Abstract

The thermal and optical properties of multilayered dental tissue structure, the result of the surface-grown prismless layer on enamel, were evaluated simultaneously using multiparameter fits of photothermal radiometry frequency responses. The photothermal field generated in a tooth sample with near-infrared laser excitation was described using a coupled diffuse-photon-density and thermal wave model. The optical (absorption and scattering) coefficients and thermal parameters (spectrally averaged infrared emissivity, thermal diffusivity and conductivity) of each layer, as well as the thickness of the upper prismless enamel layer, were fitted using a multiparameter simplex downhill minimization algorithm. The results show that the proposed fitting approach can increase robustness of the multiparameter estimation of tissue properties in the case of ill-defined multiparameter fits, which are unavoidable in in vivo tissue evaluation. The described method can readily be used for noninvasive in vitro or in vivo characterization of a wide range of layered biological tissues.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2009 (2)

A. Matvienko, A. Mandelis, R. J. Jeon, and S. H. Abrams, “Theoretical analysis of coupled diffuse-photon-density and thermal-wave field depth profiles photothermally generated in layered turbid dental structures,” J. Appl. Phys. 105, 102022 (2009).
[CrossRef]

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

2006 (1)

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

2005 (1)

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

1999 (1)

1998 (3)

1997 (2)

A. A. Oraevsky, S. L. Jacques, and F. K. Tittel, “Measurements of tissue optical properties by time-resolved detection of laser-induced transient stress,” Appl. Opt. 36, 402-415(1997).
[CrossRef] [PubMed]

M. Fava, I. S. Watanabe, F. Fava-de-Moraes, and L. R. R. S. da Costa, “Prismless enamel in human non-erapted deciduous molar teeth: a scanning electron microscopy study,” Rev. Odontol. Univ. Sao Paulo 11, 239-243(1997).

1995 (1)

1994 (1)

D. L. Buckley, R. W. Kerslake, S. J. Blackband, and A. Horsman, “Quantitative analysis of multi-slice Gd-DTPA enhanced dynamic MR images using an automated Simplex minimization procedure,” Magn. Res. Med. 32, 646-651(1994).
[CrossRef]

1992 (1)

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

1989 (3)

1983 (1)

1970 (1)

W. S. Brown, W. A. Dewey, and H. R. Jacob, “Thermal properties of teeth,” J. Dent. Res. 49, 752-755 (1970).
[CrossRef] [PubMed]

1964 (1)

M. Braden, “Heat conduction in normal human teeth,” Arch. Oral Biol. 9, 479-486 (1964).
[CrossRef] [PubMed]

Abrams, S. H.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

A. Matvienko, A. Mandelis, R. J. Jeon, and S. H. Abrams, “Theoretical analysis of coupled diffuse-photon-density and thermal-wave field depth profiles photothermally generated in layered turbid dental structures,” J. Appl. Phys. 105, 102022 (2009).
[CrossRef]

L. Nicoalides, C. Feng, A. Mandelis, and S. H. Abrams, “Quantitative dental measurements by use of simultaneous frequency-domain laser infrared photothermal radiometry and luminescence,” Appl. Opt. 41, 768-777 (2002).
[CrossRef]

Alexandrakis, G.

Alvarado-Gil, J. J.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

Amaechi, B. T.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

Anderson, R. R.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. L. Jacques, and J. A. Parrish, “Pulsed photothermal radiometry in turbid media: internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256-2262 (1989).
[CrossRef] [PubMed]

Bays, R.

Beck, H.

Bevilacqua, F.

Blackband, S. J.

D. L. Buckley, R. W. Kerslake, S. J. Blackband, and A. Horsman, “Quantitative analysis of multi-slice Gd-DTPA enhanced dynamic MR images using an automated Simplex minimization procedure,” Magn. Res. Med. 32, 646-651(1994).
[CrossRef]

Braden, M.

M. Braden, “Heat conduction in normal human teeth,” Arch. Oral Biol. 9, 479-486 (1964).
[CrossRef] [PubMed]

Brown, W. S.

W. S. Brown, W. A. Dewey, and H. R. Jacob, “Thermal properties of teeth,” J. Dent. Res. 49, 752-755 (1970).
[CrossRef] [PubMed]

Bruggemann, U.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. L. Jacques, and J. A. Parrish, “Pulsed photothermal radiometry in turbid media: internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256-2262 (1989).
[CrossRef] [PubMed]

Buckley, D. L.

D. L. Buckley, R. W. Kerslake, S. J. Blackband, and A. Horsman, “Quantitative analysis of multi-slice Gd-DTPA enhanced dynamic MR images using an automated Simplex minimization procedure,” Magn. Res. Med. 32, 646-651(1994).
[CrossRef]

Chance, B.

Cholico, P.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

da Costa, L. R. R. S.

M. Fava, I. S. Watanabe, F. Fava-de-Moraes, and L. R. R. S. da Costa, “Prismless enamel in human non-erapted deciduous molar teeth: a scanning electron microscopy study,” Rev. Odontol. Univ. Sao Paulo 11, 239-243(1997).

Del Bianco, S.

Depeursinge, C.

Dewey, W. A.

W. S. Brown, W. A. Dewey, and H. R. Jacob, “Thermal properties of teeth,” J. Dent. Res. 49, 752-755 (1970).
[CrossRef] [PubMed]

Dognitz, N.

Essenpreis, M.

Farinelli, W.

Farrell, T. J.

Fava, M.

M. Fava, I. S. Watanabe, F. Fava-de-Moraes, and L. R. R. S. da Costa, “Prismless enamel in human non-erapted deciduous molar teeth: a scanning electron microscopy study,” Rev. Odontol. Univ. Sao Paulo 11, 239-243(1997).

Fava-de-Moraes, F.

M. Fava, I. S. Watanabe, F. Fava-de-Moraes, and L. R. R. S. da Costa, “Prismless enamel in human non-erapted deciduous molar teeth: a scanning electron microscopy study,” Rev. Odontol. Univ. Sao Paulo 11, 239-243(1997).

Featherstone, J. D. B.

Feng, C.

Ferrell, R. J.

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

Ferwerda, H. A.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University Press, 1988).

Fried, D.

Glena, R. E.

Goodman, D. M.

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

Groenhuis, Z. A. J.

Gross, J. D.

Gutierrez-Juarez, G.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

Hellen, A.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

Horsman, A.

D. L. Buckley, R. W. Kerslake, S. J. Blackband, and A. Horsman, “Quantitative analysis of multi-slice Gd-DTPA enhanced dynamic MR images using an automated Simplex minimization procedure,” Magn. Res. Med. 32, 646-651(1994).
[CrossRef]

Hublin, J. J.

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

Huerta-Franco, R.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

Ishimaru, A.

Jacob, H. R.

W. S. Brown, W. A. Dewey, and H. R. Jacob, “Thermal properties of teeth,” J. Dent. Res. 49, 752-755 (1970).
[CrossRef] [PubMed]

Jacques, S. L.

Jeon, R. J.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

A. Matvienko, A. Mandelis, R. J. Jeon, and S. H. Abrams, “Theoretical analysis of coupled diffuse-photon-density and thermal-wave field depth profiles photothermally generated in layered turbid dental structures,” J. Appl. Phys. 105, 102022 (2009).
[CrossRef]

Kakaboura, A.

A. Kakaboura and L. Papagiannoulis, “Bonding of resinous materials on primary enamel, in dental hard tissues and bonding,” in Interfacial Phenomena and Related Properties, T. Eliades and C. Watts, eds. (Springer, 2005) pp. 35-51.

Kerslake, R. W.

D. L. Buckley, R. W. Kerslake, S. J. Blackband, and A. Horsman, “Quantitative analysis of multi-slice Gd-DTPA enhanced dynamic MR images using an automated Simplex minimization procedure,” Magn. Res. Med. 32, 646-651(1994).
[CrossRef]

Kienle, A.

Mandelis, A.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

A. Matvienko, A. Mandelis, R. J. Jeon, and S. H. Abrams, “Theoretical analysis of coupled diffuse-photon-density and thermal-wave field depth profiles photothermally generated in layered turbid dental structures,” J. Appl. Phys. 105, 102022 (2009).
[CrossRef]

L. Nicoalides, C. Feng, A. Mandelis, and S. H. Abrams, “Quantitative dental measurements by use of simultaneous frequency-domain laser infrared photothermal radiometry and luminescence,” Appl. Opt. 41, 768-777 (2002).
[CrossRef]

A. Mandelis, Diffusion Wave Fields: Mathematical Methods and Green Functions (Springer, 2001), Chap. 10.

Marquet, P.

Martelli, F.

Matvienko, A.

A. Matvienko, A. Mandelis, A. Hellen, R. J. Jeon, S. H. Abrams, and B. T. Amaechi, “Quantitative analysis of incipient mineral loss in hard tissues,” Proc. SPIE 7166, 71660C (2009).
[CrossRef]

A. Matvienko, A. Mandelis, R. J. Jeon, and S. H. Abrams, “Theoretical analysis of coupled diffuse-photon-density and thermal-wave field depth profiles photothermally generated in layered turbid dental structures,” J. Appl. Phys. 105, 102022 (2009).
[CrossRef]

Milner, T. E.

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

Nicoalides, L.

Olejniczak, A. J.

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

Oraevsky, A. A.

Papagiannoulis, L.

A. Kakaboura and L. Papagiannoulis, “Bonding of resinous materials on primary enamel, in dental hard tissues and bonding,” in Interfacial Phenomena and Related Properties, T. Eliades and C. Watts, eds. (Springer, 2005) pp. 35-51.

Parrish, J. A.

Patterson, M. S.

Pichardo-Molina, J. L.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

Piguet, D.

Prahl, S. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University Press, 1988).

Reid, D. J.

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

Seka, W.

Smith, T. M.

T. M. Smith, A. J. Olejniczak, D. J. Reid, R. J. Ferrell, and J. J. Hublin, “Modern human molar enamel thickness and enamel-dentin junction shape,” Arch. Oral Biol. 51, 974-995(2006).
[CrossRef] [PubMed]

Telenkov, S. A.

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

Ten Bosch, J. J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University Press, 1988).

Tittel, F. K.

Tromberg, B. J.

van den Bergh, H.

Vargas-Luna, M.

J. L. Pichardo-Molina, G. Gutierrez-Juarez, R. Huerta-Franco, M. Vargas-Luna, P. Cholico, and J. J. Alvarado-Gil, “Open photoacoustic cell technique as a tool for thermal and thermo-mechanical characterization of teeth and their restorative materials,” Int. J. Thermophys. 26, 243-253 (2005).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge University Press, 1988).

Vitkin, I. A.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

Wagnieres, G.

Watanabe, I. S.

M. Fava, I. S. Watanabe, F. Fava-de-Moraes, and L. R. R. S. da Costa, “Prismless enamel in human non-erapted deciduous molar teeth: a scanning electron microscopy study,” Rev. Odontol. Univ. Sao Paulo 11, 239-243(1997).

Welch, A. J.

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

Wilson, B. C.

S. A. Prahl, I. A. Vitkin, U. Bruggemann, B. C. Wilson, and R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203-1217 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331-2336(1989).
[CrossRef] [PubMed]

Youn, J.-I.

S. A. Telenkov, J.-I. Youn, D. M. Goodman, A. J. Welch, and T. E. Milner, “Non-contact measurement of thermal diffusivity in tissue,” Phys. Med. Biol. 46, 551-558 (2001).
[CrossRef] [PubMed]

Zaccanti, G.

Appl. Opt. (11)

Z. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456-2462(1983).
[CrossRef] [PubMed]

A. Ishimaru, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210-2215 (1989).
[CrossRef] [PubMed]

R. R. Anderson, H. Beck, U. Bruggemann, W. Farinelli, S. L. Jacques, and J. A. Parrish, “Pulsed photothermal radiometry in turbid media: internal reflection of backscattered radiation strongly influences optical dosimetry,” Appl. Opt. 28, 2256-2262 (1989).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331-2336(1989).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958-1972 (1998).
[CrossRef]

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401-7409 (1998).
[CrossRef]

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[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Effective layered tooth structure for the PTR model.

Fig. 2
Fig. 2

PTR (a) amplitude and (b) phase calculated for various enamel thicknesses.

Fig. 3
Fig. 3

Experimental setup.

Fig. 4
Fig. 4

Algorithm chart.

Fig. 5
Fig. 5

Resulting residual versus tolerance of the simplex for various numbers of intervals N.

Fig. 6
Fig. 6

Calculation time versus tolerance for various numbers of intervals N.

Fig. 7
Fig. 7

Number of nonconverged fitting attempts versus tolerance of the fits. The curve corresponding to N = 23 is shown by a dashed line.

Fig. 8
Fig. 8

Optical absorption and scattering coefficients versus tolerance. N = 23 .

Fig. 9
Fig. 9

Thermal diffusivities, thermal conductivities, and mean infrared absorption/emission coefficient versus tolerance. N = 23 .

Fig. 10
Fig. 10

Thickness, L 1 , of the prismless layer and various optical and thermal coefficients versus tolerance. N = 23 .

Fig. 11
Fig. 11

Absorption and scattering coefficients versus number of intervals between the limits N.

Fig. 12
Fig. 12

Thermal diffusivities, thermal conductivities, and mean infrared absorption/emission coefficient versus number of intervals between the limits N.

Fig. 13
Fig. 13

Thickness, L 1 , of the prismless layer versus number of intervals between the limits N.

Fig. 14
Fig. 14

Resulting residual versus number of intervals between the limits N.

Fig. 15
Fig. 15

Amplitude and phase for the PTR signal generated by a healthy tooth. Experimental data are represented with symbols. Calculated data are shown by a solid line.

Tables (2)

Tables Icon

Table 1 Upper and Lower Limits for the Initial Guess of Parameters (Bulk Enamel) [15, 16, 22, 23, 24]

Tables Icon

Table 2 Results of the Fits Based on Calculations with the Number of Intervals N from 20 to 30, Simplex Downhill Tolerance t = 10 5

Equations (29)

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Ψ t i ( z ; ω ) = Ψ c i ( z ; ω ) + Ψ d i ( z ; ω ) ,
Ψ c 1 = I 0 ( 1 R 1 ) { exp [ μ t 1 z ] + R 2 exp [ μ t 1 ( 2 L 1 z ) ] } 1 R 1 R 2 exp [ 2 μ t 1 L 1 ] , Ψ c 2 = I 0 ( 1 R 1 ) ( 1 + R 2 ) exp [ μ t 1 L 1 ] exp [ μ t 2 ( z L 2 ) ] 1 R 1 R 2 exp [ 2 μ t 1 L 1 ] , Ψ c 3 = I 0 ( 1 R 1 ) ( 1 + R 2 ) exp [ ( μ t 1 L 1 + μ t 2 L 2 ) ] exp { μ t 3 [ z ( L 1 + L 2 ) ] } 1 R 1 R 2 exp [ 2 μ t 1 L 1 ] ,
μ t i = μ a i + μ s i .
d 2 d z 2 Ψ d i ( z ) 3 μ a i μ t i Ψ d i ( z ) = 1 D i G i ( z ) .
μ t = μ a + ( 1 g ) μ s
G i ( z ) = μ s i ( μ t i + g i μ a i μ t i g μ s i ) Ψ c i .
Ψ t 1 ( z ) = a 1 exp ( Q 1 z ) + b 1 exp ( Q 1 z ) + I eff ( 1 + C μ 1 ) { exp [ μ t 1 z ] + R 2 exp [ μ t 1 ( 2 L 1 z ) ] } ,
Ψ t 2 ( z ) = a 2 exp [ Q 2 ( z L 1 ) ] + b 2 exp [ Q 2 ( z L 1 ) ] + I eff ( 1 + R 2 ) ( 1 + C μ 2 ) exp ( μ t 1 L 1 ) exp [ μ t 2 ( z L 1 ) ] ,
Ψ t 3 ( z ) = b 3 exp { Q 3 [ z ( L 1 + L 2 ) ] } + I eff ( 1 + R 2 ) ( 1 + C μ 3 ) exp [ ( μ t 1 L 1 + μ t 2 L 2 ) ] exp { μ t 3 [ z ( L 1 + L 2 ) ] } ,
C μ i = 3 μ s i ( μ t i + g μ a i ) 3 μ a i μ t i μ t i 2 , I eff = I 0 ( 1 R 1 ) 1 R 1 R 2 exp ( 2 μ t 1 L 1 ) .
Ψ d 1 ( 0 ) = A d d z Ψ d 1 ( z ) | z = 0 , Ψ d 1 ( L 1 ) = Ψ d 2 ( L 1 ) , D 1 d d z Ψ d 1 ( z ) | z = L 1 = D 2 d d z Ψ d 2 ( z ) | z = L 1 , Ψ d 2 ( L 1 + L 2 ) = Ψ d 3 ( L 1 + L 2 ) , D 2 d d z Ψ d 2 ( z ) | z = L 1 + L 2 = D 3 d d z Ψ d 3 ( z ) | z = L 1 + L 2 .
A = 2 D ( 1 + r 1 r ) ,
a 1 = d 1 P f 1 N exp ( 2 μ t 1 L 1 ) ( 2 V F + G ) exp ( Q 1 L 1 ) ( 1 + X 12 2 V X 12 ) M ( 1 X 12 + 2 V X 12 ) exp ( 2 Q 1 L 1 ) ( 1 + X 12 2 V X 12 ) , b 1 = a 1 M d 1 P f 1 N exp ( 2 μ t 1 L 1 ) , a 2 = b 2 + d 2 Y 22 + X 12 a 1 exp ( Q 1 L 1 ) X 12 b 1 exp ( Q 1 L 1 ) + Y 12 ( f 1 d 1 ) exp ( μ t 1 L 1 ) , b 2 = V F V X 12 a 1 exp ( Q 1 L 1 ) + V X 12 b 1 exp ( Q 1 L 1 ) , b 3 = a 2 X 23 exp ( Q 1 L 1 ) + b 2 X 23 exp ( Q 1 L 1 ) + Y 23 d 2 exp ( μ t 2 L 2 ) Y 33 d 3 .
M 1 Q 1 A 1 + Q 1 A , N 1 μ t 1 A 1 + Q 1 A , P = 1 + μ t 1 A 1 + Q 1 A , X i j D i Q i D j Q j , Y i j D i μ t i D j Q j , d 1 = C μ 1 I eff , f 1 = d 1 R 2 , d 2 = C μ 2 I eff ( 1 + R 2 ) exp ( μ t 1 L 1 ) , d 3 = C μ 3 I eff ( 1 + R 2 ) exp [ ( μ t 1 L 1 + μ t 2 L 2 ) ] .
F = d 2 exp ( μ t 2 L 2 ) ( Y 23 1 ) exp ( Q 2 L 2 ) ( X 23 + 1 ) + d 3 exp ( 1 Y 33 ) exp ( Q 2 L 2 ) ( X 23 + 1 ) d 2 Y 22 ( f 1 d 1 ) Y 12 exp ( μ t 1 L 1 ) ; G = ( f 1 + d 1 ) exp ( μ t 1 L 1 ) + d 1 + d 2 Y 22 + ( f 1 d 1 ) Y 12 exp ( μ t 1 L 1 ) , V = 1 1 ( X 23 1 ) ( X 23 + 1 ) exp ( 2 Q 2 L 2 ) .
d 2 d z 2 T i ( z ; ω ) σ i 2 T i ( z ; ω ) = η NR μ a i κ i Ψ t i ( z ; ω ) , i = 1 , 2 , 3 ,
σ i = i ω α i
T 1 ( z ; ω ) = A 1 exp ( σ 1 z ) + B 1 exp ( σ 1 z ) + C 1 exp ( Q 1 z ) + D 1 exp ( Q 1 z ) + E 1 exp ( μ t 1 z ) + F 1 exp [ μ t 1 ( 2 L 1 z ) ] ,
T 2 ( z ; ω ) = A 2 exp [ σ 2 ( z L 1 ) ] + B 2 exp [ σ 2 ( z L 1 ) ] + C 2 exp [ Q 2 ( z L 1 ) ] + D 2 exp [ Q 2 ( z L 1 ) ] + E 2 exp [ μ t 2 ( z L 1 ) ] ,
T 3 ( z ; ω ) = B 3 exp { σ 3 [ z ( L 1 + L 2 ) ] } + D 3 exp { Q 3 [ z ( L 1 + L 2 ) ] } + E 3 exp { μ t 3 [ z ( L 1 + L 2 ) ] } .
C i = η NR i μ a i κ i ( Q i 2 σ i 2 ) a i , i = 1 , 2 ; D i = η NR i μ a i κ i ( Q i 2 σ i 2 ) b i , i = 1 , 2 , 3 ; E i = η NR i μ a i ( 1 + C μ i ) κ i ( μ t i 2 σ i 2 ) C μ i d i , i = 1 , 2 , 3 ; F i = η NR i μ a i ( 1 + C μ i ) κ i ( μ t i 2 σ i 2 ) C μ i f 1 .
κ 1 d T 1 ( z , ω ) d z | z = 0 = H T 1 ( 0 ; ω ) , T 1 ( L 1 , ω ) = T 2 ( L 1 , ω ) , κ 1 d T 1 ( z , ω ) d z | z = L 1 = κ 2 d T 2 ( z , ω ) d z | z = L 1 ; T 2 ( L 1 + L 2 , ω ) = T 3 ( L 1 + L 2 , ω ) , κ 2 d T 2 ( z , ω ) d z | z = L 1 + L 2 = κ 3 d T 3 ( z , ω ) d z | z = L 1 + L 2 .
A 1 ( 1 b 01 ) B 1 ( 1 + b 01 ) = C 1 ( b 01 q 11 ) + D 1 ( b 01 + q 11 ) + E 1 ( b 01 + m 11 ) + F 1 exp ( μ t 1 L 1 ) ( b 01 m 11 ) , A 1 exp ( σ 1 L 1 ) + B 1 exp ( σ 1 L 1 ) A 2 B 2 = C 2 + D 2 + E 2 C 1 exp ( Q 1 L 1 ) D 1 exp ( Q 1 L 1 ) ( E 1 + F 1 ) exp ( μ t 1 L 1 ) ; b 12 A 1 exp ( σ 1 L 1 ) b 12 B 1 exp ( σ 1 L 1 ) A 2 + B 2 = q 22 C 2 q 22 D 2 m 22 E 2 q 12 C 1 exp ( Q 1 L 1 ) + q 12 D 1 exp ( Q 1 L 1 ) m 12 ( F 1 E 1 ) exp ( μ t 1 L 1 ) ; A 2 exp ( σ 2 L 2 ) + B 2 exp ( σ 2 L 2 ) + B 3 = C 2 exp ( Q 2 L 2 ) D 2 exp ( Q 2 L 2 ) E 2 exp ( μ t 2 L 2 ) D 3 E 3 , q 23 A 2 exp ( σ 2 L 2 ) q 23 B 2 exp ( σ 2 L 2 ) + B 3 = q 23 C 2 exp ( Q 2 L 2 ) + q 23 D 2 exp ( Q 2 L 2 ) + m 22 E 2 exp ( μ t 2 L 2 ) q 33 D 3 m 33 E 3 ,
b i j κ i σ i κ j σ j , q i j κ i Q i κ j σ j , m i j κ i μ t i κ j σ j .
V PTR ( ω ) = C ( ω ) μ IR [ 0 L 1 T 1 ( z , ω ) exp ( μ I R z ) d z + L 1 L 2 T 2 ( z , ω ) exp ( μ I R z ) d z + L 2 T 3 ( z , ω ) exp ( μ I R z ) d z ] .
V carbon ( ω ) = C ( ω ) 0 T carbon ( z , ω ) d z = C ( ω ) 0 I 0 2 ( 1 + k 0 σ 0 k s σ s ) k s σ s exp ( σ s z ) d z ,
V PTR ( ω ) = | V PTR ( ω ) | exp [ i φ P T R ( ω ) ] ,
Amp PTR ( ω ) = | V PTR ( ω ) | , Phase PTR ( ω ) = φ PTR ( ω ) .
Res = n = 1 n max [ log 10 ( Amp Exp ) log 10 ( Amp Theor ) ] 2 n = 1 n max [ log 10 ( Amp Exp ) ] 2 + n = 1 n max [ Phase Exp Phase Theor ] 2 n = 1 n max [ Phase Exp ] 2 ,

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