## Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method |

Biomedical Optics Express, Vol. 2, Issue 5, pp. 1169-1183 (2011)

http://dx.doi.org/10.1364/BOE.2.001169

Acrobat PDF (1497 KB)

### 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

## 1. Introduction

7. 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]

10. 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]

11. 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]

*ε*and

*μ*are the permittivity and permeability of the media, respectively. Δ

*t*, Δ

*x*, Δ

*y*and Δ

*z*are the time step and spatial steps in

*x*,

*y*and

*z*-directions, respectively. The above equation which sets the numerical stability bound for classical FDTD was first derived and published in [13

13. 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]

14. 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]

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). [CrossRef]

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

## 2. Complex relative permittivity model for hemoglobin

### 2.1. Complex refractive index data

*n*and (b) extinction coefficient,

_{r}*k*, respectively, versus wavelength,

*λ*for various hemoglobin concentrations. It can be seen that

*n*increases with higher hemoglobin concentration across all wavelengths. The same also holds true for

_{r}*k*, which indicates that higher hemoglobin concentration results in higher absorption (more loss). Another point to note is that when the hemoglobin concentration is at 0 g/dL, its

*n*corresponds to that of water [18

_{r}18. G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 um wavelength region,” Appl. Opt. **12**, 555–563 (1973). [CrossRef] [PubMed]

*k*value is also reduced to zero. This is based on the fact that water has a “transparency window” at the visible light and near infrared region, in which it has negligible absorption or losses.

### 2.2. Complex conjugate pole-residue pair model

## 3. Implementation using fundamental ADI-FDTD (FADI-FDTD) method

*χ*

^{′}= Re(

*χ*) and

*χ*

^{″}= Im(

*χ*). For simplicity, we have assumed single complex conjugate pole-residue pair for now.

**I**

*and*

_{r}**O**

*represent identity and null matrices with dimension*

_{r}*r*×

*r*. We have also included electric current sources

*s*,

_{ex}*s*and

_{ey}*s*in the formulation.

_{ez}**A**and

**B**which incur considerable arithmetic operations and memory indexing, especially when one is solving the implicit update equations. Efforts have been made to further reduce such overheads by means of a more efficient ADI-FDTD method in lossless media proposed in [16

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

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

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

**A**and

**B**on the RHS of Eq. (25) . The source excitation is also only needed at the first substep [21

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]

*E*implicitly, Eq. (27j) is substituted into Eq. (27e) to yield

_{x}- Implicit update equation of FADI-FDTD, c.f. Eq. (28) has much lesser spatial differential operators than that of conventional ADI-FDTD, c.f. Eq. (15). Thus, applying finite difference approximation to these spatial differential operators will incur less overheads in FADI-FDTD compared to conventional ADI-FDTD.
- The overall number of RHS terms in FADI-FDTD are lesser than that of conventional ADI-FDTD. This reduces the amount of arithmetic operations, memory indexing, and results in higher efficiency.

## 4. Numerical results

### 4.1. Transmission and reflection coefficients

*λ*≈

*λ*

_{0}/1.5, where

*λ*

_{0}is the wavelength in free space. The shortest wavelenght considered in free space is 250 nm). The time step Δ

*t*is always specified as Courant-Friedrich-Lewy number (CFLN), which is relative to the Courant-Friedrich-Lewy time step limit in the explicit Yee’s FDTD method, i.e.

*CFLN*= Δ

*t*/Δ

*t*where Δ

_{CFL}*t*is the maximum allowed time step of explicit FDTD method in Eq. (1).

_{CFL}### 4.2. Specific absorption rate (SAR)

*s*is the power loss density in W/m

_{l}^{3}and

*ρ*is the density of the biological media in kg/m

^{3}, which gives the SAR the overall unit of W/kg. In most of the previous SAR analysis literatures [22

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). [CrossRef] [PubMed]

**Ẽ**is the phasor form (without time dependence) of the electric field components. In time domain methods, such as the FDTD methods, the significant benefit is such that a narrow pulse can be injected to provide a wide range of frequency points using only a single run of simulation. In this case,

**Ẽ**can be the Fourier transform of the recorded electric field components, which gives us the information across a range of frequency resulted from illumination by any sinusoidal (monochromatic) source. This however, can only be performed if the medium of interest is modeled as the simplest case of conductive media, described by only a single conductivity value across the frequency range of interest. This is usually not the case in most biological media, such as the human blood described in this paper, where the conductivity changes with frequency. To use Eq. (30), the simulation has to be carried out point by point across the frequency range of interest using sinusoidal excitation, along with the corresponding conductivity values at different frequency. The advantage of FDTD method is thus not being exploited fully.

*ε*=

*ε*

^{′}–

*jε*

^{″}, where

*ε*

^{′}and

*ε*

^{″}are the real and negative imaginary (dissipative) parts, respectively, the power loss density can be found using [26, 27] Note that for conductive media with single conductivity value,

*ε*

^{″}=

*σ/ω*and Eq. (31) conforms to Eq. (30), which is considered as a special case for simplest conductive media. Hence, Eq. (31) is the generalization of power loss density for dispersive media. Using Eq. (31), we are now able to obtain the information across a range of frequency with only a single run of simulation, which is highly efficient.

*μ*m. The straight long capillary is oriented along

*z*-direction and is surrounded by tissue, which is assumed to be made up of mostly water and thus, having similar dielectric properties of water. Since the water has negligible absorption in the visible light and near infrared region, the refractive index is taken as a constant value of 1.33. Both models of 15 g/dL and 33 g/dL hemoglobin concentrations are applied to the capillary. The cell size is chosen as 11.1 nm which corresponds to CPW of 15 at shortest wavelength in the human blood and the time step is chosen at CFLN=4. A

*z*-directed line current source of the same modulated Gaussian pulse as the previous subsection illuminates upon the capillary, exciting the frequency contents from 300 THz to 1200 THz. The perfectly matched layer [28

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]

*ρ*of the human blood is 1060 kg/m

^{3}[25]. The SAR is computed at frequencies of 440 THz, 550 THz and 720 THz. These frequencies correspond to free space wavelengths of 680 nm, 545 mm and 417 nm. Refering to Fig. 3, wavelength at 417 nm is where the highest absorption peak occurs, followed by 545 nm. The lowest absorption occurs at 680 nm. Figs. 5, 6 and 7 illustrate the SAR values (in W/kg) at different locations of the capillary at 440 THz, 550 THz and 720 THz, respectively. All the SAR values are plotted in logarithmic scale (log

_{10}

*SAR*). All figures (a) refer to hemoglobin concentration of 15 g/dL while figures (b) refer to hemoglobin concentration of 33 g/dL. The SAR values at each location have been scaled to correspond to incident power density

^{2}in free space. Note that the source illuminates from the left to right of the figures. We first notice that frequency at 720 THz (417 nm), c.f. Fig. 7 gives rise to the highest SAR values, followed by 550 THz (545 nm), c.f. Fig. 6 and 440 THz (680 nm), c.f. Fig. 5. This is in consistent with the absorption coefficient profile in Fig. 3. Another interesting observation is that at only 720 THz (417 nm), the SAR in the capillary shows a descending trend from the left to right along the wave travelling direction, while other frequencies show rather uniform SAR across the whole capillary region. This indicates that at the absorption peak of 720 THz (417 nm), the bulk of the power is absorbed near the left side, which results in lesser power penetration across the capillary. On the other hand, the absorbed power is well distributed across the whole region of the capillary at other frequencies due to weaker absorption. The standing wave effect may be attributed to the closed volume of the capillary. At the same time, it is observed that capillary with hemoglobin concentration of 33 g/dL generally exhibits higher SAR values across all frequencies. This concentration represents that of only the red blood cells, and could possibly model blood clot inside a capillary, which is formed as a result of entrapment of red blood cells [29

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). [CrossRef]

## 5. Conclusion

## References and links

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. |

2. | 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. |

3. | W. G. Zijlstra, A. Buursma, and O. W. van Assendelft, |

4. | M. Cope, “The application of near infrared spectroscopy to non invasive monitoring of cerebral oxygenation in the newborn infant,” PhD Dissertation, (1991). |

5. | S. A. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water,” http://omlc.ogi.edu/spectra/hemoglobin/summary.html (1998). |

6. | J. G. Kim, M. Xia, and H. Liu, “Extinction coefficients of hemoglobin for near-infrared spectroscopy of tissue,” IEEE Eng. Med. Biol. Mag. |

7. | 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. |

8. | M. Meinke and M. Friebel, “Complex refractive index of hemoglobin in the wavelength range from 250 to 1100 nm,” Proc. SPIE |

9. | 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. | 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. |

11. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. |

12. | A. Taflove and S. C. Hagness, |

13. | 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. |

14. | 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. |

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. |

16. | E. L. Tan, “Efficient algorithm for the unconditionally stable 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. |

17. | E. L. Tan, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. |

18. | G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 um wavelength region,” Appl. Opt. |

19. | B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery |

20. | D. Y. Heh and E. L. Tan, “Corrected impulse invariance method in z-transform theory for frequency-dependent FDTD methods,” IEEE Trans. Antennas Propag. |

21. | E. L. Tan, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. |

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. |

23. | S. Paker and L. Sevgi, “FDTD evaluation of the SAR distribution in a human head near a mobile cellular phone,” Elektrik |

24. | I. Laakso, “FDTD method in assessment of human exposure to base station radiation,” Masters Dissertation, (2007). |

25. | 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. , |

26. | J. D. Jackson, |

27. | D. M. Pozar, |

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 |

29. | K. Boryczko, W. Dzewinel, and D. A. Yuen, “Modeling fibrin aggregation in blood flow with discrete particles,” Comput. Methods Prog. Biomed. |

**OCIS Codes**

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: January 4, 2011

Revised Manuscript: March 18, 2011

Manuscript Accepted: March 22, 2011

Published: April 12, 2011

**Citation**

Ding Yu Heh and Eng Leong Tan, "Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method," Biomed. Opt. Express **2**, 1169-1183 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-5-1169

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### References

- 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).
- 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 O. W. van Assendelft, Visible and Near Infrared Absorption Spectra of Human and Animal Haemoglobin: Determination and Application , (VSP, Zeist, 2000).
- M. Cope, “The application of near infrared spectroscopy to non invasive monitoring of cerebral oxygenation in the newborn infant,” PhD Dissertation, (1991).
- S. A. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water,” http://omlc.ogi.edu/spectra/hemoglobin/summary.html (1998).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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, “Fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain methods,” IEEE Trans. Antennas Propag. 56, 170–177 (2008). [CrossRef]
- G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 um wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef] [PubMed]
- B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Delivery 14, 1052–1061 (1999). [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, “Concise current source implementation for efficient 3-D ADI-FDTD method,” IEEE Microw. Wireless Comp. Lett. 17, 748–750 (2007). [CrossRef]
- 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]
- 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).
- I. Laakso, “FDTD method in assessment of human exposure to base station radiation,” Masters Dissertation, (2007).
- 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).
- J. D. Jackson, Classical Electrodynamics , 3rd ed. (John Wiley & Sons, 1998).
- D. M. Pozar, Microwave Engineering , 3rd ed. (Wiley, 2005).
- 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]
- 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]

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