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Measurements and modelling of the influence of dentine colour and enamel on tooth colour
journal of dentistry 43 (2015) 373–381
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Measurements and modelling of the influence of dentine colour and enamel on tooth colour Paul D. Battersby 1, Stephen J. Battersby * Philips Research Laboratories, 101 Cambridge Science Park, Milton Road, Cambridge CB4 0FY, UK
article info
abstract
Article history:
Objectives: We provide a quantitative predictive model for the extent to which coloured
Received 5 February 2014
dentine, visible through the enamel, contributes to tooth colour. Our model uses ðL ; a ; b Þ
Received in revised form
measurements rather than spectral measurements.
21 October 2014
Methods: We have used a model system, composed of a slice of bovine enamel placed on top
Accepted 7 November 2014
of coloured paper. We have measured the colour of the enamel–paper combination, as an analogue for a tooth, and have related this to the colour of the paper, as an analogue for dentine. By changing the paper colour, we have been able to explore how the colour of
Keywords:
dentine determines tooth colour, according to our model system. We have also compared
Tooth colour
hydrated and desiccated samples.
Colour appearance model
Results: In qualitative terms, superimposing the enamel on top of the paper increases the
Dentine colour
‘‘lightness’’ for all colours tested except white while simultaneously reducing the chroma-
Enamel colour
ticity, a measure of the extent to which the colour differs from grey. Desiccated enamel is
Tooth discolouration
much more effective at increasing the lightness and reducing the chromaticity than
Tooth whitening
hydrated enamel. Quantitatively, our measurements are reproduced by the mathematical model we have developed to within 2% in ‘‘lightness’’ and about 8% in chromaticity. Conclusions: We are able to predict the colour of an analogue for a tooth, composed of bovine enamel and coloured paper, from the colour of an analogue for the dentine, the coloured paper alone, with good accuracy. This understanding provides insights into the role of dentine colour in determining tooth colour. Clinical Significance: Our work helps quantify the importance of dentine colour, compared to other, extrinsic causes of colour, such as staining, in determining the visible colour of teeth. Our predicted colours represent a baseline to which extrinsic sources will add. # 2015 Published by Elsevier Ltd.
1.
Introduction
There are many potential causes of tooth discolouration, including extrinsic causes such as surface staining and enamel discoloration through smoking or drinking tea, coffee or wine, and intrinsic causes such as a degradation in the colour of the
dentine which is visible externally through the enamel.1,2,3 Some of these discoloration mechanisms are short term and reversible with suitable cleaning.4 However, other causes are semi-permanent, reversible only through tooth bleaching procedures using, for example, carbamide peroxide and commonly known as tooth whitening.5,6 This paper focuses on the role of dentine colour in determining tooth colour. We are
* Corresponding author. Tel.: +44 1223 427500; fax: +44 1223 452550. E-mail address: [email protected] (S.J. Battersby). 1 Current address: School of Dentistry, Cardiff University, Heath Park, Cardiff, Wales CF14 4XY, UK. http://dx.doi.org/10.1016/j.jdent.2014.11.003 0300-5712/# 2015 Published by Elsevier Ltd.
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journal of dentistry 43 (2015) 373–381
able to predict tooth colour due to dentine colour alone, with both colours measured using simple colourimeter measurements such as ðL ; a ; b Þ, rather than full spectral measurements. This predicted tooth colour represents a baseline to which extrinsic causes of discolouration will add. As the extent of discolouration due to the latter cannot easily be quantified, we cannot fully resolve the debate over the importance of dentine colour in the overall appearance of human dentition. However, our baseline colour predictions should allow practitioners and researchers to develop an understanding of the relative importance of intrinsic and extrinsic colouration sources in that overall appearance. Enamel is a translucent scattering medium which, at the thicknesses present in human teeth, does not fully obscure the colour of the underlying dentine. For example, an in vitro study of the colour of 28 teeth from different patients showed strong correlation with dentine colour when the overlying enamel was removed.7 Degradation in the colour of dentine, for example with age, can therefore lead to tooth discolouration.8,9 This is exacerbated when thinning of the enamel takes place or when gum recession occurs, thereby exposing tooth surface closer to the cementoenamel junction and hence covered in thinner enamel.7 An extreme example of the role of dentine colour occurs when blood escapes from the blood vessels in the tooth and perfuses the dentine, leading to severe discolouration of the dentine. Here, the external appearance of the tooth is also considerably darkened. In order to understand these effects, a theoretical model, able to predict tooth colour from dentine colour and enamel characteristics, is desirable. However, such a model faces three challenges, two theoretical and one practical. Firstly, it is necessary to identify a model for the wavelength dependent scattering of light in the enamel and hence be able to predict the spectrum of light emerging from the enameldentine combination from that emerging from the dentine alone. Secondly, it is necessary to relate those spectra to the colour perceived by the human visual system, with its associated complexities. Finally, for practical application, it is necessary to recognise that the measurement of spectral reflectivities using a spectrophotometer, whilst possible in the laboratory, is unlikely in the dental surgery/office and that the terminology used in optical physics is alien to clinical dentistry. A more straightforward measurement, using the colourimeters already in routine use in the surgery/office, is to be preferred. Our work addresses all three challenges. O’Brien et al.10 tackled the first two challenges using a model system composed of dental porcelain to represent the enamel and coloured opaque backing layers to represent the dentine. They used the so-called two flux model of Kubelka and Munk11 (K-M model) to represent scattering in the porcelain but measured the full reflectance spectra for both the porcelain and the backing layers using a spectrophotometer. They then compared the measured and predicted CIE colour parameters of luminous reflectance, dominant wavelength and excitation purity.12 However, as noted above, measuring these spectra is impractical in the dental surgery/ office and inferring it from measured L*a*b* values is impossible due to the weighted averaging used to derive these values. Fortunately, the colour saturation levels typically exhibited by human dentine and enamel are sufficiently low that an approximation13 can be used to relate directly the L*a*b* values
for a coloured surface such as dentine to the equivalent values when the coloured surface is covered by a scattering layer such as enamel. The aim of our work is to show that it is possible to predict tooth colour measured directly with a colourimeter from equivalent measurements on the dentine, without the need for full spectral measurements, by combining a physical model for light scattering, a means of predicting perceived colour and a mathematical approximation method. We also use a model system, like that of O’Brien et al., but with a slice of bovine enamel as the scattering layer and coloured paper as the backing. The paper is an analogue for the dentine while its combination with the bovine enamel is an analogue for a tooth. Using a wide range of paper colours, whilst keeping the enamel slice the same, we have been able to measure the influence of the colour of the dentine analogue on the colour of the tooth analogue. This model system has a very abrupt boundary between enamel and dentine. Fortunately, this is also the situation for human teeth.14 In Section 2, we describe our sample preparation and measurement method while, in Section 3, we summarise the key results of our theoretical analysis.13 In Section 4, we present our comparison between the measured L*a*b* values for the dentine analogue and the tooth analogue and the calculated L*a*b* values for the tooth analogue based on our theoretical work. Sections 5 and 6 discuss the results and present conclusions.
2.
Materials and methods
A bovine tooth was provided by the Academic Centre for Dentistry (ACTA), Amsterdam. A section of enamel was cut from the labial surface of the tooth, which had been previously bleached for 50 min using 23% carbamide peroxide in a proprietary preparation to minimise the effect of enamel colour, using a mechanical microtome at a depth of 3–4 mm from the surface. The section was then thinned to about 1 mm, chosen to represent the thickest enamel typically observed in human teeth. A thinner slice would have been desirable but was judged to be mechanically too fragile to prepare. The cut surface was then polished using silicon carbide grits of diminishing size, then 6 mm water-based diamond suspension using a shortnap polishing cloth and finally 1 mm water-based diamond suspension. The other surface was the natural surface of the tooth and hence was curved. Over the measurement area, the thickness of enamel varied from 1 mm at the centre to 0.7 mm at the edge. This single enamel section was used in all the measurements of the tooth analogue. Our coloured backing layers, used as the dentine analogue, were prepared using Microsoft Word’s colour palette to determine the RGB colour and then printed in 4 cm 4 cm squares on heavy weight gloss photo paper using a laser jet printer set to high quality mode. Gloss paper was used as it is largely colour safe when wet, a necessary feature as we wish to compare both desiccated and hydrated tooth analogues. For measurements when desiccated, the printed paper was used ‘‘as is’’, direct from the printer, while the enamel was dried by storing overnight in a ventilated incubator at 37 8C. For hydrated analogues, both enamel and paper were hydrated as follows: the enamel was stored overnight in deionised
journal of dentistry 43 (2015) 373–381
water while the paper was hydrated by adding deionised water in single drops to the rear (non-gloss) surface until the colour darkened uniformly. This slight darkening on hydration is due to reduced light scattering in the paper when the pores between the fibres are filled with water rather than air. This observation will be repeated for hydrated tooth enamel, as we see later. Printed colours are notoriously difficult to reproduce so we do not quote RGB values, choosing instead to label the measured data points with an approximate colour name but to use the ðL ; a ; b Þ values measured on the paper itself as the definition of the actual colour. All colours were measured using a CR-400 chromameter and a DP-400 data processor from Konica-Minolta. Output was directly in ðL ; a ; b Þ format. The measurements used a pulsed Xenon lamp with a type C illuminant characteristic, representing average daylight, and the system illuminated the measurement surface with white light at all angles and collected the reflected light within an 88 cone. For each colour of backing layer and for each combination of enamel and backing layer, the instrument was set to take 3 repeated colour measurements. The sample was then rotated 908 clockwise and a further 3 colour measurements acquired, followed by two further rotations and two further sets of 3 colour measurements. The resulting 12 measurements were then averaged to give a single colour measurement. Standard deviations in measured L* values were typically 0.4 and no worse than 0.8. Standard deviations in a* and b* were typically 0.2 and no worse than 0.6.
3.
Theoretical analysis
As noted above, in order to calculate tooth analogue colour from dentine analogue colour, we require a) a model for the wavelength dependent scattering of light in the bovine enamel and b) a method to relate the spectrum of light reflected by the dentine analogue, and that reflected by the tooth analogue, to the colour perceived by the human visual system and represented in this work by ðL ; a ; b Þ values. This analysis is reported elsewhere13 but is summarised below. Wavelength dependent variables are in bold while constants, or terms with other dependencies, are in normal face.
3.1.
Scattering of light in the enamel
We use the theory of Kubelka and Munk11 to represent scattering in the enamel. They showed that the wavelength-dependent reflectivity, R0 , of a scattering layer on top of an opaque layer of wavelength dependent reflectivity, R, is given by: R0 ¼
ð1 a RÞsinhðb SDÞ þ b Rcoshðb SDÞ ða RÞsinhðb SDÞ þ bcoshðb SDÞ
(1)
where a¼1þ b¼
A S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1
(2) (3)
where S is scattering coefficient for diffuse light (m1); A is absorption coefficient for diffuse light (m1), and D is thickness of the scattering layer.
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O’Brien et al. used the above expression for R0 in their work. However, we find it easier to represent this reflection coefficient using the following identical but alternative definition: 2
R0 ¼ R00 þ
T00 R ð1 R00 RÞ
(4)
R00 ¼
sinhðb SDÞ asinhðb SDÞ þ bcoshðb SDÞ
(5)
T00 ¼
b asinhðb SDÞ þ bcoshðb SDÞ
(6)
Here, R00 and T00 are the wavelength dependent reflection and transmission coefficients of the scattering (bovine enamel) layer. We prefer to use Eq. (4) rather than Eq. (1) as the former is expressed in macroscopic quantities rather than microscopic quantities. Our model for tooth colour is expressed in terms of weighted averages of the reflectivity and transmissivity of the scattering layer, averages which are more difficult to define when using microscopic quantities. However, we can derive equivalent weighted averages of S and A by substituting our results for the weighted averages of R00 and T00 into Eqs. (5) and (6) and solving. We do this in Section 4.1.
3.2. Relationship between perceived colour of the dentine analogue and of the tooth analogue, based on our weak colour approximation Before we can present the summary of our model, it is necessary to discuss how the colour of a surface is perceived by the human visual system and how it can be represented quantitatively. The complexity of our visual system can be illustrated by the ability of our TVs or computer monitors to reproduce vibrant yellow colours while a close inspection of the screen shows that just red, green and blue (RGB) pixels are present. In this paper, it is not possible to describe in depth how the colour that the brain perceives is related to the spectrum of light entering the eye. However, we shall cover enough to understand our results. Practitioners who use the Vita 3D Master shade guide system are already using a system to quantify surface colour, summarised by codes such as 3M2 or 4L1. The first number, called ‘‘Value’’ by Vita, represents the lightness of the colour. The letter, L, M or R, called ‘‘Hue’’ by Vita, represents the extent to which the colour is more yellow than average, average or more red than average respectively. Finally, the second number, called ‘‘Chroma’’ by Vita, represents the vibrancy of the colour, with higher numbers being more vibrant and lower numbers less vibrant. Perceived colour is always represented by three quantities, like Value, Hue, Chroma or RGB for a colour computer monitor or the CIE ðL ; a ; b Þ triplet, because the eye has only 3 types of colour receptors and only three stimuli go to the brain for it to interpret into a colour. As we live in a three dimensional world, colours are almost always represented as positions in ‘‘colour space’’. The fraction of colour space which is required to represent typical human dentition colour is very small compared to the space which represents all possible colours visible to the human eye. Vita’s 3D Master shade guide therefore represents those colours qualitatively in terms of
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deviations from an average. In order to encompass the full range of visible colours, colour scientists use a different measurement system known as Lab colour space. Practitioners using the Vita Easyshade colour measurement device can set their units to give readout in terms of ðL ; a ; b Þ instead of the 3D Master values, thereby showing that there is a oneto-one correspondence between a 3D Master shade value and a ðL ; a ; b Þ value. However, the range of possible ðL ; a ; b Þ values for all visible colours is vastly greater than the range of values observed in human dentition. In our work, we have deliberately used colours which cover a very much wider range than those encountered in natural dentition on the basis that the understanding we have developed for this wide range of colours will apply even more accurately for the small range of colours encountered naturally. Lab colour space is represented in Fig. 1, with axes for L*, a* and b* each at right angles. A colour is represented by how light it is, given by its position along the vertical L* axis from black at the bottom to white at the top, and by the direction and extent of its deviation away from that vertical L* axis, given by the amount of deviation along the a* and b* axes. The direction of the deviation represents the perceived colour and is indicated by a colour wheel divided into six sectors, each labelled with a colour. Colour variation around the wheel is smooth and continuous and our colour labels represent the colour in the centre of each of the six sectors. For this reason, we have called the sectors ‘‘Reds’’, ‘‘Greens’’ etc. to emphasise that they are a progression of colours not a single colour. Vita’s ‘‘Value’’ axis is qualitatively the same as the L* axis, while their ‘‘Hue’’ and ‘‘Chroma’’ quantities are qualitatively similar to mathematically defined measures with the same names shown in Fig. 1, with hue being the angle at the L* axis enclosed by a right angle triangle with a* and b* as its two edges, while chroma is the length of the hypotenuse. Vita’s ‘‘M’’ Hue is a population average hue while ‘‘L’’ is slightly towards yellow (higher hue angles) and ‘‘R’’ slightly towards red (lower hue angles).
However, as we need to define colour quantitatively, we must use the ðL ; a ; b Þ values themselves. L* is a measure of the lightness of a colour, defined in a way that is independent of hue and chroma. Hue and chroma, or alternatively a* and b*, define the shade and vibrancy of the colour and can be mathematically derived from the spectrum of reflected light, as discussed elsewhere13. a* and b* are collectively called the chromaticity. Rather confusingly, a* and b* are different from a and b in Eqs. (1)–(6), where they are simply combinations of S and A. However, we keep this symbology because each is conventional within its own specialism. With the above understanding of ðL ; a ; b Þ as the quantitative measure of perceived colour, L* values of both tooth and dentine analogues are related as follows, according to our Weak Colour Approximation,13 which applies when the colour of both the enamel and the dentine analogue backing layer are not strongly saturated: L ¼ 116R1=3 16 2 T00 R R0 ¼ R00 þ ð1 R00 RÞ 1=3 0 L ¼ 116R0 16
(7)
where L* is lightness of the dentine analogue (coloured paper), L0 is lightness of the tooth analogue (bovine enamel with coloured paper backing), R is a weighted average of R, R0 is a weighted average of R0 , R00 is a weighted average of R00 , and T00 is a weighted average of T00 and where the exact weighted averaging method is defined elsewhere.13 Here, we treat R00 and T00 as fitting parameters. Values of chromaticity for the tooth analogue, ða0 ; b0 Þ, derived from that for the dentine analogue, ða ; b Þ, are given by:
a0 b0
¼
2=3 2 R T00 a 0 2 b R ð1 R00 RÞ 00 2=3 2 00 R T00 R a 1 þ 0 00 b00 R 1R R
(8)
where ða00 ; b00 Þ is the chromaticity of the enamel slice measured with a black backing layer and where we have used a matrix representation of the equations to emphasise the identical nature of the coefficients in expressions for a*0 and b*0 .
4.
Fig. 1 – Representation of ðL0 a0 b0 Þ and (L*, a*, b*) in L*,a*,b* space, with a colour wheel of 6 sectors to indicate hue in terms of the position around the colour wheel and chroma in terms of the radial distance away from the L axis.
Results
Fig. 1 shows points in Lab space representing the colour of the dentine analogue, ðL ; a ; b Þ, and of the tooth analogue, ðL0 ; a0 ; b0 Þ. For all colours used except white, placing the slice of bovine enamel on top of the coloured paper causes the change shown in Fig. 1: (a) the lightness (L*) increases giving an external appearance which is lighter than the underlying dentine analogue, (b) the vibrancy (chroma) of the colour, represented by the distance from the L* axis, reduces giving a whiter appearance while (c) the hue, represented by the position around the colour wheel, is most often broadly unchanged, though it can sometimes change quite markedly, with some yellow dentine analogue colours changing to blue tooth analogue colours, for example, when the enamel is place on top. A strength of our theoretical analysis is its ability to predict even these dramatic colour shifts with accuracy.
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In the next two sub-sections, we discuss the results for lightness, L*, and chromaticity, (a*,b*), respectively.
4.1.
Lightness (L )
Measured values of L0 are plotted against measured values of L* (crosses) for both hydrated and desiccated samples of bovine enamel and coloured paper in Fig. 2 (upper curves, left hand scale). Numerical values for ðL0 a0 b0 Þ and ðL ; a ; b Þ are given in tabular form elsewhere13 and are not repeated here. In spite of the wide range of colours used, many of which were quite vibrant, the points all lie on a smooth curve for each of the hydrated and desiccated samples. The smooth curve has been fitted using Eq. (7) using fitting parameters, with R00 representing the averaged reflectivity of the enamel layer, and a quantity, L ¼ 1 R00 T00 , representing the averaged fraction of light loss in the enamel layer. The errors between the measured data values and the theoretical fit are no more than 2% for any of the points shown in Fig. 2. The results of the theory alone are also plotted in Fig. 2 (lower curves, right hand scale) to illustrate the range of predicted relationships between L0 and L* for three different values for the averaged reflectivity of the enamel layer (R00 ) and two different values for the averaged fraction of light loss in the enamel layer (L). The averaged reflectivity is dominant in determining the lightness of the tooth analogue when the dentine analogue is very dark, as the dentine itself makes only a small contribution to the overall lightness. The fraction of light loss in the enamel is dominant in determining the lightness of the tooth analogue when the dentine analogue is
very light. This can be understood in the following way. If the dentine itself is a perfect reflector (L ¼ 100) and, simultaneously, the enamel is also lossless (L = 0), then the combination must also be a perfect reflector (L0 ¼ 100) as neither the enamel nor the dentine causes light loss. This conclusion is valid for all values of the averaged enamel reflectivity, R00 . Hence, all curves with L = 0 go through the point (100, 100), independent of the averaged reflectivity. It is then easy to see that an increase in loss predominantly affects the tooth lightness at high dentine lightness as the reflectivity itself has no effect. These curves also illustrate the simplicity of the fitting process: R00 predominantly comes from fitting data points with dark dentine and L from fitting data points with light dentine. The fitting parameters used in Fig. 2 are given in Table 1 below. These values can be substituted into equations equivalent to Eqs. (5) and (6), but expressed in terms of suitable weighted averages of A and S, and a 2D Newton– Raphson method used to find the resulting solutions, A and S, using an average D of 0.9 mm. These values are given in Table 2. The absorption coefficient compares well to that measured by Spitzer and ten Bosch15, who measured a decadic absorption coefficient of about 25 m1 for a 0.17 mm thick slice of hydrated bovine enamel at 550 nm. This wavelength is close to the maximum in the weighting function used to define A and S13 and hence a reasonable comparison to our own values. Their decadic coefficient is equivalent to a Napierian absorption coefficient, used here, of 58 m1, a value close to our own observations. However, agreement is less good for the scattering coefficient as they measured a decadic scattering coefficient of about 1700 m1 for the same slice of bovine enamel, also at 550 nm. This is equivalent to a Napierian scattering coefficient of 3900 m1, almost ten times our own measurement. However, the differences appear justifiable as natural variations between enamel characteristic as the slice use by Spitzer and ten Bosch had a reflectivity of around 0.4 even for a slice as thin as 0.17 mm while our much thicker slice has a reflectivity with black backing of 0.265. (Here we assume that our measurements with black backing are comparable to Spitzer and ten Bosch’s measurements without backing at all. This is correct provided light passing through the enamel of Spitzer and ten Bosch did not make its way back to the detector in some way. Without detailed knowledge of their experimental set up, however, this is impossible to judge.)
Table 1 – Fitting parameters used to derive the solid lines in Fig.2. Fig. 2 – Upper plot, left hand axis: Measured L0 against measured L for desiccated and hydrated samples (crosses). Labels indicate w = white, y = yellow, o = orange, r = red, g = green, b = blue, v = violet, i = indigo and bl = black. Numerical values are given elsewhere.13 Theoretical fitted lines (solid) use averaged reflectivity, loss and transmissivity fitting parameters, R00 , L and T00 , given below the line labels. Lower plot, right hand axis: Theoretical relationship between L0 and L from Eq. (7) using 3 different values of R00 and loss factors of 0 (solid lines) and 0.1 (dashed lines).
Hydrated (Fig. 2) Desiccated (Fig. 2)
R00
T00
L ¼ 1 R00 T00
0.265 0.605
0.665 0.32
0.07 0.075
Table 2 – Napierian scattering and absorption coefficients for the bovine enamel.
Hydrated (Fig. 2) Desiccated (Fig. 2)
S(m1)
A(m1)
438 2001
81 89
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4.2.
journal of dentistry 43 (2015) 373–381
Chromaticity (a*,b*)
a* and b* define the hue and vibrancy (chroma) of the dentine and tooth analogues. In Fig. 3, measured values of ða0 b0 Þ (pluses) and ða ; b Þ (circles) for hydrated samples only are plotted on the a*, b* plane for a different set of dentine analogue colours from those shown in Fig. 2. Arrows link pairs of data points using the same backing layer colour and labels close to the ða ; b Þ point indicate the approximate colour. Numerical values are again given elsewhere13 and are not repeated here. It is clear from Fig. 3 that there is a general inward radial movement from the dentine analogue to the tooth analogue. However, rather than directly towards the origin, the radial movement is broadly towards a point located at roughly (2,6) with the result that some backing layer colours, such as light yellow (ly), actually acquire a blue hue when the bovine enamel is superimposed. Furthermore, some colours, such as cherry and magenta (ch and m) seem to move towards a point much closer to the origin than the other colours. Our theory13 is capable of predicting these changes in chromaticity to good accuracy with no additional fitting parameters, other than the ones already used to fit the L* data. This theory shows that the a* value for the tooth analogue is simply the sum of the a* for the dentine analogue and a fixed a* value for the enamel, with each multiplied by a weighting factor between 0 and 1. Here, the a* value for the enamel is measured when placed on a perfectly black surface. The b* value for the tooth analogue is related to the b* values for the dentine analogue and for the enamel in exactly the same way with exactly the same weighting factors. These
Fig. 3 – Measured ða0 b0 Þ (pluses) and measured ða ; b Þ (circles) for hydrated samples only for a different set of dentine analogue backing colours from those of Fig. 2. Arrows link pairs of data points using the same dentine analogue colours, and labels close to the ða ; b Þ point indicate approximate colour using y = yellow, o = orange, r = red, ch = cherry, m = magenta, i = indigo, b = blue, ccyan, pg = pea green, g = green and l in front of these labels are similar hues but with lower chromaticity. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
2=3
2
2
weighting factors are the coefficients ðR=R0 Þ ðT00 =ð1 R00 RÞ Þ 2=3 2 and ðR00 =R0 Þ ð1 ðT00 R=ð1 ðR00 RÞÞÞ in Eq. (8) and are given in graphical form in Fig. 4 for a fixed value of loss factor, L, of 0.1, very close to the measured values of 0.07 and 0.075. In order to understand how the weighting factors help us determine what happens to the observed colour when the bovine enamel is placed on top of the coloured backing layer, we need measured values for ða00 ; b00 Þ. These were measured with the enamel slice superimposed on a matt black surface to be (2.2,4.9) when the enamel is hydrated and (1.7,3.1) when the enamel is desiccated. A comparison to the points marked with circles in Fig. 3 shows that ða00 ; b00 Þ is much smaller than ða ; b Þ for all the colours used. With this in mind, we can use Fig. 4 to make the following qualitative statements (and here we drop the use of the word analogue as these statements should also apply in vivo):
When the dentine is light (L* is high), the tooth chromaticity is dominated by that of the dentine. This is because the weighting factor for the dentine chromaticity is much greater than the weighting factor for the enamel chromaticity, which is itself small. An increase in the reflectivity of the enamel, for example by dehydrating the tooth, rapidly reduces the contribution of dentine chromaticity to that of the tooth when the dentine itself is light. This is because the weighting factor for dentine chromaticity reduces quickly as R00 increases. As the dentine darkens (L* reduces), the chromaticity of the tooth decreases at fixed dentine chromaticity. This occurs more quickly when the enamel reflectivity is high, for example with dehydrated enamel, than when the reflectivity is low, for example with hydrated enamel. This is because the slope of the weighting factor for dentine
Fig. 4 – Weighting factors for the dentine chromaticity (solid lines) and the enamel chromaticity (dashed lines) which, when applied to each chromaticity value respectively and summed, gives the chromaticity for the tooth. The lines are for a constant fraction of light loss = 0.1, with the averaged reflectivity, R00 , varied from 0.1 (extreme left hand curves), in steps of 0.1, to 0.7 (extreme right hand curves). The associated averaged transmissivity varies from 0.8 to 0.2.
journal of dentistry 43 (2015) 373–381
chromaticity in overall tooth chromaticity is steep for high R00 but more gradual when R00 is low). The relative contribution of enamel chromaticity and dentine chromaticity to the overall tooth chromaticity when the dentine is dark (low L*) cannot easily be judged without an actual calculation. Although the weighting factor for the enamel chromaticity is much greater than for the dentine chromaticity, the enamel chromaticity itself is small compared to the dentine chromaticity. It is for this reason that, in Fig. 3, while we see many of the chromaticity values move towards (2,6) when the enamel is superimposed on the backing layer, some colours, notably cherry and magenta, move in a slightly different direction. In some cases, the chromaticity of the dentine remains big enough for the tooth to retain a similar hue to the dentine but in other cases, notably light yellow (ly in Fig. 3), the contribution from the chromaticity of the enamel is sufficient to move the tooth chromaticity to a completely different hue, so the tooth has a blue hue in spite of the dentine being yellow. Our theory reproduces all these qualitative observations quantitatively. A comparison between measured (+) and predicted (x) values of ða0 b0 Þ is shown in Fig. 5. Here, predicted values are based only on measured values of ða ; b Þ and ða00 ; b00 Þ and on values for the weighting factors similar to those in Fig. 4 but with L = 0.06 rather than 0.1. Values of R00 , L and hence also of T00 are found to be 0.29, 0.06 and 0.65, respectively13 by fitting Eq. (7) to L* data similar to Fig. 2 and are
Fig. 5 – Pairs of measured (+) and predicted (x) values of ða0 b0 Þ where the pairs are indicated by the oval. Predicted values are based on measured values for ða ; b Þ and ða00 ; b00 Þ and the weighting factors given in Fig. 4 (with L = 0.06). The appropriate values of R00 and L are chosen from fitting our theory to the L data in a way similar to Fig. 2. Labels indicate the approximate colour using y = yellow, o = orange, r = red, ch = cherry, m = magenta, i = indigo, b = blue, c-cyan, pg = pea green, g = green and points with l in front of these labels are similar hues but with lower chromaticity. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
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used without further adjustment. In Fig. 5, pairs of measured and predicted values of ða0 b0 Þ are grouped by ovals. Differences between measured and predicted values of ða0 b0 Þ are small, at about 8% of the magnitude of ða ; b Þ. 0 0 Absolute a and b values for the tooth analogue are predicted typically to within less than 3 and no worse than 5. Even the colours such as light yellow, cherry and magenta, picked out earlier as having apparently abnormal behaviour, are modelled to this accuracy by our weighting factors. Finally, Fig. 6 shows the effect on chromaticity of desiccating the samples. Here, the colours are the same as those shown in Fig. 2, where we saw that desiccating the enamel led to higher reflectivity and hence a much lighter colour. From Fig. 6, we can see how this higher reflectivity also reduces the measured chromaticity of the tooth much more than when the enamel reflectivity is less. We noted above how the weighting factors change when the enamel reflectivity changes and we can see from the similarity between measured and predicted results for all colours, whether desiccated or hydrated, that our model reproduces this behaviour. (The degree of agreement between measured and predicted values of chromaticity in Fig. 6 is not as good as in Fig. 3. This is actually expected from our theoretical work because the colours used in Fig. 6 are more strongly saturated. This is discussed in more detail elsewhere.13) The model therefore not only predicts the measured colours with hydrated enamel but can also accurately predict the effect of using desiccated enamel.
Fig. 6 – Pairs of measured (+) and predicted (x) values of ða0 b0 Þ where the pairs are indicated by the oval. Solid ovals are for hydrated samples while the dashed oval indicates the samples have been desiccated. Predicted values are based on measured values for ða ; b Þ and ða00 ; b00 Þ and the weighting factors given in Fig. 4 (with L = 0.07 for hydrated samples and 0.075 for desiccated samples). The appropriate values of R00 and L are those given in Fig. 2, as these results are for the same samples as in that figure. Labels indicate the approximate colour using y = yellow, o = orange, r = red, g = green, b = blue, v = violet, i = indigo. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
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5.
journal of dentistry 43 (2015) 373–381
Discussion
Unless caused by trauma, no one has cherry or magenta coloured dentine and no practitioner has ever encountered cyan coloured dentine. The reason why we have used such a wide range of dentine analogue colours for our backing layers is to test, with much more extreme colours than practitioners will encounter in practice, the predictive accuracy of our theoretical model, of the reflectivity and loss fractions derived from the fitting process shown in Fig. 2 and of the weighting factors shown in Fig. 4. This should give confidence in their use with the much narrower range of colours typically found in human dentine. Indeed, our model is based on an approximation which becomes more accurate the less vibrant the colour of dentine analogue backing layer used. (This is the reason why the agreement between measured and predicted tooth analogue colours in Fig. 6 is worse than in Fig. 3 as the colours used in Fig. 6 are more vibrant than those in Fig. 3.) So, our model can be expected to be even more accurate when applied to natural dentine colours than to the more extreme colours used here. More needs to be done to make the model we have developed find practical uses. Our aim has been to develop a method to predict the baseline tooth colour in humans arising from coloured dentine visible through the translucent enamel and to which extrinsic sources of discolouration will add. Our work therefore needs to be repeated using human tooth enamel rather than bovine enamel and with a range of different enamel thicknesses so that we can test the validity of the K-M Model. With these in place, it should be possible to take a single measurement of tooth colour and, based on the expected enamel thickness at the point of measurement, work out the expected colour of dentine under the assumption that extrinsic causes of colour are negligible. Furthermore, it should be possible to measure dentine colour, when exposed during restorative dentistry for example, and infer the likely tooth colour from that dentine alone across the rest of the dentition under the assumption that the exposed dentine colour was typical for all teeth in that person. Finally, in a different context, it should be possible to prepare porcelain veneers which accurately match the surrounding dentition, even though some of the colour from the underlying tooth or bonding material is actually visible through the porcelain, by accurately predicting the colour which would be seen through the porcelain and hence adjusting the colour of the porcelain itself to correct for this. Nonetheless, our work already highlights trends which are clinically relevant and can be helpful to both the practitioner and clinical researcher and we highlight these here (once again dropping the word analogue).
1. Fig. 2 shows how the greater enamel reflectivity which occurs when the enamel is desiccated results in a higher lightness value for a given colour of dentine. Any tooth whitening technique which partially dries the surface enamel of the tooth can therefore result in elevated lightness values which quickly fall as the enamel rehydrates. In-surgery measurements of post-treatment light-
ness, when there is a risk of enamel dehydration during treatment, should therefore be made only after a suitable delay to allow rehydration to take place. 2. During whitening, the rate at which the lightness of the tooth increases as the dentine itself gets lighter, shown in the data for hydrated enamel in Fig. 2, is steeper at light dentine colours than at darker dentine colours. This suggests that teeth with dentine which is already light in colour are easier to whiten, at least in terms of the change in dentine lightness required to achieve a given change in tooth lightness, than teeth with darker dentine. This does not necessarily mean that lighter coloured teeth require less bleaching time to lighten them than darker teeth, as the chemistry of the bleaching process might be faster with dark dentine, thereby compensating for the effect we point out here. Nevertheless, it is possible that early intervention with less severe whitening regimes may be more effective in delivering whitened teeth than later intervention, even with more severe regimes. 3. Although we have worked only with a single thickness of enamel, Eq. (5) shows that the reflectivity of the enamel will be greater with thicker enamel and less with thinner enamel. The same will be true for the loss from Eq. (6). Enamel thickness varies in human teeth, being thickest in the middle of a face and towards the tip or crown and thinning both towards the cementoenamel junction (where it falls to zero) and both mesially and distally. Our understanding of the influence of enamel reflectivity on tooth lightness suggests that, provided intrinsic sources of colour from the dentine dominate, the lightness of the tooth itself should have a similar pattern, being lightest where the enamel is thick and darkening in the directions in which the enamel thins. This type of lightness distribution, darkening towards the cementoenamel junction and both mesially and distally, is indeed seen in practice. 4. The weighting factors in Fig. 4 show how the tooth chromaticity for fixed dentine lightness decreases with increased enamel reflectivity (at least when the dentine colour is still the dominant factor in determining tooth colour). This suggests that the chromaticity over the tooth surface should also follow the same pattern as the lightness, being least coloured (i.e. closest to white) where the enamel is thickest and more coloured where the enamel is thin. Once again, this pattern is observed in practice.
6.
Conclusions
We have developed a theoretical model which can predict the colour of a tooth, when extrinsic sources of colouration are negligible, from the light scattering (reflectivity) and loss fractions in the enamel and from the colour of the dentine. The actual theory is complex, owing to the complexity of the human visual system. Quantitative prediction of how the human visual system perceives colours, and how those perceived colours change when scattering in the enamel changes the spectrum of light incident on the eye, requires both a knowledge of colour science and the use of mathematics which is likely to obscure more than enlighten. We have therefore chosen to publish our theoretical model elsewhere13
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and use this paper to highlight (a) the evidence of its predictive accuracy and (b) the clinical relevance of the predictions it makes. Further measurements on human enamel and with a range of different enamel thicknesses are required in order to make the model we have developed more applicable in practical situations. Nevertheless, the predicted trends in lightness and chromaticity that the model provides have been used in this paper to highlight some clinically relevant outcomes.
Acknowledgements The authors would like to acknowledge Dr. Nigel Young for suggesting the model system we have used for our measurements and Prof. Dr. Ingrid Heynderickx for her careful coaching on the human visual system and on colour perception. Figures 3, 5 and 6 are reprinted by permission of John Wiley & Sons, Inc. from ref. 13. Copyright # 2014 by John Wiley & Sons, Inc.
references
1. Watts A, Addy M. Tooth discolouration and staining: a review of the literature. British Dental Journal 2001:309–16. 2. Joiner A. Tooth colour: a review of the literature. Journal of Dentistry 2004;32:3–12. 3. Joiner A, Hopkinson I, Deng Y, Westland S. A review of tooth colour and whiteness. Journal of Dentistry 2008;36S:S2–S7. 4. Joiner A. Review of the extrinsic stain removal and enamel/ dentine abrasion by a calcium carbonate and perlite containing whitening toothpaste. International Dental Journal 2006:175–80.
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5. Leonard R, Sharma A, Haywood V. Use of different concentrations of carbamide peroxide for bleaching teeth: an in vitro study. Quintessence International 1998:503–7. 6. Lin C, Chou T, Chen J, Chen J, Chuang F, Lee H, et al. Evaluation of the effect of laser tooth whitening. International Journal of Prosthodontology 2008:415–8. 7. ten Bosch JJ, Coops JC. Tooth colour and reflectance as related to light scattering and enamel hardness. Journal of Dental Research 1995;74:374–80. 8. Jahangiri L, Reinhardt SB, Mehra RV, Matheson PB. Relationship between tooth shade value and skin colour: an observational study. Journal of Prosthetic Dentistry 2002;87: 149–52. 9. Goodkind RJ, Keenan K, Schwabacher WB. Use of a fibreoptic colorimeter for an in vivo colour measurement of 2830 anterior teeth. Journal of Prosthetic Dentistry 1987;58: 535–42. 10. O’Brien WJ, Johnston WM, Fanian F. Double-layer color effects in porcelain systems. Journal of Dental Research 1985;64:940–3. 11. Kubelka P. New contributions to the optics of intensely light scattering materials. Part 1. Journal of the Optical Society of America 1948;38:448–57. 12. Commission Internationale de l’Eclairage. Colorimetry, Official Recommendations of the International Commission on Illumination. Publication CIE No. 15 (E-1.3.1) Paris, France: Bureau Centrale de la CIE; 1971. 13. Battersby S, Battersby P. An optical model relating (L*a*b*) values for a scattering surface covered with a translucent layer to (L*a*b*) values for the uncovered surface and its application to tooth colour. Color Research and Application 2014. 14. Cho YW, Cho JH, Lee KB. A study on enamel thickness of maxillary incisors using X-ray micro computed tomography. The Journal of Korean Academy of Prosthodontic 2010;48:301–7. 15. Spitzer D, ten Bosch JJ. The absorption and scattering of light in bovine and human dental enamel. Calcified Tissue Research 1975;17:129–37.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.intl.elsevierhealth.com/journals/jden
Measurements and modelling of the influence of dentine colour and enamel on tooth colour Paul D. Battersby 1, Stephen J. Battersby * Philips Research Laboratories, 101 Cambridge Science Park, Milton Road, Cambridge CB4 0FY, UK
article info
abstract
Article history:
Objectives: We provide a quantitative predictive model for the extent to which coloured
Received 5 February 2014
dentine, visible through the enamel, contributes to tooth colour. Our model uses ðL ; a ; b Þ
Received in revised form
measurements rather than spectral measurements.
21 October 2014
Methods: We have used a model system, composed of a slice of bovine enamel placed on top
Accepted 7 November 2014
of coloured paper. We have measured the colour of the enamel–paper combination, as an analogue for a tooth, and have related this to the colour of the paper, as an analogue for dentine. By changing the paper colour, we have been able to explore how the colour of
Keywords:
dentine determines tooth colour, according to our model system. We have also compared
Tooth colour
hydrated and desiccated samples.
Colour appearance model
Results: In qualitative terms, superimposing the enamel on top of the paper increases the
Dentine colour
‘‘lightness’’ for all colours tested except white while simultaneously reducing the chroma-
Enamel colour
ticity, a measure of the extent to which the colour differs from grey. Desiccated enamel is
Tooth discolouration
much more effective at increasing the lightness and reducing the chromaticity than
Tooth whitening
hydrated enamel. Quantitatively, our measurements are reproduced by the mathematical model we have developed to within 2% in ‘‘lightness’’ and about 8% in chromaticity. Conclusions: We are able to predict the colour of an analogue for a tooth, composed of bovine enamel and coloured paper, from the colour of an analogue for the dentine, the coloured paper alone, with good accuracy. This understanding provides insights into the role of dentine colour in determining tooth colour. Clinical Significance: Our work helps quantify the importance of dentine colour, compared to other, extrinsic causes of colour, such as staining, in determining the visible colour of teeth. Our predicted colours represent a baseline to which extrinsic sources will add. # 2015 Published by Elsevier Ltd.
1.
Introduction
There are many potential causes of tooth discolouration, including extrinsic causes such as surface staining and enamel discoloration through smoking or drinking tea, coffee or wine, and intrinsic causes such as a degradation in the colour of the
dentine which is visible externally through the enamel.1,2,3 Some of these discoloration mechanisms are short term and reversible with suitable cleaning.4 However, other causes are semi-permanent, reversible only through tooth bleaching procedures using, for example, carbamide peroxide and commonly known as tooth whitening.5,6 This paper focuses on the role of dentine colour in determining tooth colour. We are
* Corresponding author. Tel.: +44 1223 427500; fax: +44 1223 452550. E-mail address: [email protected] (S.J. Battersby). 1 Current address: School of Dentistry, Cardiff University, Heath Park, Cardiff, Wales CF14 4XY, UK. http://dx.doi.org/10.1016/j.jdent.2014.11.003 0300-5712/# 2015 Published by Elsevier Ltd.
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journal of dentistry 43 (2015) 373–381
able to predict tooth colour due to dentine colour alone, with both colours measured using simple colourimeter measurements such as ðL ; a ; b Þ, rather than full spectral measurements. This predicted tooth colour represents a baseline to which extrinsic causes of discolouration will add. As the extent of discolouration due to the latter cannot easily be quantified, we cannot fully resolve the debate over the importance of dentine colour in the overall appearance of human dentition. However, our baseline colour predictions should allow practitioners and researchers to develop an understanding of the relative importance of intrinsic and extrinsic colouration sources in that overall appearance. Enamel is a translucent scattering medium which, at the thicknesses present in human teeth, does not fully obscure the colour of the underlying dentine. For example, an in vitro study of the colour of 28 teeth from different patients showed strong correlation with dentine colour when the overlying enamel was removed.7 Degradation in the colour of dentine, for example with age, can therefore lead to tooth discolouration.8,9 This is exacerbated when thinning of the enamel takes place or when gum recession occurs, thereby exposing tooth surface closer to the cementoenamel junction and hence covered in thinner enamel.7 An extreme example of the role of dentine colour occurs when blood escapes from the blood vessels in the tooth and perfuses the dentine, leading to severe discolouration of the dentine. Here, the external appearance of the tooth is also considerably darkened. In order to understand these effects, a theoretical model, able to predict tooth colour from dentine colour and enamel characteristics, is desirable. However, such a model faces three challenges, two theoretical and one practical. Firstly, it is necessary to identify a model for the wavelength dependent scattering of light in the enamel and hence be able to predict the spectrum of light emerging from the enameldentine combination from that emerging from the dentine alone. Secondly, it is necessary to relate those spectra to the colour perceived by the human visual system, with its associated complexities. Finally, for practical application, it is necessary to recognise that the measurement of spectral reflectivities using a spectrophotometer, whilst possible in the laboratory, is unlikely in the dental surgery/office and that the terminology used in optical physics is alien to clinical dentistry. A more straightforward measurement, using the colourimeters already in routine use in the surgery/office, is to be preferred. Our work addresses all three challenges. O’Brien et al.10 tackled the first two challenges using a model system composed of dental porcelain to represent the enamel and coloured opaque backing layers to represent the dentine. They used the so-called two flux model of Kubelka and Munk11 (K-M model) to represent scattering in the porcelain but measured the full reflectance spectra for both the porcelain and the backing layers using a spectrophotometer. They then compared the measured and predicted CIE colour parameters of luminous reflectance, dominant wavelength and excitation purity.12 However, as noted above, measuring these spectra is impractical in the dental surgery/ office and inferring it from measured L*a*b* values is impossible due to the weighted averaging used to derive these values. Fortunately, the colour saturation levels typically exhibited by human dentine and enamel are sufficiently low that an approximation13 can be used to relate directly the L*a*b* values
for a coloured surface such as dentine to the equivalent values when the coloured surface is covered by a scattering layer such as enamel. The aim of our work is to show that it is possible to predict tooth colour measured directly with a colourimeter from equivalent measurements on the dentine, without the need for full spectral measurements, by combining a physical model for light scattering, a means of predicting perceived colour and a mathematical approximation method. We also use a model system, like that of O’Brien et al., but with a slice of bovine enamel as the scattering layer and coloured paper as the backing. The paper is an analogue for the dentine while its combination with the bovine enamel is an analogue for a tooth. Using a wide range of paper colours, whilst keeping the enamel slice the same, we have been able to measure the influence of the colour of the dentine analogue on the colour of the tooth analogue. This model system has a very abrupt boundary between enamel and dentine. Fortunately, this is also the situation for human teeth.14 In Section 2, we describe our sample preparation and measurement method while, in Section 3, we summarise the key results of our theoretical analysis.13 In Section 4, we present our comparison between the measured L*a*b* values for the dentine analogue and the tooth analogue and the calculated L*a*b* values for the tooth analogue based on our theoretical work. Sections 5 and 6 discuss the results and present conclusions.
2.
Materials and methods
A bovine tooth was provided by the Academic Centre for Dentistry (ACTA), Amsterdam. A section of enamel was cut from the labial surface of the tooth, which had been previously bleached for 50 min using 23% carbamide peroxide in a proprietary preparation to minimise the effect of enamel colour, using a mechanical microtome at a depth of 3–4 mm from the surface. The section was then thinned to about 1 mm, chosen to represent the thickest enamel typically observed in human teeth. A thinner slice would have been desirable but was judged to be mechanically too fragile to prepare. The cut surface was then polished using silicon carbide grits of diminishing size, then 6 mm water-based diamond suspension using a shortnap polishing cloth and finally 1 mm water-based diamond suspension. The other surface was the natural surface of the tooth and hence was curved. Over the measurement area, the thickness of enamel varied from 1 mm at the centre to 0.7 mm at the edge. This single enamel section was used in all the measurements of the tooth analogue. Our coloured backing layers, used as the dentine analogue, were prepared using Microsoft Word’s colour palette to determine the RGB colour and then printed in 4 cm 4 cm squares on heavy weight gloss photo paper using a laser jet printer set to high quality mode. Gloss paper was used as it is largely colour safe when wet, a necessary feature as we wish to compare both desiccated and hydrated tooth analogues. For measurements when desiccated, the printed paper was used ‘‘as is’’, direct from the printer, while the enamel was dried by storing overnight in a ventilated incubator at 37 8C. For hydrated analogues, both enamel and paper were hydrated as follows: the enamel was stored overnight in deionised
journal of dentistry 43 (2015) 373–381
water while the paper was hydrated by adding deionised water in single drops to the rear (non-gloss) surface until the colour darkened uniformly. This slight darkening on hydration is due to reduced light scattering in the paper when the pores between the fibres are filled with water rather than air. This observation will be repeated for hydrated tooth enamel, as we see later. Printed colours are notoriously difficult to reproduce so we do not quote RGB values, choosing instead to label the measured data points with an approximate colour name but to use the ðL ; a ; b Þ values measured on the paper itself as the definition of the actual colour. All colours were measured using a CR-400 chromameter and a DP-400 data processor from Konica-Minolta. Output was directly in ðL ; a ; b Þ format. The measurements used a pulsed Xenon lamp with a type C illuminant characteristic, representing average daylight, and the system illuminated the measurement surface with white light at all angles and collected the reflected light within an 88 cone. For each colour of backing layer and for each combination of enamel and backing layer, the instrument was set to take 3 repeated colour measurements. The sample was then rotated 908 clockwise and a further 3 colour measurements acquired, followed by two further rotations and two further sets of 3 colour measurements. The resulting 12 measurements were then averaged to give a single colour measurement. Standard deviations in measured L* values were typically 0.4 and no worse than 0.8. Standard deviations in a* and b* were typically 0.2 and no worse than 0.6.
3.
Theoretical analysis
As noted above, in order to calculate tooth analogue colour from dentine analogue colour, we require a) a model for the wavelength dependent scattering of light in the bovine enamel and b) a method to relate the spectrum of light reflected by the dentine analogue, and that reflected by the tooth analogue, to the colour perceived by the human visual system and represented in this work by ðL ; a ; b Þ values. This analysis is reported elsewhere13 but is summarised below. Wavelength dependent variables are in bold while constants, or terms with other dependencies, are in normal face.
3.1.
Scattering of light in the enamel
We use the theory of Kubelka and Munk11 to represent scattering in the enamel. They showed that the wavelength-dependent reflectivity, R0 , of a scattering layer on top of an opaque layer of wavelength dependent reflectivity, R, is given by: R0 ¼
ð1 a RÞsinhðb SDÞ þ b Rcoshðb SDÞ ða RÞsinhðb SDÞ þ bcoshðb SDÞ
(1)
where a¼1þ b¼
A S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 1
(2) (3)
where S is scattering coefficient for diffuse light (m1); A is absorption coefficient for diffuse light (m1), and D is thickness of the scattering layer.
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O’Brien et al. used the above expression for R0 in their work. However, we find it easier to represent this reflection coefficient using the following identical but alternative definition: 2
R0 ¼ R00 þ
T00 R ð1 R00 RÞ
(4)
R00 ¼
sinhðb SDÞ asinhðb SDÞ þ bcoshðb SDÞ
(5)
T00 ¼
b asinhðb SDÞ þ bcoshðb SDÞ
(6)
Here, R00 and T00 are the wavelength dependent reflection and transmission coefficients of the scattering (bovine enamel) layer. We prefer to use Eq. (4) rather than Eq. (1) as the former is expressed in macroscopic quantities rather than microscopic quantities. Our model for tooth colour is expressed in terms of weighted averages of the reflectivity and transmissivity of the scattering layer, averages which are more difficult to define when using microscopic quantities. However, we can derive equivalent weighted averages of S and A by substituting our results for the weighted averages of R00 and T00 into Eqs. (5) and (6) and solving. We do this in Section 4.1.
3.2. Relationship between perceived colour of the dentine analogue and of the tooth analogue, based on our weak colour approximation Before we can present the summary of our model, it is necessary to discuss how the colour of a surface is perceived by the human visual system and how it can be represented quantitatively. The complexity of our visual system can be illustrated by the ability of our TVs or computer monitors to reproduce vibrant yellow colours while a close inspection of the screen shows that just red, green and blue (RGB) pixels are present. In this paper, it is not possible to describe in depth how the colour that the brain perceives is related to the spectrum of light entering the eye. However, we shall cover enough to understand our results. Practitioners who use the Vita 3D Master shade guide system are already using a system to quantify surface colour, summarised by codes such as 3M2 or 4L1. The first number, called ‘‘Value’’ by Vita, represents the lightness of the colour. The letter, L, M or R, called ‘‘Hue’’ by Vita, represents the extent to which the colour is more yellow than average, average or more red than average respectively. Finally, the second number, called ‘‘Chroma’’ by Vita, represents the vibrancy of the colour, with higher numbers being more vibrant and lower numbers less vibrant. Perceived colour is always represented by three quantities, like Value, Hue, Chroma or RGB for a colour computer monitor or the CIE ðL ; a ; b Þ triplet, because the eye has only 3 types of colour receptors and only three stimuli go to the brain for it to interpret into a colour. As we live in a three dimensional world, colours are almost always represented as positions in ‘‘colour space’’. The fraction of colour space which is required to represent typical human dentition colour is very small compared to the space which represents all possible colours visible to the human eye. Vita’s 3D Master shade guide therefore represents those colours qualitatively in terms of
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deviations from an average. In order to encompass the full range of visible colours, colour scientists use a different measurement system known as Lab colour space. Practitioners using the Vita Easyshade colour measurement device can set their units to give readout in terms of ðL ; a ; b Þ instead of the 3D Master values, thereby showing that there is a oneto-one correspondence between a 3D Master shade value and a ðL ; a ; b Þ value. However, the range of possible ðL ; a ; b Þ values for all visible colours is vastly greater than the range of values observed in human dentition. In our work, we have deliberately used colours which cover a very much wider range than those encountered in natural dentition on the basis that the understanding we have developed for this wide range of colours will apply even more accurately for the small range of colours encountered naturally. Lab colour space is represented in Fig. 1, with axes for L*, a* and b* each at right angles. A colour is represented by how light it is, given by its position along the vertical L* axis from black at the bottom to white at the top, and by the direction and extent of its deviation away from that vertical L* axis, given by the amount of deviation along the a* and b* axes. The direction of the deviation represents the perceived colour and is indicated by a colour wheel divided into six sectors, each labelled with a colour. Colour variation around the wheel is smooth and continuous and our colour labels represent the colour in the centre of each of the six sectors. For this reason, we have called the sectors ‘‘Reds’’, ‘‘Greens’’ etc. to emphasise that they are a progression of colours not a single colour. Vita’s ‘‘Value’’ axis is qualitatively the same as the L* axis, while their ‘‘Hue’’ and ‘‘Chroma’’ quantities are qualitatively similar to mathematically defined measures with the same names shown in Fig. 1, with hue being the angle at the L* axis enclosed by a right angle triangle with a* and b* as its two edges, while chroma is the length of the hypotenuse. Vita’s ‘‘M’’ Hue is a population average hue while ‘‘L’’ is slightly towards yellow (higher hue angles) and ‘‘R’’ slightly towards red (lower hue angles).
However, as we need to define colour quantitatively, we must use the ðL ; a ; b Þ values themselves. L* is a measure of the lightness of a colour, defined in a way that is independent of hue and chroma. Hue and chroma, or alternatively a* and b*, define the shade and vibrancy of the colour and can be mathematically derived from the spectrum of reflected light, as discussed elsewhere13. a* and b* are collectively called the chromaticity. Rather confusingly, a* and b* are different from a and b in Eqs. (1)–(6), where they are simply combinations of S and A. However, we keep this symbology because each is conventional within its own specialism. With the above understanding of ðL ; a ; b Þ as the quantitative measure of perceived colour, L* values of both tooth and dentine analogues are related as follows, according to our Weak Colour Approximation,13 which applies when the colour of both the enamel and the dentine analogue backing layer are not strongly saturated: L ¼ 116R1=3 16 2 T00 R R0 ¼ R00 þ ð1 R00 RÞ 1=3 0 L ¼ 116R0 16
(7)
where L* is lightness of the dentine analogue (coloured paper), L0 is lightness of the tooth analogue (bovine enamel with coloured paper backing), R is a weighted average of R, R0 is a weighted average of R0 , R00 is a weighted average of R00 , and T00 is a weighted average of T00 and where the exact weighted averaging method is defined elsewhere.13 Here, we treat R00 and T00 as fitting parameters. Values of chromaticity for the tooth analogue, ða0 ; b0 Þ, derived from that for the dentine analogue, ða ; b Þ, are given by:
a0 b0
¼
2=3 2 R T00 a 0 2 b R ð1 R00 RÞ 00 2=3 2 00 R T00 R a 1 þ 0 00 b00 R 1R R
(8)
where ða00 ; b00 Þ is the chromaticity of the enamel slice measured with a black backing layer and where we have used a matrix representation of the equations to emphasise the identical nature of the coefficients in expressions for a*0 and b*0 .
4.
Fig. 1 – Representation of ðL0 a0 b0 Þ and (L*, a*, b*) in L*,a*,b* space, with a colour wheel of 6 sectors to indicate hue in terms of the position around the colour wheel and chroma in terms of the radial distance away from the L axis.
Results
Fig. 1 shows points in Lab space representing the colour of the dentine analogue, ðL ; a ; b Þ, and of the tooth analogue, ðL0 ; a0 ; b0 Þ. For all colours used except white, placing the slice of bovine enamel on top of the coloured paper causes the change shown in Fig. 1: (a) the lightness (L*) increases giving an external appearance which is lighter than the underlying dentine analogue, (b) the vibrancy (chroma) of the colour, represented by the distance from the L* axis, reduces giving a whiter appearance while (c) the hue, represented by the position around the colour wheel, is most often broadly unchanged, though it can sometimes change quite markedly, with some yellow dentine analogue colours changing to blue tooth analogue colours, for example, when the enamel is place on top. A strength of our theoretical analysis is its ability to predict even these dramatic colour shifts with accuracy.
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In the next two sub-sections, we discuss the results for lightness, L*, and chromaticity, (a*,b*), respectively.
4.1.
Lightness (L )
Measured values of L0 are plotted against measured values of L* (crosses) for both hydrated and desiccated samples of bovine enamel and coloured paper in Fig. 2 (upper curves, left hand scale). Numerical values for ðL0 a0 b0 Þ and ðL ; a ; b Þ are given in tabular form elsewhere13 and are not repeated here. In spite of the wide range of colours used, many of which were quite vibrant, the points all lie on a smooth curve for each of the hydrated and desiccated samples. The smooth curve has been fitted using Eq. (7) using fitting parameters, with R00 representing the averaged reflectivity of the enamel layer, and a quantity, L ¼ 1 R00 T00 , representing the averaged fraction of light loss in the enamel layer. The errors between the measured data values and the theoretical fit are no more than 2% for any of the points shown in Fig. 2. The results of the theory alone are also plotted in Fig. 2 (lower curves, right hand scale) to illustrate the range of predicted relationships between L0 and L* for three different values for the averaged reflectivity of the enamel layer (R00 ) and two different values for the averaged fraction of light loss in the enamel layer (L). The averaged reflectivity is dominant in determining the lightness of the tooth analogue when the dentine analogue is very dark, as the dentine itself makes only a small contribution to the overall lightness. The fraction of light loss in the enamel is dominant in determining the lightness of the tooth analogue when the dentine analogue is
very light. This can be understood in the following way. If the dentine itself is a perfect reflector (L ¼ 100) and, simultaneously, the enamel is also lossless (L = 0), then the combination must also be a perfect reflector (L0 ¼ 100) as neither the enamel nor the dentine causes light loss. This conclusion is valid for all values of the averaged enamel reflectivity, R00 . Hence, all curves with L = 0 go through the point (100, 100), independent of the averaged reflectivity. It is then easy to see that an increase in loss predominantly affects the tooth lightness at high dentine lightness as the reflectivity itself has no effect. These curves also illustrate the simplicity of the fitting process: R00 predominantly comes from fitting data points with dark dentine and L from fitting data points with light dentine. The fitting parameters used in Fig. 2 are given in Table 1 below. These values can be substituted into equations equivalent to Eqs. (5) and (6), but expressed in terms of suitable weighted averages of A and S, and a 2D Newton– Raphson method used to find the resulting solutions, A and S, using an average D of 0.9 mm. These values are given in Table 2. The absorption coefficient compares well to that measured by Spitzer and ten Bosch15, who measured a decadic absorption coefficient of about 25 m1 for a 0.17 mm thick slice of hydrated bovine enamel at 550 nm. This wavelength is close to the maximum in the weighting function used to define A and S13 and hence a reasonable comparison to our own values. Their decadic coefficient is equivalent to a Napierian absorption coefficient, used here, of 58 m1, a value close to our own observations. However, agreement is less good for the scattering coefficient as they measured a decadic scattering coefficient of about 1700 m1 for the same slice of bovine enamel, also at 550 nm. This is equivalent to a Napierian scattering coefficient of 3900 m1, almost ten times our own measurement. However, the differences appear justifiable as natural variations between enamel characteristic as the slice use by Spitzer and ten Bosch had a reflectivity of around 0.4 even for a slice as thin as 0.17 mm while our much thicker slice has a reflectivity with black backing of 0.265. (Here we assume that our measurements with black backing are comparable to Spitzer and ten Bosch’s measurements without backing at all. This is correct provided light passing through the enamel of Spitzer and ten Bosch did not make its way back to the detector in some way. Without detailed knowledge of their experimental set up, however, this is impossible to judge.)
Table 1 – Fitting parameters used to derive the solid lines in Fig.2. Fig. 2 – Upper plot, left hand axis: Measured L0 against measured L for desiccated and hydrated samples (crosses). Labels indicate w = white, y = yellow, o = orange, r = red, g = green, b = blue, v = violet, i = indigo and bl = black. Numerical values are given elsewhere.13 Theoretical fitted lines (solid) use averaged reflectivity, loss and transmissivity fitting parameters, R00 , L and T00 , given below the line labels. Lower plot, right hand axis: Theoretical relationship between L0 and L from Eq. (7) using 3 different values of R00 and loss factors of 0 (solid lines) and 0.1 (dashed lines).
Hydrated (Fig. 2) Desiccated (Fig. 2)
R00
T00
L ¼ 1 R00 T00
0.265 0.605
0.665 0.32
0.07 0.075
Table 2 – Napierian scattering and absorption coefficients for the bovine enamel.
Hydrated (Fig. 2) Desiccated (Fig. 2)
S(m1)
A(m1)
438 2001
81 89
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4.2.
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Chromaticity (a*,b*)
a* and b* define the hue and vibrancy (chroma) of the dentine and tooth analogues. In Fig. 3, measured values of ða0 b0 Þ (pluses) and ða ; b Þ (circles) for hydrated samples only are plotted on the a*, b* plane for a different set of dentine analogue colours from those shown in Fig. 2. Arrows link pairs of data points using the same backing layer colour and labels close to the ða ; b Þ point indicate the approximate colour. Numerical values are again given elsewhere13 and are not repeated here. It is clear from Fig. 3 that there is a general inward radial movement from the dentine analogue to the tooth analogue. However, rather than directly towards the origin, the radial movement is broadly towards a point located at roughly (2,6) with the result that some backing layer colours, such as light yellow (ly), actually acquire a blue hue when the bovine enamel is superimposed. Furthermore, some colours, such as cherry and magenta (ch and m) seem to move towards a point much closer to the origin than the other colours. Our theory13 is capable of predicting these changes in chromaticity to good accuracy with no additional fitting parameters, other than the ones already used to fit the L* data. This theory shows that the a* value for the tooth analogue is simply the sum of the a* for the dentine analogue and a fixed a* value for the enamel, with each multiplied by a weighting factor between 0 and 1. Here, the a* value for the enamel is measured when placed on a perfectly black surface. The b* value for the tooth analogue is related to the b* values for the dentine analogue and for the enamel in exactly the same way with exactly the same weighting factors. These
Fig. 3 – Measured ða0 b0 Þ (pluses) and measured ða ; b Þ (circles) for hydrated samples only for a different set of dentine analogue backing colours from those of Fig. 2. Arrows link pairs of data points using the same dentine analogue colours, and labels close to the ða ; b Þ point indicate approximate colour using y = yellow, o = orange, r = red, ch = cherry, m = magenta, i = indigo, b = blue, ccyan, pg = pea green, g = green and l in front of these labels are similar hues but with lower chromaticity. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
2=3
2
2
weighting factors are the coefficients ðR=R0 Þ ðT00 =ð1 R00 RÞ Þ 2=3 2 and ðR00 =R0 Þ ð1 ðT00 R=ð1 ðR00 RÞÞÞ in Eq. (8) and are given in graphical form in Fig. 4 for a fixed value of loss factor, L, of 0.1, very close to the measured values of 0.07 and 0.075. In order to understand how the weighting factors help us determine what happens to the observed colour when the bovine enamel is placed on top of the coloured backing layer, we need measured values for ða00 ; b00 Þ. These were measured with the enamel slice superimposed on a matt black surface to be (2.2,4.9) when the enamel is hydrated and (1.7,3.1) when the enamel is desiccated. A comparison to the points marked with circles in Fig. 3 shows that ða00 ; b00 Þ is much smaller than ða ; b Þ for all the colours used. With this in mind, we can use Fig. 4 to make the following qualitative statements (and here we drop the use of the word analogue as these statements should also apply in vivo):
When the dentine is light (L* is high), the tooth chromaticity is dominated by that of the dentine. This is because the weighting factor for the dentine chromaticity is much greater than the weighting factor for the enamel chromaticity, which is itself small. An increase in the reflectivity of the enamel, for example by dehydrating the tooth, rapidly reduces the contribution of dentine chromaticity to that of the tooth when the dentine itself is light. This is because the weighting factor for dentine chromaticity reduces quickly as R00 increases. As the dentine darkens (L* reduces), the chromaticity of the tooth decreases at fixed dentine chromaticity. This occurs more quickly when the enamel reflectivity is high, for example with dehydrated enamel, than when the reflectivity is low, for example with hydrated enamel. This is because the slope of the weighting factor for dentine
Fig. 4 – Weighting factors for the dentine chromaticity (solid lines) and the enamel chromaticity (dashed lines) which, when applied to each chromaticity value respectively and summed, gives the chromaticity for the tooth. The lines are for a constant fraction of light loss = 0.1, with the averaged reflectivity, R00 , varied from 0.1 (extreme left hand curves), in steps of 0.1, to 0.7 (extreme right hand curves). The associated averaged transmissivity varies from 0.8 to 0.2.
journal of dentistry 43 (2015) 373–381
chromaticity in overall tooth chromaticity is steep for high R00 but more gradual when R00 is low). The relative contribution of enamel chromaticity and dentine chromaticity to the overall tooth chromaticity when the dentine is dark (low L*) cannot easily be judged without an actual calculation. Although the weighting factor for the enamel chromaticity is much greater than for the dentine chromaticity, the enamel chromaticity itself is small compared to the dentine chromaticity. It is for this reason that, in Fig. 3, while we see many of the chromaticity values move towards (2,6) when the enamel is superimposed on the backing layer, some colours, notably cherry and magenta, move in a slightly different direction. In some cases, the chromaticity of the dentine remains big enough for the tooth to retain a similar hue to the dentine but in other cases, notably light yellow (ly in Fig. 3), the contribution from the chromaticity of the enamel is sufficient to move the tooth chromaticity to a completely different hue, so the tooth has a blue hue in spite of the dentine being yellow. Our theory reproduces all these qualitative observations quantitatively. A comparison between measured (+) and predicted (x) values of ða0 b0 Þ is shown in Fig. 5. Here, predicted values are based only on measured values of ða ; b Þ and ða00 ; b00 Þ and on values for the weighting factors similar to those in Fig. 4 but with L = 0.06 rather than 0.1. Values of R00 , L and hence also of T00 are found to be 0.29, 0.06 and 0.65, respectively13 by fitting Eq. (7) to L* data similar to Fig. 2 and are
Fig. 5 – Pairs of measured (+) and predicted (x) values of ða0 b0 Þ where the pairs are indicated by the oval. Predicted values are based on measured values for ða ; b Þ and ða00 ; b00 Þ and the weighting factors given in Fig. 4 (with L = 0.06). The appropriate values of R00 and L are chosen from fitting our theory to the L data in a way similar to Fig. 2. Labels indicate the approximate colour using y = yellow, o = orange, r = red, ch = cherry, m = magenta, i = indigo, b = blue, c-cyan, pg = pea green, g = green and points with l in front of these labels are similar hues but with lower chromaticity. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
379
used without further adjustment. In Fig. 5, pairs of measured and predicted values of ða0 b0 Þ are grouped by ovals. Differences between measured and predicted values of ða0 b0 Þ are small, at about 8% of the magnitude of ða ; b Þ. 0 0 Absolute a and b values for the tooth analogue are predicted typically to within less than 3 and no worse than 5. Even the colours such as light yellow, cherry and magenta, picked out earlier as having apparently abnormal behaviour, are modelled to this accuracy by our weighting factors. Finally, Fig. 6 shows the effect on chromaticity of desiccating the samples. Here, the colours are the same as those shown in Fig. 2, where we saw that desiccating the enamel led to higher reflectivity and hence a much lighter colour. From Fig. 6, we can see how this higher reflectivity also reduces the measured chromaticity of the tooth much more than when the enamel reflectivity is less. We noted above how the weighting factors change when the enamel reflectivity changes and we can see from the similarity between measured and predicted results for all colours, whether desiccated or hydrated, that our model reproduces this behaviour. (The degree of agreement between measured and predicted values of chromaticity in Fig. 6 is not as good as in Fig. 3. This is actually expected from our theoretical work because the colours used in Fig. 6 are more strongly saturated. This is discussed in more detail elsewhere.13) The model therefore not only predicts the measured colours with hydrated enamel but can also accurately predict the effect of using desiccated enamel.
Fig. 6 – Pairs of measured (+) and predicted (x) values of ða0 b0 Þ where the pairs are indicated by the oval. Solid ovals are for hydrated samples while the dashed oval indicates the samples have been desiccated. Predicted values are based on measured values for ða ; b Þ and ða00 ; b00 Þ and the weighting factors given in Fig. 4 (with L = 0.07 for hydrated samples and 0.075 for desiccated samples). The appropriate values of R00 and L are those given in Fig. 2, as these results are for the same samples as in that figure. Labels indicate the approximate colour using y = yellow, o = orange, r = red, g = green, b = blue, v = violet, i = indigo. Black and white are omitted as they have zero chromaticity. Numerical values are again given elsewhere.13
380
5.
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Discussion
Unless caused by trauma, no one has cherry or magenta coloured dentine and no practitioner has ever encountered cyan coloured dentine. The reason why we have used such a wide range of dentine analogue colours for our backing layers is to test, with much more extreme colours than practitioners will encounter in practice, the predictive accuracy of our theoretical model, of the reflectivity and loss fractions derived from the fitting process shown in Fig. 2 and of the weighting factors shown in Fig. 4. This should give confidence in their use with the much narrower range of colours typically found in human dentine. Indeed, our model is based on an approximation which becomes more accurate the less vibrant the colour of dentine analogue backing layer used. (This is the reason why the agreement between measured and predicted tooth analogue colours in Fig. 6 is worse than in Fig. 3 as the colours used in Fig. 6 are more vibrant than those in Fig. 3.) So, our model can be expected to be even more accurate when applied to natural dentine colours than to the more extreme colours used here. More needs to be done to make the model we have developed find practical uses. Our aim has been to develop a method to predict the baseline tooth colour in humans arising from coloured dentine visible through the translucent enamel and to which extrinsic sources of discolouration will add. Our work therefore needs to be repeated using human tooth enamel rather than bovine enamel and with a range of different enamel thicknesses so that we can test the validity of the K-M Model. With these in place, it should be possible to take a single measurement of tooth colour and, based on the expected enamel thickness at the point of measurement, work out the expected colour of dentine under the assumption that extrinsic causes of colour are negligible. Furthermore, it should be possible to measure dentine colour, when exposed during restorative dentistry for example, and infer the likely tooth colour from that dentine alone across the rest of the dentition under the assumption that the exposed dentine colour was typical for all teeth in that person. Finally, in a different context, it should be possible to prepare porcelain veneers which accurately match the surrounding dentition, even though some of the colour from the underlying tooth or bonding material is actually visible through the porcelain, by accurately predicting the colour which would be seen through the porcelain and hence adjusting the colour of the porcelain itself to correct for this. Nonetheless, our work already highlights trends which are clinically relevant and can be helpful to both the practitioner and clinical researcher and we highlight these here (once again dropping the word analogue).
1. Fig. 2 shows how the greater enamel reflectivity which occurs when the enamel is desiccated results in a higher lightness value for a given colour of dentine. Any tooth whitening technique which partially dries the surface enamel of the tooth can therefore result in elevated lightness values which quickly fall as the enamel rehydrates. In-surgery measurements of post-treatment light-
ness, when there is a risk of enamel dehydration during treatment, should therefore be made only after a suitable delay to allow rehydration to take place. 2. During whitening, the rate at which the lightness of the tooth increases as the dentine itself gets lighter, shown in the data for hydrated enamel in Fig. 2, is steeper at light dentine colours than at darker dentine colours. This suggests that teeth with dentine which is already light in colour are easier to whiten, at least in terms of the change in dentine lightness required to achieve a given change in tooth lightness, than teeth with darker dentine. This does not necessarily mean that lighter coloured teeth require less bleaching time to lighten them than darker teeth, as the chemistry of the bleaching process might be faster with dark dentine, thereby compensating for the effect we point out here. Nevertheless, it is possible that early intervention with less severe whitening regimes may be more effective in delivering whitened teeth than later intervention, even with more severe regimes. 3. Although we have worked only with a single thickness of enamel, Eq. (5) shows that the reflectivity of the enamel will be greater with thicker enamel and less with thinner enamel. The same will be true for the loss from Eq. (6). Enamel thickness varies in human teeth, being thickest in the middle of a face and towards the tip or crown and thinning both towards the cementoenamel junction (where it falls to zero) and both mesially and distally. Our understanding of the influence of enamel reflectivity on tooth lightness suggests that, provided intrinsic sources of colour from the dentine dominate, the lightness of the tooth itself should have a similar pattern, being lightest where the enamel is thick and darkening in the directions in which the enamel thins. This type of lightness distribution, darkening towards the cementoenamel junction and both mesially and distally, is indeed seen in practice. 4. The weighting factors in Fig. 4 show how the tooth chromaticity for fixed dentine lightness decreases with increased enamel reflectivity (at least when the dentine colour is still the dominant factor in determining tooth colour). This suggests that the chromaticity over the tooth surface should also follow the same pattern as the lightness, being least coloured (i.e. closest to white) where the enamel is thickest and more coloured where the enamel is thin. Once again, this pattern is observed in practice.
6.
Conclusions
We have developed a theoretical model which can predict the colour of a tooth, when extrinsic sources of colouration are negligible, from the light scattering (reflectivity) and loss fractions in the enamel and from the colour of the dentine. The actual theory is complex, owing to the complexity of the human visual system. Quantitative prediction of how the human visual system perceives colours, and how those perceived colours change when scattering in the enamel changes the spectrum of light incident on the eye, requires both a knowledge of colour science and the use of mathematics which is likely to obscure more than enlighten. We have therefore chosen to publish our theoretical model elsewhere13
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and use this paper to highlight (a) the evidence of its predictive accuracy and (b) the clinical relevance of the predictions it makes. Further measurements on human enamel and with a range of different enamel thicknesses are required in order to make the model we have developed more applicable in practical situations. Nevertheless, the predicted trends in lightness and chromaticity that the model provides have been used in this paper to highlight some clinically relevant outcomes.
Acknowledgements The authors would like to acknowledge Dr. Nigel Young for suggesting the model system we have used for our measurements and Prof. Dr. Ingrid Heynderickx for her careful coaching on the human visual system and on colour perception. Figures 3, 5 and 6 are reprinted by permission of John Wiley & Sons, Inc. from ref. 13. Copyright # 2014 by John Wiley & Sons, Inc.
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