic materials, TP values measured by the spectroradiometer and the spectrophotometer were found to be significantly different but highly correlated. A significant correlation between the translucency of ceramic materials and polymerization efficiency of underlying luting composites was found using TP determinations. TP determinations were used to demonstrate the significant effect of bleaching on human enamel. Data recommended for use as reference in the development of esthetic restorative materials and clinical shade matching were presented using both CR and TP determinations.

In summary, colorimetric and optical evaluation of dental tissues and dental materials using established and standardized methods are very important for continuing development of materials and techniques used in esthetic dentistry, assisting to improve the successful rate of esthetic dentistry procedures, where color appearance and perception are crucial, and, therefore, benefiting patients and the dental profession.