Abstract

This paper presents the modeling of hemoglobin at optical frequency (250 nm – 1000 nm) using the unconditionally stable fundamental alternating-direction-implicit finite-difference time-domain (FADI-FDTD) method. An accurate model based on complex conjugate pole-residue pairs is proposed to model the complex permittivity of hemoglobin at optical frequency. Two hemoglobin concentrations at 15 g/dL and 33 g/dL are considered. The model is then incorporated into the FADI-FDTD method for solving electromagnetic problems involving interaction of light with hemoglobin. The computation of transmission and reflection coefficients of a half space hemoglobin medium using the FADI-FDTD validates the accuracy of our model and method. The specific absorption rate (SAR) distribution of human capillary at optical frequency is also shown. While maintaining accuracy, the unconditionally stable FADI-FDTD method exhibits high efficiency in modeling hemoglobin.

© 2011 OSA

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  15. T. Namiki, “3-D ADI-FDTD method-Unconditionally stable time domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 48, 1743–1748 (2000).
    [CrossRef]
  16. E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 7–9 (2007).
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  17. E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. 56, 170–177 (2008).
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  21. E. L. Tan, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 748–750 (2007).
    [CrossRef]
  22. C. M. Furse and O. P. Gahdhi, “A memory efficient method of calculating specific absorption rate in CW FDTD simulations,” IEEE Trans. Biomed. Eng. 43, 558–560 (1996).
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  26. J. D. Jackson, Classical Electrodynamics , 3rd ed. (John Wiley & Sons, 1998).
  27. D. M. Pozar, Microwave Engineering , 3rd ed. (Wiley, 2005).
  28. W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
    [CrossRef]
  29. K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
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2010 (1)

W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
[CrossRef]

2009 (1)

D. Y. Heh and E. L. Tan, “Corrected impulse invariance method in z-transform theory for frequency-dependent FDTD methods,” IEEE Trans. Antennas Propag. 57, 2683–2690 (2009).
[CrossRef]

2008 (1)

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. 56, 170–177 (2008).
[CrossRef]

2007 (4)

E. L. Tan, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 748–750 (2007).
[CrossRef]

J. G. Kim and H. Liu, “Variation of haemoglobin extinction coefficients can cause errors in the determination of haemoglobin concentration measured by near-infrared spectroscopy,” Phys. Med. Biol. 52, 6295–6332 (2007).
[CrossRef] [PubMed]

E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 7–9 (2007).
[CrossRef]

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

2006 (1)

2005 (5)

D. M. Pozar, Microwave Engineering , 3rd ed. (Wiley, 2005).

M. Meinke and M. Friebel, “Complex refractive index of hemoglobin in the wavelength range from 250 to 1100 nm,” Proc. SPIE 5862, 1–7 (2005).

M. Meinke and M. Friebel, “Determination of the complex refractive index of highly concentrated hemoglobin solutions using transmittance and reflectance measurements,” J. Biomed. Opt. 10, 064019 (2005).
[CrossRef]

J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. 24, 118–121 (2005).
[CrossRef] [PubMed]

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method , (Artech House, 2005).

2004 (1)

K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
[CrossRef]

2000 (3)

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000).
[CrossRef]

T. Namiki, “3-D ADI-FDTD method-Unconditionally stable time domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 48, 1743–1748 (2000).
[CrossRef]

W. G. Zijlstra, A. Buursma, and O. W. van Assendelft, Visible and Near Infrared Absorption Spectra of Human and Animal Haemoglobin: Determination and Application , (VSP, Zeist, 2000).

1999 (1)

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[CrossRef]

1998 (2)

J. D. Jackson, Classical Electrodynamics , 3rd ed. (John Wiley & Sons, 1998).

S. Paker and L. Sevgi, “FDTD evaluation of the SAR distribution in a human head near a mobile cellular phone,” Elektrik 6, 227–242 (1998).

1996 (1)

C. M. Furse and O. P. Gahdhi, “A memory efficient method of calculating specific absorption rate in CW FDTD simulations,” IEEE Trans. Biomed. Eng. 43, 558–560 (1996).
[CrossRef] [PubMed]

1994 (1)

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

1991 (1)

W. G. Zijlstra, A. Buursma, and W. P. Meeuwsen-van der Roest, “Absorption spectra of human fetal and adult oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Clin. Chem. 37, 1633–1668 (1991).

1975 (1)

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[CrossRef]

1973 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Baranauskas, V.

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

Bonani, L. H.

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

Boryczko, K.

K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
[CrossRef]

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[CrossRef]

Buursma, A.

W. G. Zijlstra, A. Buursma, and O. W. van Assendelft, Visible and Near Infrared Absorption Spectra of Human and Animal Haemoglobin: Determination and Application , (VSP, Zeist, 2000).

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

W. G. Zijlstra, A. Buursma, and W. P. Meeuwsen-van der Roest, “Absorption spectra of human fetal and adult oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Clin. Chem. 37, 1633–1668 (1991).

Catsburg, J. F.

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

Chen, Z.

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000).
[CrossRef]

Dzewinel, W.

K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
[CrossRef]

Falke, H. E.

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

Friebel, M.

M. Meinke and M. Friebel, “Model function to calculate the refractive index of native hemoglobin in the wavelength range of 250 nm to 1100 nm dependent on concentration,” Appl. Opt. 45, 2838–2842 (2006).
[CrossRef] [PubMed]

M. Meinke and M. Friebel, “Determination of the complex refractive index of highly concentrated hemoglobin solutions using transmittance and reflectance measurements,” J. Biomed. Opt. 10, 064019 (2005).
[CrossRef]

M. Meinke and M. Friebel, “Complex refractive index of hemoglobin in the wavelength range from 250 to 1100 nm,” Proc. SPIE 5862, 1–7 (2005).

Furse, C. M.

C. M. Furse and O. P. Gahdhi, “A memory efficient method of calculating specific absorption rate in CW FDTD simulations,” IEEE Trans. Biomed. Eng. 43, 558–560 (1996).
[CrossRef] [PubMed]

Gahdhi, O. P.

C. M. Furse and O. P. Gahdhi, “A memory efficient method of calculating specific absorption rate in CW FDTD simulations,” IEEE Trans. Biomed. Eng. 43, 558–560 (1996).
[CrossRef] [PubMed]

Galan, C. F.

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

Garcez, S. G.

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

Gustavsen, B.

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method , (Artech House, 2005).

Hale, G. M.

Heh, D. Y.

W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
[CrossRef]

D. Y. Heh and E. L. Tan, “Corrected impulse invariance method in z-transform theory for frequency-dependent FDTD methods,” IEEE Trans. Antennas Propag. 57, 2683–2690 (2009).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics , 3rd ed. (John Wiley & Sons, 1998).

Kim, J. G.

J. G. Kim and H. Liu, “Variation of haemoglobin extinction coefficients can cause errors in the determination of haemoglobin concentration measured by near-infrared spectroscopy,” Phys. Med. Biol. 52, 6295–6332 (2007).
[CrossRef] [PubMed]

J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. 24, 118–121 (2005).
[CrossRef] [PubMed]

Liu, H.

J. G. Kim and H. Liu, “Variation of haemoglobin extinction coefficients can cause errors in the determination of haemoglobin concentration measured by near-infrared spectroscopy,” Phys. Med. Biol. 52, 6295–6332 (2007).
[CrossRef] [PubMed]

J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. 24, 118–121 (2005).
[CrossRef] [PubMed]

Meeuwsen-van der Roest, W. P.

W. G. Zijlstra, A. Buursma, and W. P. Meeuwsen-van der Roest, “Absorption spectra of human fetal and adult oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Clin. Chem. 37, 1633–1668 (1991).

Meinke, M.

M. Meinke and M. Friebel, “Model function to calculate the refractive index of native hemoglobin in the wavelength range of 250 nm to 1100 nm dependent on concentration,” Appl. Opt. 45, 2838–2842 (2006).
[CrossRef] [PubMed]

M. Meinke and M. Friebel, “Complex refractive index of hemoglobin in the wavelength range from 250 to 1100 nm,” Proc. SPIE 5862, 1–7 (2005).

M. Meinke and M. Friebel, “Determination of the complex refractive index of highly concentrated hemoglobin solutions using transmittance and reflectance measurements,” J. Biomed. Opt. 10, 064019 (2005).
[CrossRef]

Namiki, T.

T. Namiki, “3-D ADI-FDTD method-Unconditionally stable time domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 48, 1743–1748 (2000).
[CrossRef]

Paker, S.

S. Paker and L. Sevgi, “FDTD evaluation of the SAR distribution in a human head near a mobile cellular phone,” Elektrik 6, 227–242 (1998).

Pozar, D. M.

D. M. Pozar, Microwave Engineering , 3rd ed. (Wiley, 2005).

Querry, M. R.

Semlyen, A.

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[CrossRef]

Sevgi, L.

S. Paker and L. Sevgi, “FDTD evaluation of the SAR distribution in a human head near a mobile cellular phone,” Elektrik 6, 227–242 (1998).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method , (Artech House, 2005).

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[CrossRef]

Tan, E. L.

W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
[CrossRef]

D. Y. Heh and E. L. Tan, “Corrected impulse invariance method in z-transform theory for frequency-dependent FDTD methods,” IEEE Trans. Antennas Propag. 57, 2683–2690 (2009).
[CrossRef]

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. 56, 170–177 (2008).
[CrossRef]

E. L. Tan, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 748–750 (2007).
[CrossRef]

E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 7–9 (2007).
[CrossRef]

Tay, W. C.

W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
[CrossRef]

van Assendelft, O. W.

W. G. Zijlstra, A. Buursma, and O. W. van Assendelft, Visible and Near Infrared Absorption Spectra of Human and Animal Haemoglobin: Determination and Application , (VSP, Zeist, 2000).

Xia, M.

J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. 24, 118–121 (2005).
[CrossRef] [PubMed]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Yuen, D. A.

K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
[CrossRef]

Zhang, J.

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000).
[CrossRef]

Zheng, F.

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000).
[CrossRef]

Zijlstra, W. G.

W. G. Zijlstra, A. Buursma, and O. W. van Assendelft, Visible and Near Infrared Absorption Spectra of Human and Animal Haemoglobin: Determination and Application , (VSP, Zeist, 2000).

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

W. G. Zijlstra, A. Buursma, and W. P. Meeuwsen-van der Roest, “Absorption spectra of human fetal and adult oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Clin. Chem. 37, 1633–1668 (1991).

Appl. Opt. (2)

Clin. Chem. (1)

W. G. Zijlstra, A. Buursma, and W. P. Meeuwsen-van der Roest, “Absorption spectra of human fetal and adult oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Clin. Chem. 37, 1633–1668 (1991).

Comput. Biochem. Physiol. (1)

W. G. Zijlstra, A. Buursma, H. E. Falke, and J. F. Catsburg, “Spectrophotometry of hemoglobin: absorption spectra of rat oxyhemoglobin, deoxyhemoglobin, carboxyhemoglobin, and methemoglobin,” Comput. Biochem. Physiol. 107B, 161–166 (1994).
[CrossRef]

Comput. Methods Prog. Biomed. (1)

K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. 75, 181–194 (2004).
[CrossRef]

Elektrik (1)

S. Paker and L. Sevgi, “FDTD evaluation of the SAR distribution in a human head near a mobile cellular phone,” Elektrik 6, 227–242 (1998).

IEEE Eng. Med. Biol. Mag. (1)

J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. 24, 118–121 (2005).
[CrossRef] [PubMed]

IEEE Microw. Wireless Comp. Lett. (2)

E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 7–9 (2007).
[CrossRef]

E. L. Tan, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 748–750 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

D. Y. Heh and E. L. Tan, “Corrected impulse invariance method in z-transform theory for frequency-dependent FDTD methods,” IEEE Trans. Antennas Propag. 57, 2683–2690 (2009).
[CrossRef]

E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. 56, 170–177 (2008).
[CrossRef]

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

C. M. Furse and O. P. Gahdhi, “A memory efficient method of calculating specific absorption rate in CW FDTD simulations,” IEEE Trans. Biomed. Eng. 43, 558–560 (1996).
[CrossRef] [PubMed]

IEEE Trans. Microw. Theory Tech. (3)

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 23, 623–630 (1975).
[CrossRef]

F. Zheng, Z. Chen, and J. Zhang, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech. 48, 1550–1558 (2000).
[CrossRef]

T. Namiki, “3-D ADI-FDTD method-Unconditionally stable time domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. 48, 1743–1748 (2000).
[CrossRef]

IEEE Trans. Power Delivery (1)

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999).
[CrossRef]

J. Biomed. Opt. (1)

M. Meinke and M. Friebel, “Determination of the complex refractive index of highly concentrated hemoglobin solutions using transmittance and reflectance measurements,” J. Biomed. Opt. 10, 064019 (2005).
[CrossRef]

Phys. Med. Biol. (1)

J. G. Kim and H. Liu, “Variation of haemoglobin extinction coefficients can cause errors in the determination of haemoglobin concentration measured by near-infrared spectroscopy,” Phys. Med. Biol. 52, 6295–6332 (2007).
[CrossRef] [PubMed]

PIER M (1)

W. C. Tay, D. Y. Heh, and E. L. Tan, “GPU-accelerated fundamental ADI-FDTD with complex frequency shifted convolutional perfectly matched layer,” PIER M 14, 177–192 (2010).
[CrossRef]

Proc. SPIE (1)

M. Meinke and M. Friebel, “Complex refractive index of hemoglobin in the wavelength range from 250 to 1100 nm,” Proc. SPIE 5862, 1–7 (2005).

SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. (1)

S. G. Garcez, C. F. Galan, L. H. Bonani, and V. Baranauskas, “Estimating the electromagnetic field effects in biological tissues through the finite-difference time-domain method,” SBMO/IEEE MTT-S Int. Microw. and Optoelectronics Conf. Proc. , 43, 58–62 (2007).

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Figures (7)

Fig. 1
Fig. 1

(a) Real part of complex refractive index, nr and (b) extinction coefficient, k versus wavelength, λ for different hemoglobin concentrations. Both nr and k increase with higher hemoglobin concentration across all wavelengths.

Fig. 2
Fig. 2

(a) Real part, εr and (b) imaginary part, εi of complex relative permittivity versus wavelength, λ : model and experimental data for hemoglobin concentrations of 15 g/dL and 33 g/dL. A good fit is observed between our proposed model and experimental data across all wavelengths.

Fig. 3
Fig. 3

Absorption coefficient, μa versus wavelength, λ : model and experimental data for hemoglobin concentrations of 15 g/dL and 33 g/dL.

Fig. 4
Fig. 4

Magnitude of (a) transmission and (b) reflection coefficients versus frequency for various CFLNs.

Fig. 5
Fig. 5

SAR (in logarithmic scale) at different locations of capillary at 440 THz (680 nm) for hemoglobin concentrations of (a) 15 g/dL and (b) 33 g/dL. The wave illuminates from left to right.

Fig. 6
Fig. 6

SAR (in logarithmic scale) at different locations of capillary at 550 THz (545 nm) for hemoglobin concentrations of (a) 15 g/dL and (b) 33 g/dL. The wave illuminates from left to right.

Fig. 7
Fig. 7

SAR (in logarithmic scale) at different locations of capillary at 720 THz (417 nm) for hemoglobin concentrations of (a) 15 g/dL and (b) 33 g/dL. The wave illuminates from left to right.

Tables (4)

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Table 1 Fitted Values of ap , rp and ε for Hemoglobin Concentration of 15 g/dL

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Table 2 Fitted Values of ap , rp and ε for Hemoglobin Concentration of 33 g/dL

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Table 3 Maximum and Average SAR Values Computed using FADI-FDTD and Explicit FDTD at Hemoglobin Concentration of 15 g/dL

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Table 4 Maximum and Average SAR Values Computed Using FADI-FDTD and Explicit FDTD at Hemoglobin Concentration of 33 g/dL

Equations (53)

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Δ t μ ɛ 1 Δ x 2 + 1 Δ y 2 + 1 Δ z 2
n ( λ ) = n r ( λ ) j k ( λ )
k = μ a λ 4 π
μ a = ln ( 10 ) × e × C 64500
n r = n w r ( β c + 1 )
ɛ ( ω ) = n ( ω ) 2 = n r ( ω ) 2 k ( ω ) 2 j 2 n r ( ω ) k ( ω )
ɛ ( ω ) = ɛ + p ( r p j ω a p + r p * j ω a p * )
× H ( t ) = ɛ 0 ɛ t E ( t ) + σ E ( t ) + ɛ 0 p 1 { j ω ( r p j ω a p + r p * j ω a p * ) E ( ω ) } = ɛ 0 ɛ t E ( t ) + σ E ( t ) + p ( J p ( t ) + J ^ p ( t ) )
× E ( t ) = μ 0 μ t H ( t ) + σ * H ( t ) + μ 0 p 1 { j ω ( q p j ω b p + q p * j ω b p * ) H ( ω ) } = μ 0 μ t H ( t ) + σ * H ( t ) + p ( M p ( t ) + M ^ p ( t ) )
t J p ( t ) a p J p ( t ) = ɛ 0 r p t E ( t )
t J ^ p ( t ) a p * J ^ p ( t ) = ɛ 0 r p * t E ( t )
t M p ( t ) b p M p ( t ) = μ 0 q p t H ( t )
t M ^ p ( t ) b p * M ^ p ( t ) = μ 0 q p * t H ( t ) .
× H ( t ) = ɛ 0 ɛ t E ( t ) + σ E ( t ) + p 2 Re { J p ( t ) }
× E ( t ) = μ 0 μ t H ( t ) + σ * H ( t ) + p 2 Re { M p ( t ) } .
J p n + 1 2 = 1 + a p Δ t 4 1 a p Δ t 4 J p n + r p 1 a p Δ t 4 ( E n + 1 2 E n ) = k 1 p J p n + k 2 p ( E n + 1 2 E n )
M p n + 1 2 = 1 + b p Δ t 4 1 b p Δ t 4 M p n + q p 1 b p Δ t 4 ( H n + 1 2 H n ) = l 1 p M p n + l 2 p ( H n + 1 2 H n ) .
E x n + 1 2 c 2 y H z n + 1 2 = c 1 E x n c 2 z H y n c 2 p Re { ( 1 + k 1 p ) J x p n }
H z n + 1 2 d 2 y E x n + 1 2 = d 1 H z n d 2 x E y n d 2 p Re { ( 1 + l 1 p ) M x p n }
c 1 = 1 σ Δ t 4 ɛ 0 ɛ + Δ t 4 ɛ 0 ɛ p 2 Re ( k 2 p ) 1 + σ Δ t 4 ɛ 0 ɛ + Δ t 4 ɛ 0 ɛ p 2 Re ( k 2 p ) , c 2 = Δ t 2 ɛ 0 ɛ 1 + σ Δ t 4 ɛ 0 ɛ + Δ t 4 ɛ 0 ɛ p 2 Re ( k 2 p ) ,
d 1 = 1 σ * Δ t 4 μ 0 μ + Δ t 4 μ 0 μ p 2 Re ( l 2 p ) 1 + σ * Δ t 4 μ 0 μ + Δ t 4 μ 0 μ p 2 Re ( l 2 p ) , d 2 = Δ t 2 μ 0 μ 1 + σ * Δ t 4 μ 0 μ + Δ t 4 μ 0 μ p 2 Re ( l 2 p ) .
E x n + 1 2 c 2 y ( d 2 y E x n + 1 2 ) = c 1 E x n c 2 z H y n c 2 p Re { ( 1 + k 1 p ) J x p n } + c 2 y ( d 1 H z n ) c 2 y ( d 2 x E y n ) c 2 y ( d 2 p Re { ( 1 + l 1 p ) M x p n } ) .
( I 18 D 2 A + F l ) u n + 1 2 = ( D 1 + D 2 B + F r ) u n + D 2 s n + 1 2
( I 18 D 2 B + F l ) u n + 1 = ( D 1 + D 2 A + F r ) u n + 1 2 + D 2 s n + 1 2
u = [ E x , E y , E z , H x , H y , H z , J x , J y , J z , J x , J y , J z , M x , M y , M z , M x , M y , M z ] T
s = [ s e x , s e y , s e z , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ] T
A = [ O 3 K 1 O 3 O 3 O 3 O 3 K 2 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 ] , B = [ O 3 K 2 O 3 O 3 O 3 O 3 K 1 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 ]
K 1 = [ 0 0 y z 0 0 0 x 0 ] , K 2 = [ 0 z 0 0 0 x y 0 0 ]
D 1 = diag ( c 1 , c 1 , c 1 , d 1 , d 1 , d 1 , k 1 , k 1 , k 1 , k 1 , k 1 , k 1 , l 1 , l 1 , l 1 , l 1 , l 1 , l 1 )
D 2 = diag ( c 2 , c 2 , c 2 , d 2 , d 2 , d 2 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 )
F l = [ O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 O 3 k 2 I 3 O 3 O 3 O 3 O 3 O 3 k 2 I 3 O 3 O 3 O 3 O 3 O 3 O 3 l 2 I 3 O 3 O 3 O 3 O 3 O 3 l 2 I 3 O 3 O 3 O 3 O 3 ]
F r = [ O 3 O 3 c 2 ( 1 + k 1 ) I 3 c 2 k 1 I 3 O 3 O 3 O 3 O 3 O 3 O 3 d 2 ( 1 + l 1 ) I 3 d 2 l 1 I 3 k 2 I 3 O 3 O 3 k 1 I 3 O 3 O 3 k 2 I 3 O 3 k 1 I 3 O 3 O 3 O 3 O 3 l 2 I 3 O 3 O 3 O 3 l 1 I 3 O 3 l 2 I 3 O 3 O 3 l 1 I 3 O 3 ] .
v ˜ n = [ ( I 12 + D 1 ) + F l + F r ] u n v ˜ n 1 2
( I 12 D 2 A + F l ) u n + 1 2 = v ˜ n + D 2 s n + 1 2
v ˜ n + 1 2 = [ ( I 12 + D 1 ) + F l + F r ] u n + 1 2 v ˜ n
( I 12 D 2 B + F l ) u n + 1 = v ˜ n + 1 2
v ˜ = [ e ˜ x , e ˜ y , e ˜ z , h ˜ x , h ˜ y , h ˜ z , j ˜ x , j ˜ y , j ˜ z , j ˜ x , j ˜ y , j ˜ z , m ˜ x , m ˜ y , m ˜ z , m ˜ x , m ˜ y , m ˜ z ] T ,
e ˜ ξ n = ( 1 + c 2 ) E ξ n e ˜ x n 1 2 c 2 p Re { ( 1 + k 1 p ) J ξ p n } , ξ = x , y , z
h ˜ ξ n = ( 1 + d 2 ) H ξ n h ˜ x n 1 2 d 2 p Re { ( 1 + l 1 p ) M ξ p n } , ξ = x , y , z
j ˜ ξ p n = ( 1 + k 1 p ) J ξ p n j ˜ ξ p n 1 2 2 k 2 p E ξ n , p , ξ = x , y , z
m ˜ ξ p n = ( 1 + l 1 p ) M ξ p n m ˜ ξ p n 1 2 2 l 2 p H ξ n , p , ξ = x , y , z
E x n + 1 2 c 2 y H z n + 1 2 = e ˜ x n c 2 s e x
E y n + 1 2 c 2 z H x n + 1 2 = e ˜ y n c 2 s e y
E z n + 1 2 c 2 x H y n + 1 2 = e ˜ z n c 2 s e z
H x n + 1 2 d 2 z E y n + 1 2 = h ˜ x n
H y n + 1 2 d 2 x E z n + 1 2 = h ˜ y n
H z n + 1 2 d 2 y E x n + 1 2 = h ˜ z n
J ξ p n + 1 2 = k 2 p E ξ n + 1 2 + j ˜ ξ p n , p , ξ = x , y , z
M ξ p n + 1 2 = l 2 p H ξ n + 1 2 + m ˜ ξ p n , p , ξ = x , y , z .
E x n + 1 2 c 2 y ( d 2 y E x n + 1 2 ) = e ˜ x n c 2 s e x + c 2 y h ˜ x n .
SAR = s l ρ
s l = 1 2 σ | E ˜ | 2
s l = 1 2 ω ɛ | E ˜ | 2 .

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