Abstract

Noninvasive dental diagnostics is a growing discipline since it has been established that early detection and quantification of tooth mineral loss can reverse caries lesions in their incipient state. A theoretical coupled diffuse photon density and thermal-wave model was developed and applied to photothermal radiometric frequency responses, fitted to experimental data using a multiparameter simplex downhill minimization algorithm for the extraction of optothermophysical properties from artificially demineralized human enamel. The aim of this study was to evaluate the reliability and robustness of the advanced fitting algorithm. The results showed a select group of optical and thermal transport parameters and thicknesses were reliably extracted from the computational fitting algorithm. Theoretically derived thicknesses were accurately predicted, within about 20% error, while the estimated error in the optical and thermal property evaluation was within the values determined from early studies using destructive analyses. The high fidelity of the theoretical model illustrates its efficacy, reliability, and applicability toward the nondestructive characterization of depthwise inhomogeneous sound enamel and complex enamel caries lesions.

© 2010 Optical Society of America

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2009

A. Matvienko, A. Mandelis, and S. H. Abrams, “Robust multiparameter method of evaluating the optical and thermal properties of a layered tissue structure using photothermal radiometry,” Appl. Opt. 48, 3193–3204 (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]

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]

2008

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

2007

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

R. H. Selwitz, A. I. Ismail, and N. B. Pitts, “Dental caries,” Lancet 369, 51–59 (2007).
[CrossRef] [PubMed]

A. J. Panas, M. Preiskorn, M. Dabrowski, and S. Żmuda, “Validation of hard tooth tissue thermal diffusivity measurements applying an infrared camera,” Infrared Phys. Technol. 49, 302–305 (2007).
[CrossRef]

2006

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

C. Darling, G. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310nm,” J. Biomed. Opt. 11, 034023 (2006).
[CrossRef]

2004

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

R. J. Jeon, A. Mandelis, V. Sanchez, and S. H. Abrams, “Non-intrusive, non-contacting frequency-domain photothermal radiometry and luminescence depth profilometry of natural carious and artificial sub-surface lesions in human teeth,” J. Biomed. Opt. 9, 804–819 (2004).
[CrossRef] [PubMed]

2003

T. Kodaka, “Scanning electron microscopic observations of surface prismless enamel formed by minute crystals in some human permanent teeth,” Anat. Sci. Int. 78, 79–84 (2003).
[CrossRef] [PubMed]

A. J. Panas, S. Żmuda, J. Terpiłowski, and M. Preiskorn, “Investigation of the thermal diffusivity of human tooth hard tissue.” Int. J. Thermophys. 24, 837–848 (2003).
[CrossRef]

C. Mujat, M. H. van der Veen, J. L. Ruben, J. J. ten Bosch, and A. Dogariu, “Optical pathlength spectroscopy of incipient caries lesions in relation to quantitative light fluorescence and lesion characteristics,” Appl. Opt. 42, 2979–2986 (2003).
[CrossRef] [PubMed]

2002

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

M. H. van der Veen, M. Ando, G. K. Stookey, and E. de Josselin de Jong, “A Monte Carlo simulation of the influence of sound enamel scattering coefficient on lesion visibility in light-induced fluorescence,” Caries Res. 36, 10–18 (2002).
[CrossRef] [PubMed]

2001

J. C. Ragain and W. M. Johnston, “Accuracy of Kubelka–Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449–452 (2001).
[CrossRef] [PubMed]

A. Mandelis, L. Nicolaides, and Y. Chen, “Structure and the reflectionless/refractionless nature of parabolic diffusion wave fields,” Phys. Rev. Lett. 87, 020801 (2001).
[CrossRef]

L. Nicolaides, Y. Chen, A. Mandelis, and I. A. Vitkin, “Theoretical, experimental, and computational aspects of optical property determination of turbid media by using frequency-domain laser infrared photothermal radiometry,” J. Opt. Soc. Am. A 18, 2548–2556 (2001).
[CrossRef]

2000

C. C. Ko, D. Tantbirojn, T. Wang, and W. H. Douglas, “Optical scattering power for characterisation of mineral loss,” J. Dent. Res. 79, 1584–1589 (2000).
[CrossRef] [PubMed]

1998

B. T. Amaechi, S. M. Higham, and W. M. Edgar, “Factors affecting the development of carious lesions in bovine teeth in vitro,” Arch. Oral Biol. 43, 619–628 (1998).
[CrossRef] [PubMed]

1997

J. J. M. Damen, R. A. M. Exterkate, and J. M. ten Cate, “Reproducibility of TMR for the determination of longitudinal mineral changes in dental hard tissues,” Adv. Dent. Res. 11, 415–419 (1997).
[CrossRef]

1995

1993

G. P. Chebotareva, A. P. Nikitin, B. V. Zubov, and A. P. Chebotarev, “Investigation of teeth absorption in the IR range by the pulsed photothermal radiometry,” Proc. SPIE 2080, 117–128 (1993).
[CrossRef]

G. A. Macho and M. A. Berner, “Enamel thickness of human maxillary molars reconsidered,” Am. J. Phys. Anthropol. 92, 189–200 (1993).
[CrossRef] [PubMed]

1991

T. Kodaka, M. Kuroiwa, and S. Higashi, “Structural and distribution patterns of surface ‘prismless’ enamel in human permanent teeth,” Caries Res. 25, 7–20 (1991).
[CrossRef] [PubMed]

J. R. Zijp and J. J. ten Bosch, “Angular dependence of HeNe-laser light scattering by bovine and human dentine,” Arch. Oral Biol. 36, 283–289 (1991).
[CrossRef] [PubMed]

1990

Y. Minesaki, “Thermal properties of human teeth and dental cements,” Shika Zairyo Kikai 9, 633–646 (1990).
[PubMed]

1987

B. Angmar-Månsson and J. J. ten Bosch, “Optical methods for the detection and quantification of caries,” Adv. Dent. Res. 1, 14–20 (1987).
[CrossRef] [PubMed]

E. de Josselin de Jong, A. H. I. M. Linden, and J. J. ten Bosch, “Longitudinal microradiography: a non-destructive automated quantitative method to follow mineral changes in mineralised tissue slices,” Phys. Med. Biol. 32, 1209–1220 (1987).
[CrossRef] [PubMed]

1977

D. Spitzer and J. J. ten Bosch, “Luminescence quantum yields of sound and carious dental enamel,” Calcif. Tissue Res. 24, 249–251 (1977).
[CrossRef] [PubMed]

1975

A. Groeneveld, D. J. Purdell-Lewis, and J. Arends, “Influence of the mineral content of enamel on caries-like lesions produced in hydroxyethylcellulose buffer solutions,” Caries Res. 9, 127–138 (1975).
[CrossRef] [PubMed]

D. Spitzer and J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

1970

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

1967

A. J. Gwinnett, “The ultrastructure of the “prismless” enamel of permanent human teeth,” Arch. Oral Biol. 12, 381–387 (1967).
[CrossRef] [PubMed]

1966

L. W. Ripa, A. J. Gwinnett, and M. G. Buonocore, “The “prismless” outer layer of deciduous and permanent enamel,” Arch. Oral Biol. 11, 41–48 (1966).
[CrossRef] [PubMed]

1964

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

1962

J. A. Gray, “Kinetics of the dissolution of human dental enamel in acid,” J. Dent. Res. 41, 633–645 (1962).
[CrossRef] [PubMed]

1961

R. G. Craig and F. A. Peyton, “Thermal conductivity of teeth structures, dentin cements, and amalgam,” J. Dent. Res. 40, 411–418 (1961).
[CrossRef]

Abrams, S. H.

A. Matvienko, A. Mandelis, and S. H. Abrams, “Robust multiparameter method of evaluating the optical and thermal properties of a layered tissue structure using photothermal radiometry,” Appl. Opt. 48, 3193–3204 (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]

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]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

R. J. Jeon, A. Mandelis, V. Sanchez, and S. H. Abrams, “Non-intrusive, non-contacting frequency-domain photothermal radiometry and luminescence depth profilometry of natural carious and artificial sub-surface lesions in human teeth,” J. Biomed. Opt. 9, 804–819 (2004).
[CrossRef] [PubMed]

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

Al-Khateeb, S.

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

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]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

B. T. Amaechi, S. M. Higham, and W. M. Edgar, “Factors affecting the development of carious lesions in bovine teeth in vitro,” Arch. Oral Biol. 43, 619–628 (1998).
[CrossRef] [PubMed]

Ando, M.

M. H. van der Veen, M. Ando, G. K. Stookey, and E. de Josselin de Jong, “A Monte Carlo simulation of the influence of sound enamel scattering coefficient on lesion visibility in light-induced fluorescence,” Caries Res. 36, 10–18 (2002).
[CrossRef] [PubMed]

Angmar-Månsson, B.

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

B. Angmar-Månsson and J. J. ten Bosch, “Optical methods for the detection and quantification of caries,” Adv. Dent. Res. 1, 14–20 (1987).
[CrossRef] [PubMed]

Arends, J.

A. Groeneveld, D. J. Purdell-Lewis, and J. Arends, “Influence of the mineral content of enamel on caries-like lesions produced in hydroxyethylcellulose buffer solutions,” Caries Res. 9, 127–138 (1975).
[CrossRef] [PubMed]

Berner, M. A.

G. A. Macho and M. A. Berner, “Enamel thickness of human maxillary molars reconsidered,” Am. J. Phys. Anthropol. 92, 189–200 (1993).
[CrossRef] [PubMed]

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]

Buonocore, M. G.

L. W. Ripa, A. J. Gwinnett, and M. G. Buonocore, “The “prismless” outer layer of deciduous and permanent enamel,” Arch. Oral Biol. 11, 41–48 (1966).
[CrossRef] [PubMed]

Chebotarev, A. P.

G. P. Chebotareva, A. P. Nikitin, B. V. Zubov, and A. P. Chebotarev, “Investigation of teeth absorption in the IR range by the pulsed photothermal radiometry,” Proc. SPIE 2080, 117–128 (1993).
[CrossRef]

Chebotareva, G. P.

G. P. Chebotareva, A. P. Nikitin, B. V. Zubov, and A. P. Chebotarev, “Investigation of teeth absorption in the IR range by the pulsed photothermal radiometry,” Proc. SPIE 2080, 117–128 (1993).
[CrossRef]

Chen, Y.

Craig, R. G.

R. G. Craig and F. A. Peyton, “Thermal conductivity of teeth structures, dentin cements, and amalgam,” J. Dent. Res. 40, 411–418 (1961).
[CrossRef]

Dabrowski, M.

A. J. Panas, M. Preiskorn, M. Dabrowski, and S. Żmuda, “Validation of hard tooth tissue thermal diffusivity measurements applying an infrared camera,” Infrared Phys. Technol. 49, 302–305 (2007).
[CrossRef]

Damen, J. J. M.

J. J. M. Damen, R. A. M. Exterkate, and J. M. ten Cate, “Reproducibility of TMR for the determination of longitudinal mineral changes in dental hard tissues,” Adv. Dent. Res. 11, 415–419 (1997).
[CrossRef]

Darling, C.

C. Darling, G. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310nm,” J. Biomed. Opt. 11, 034023 (2006).
[CrossRef]

de Josselin de Jong, E.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

M. H. van der Veen, M. Ando, G. K. Stookey, and E. de Josselin de Jong, “A Monte Carlo simulation of the influence of sound enamel scattering coefficient on lesion visibility in light-induced fluorescence,” Caries Res. 36, 10–18 (2002).
[CrossRef] [PubMed]

E. de Josselin de Jong, A. H. I. M. Linden, and J. J. ten Bosch, “Longitudinal microradiography: a non-destructive automated quantitative method to follow mineral changes in mineralised tissue slices,” Phys. Med. Biol. 32, 1209–1220 (1987).
[CrossRef] [PubMed]

E. de Josselin de Jong, A. F. Hall, and M. H. van der Veen, “Quantitative light-induced fluorescence detection method: a Monte Carlo simulation model,” in Proceedings of the 1st Annual Indiana Conference. Early Detection of Dental Caries, G.K.Stookey, ed. (Indiana University, 1996), pp. 91–104.

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]

Dogariu, A.

Douglas, W. H.

C. C. Ko, D. Tantbirojn, T. Wang, and W. H. Douglas, “Optical scattering power for characterisation of mineral loss,” J. Dent. Res. 79, 1584–1589 (2000).
[CrossRef] [PubMed]

Edgar, W. M.

B. T. Amaechi, S. M. Higham, and W. M. Edgar, “Factors affecting the development of carious lesions in bovine teeth in vitro,” Arch. Oral Biol. 43, 619–628 (1998).
[CrossRef] [PubMed]

Exterkate, R. A. M.

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

J. J. M. Damen, R. A. M. Exterkate, and J. M. ten Cate, “Reproducibility of TMR for the determination of longitudinal mineral changes in dental hard tissues,” Adv. Dent. Res. 11, 415–419 (1997).
[CrossRef]

Featherstone, J. D. B.

Flannery, B. P.

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

Fried, D.

C. Darling, G. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310nm,” J. Biomed. Opt. 11, 034023 (2006).
[CrossRef]

D. Fried, R. E. Glena, J. D. B. Featherstone, and W. Seka, “Nature of light scattering in dental enamel and dentin at visible and near-infrared wavelengths,” Appl. Opt. 34, 1278–1285(1995).
[CrossRef] [PubMed]

Giertsen, E.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

Glena, R. E.

Gmur, R.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

Gray, J. A.

J. A. Gray, “Kinetics of the dissolution of human dental enamel in acid,” J. Dent. Res. 41, 633–645 (1962).
[CrossRef] [PubMed]

Groeneveld, A.

A. Groeneveld, D. J. Purdell-Lewis, and J. Arends, “Influence of the mineral content of enamel on caries-like lesions produced in hydroxyethylcellulose buffer solutions,” Caries Res. 9, 127–138 (1975).
[CrossRef] [PubMed]

Guggenheim, B.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

Gwinnett, A. J.

A. J. Gwinnett, “The ultrastructure of the “prismless” enamel of permanent human teeth,” Arch. Oral Biol. 12, 381–387 (1967).
[CrossRef] [PubMed]

L. W. Ripa, A. J. Gwinnett, and M. G. Buonocore, “The “prismless” outer layer of deciduous and permanent enamel,” Arch. Oral Biol. 11, 41–48 (1966).
[CrossRef] [PubMed]

Hall, A. F.

E. de Josselin de Jong, A. F. Hall, and M. H. van der Veen, “Quantitative light-induced fluorescence detection method: a Monte Carlo simulation model,” in Proceedings of the 1st Annual Indiana Conference. Early Detection of Dental Caries, G.K.Stookey, ed. (Indiana University, 1996), pp. 91–104.

Han, C.

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

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]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

Higashi, S.

T. Kodaka, M. Kuroiwa, and S. Higashi, “Structural and distribution patterns of surface ‘prismless’ enamel in human permanent teeth,” Caries Res. 25, 7–20 (1991).
[CrossRef] [PubMed]

Higham, S. M.

B. T. Amaechi, S. M. Higham, and W. M. Edgar, “Factors affecting the development of carious lesions in bovine teeth in vitro,” Arch. Oral Biol. 43, 619–628 (1998).
[CrossRef] [PubMed]

Huynh, G.

C. Darling, G. Huynh, and D. Fried, “Light scattering properties of natural and artificially demineralized dental enamel at 1310nm,” J. Biomed. Opt. 11, 034023 (2006).
[CrossRef]

Ismail, A. I.

R. H. Selwitz, A. I. Ismail, and N. B. Pitts, “Dental caries,” Lancet 369, 51–59 (2007).
[CrossRef] [PubMed]

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]

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]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

R. J. Jeon, A. Mandelis, V. Sanchez, and S. H. Abrams, “Non-intrusive, non-contacting frequency-domain photothermal radiometry and luminescence depth profilometry of natural carious and artificial sub-surface lesions in human teeth,” J. Biomed. Opt. 9, 804–819 (2004).
[CrossRef] [PubMed]

Johnston, W. M.

J. C. Ragain and W. M. Johnston, “Accuracy of Kubelka–Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449–452 (2001).
[CrossRef] [PubMed]

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.

Ko, C. C.

C. C. Ko, D. Tantbirojn, T. Wang, and W. H. Douglas, “Optical scattering power for characterisation of mineral loss,” J. Dent. Res. 79, 1584–1589 (2000).
[CrossRef] [PubMed]

Kodaka, T.

T. Kodaka, “Scanning electron microscopic observations of surface prismless enamel formed by minute crystals in some human permanent teeth,” Anat. Sci. Int. 78, 79–84 (2003).
[CrossRef] [PubMed]

T. Kodaka, M. Kuroiwa, and S. Higashi, “Structural and distribution patterns of surface ‘prismless’ enamel in human permanent teeth,” Caries Res. 25, 7–20 (1991).
[CrossRef] [PubMed]

Kulkarni, G.

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

Kuroiwa, M.

T. Kodaka, M. Kuroiwa, and S. Higashi, “Structural and distribution patterns of surface ‘prismless’ enamel in human permanent teeth,” Caries Res. 25, 7–20 (1991).
[CrossRef] [PubMed]

Linden, A. H. I. M.

E. de Josselin de Jong, A. H. I. M. Linden, and J. J. ten Bosch, “Longitudinal microradiography: a non-destructive automated quantitative method to follow mineral changes in mineralised tissue slices,” Phys. Med. Biol. 32, 1209–1220 (1987).
[CrossRef] [PubMed]

Macho, G. A.

G. A. Macho and M. A. Berner, “Enamel thickness of human maxillary molars reconsidered,” Am. J. Phys. Anthropol. 92, 189–200 (1993).
[CrossRef] [PubMed]

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]

A. Matvienko, A. Mandelis, and S. H. Abrams, “Robust multiparameter method of evaluating the optical and thermal properties of a layered tissue structure using photothermal radiometry,” Appl. Opt. 48, 3193–3204 (2009).
[CrossRef]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

R. J. Jeon, A. Mandelis, V. Sanchez, and S. H. Abrams, “Non-intrusive, non-contacting frequency-domain photothermal radiometry and luminescence depth profilometry of natural carious and artificial sub-surface lesions in human teeth,” J. Biomed. Opt. 9, 804–819 (2004).
[CrossRef] [PubMed]

A. Mandelis, L. Nicolaides, and Y. Chen, “Structure and the reflectionless/refractionless nature of parabolic diffusion wave fields,” Phys. Rev. Lett. 87, 020801 (2001).
[CrossRef]

L. Nicolaides, Y. Chen, A. Mandelis, and I. A. Vitkin, “Theoretical, experimental, and computational aspects of optical property determination of turbid media by using frequency-domain laser infrared photothermal radiometry,” J. Opt. Soc. Am. A 18, 2548–2556 (2001).
[CrossRef]

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

Matvienko, A.

A. Matvienko, A. Mandelis, and S. H. Abrams, “Robust multiparameter method of evaluating the optical and thermal properties of a layered tissue structure using photothermal radiometry,” Appl. Opt. 48, 3193–3204 (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]

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]

R. J. Jeon, A. Hellen, A. Matvienko, A. Mandelis, S. H. Abrams, and B. T. Amaechi, “In vitro detection and quantification of enamel and root caries using infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 13, 034025 (2008).
[CrossRef] [PubMed]

R. J. Jeon, A. Matvienko, A. Mandelis, S. H. Abrams, B. T. Amaechi, and G. Kulkarni, “Detection of interproximal demineralized lesions on human teeth in vitro using frequency-domain infrared photothermal radiometry and modulated luminescence,” J. Biomed. Opt. 12, 034028 (2007).
[CrossRef] [PubMed]

Minesaki, Y.

Y. Minesaki, “Thermal properties of human teeth and dental cements,” Shika Zairyo Kikai 9, 633–646 (1990).
[PubMed]

Mujat, C.

Nicolaides, L.

Nikitin, A. P.

G. P. Chebotareva, A. P. Nikitin, B. V. Zubov, and A. P. Chebotarev, “Investigation of teeth absorption in the IR range by the pulsed photothermal radiometry,” Proc. SPIE 2080, 117–128 (1993).
[CrossRef]

Panas, A. J.

A. J. Panas, M. Preiskorn, M. Dabrowski, and S. Żmuda, “Validation of hard tooth tissue thermal diffusivity measurements applying an infrared camera,” Infrared Phys. Technol. 49, 302–305 (2007).
[CrossRef]

A. J. Panas, S. Żmuda, J. Terpiłowski, and M. Preiskorn, “Investigation of the thermal diffusivity of human tooth hard tissue.” Int. J. Thermophys. 24, 837–848 (2003).
[CrossRef]

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.

Peyton, F. A.

R. G. Craig and F. A. Peyton, “Thermal conductivity of teeth structures, dentin cements, and amalgam,” J. Dent. Res. 40, 411–418 (1961).
[CrossRef]

Pitts, N. B.

R. H. Selwitz, A. I. Ismail, and N. B. Pitts, “Dental caries,” Lancet 369, 51–59 (2007).
[CrossRef] [PubMed]

Preiskorn, M.

A. J. Panas, M. Preiskorn, M. Dabrowski, and S. Żmuda, “Validation of hard tooth tissue thermal diffusivity measurements applying an infrared camera,” Infrared Phys. Technol. 49, 302–305 (2007).
[CrossRef]

A. J. Panas, S. Żmuda, J. Terpiłowski, and M. Preiskorn, “Investigation of the thermal diffusivity of human tooth hard tissue.” Int. J. Thermophys. 24, 837–848 (2003).
[CrossRef]

Press, W. H.

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

Purdell-Lewis, D. J.

A. Groeneveld, D. J. Purdell-Lewis, and J. Arends, “Influence of the mineral content of enamel on caries-like lesions produced in hydroxyethylcellulose buffer solutions,” Caries Res. 9, 127–138 (1975).
[CrossRef] [PubMed]

Ragain, J. C.

J. C. Ragain and W. M. Johnston, “Accuracy of Kubelka–Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449–452 (2001).
[CrossRef] [PubMed]

Ripa, L. W.

L. W. Ripa, A. J. Gwinnett, and M. G. Buonocore, “The “prismless” outer layer of deciduous and permanent enamel,” Arch. Oral Biol. 11, 41–48 (1966).
[CrossRef] [PubMed]

Ruben, J. L.

Sanchez, V.

R. J. Jeon, A. Mandelis, V. Sanchez, and S. H. Abrams, “Non-intrusive, non-contacting frequency-domain photothermal radiometry and luminescence depth profilometry of natural carious and artificial sub-surface lesions in human teeth,” J. Biomed. Opt. 9, 804–819 (2004).
[CrossRef] [PubMed]

R. J. Jeon, C. Han, A. Mandelis, V. Sanchez, and S. H. Abrams, “Diagnosis of pit and fissure caries using frequency-domain infrared photothermal radiometry and modulated laser luminescence,” Caries Res. 38, 497–513 (2004).
[CrossRef] [PubMed]

Seka, W.

Selwitz, R. H.

R. H. Selwitz, A. I. Ismail, and N. B. Pitts, “Dental caries,” Lancet 369, 51–59 (2007).
[CrossRef] [PubMed]

Spitzer, D.

D. Spitzer and J. J. ten Bosch, “Luminescence quantum yields of sound and carious dental enamel,” Calcif. Tissue Res. 24, 249–251 (1977).
[CrossRef] [PubMed]

D. Spitzer and J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

Stookey, G. K.

M. H. van der Veen, M. Ando, G. K. Stookey, and E. de Josselin de Jong, “A Monte Carlo simulation of the influence of sound enamel scattering coefficient on lesion visibility in light-induced fluorescence,” Caries Res. 36, 10–18 (2002).
[CrossRef] [PubMed]

Tantbirojn, D.

C. C. Ko, D. Tantbirojn, T. Wang, and W. H. Douglas, “Optical scattering power for characterisation of mineral loss,” J. Dent. Res. 79, 1584–1589 (2000).
[CrossRef] [PubMed]

ten Bosch, J. J.

C. Mujat, M. H. van der Veen, J. L. Ruben, J. J. ten Bosch, and A. Dogariu, “Optical pathlength spectroscopy of incipient caries lesions in relation to quantitative light fluorescence and lesion characteristics,” Appl. Opt. 42, 2979–2986 (2003).
[CrossRef] [PubMed]

J. R. Zijp and J. J. ten Bosch, “Angular dependence of HeNe-laser light scattering by bovine and human dentine,” Arch. Oral Biol. 36, 283–289 (1991).
[CrossRef] [PubMed]

B. Angmar-Månsson and J. J. ten Bosch, “Optical methods for the detection and quantification of caries,” Adv. Dent. Res. 1, 14–20 (1987).
[CrossRef] [PubMed]

E. de Josselin de Jong, A. H. I. M. Linden, and J. J. ten Bosch, “Longitudinal microradiography: a non-destructive automated quantitative method to follow mineral changes in mineralised tissue slices,” Phys. Med. Biol. 32, 1209–1220 (1987).
[CrossRef] [PubMed]

D. Spitzer and J. J. ten Bosch, “Luminescence quantum yields of sound and carious dental enamel,” Calcif. Tissue Res. 24, 249–251 (1977).
[CrossRef] [PubMed]

D. Spitzer and J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

ten Cate, J. M.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

S. Al-Khateeb, R. A. M. Exterkate, E. de Josselin de Jong, B. Angmar-Månsson, and J. M. ten Cate, “Light-induced fluorescence studies on dehydration of incipient enamel lesions,” Caries Res. 36, 25–30 (2002).
[CrossRef] [PubMed]

J. J. M. Damen, R. A. M. Exterkate, and J. M. ten Cate, “Reproducibility of TMR for the determination of longitudinal mineral changes in dental hard tissues,” Adv. Dent. Res. 11, 415–419 (1997).
[CrossRef]

Terpilowski, J.

A. J. Panas, S. Żmuda, J. Terpiłowski, and M. Preiskorn, “Investigation of the thermal diffusivity of human tooth hard tissue.” Int. J. Thermophys. 24, 837–848 (2003).
[CrossRef]

Teukolsky, S. A.

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

van der Veen, M. H.

R. Gmur, E. Giertsen, M. H. van der Veen, E. de Josselin de Jong, J. M. ten Cate, and B. Guggenheim, “In vitro quantitative light-induced fluorescence to measure changes in enamel mineralization,” Clin. Oral Invest. 10, 187–195 (2006).
[CrossRef]

C. Mujat, M. H. van der Veen, J. L. Ruben, J. J. ten Bosch, and A. Dogariu, “Optical pathlength spectroscopy of incipient caries lesions in relation to quantitative light fluorescence and lesion characteristics,” Appl. Opt. 42, 2979–2986 (2003).
[CrossRef] [PubMed]

M. H. van der Veen, M. Ando, G. K. Stookey, and E. de Josselin de Jong, “A Monte Carlo simulation of the influence of sound enamel scattering coefficient on lesion visibility in light-induced fluorescence,” Caries Res. 36, 10–18 (2002).
[CrossRef] [PubMed]

E. de Josselin de Jong, A. F. Hall, and M. H. van der Veen, “Quantitative light-induced fluorescence detection method: a Monte Carlo simulation model,” in Proceedings of the 1st Annual Indiana Conference. Early Detection of Dental Caries, G.K.Stookey, ed. (Indiana University, 1996), pp. 91–104.

Vetterling, W. T.

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

Vitkin, I. A.

Wang, T.

C. C. Ko, D. Tantbirojn, T. Wang, and W. H. Douglas, “Optical scattering power for characterisation of mineral loss,” J. Dent. Res. 79, 1584–1589 (2000).
[CrossRef] [PubMed]

Zijp, J. R.

J. R. Zijp and J. J. ten Bosch, “Angular dependence of HeNe-laser light scattering by bovine and human dentine,” Arch. Oral Biol. 36, 283–289 (1991).
[CrossRef] [PubMed]

J. R. Zijp, “Optical properties of dental hard tissues,” Ph.D. dissertation (University of Groningen, 2001).

Zmuda, S.

A. J. Panas, M. Preiskorn, M. Dabrowski, and S. Żmuda, “Validation of hard tooth tissue thermal diffusivity measurements applying an infrared camera,” Infrared Phys. Technol. 49, 302–305 (2007).
[CrossRef]

A. J. Panas, S. Żmuda, J. Terpiłowski, and M. Preiskorn, “Investigation of the thermal diffusivity of human tooth hard tissue.” Int. J. Thermophys. 24, 837–848 (2003).
[CrossRef]

Zubov, B. V.

G. P. Chebotareva, A. P. Nikitin, B. V. Zubov, and A. P. Chebotarev, “Investigation of teeth absorption in the IR range by the pulsed photothermal radiometry,” Proc. SPIE 2080, 117–128 (1993).
[CrossRef]

Adv. Dent. Res.

B. Angmar-Månsson and J. J. ten Bosch, “Optical methods for the detection and quantification of caries,” Adv. Dent. Res. 1, 14–20 (1987).
[CrossRef] [PubMed]

J. J. M. Damen, R. A. M. Exterkate, and J. M. ten Cate, “Reproducibility of TMR for the determination of longitudinal mineral changes in dental hard tissues,” Adv. Dent. Res. 11, 415–419 (1997).
[CrossRef]

Am. J. Phys. Anthropol.

G. A. Macho and M. A. Berner, “Enamel thickness of human maxillary molars reconsidered,” Am. J. Phys. Anthropol. 92, 189–200 (1993).
[CrossRef] [PubMed]

Anat. Sci. Int.

T. Kodaka, “Scanning electron microscopic observations of surface prismless enamel formed by minute crystals in some human permanent teeth,” Anat. Sci. Int. 78, 79–84 (2003).
[CrossRef] [PubMed]

Appl. Opt.

Arch. Oral Biol.

A. J. Gwinnett, “The ultrastructure of the “prismless” enamel of permanent human teeth,” Arch. Oral Biol. 12, 381–387 (1967).
[CrossRef] [PubMed]

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

B. T. Amaechi, S. M. Higham, and W. M. Edgar, “Factors affecting the development of carious lesions in bovine teeth in vitro,” Arch. Oral Biol. 43, 619–628 (1998).
[CrossRef] [PubMed]

L. W. Ripa, A. J. Gwinnett, and M. G. Buonocore, “The “prismless” outer layer of deciduous and permanent enamel,” Arch. Oral Biol. 11, 41–48 (1966).
[CrossRef] [PubMed]

J. R. Zijp and J. J. ten Bosch, “Angular dependence of HeNe-laser light scattering by bovine and human dentine,” Arch. Oral Biol. 36, 283–289 (1991).
[CrossRef] [PubMed]

Calcif. Tissue Res.

D. Spitzer and J. J. ten Bosch, “Luminescence quantum yields of sound and carious dental enamel,” Calcif. Tissue Res. 24, 249–251 (1977).
[CrossRef] [PubMed]

D. Spitzer and J. J. ten Bosch, “The absorption and scattering of light in bovine and human dental enamel,” Calcif. Tissue Res. 17, 129–137 (1975).
[CrossRef] [PubMed]

Caries Res.

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Figures (5)

Fig. 1
Fig. 1

PTR experimental setup.

Fig. 2
Fig. 2

Three-layer geometrical representation used for theoretical analysis and associated optical and thermal parameters of each layer. μ a j , optical absorption coefficient of layer (j); μ s j , optical scattering coefficient of layer (j); κ j , thermal conductivity of layer (j); α j , thermal diffusivity of layer (j); L j , thickness of layer (j); R j , optical reflection coefficient of layer (j).

Fig. 3
Fig. 3

Schematic geometry of effective layers used for the multiparameter fittings of (a) sound enamel and (b) demineralized enamel.

Fig. 4
Fig. 4

Schematic mineral content profile for the theoretical determination of layer thicknesses. M SL denotes the maximum mineral volume of the surface layer. M LB refers to the minimum mineral content in the subsurface lesion body. SL MAX refers to the maximum thickness of the surface layer, defined as the median between M SL and M LB . L D refers to the TMR defined lesion depth at 95% of the sound enamel calibration level at 87 vol. % . L M is the median between M LB and L 1 (intact surface layer) and L 2 (subsurface lesion).

Fig. 5
Fig. 5

PTR amplitude and phase experimental and three-layer theory plots for the (a) 10 day and (b) 40 day demineralized samples. The experimental data are represented by symbols and the calculated theory is shown as solid curves.

Tables (4)

Tables Icon

Table 1 List of Fitted Parameters for Sound and Demineralized Enamel

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Table 2 Fixed Upper and Lower Limits of the Fundamental Parameters Defined for the Multiparameter Fitting of Sound and Demineralized Enamel and the TMR Defined Limits for Layers 1 and 2 from TMR Mineral Density Profiles

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Table 3 Percentage Differences for Fitting the Final Demineralized PTR Curves of Two Samples with Closed Thickness Limits ( D CLOSED ) and Open Thickness Limits ( D OPEN ) a

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Table 4 Percentage Difference Attributed to the Standard Deviation of Experimental PTR Measurements a

Equations (34)

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V carbon ( ω ) = C ( ω ) I 0 2 ( 1 + κ 0 σ 0 κ s σ s ) κ s σ s ,
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 ) ] + R 3 exp { μ t 2 [ 2 L 2 ( z L 1 ) ] } ( 1 R 1 R 2 exp [ 2 μ t 1 L 1 ] ) ( 1 R 2 R 3 exp [ 2 μ t 2 L 2 ] ) , Ψ c 3 = I 0 ( 1 R 1 ) ( 1 R 2 ) ( 1 R 3 ) exp [ ( μ t 1 L 1 ) ] exp [ ( 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 ] ) ( 1 R 2 R 3 exp [ μ t 2 L 2 ] ) ,
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 Diff i G i ( z ) ,
G i ( z ) = μ s i ( μ t i + g i μ a i μ t i g μ s i ) Ψ c i ,
μ t = μ a + ( 1 g ) μ s .
Ψ t 1 ( z ) = a 1 exp ( Q 1 z ) + b 1 exp ( Q 1 z ) + I eff i ( 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 I eff 2 ( 1 + C μ 2 ) exp [ μ t 2 ( z L 1 ) ] + R 3 exp [ μ t 2 ( 2 L 2 ( z L 1 ) ] ,
Ψ t 3 ( z ) = b 3 exp { Q 3 [ z ( L 1 + L 2 ) ] } + I eff 1 I eff 2 I eff 3 ( 1 + C μ 3 ) 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 1 = I 0 ( 1 R 1 ) 1 R 1 R 2 exp ( 2 μ t 1 L 1 ) , I eff 2 = ( 1 R 2 ) exp ( μ t 1 L 1 ) 1 R 2 R 3 exp ( 2 μ t 2 L 2 ) , I eff 3 = ( 1 R 3 ) exp ( μ t 2 L 2 ) .
Ψ d 1 ( 0 ) = A d d z Ψ d 1 ( z ) | z = 0 ,
Ψ d 1 ( L 1 ) = Ψ d 2 ( L 1 ) ,
Diff 1 d d z Ψ d 1 ( z ) | z = L 1 = Diff 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 ) ,
Diff 2 d d z Ψ d 2 ( z ) | z = L 1 + L 2 = Diff 3 d d z Ψ d 3 ( z ) | z = L 1 + L 2 .
A = 2 Diff ( 1 + r 1 r ) ,
a 1 = 2 V F + G + ( f 1 N exp ( 2 μ t 1 L 1 ) + d 1 P ) ( 1 + X 12 2 V X 12 ) ( 1 X 12 + 2 V X 12 ) exp ( Q 1 L 1 ) M ( 1 X 12 + 2 V X 12 ) exp ( Q 1 L 1 ) , b 1 = a 1 M d 1 P f 1 N exp ( 2 μ t 1 L 1 ) , a 2 = b 2 Y 22 [ f 2 exp ( 2 μ t 2 L 2 ) d 2 ] + 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 2 L 2 ) + b 2 X 23 exp ( Q 2 L 2 ) + Y 23 d 2 exp ( μ t 2 L 2 ) Y 23 f 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 ) + f 2 exp ( μ t 2 L 2 ) ( 1 Y 23 ) exp ( Q 2 L 2 ) ( X 23 + 1 ) + d 3 ( 1 Y 33 ) exp ( Q 2 L 2 ) ( X 23 + 1 ) + Y 22 ( f 2 exp ( 2 μ t 2 L 2 ) d 2 ) Y 12 exp ( f 1 d 1 ) exp ( μ t 1 L 1 ) ; G = d 1 ( 1 + Y 12 ) exp ( μ t 1 L 1 ) f 1 ( 1 Y 12 ) exp ( μ t 1 L 1 ) + d 2 ( 1 + Y 22 ) + f 2 ( 1 Y 22 ) exp ( 2 μ t 2 L 2 ) ; 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 ) ; i = 1 , 2 , 3 ,
σ i = ( 1 + j ) ω / 2 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 ) ] + F 2 exp { μ t 2 [ 2 L 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 1 = η NR 1 μ a 1 κ 1 ( μ t 1 2 σ 1 2 ) I eff 1 ( 1 + C μ 1 ) ; E 2 = η NR 2 μ a 2 κ 2 ( μ t 2 2 σ 2 2 ) I eff 1 I eff 2 ( 1 + C μ 2 ) ; E 3 = η NR 3 μ a 3 κ 3 ( μ t 3 2 σ 3 2 ) I eff 1 I eff 2 I eff 3 ( 1 + C μ 3 ) ; F 1 = η NR 1 μ a 1 κ 1 ( μ t 1 2 σ 1 2 ) I eff 1 ( 1 + C μ 1 ) R 2 ; F 2 = η NR 2 μ a 2 κ 2 ( μ t 2 2 σ 2 2 ) I eff 1 I eff 2 ( 1 + C μ 2 ) R 3 .
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 ,
V PTR ( ω ) = C ( ω ) μ IR [ 0 L 1 T ( z , ω ) e μ IR z d z + L 1 L 2 T ( z , ω ) e μ IR z d z + L 2 T ( z , ω ) e μ IR z d z ] .
V PTR ( ω ) = | V PTR ( ω ) | exp i φ PTR ( ω ) ,
Amp PTR ( ω ) = | V PTR ( ω ) | , Phase PTR ( ω ) = φ PTR ( ω ) .

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