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Biomedical Optics Express

Biomedical Optics Express

  • Editor: Joseph A. Izatt
  • Vol. 2, Iss. 5 — May. 1, 2011
  • pp: 1169–1183
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Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method

Ding Yu Heh and Eng Leong Tan  »View Author Affiliations


Biomedical Optics Express, Vol. 2, Issue 5, pp. 1169-1183 (2011)
http://dx.doi.org/10.1364/BOE.2.001169


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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

The information on optical properties of human blood is crucial for many medical applications. In particular, the hemoglobin in red blood cells plays an important role by transporting oxygen throughout the human body. Past literatures have successfully determined the molar extinction coefficient [1

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).

7

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]

] of hemoglobin at optical frequency. The real part of its complex refractive index has also been found using transmittance and reflectance measurements [8

8. 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).

10

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]

]. With such knowledge on the complex refractive index, a better understanding of the interaction of light with the biological media is greatly desired. Owing to this, the finite-difference time-domain (FDTD) method [11

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]

, 12

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

] has been widely used for solving electromagnetic problems and modeling electromagnetic wave propagation in various media. Unfortunately, the usage of such explicit FDTD method is limited by the fact that its time step is restricted by the Courant-Friedrich-Lewy (CFL) condition, given below as
Δtμɛ1Δx2+1Δy2+1Δz2
(1)
where ε and μ are the permittivity and permeability of the media, respectively. Δt, Δx, Δy and Δz 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]

]. Note that if the spatial steps are small, this restriction may result in undesirably long simulation time.

The conventional alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method [14

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

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]

], on the other hand, has unconditional stability feature where the time step is no longer restricted by the CFL stability criterion. This has circumvented the above limitation in its own right. However, due to the fact that the algorithm is implicit in nature, its implementation is complicated. Not only are there tridiagonal systems that need to be solved, the right-hand-side (RHS) of each equations involve many update coefficients and field variables, which calls for considerable arithmetic operations and memory indexing. Recently, these overheads has been overcome by the introduction of the fundamental alternating-direction-implicit finite-difference time-domain (FADI-FDTD) method [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

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]

]. In FADI-FDTD, its algorithm is included within a family of fundamental implicit schemes, which feature similar fundamental updating structures that are in simplest forms with most efficient matrix-operator-free RHS. This leads to substantial reduction in arithmetic operations and memory indexing. Nevertheless, while the evolution of computational electromagnetic in time domain from explicit FDTD method to FADI-FDTD has been encouraging, there is still a lack of appropriate complex permittivity models for biological media (in this case, hemoglobin) at optical frequency. Even then, the implementation of such appropriate complex permittivity models into the FADI-FDTD is also highly sought after.

In this paper, we seek to address the aforementioned problems and bridge the necessary gaps. First, we propose a model based on complex conjugate pole-residue pairs to describe the electrical behaviour of hemoglobin, where it is fitted against the complex refractive index data. Once an accurate model is found, it is incorporated into FADI-FDTD method for solving electromagnetic problems involving light interactions with biological media. Finally, some numerical examples are provided to validate the proposed method and to demonstrate some electromagnetic wave interactions with human blood at optical frequency.

2. Complex relative permittivity model for hemoglobin

2.1. Complex refractive index data

Therefore, using Eq. (4) and Eq. (5), the complex refractive index of hemoglobin at optical frequency can be extracted for various concentrations conveniently. Figure 1 plots the (a) real part of complex refractive index, nr and (b) extinction coefficient, k, respectively, versus wavelength, λ for various hemoglobin concentrations. It can be seen that nr increases with higher hemoglobin concentration across all wavelengths. The same also holds true for 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 nr corresponds to that of water [18

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]

]. At this 0 g/dL concentration the 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.

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.

2.2. Complex conjugate pole-residue pair model

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

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

The full update equations of conventional ADI-FDTD can be summarized in compact matrix form (in two substeps) as
(I18D2A+Fl)un+12=(D1+D2B+Fr)un+D2sn+12
(16a)
(I18D2B+Fl)un+1=(D1+D2A+Fr)un+12+D2sn+12
(16b)
where
u=[Ex,Ey,Ez,Hx,Hy,Hz,Jx,Jy,Jz,Jx,Jy,Jz,Mx,My,Mz,Mx,My,Mz]T
(17)
s=[sex,sey,sez,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]T
(18)
A=[O3K1O3O3O3O3K2O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3],B=[O3K2O3O3O3O3K1O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3O3]
(19)
K1=[00yz000x0],K2=[0z000xy00]
(20)
D1=diag(c1,c1,c1,d1,d1,d1,k1,k1,k1,k1,k1,k1,l1,l1,l1,l1,l1,l1)
(21)
D2=diag(c2,c2,c2,d2,d2,d2,0,0,0,0,0,0,0,0,0,0,0,0)
(22)
Fl=[O3O3O3O3O3O3O3O3O3O3O3O3k2I3O3O3O3O3O3k2I3O3O3O3O3O3O3l2I3O3O3O3O3O3l2I3O3O3O3O3]
(23)
Fr=[O3O3c2(1+k1)I3c2k1I3O3O3O3O3O3O3d2(1+l1)I3d2l1I3k2I3O3O3k1I3O3O3k2I3O3k1I3O3O3O3O3l2I3O3O3O3l1I3O3l2I3O3O3l1I3O3].
(24)
Note that variables χ = Re(χ) and χ = Im(χ). For simplicity, we have assumed single complex conjugate pole-residue pair for now. I r and O r represent identity and null matrices with dimension r × r. We have also included electric current sources sex, sey and sez in the formulation.

A quick glance at Eq. (16) shows that its right-hand-side (RHS) still involves matrix operators 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

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]

]. Such efficient ADI algorithm is included within a family of fundamental implicit schemes, which feature similar fundamental updating structures that are in simplest forms with most efficient matrix-operator-free RHS. Following [17

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]

], it can be shown that Eq. (16) is reducible to the fundamental form (FADI-FDTD), which calls for the following update procedures:
v˜n=[(I12+D1)+Fl+Fr]unv˜n12
(25a)
(I12D2A+Fl)un+12=v˜n+D2sn+12
(25b)
v˜n+12=[(I12+D1)+Fl+Fr]un+12v˜n
(25c)
(I12D2B+Fl)un+1=v˜n+12
(25d)
where Eq. (25a) and Eq. (25b) form the first substep while Eq. (25c) and Eq. (25d) form the second. It is worth noting that the FADI-FDTD has matrix-operator-free RHS, i.e., there is no matrix operator 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]

] which further simplifies the implementation compared to the conventional ADI-FDTD method.

We now provide the full update equations of FADI-FDTD from Eq. (25) . Letting
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,
(26)
we have for the first substep,
e˜ξn=(1+c2)Eξne˜xn12c2pRe{(1+k1p)Jξpn},ξ=x,y,z
(27a)
h˜ξn=(1+d2)Hξnh˜xn12d2pRe{(1+l1p)Mξpn},ξ=x,y,z
(27b)
j˜ξpn=(1+k1p)Jξpnj˜ξpn122k2pEξn,p,ξ=x,y,z
(27c)
m˜ξpn=(1+l1p)Mξpnm˜ξpn122l2pHξn,p,ξ=x,y,z
(27d)
Exn+12c2yHzn+12=e˜xnc2sex
(27e)
Eyn+12c2zHxn+12=e˜ync2sey
(27f)
Ezn+12c2xHyn+12=e˜znc2sez
(27g)
Hxn+12d2zEyn+12=h˜xn
(27h)
Hyn+12d2xEzn+12=h˜yn
(27i)
Hzn+12d2yExn+12=h˜zn
(27j)
Jξpn+12=k2pEξn+12+j˜ξpn,p,ξ=x,y,z
(27k)
Mξpn+12=l2pHξn+12+m˜ξpn,p,ξ=x,y,z.
(27l)
The second substep can be similarly written down by expanding Eq. (25c) and Eq. (25d). To solve for Ex implicitly, Eq. (27j) is substituted into Eq. (27e) to yield
Exn+12c2y(d2yExn+12)=e˜xnc2sex+c2yh˜xn.
(28)

The advantages of the FADI-FDTD over the conventional ADI-FDTD can be summarized as follows:
  1. Matrix-operator-free RHS in FADI-FDTD, c.f. Eq. (25) . On the other hand, the RHS of conventional ADI-FDTD still involves matrix operators A and B, c.f. Eq. (16) .
  2. Source excitation is only needed in first substep of FADI-FDTD, c.f. Eq. (25b), compared to both substeps of conventional ADI-FDTD, c.f. Eq. (16a) and Eq. (16b).
  3. Implicit update equation of FADI-FDTD, c.f. Eq. (28) is much simpler compared to that of conventional ADI-FDTD, c.f. Eq. (15).
  4. 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.
  5. No dispersive terms J and M is required in implicit update equation of FADI-FDTD, c.f. Eq. (28). On the other hand, they are required in implicit update equation of conventional ADI-FDTD, c.f. Eq. (15).
  6. 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

To demonstrate the accuracy of our hemoglobin model in the FADI-FDTD method, we first compute the transmission and reflection coefficients of a half space human blood numerically. A modulated Gaussian pulse is launched from free space and upon hitting the human blood interface, undergoes transmission and reflection. The modulated Gaussian pulse has a bandwidth of 1500 THz around centre frequency of 750 THz, thus covering the entire frequency spectrum of the human blood model (300 THz to 1200 THz). The model of 15 g/dL hemoglobin concentration (Table 1) is used. Cell size is set at 3.125 nm which corresponds to cell per wavelength (CPW) of around 50 for the shortest wavelength in the human blood (wavelength in human blood, λλ 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 Δt 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 = ΔttCFL where ΔtCFL is the maximum allowed time step of explicit FDTD method in Eq. (1).

The transmission and reflection coefficents (in frequency domain) can be obtained numerically by taking the ratio of the discrete Fourier transform of the transmitted and reflected wave, respectively to the discrete Fourier transform of the incident wave. The numerical experiment is run for various CFLNs to validate the accuracy of the hemoglobin model under different time steps. Figure 4 graphs the magnitude of the computed (a) transmission and (b) reflection coefficients, respectively versus frequency for various CFLNs. They are compared against the reference transmission and reflection coefficients of our analytical hemoglobin model. Also shown in the figures are the transmission and reflection coefficients computed using the explicit FDTD method. Since the explicit FDTD method is not unconditionally stable, it is run at its maximum time step limit of CFLN=1. Any time step larger than that will render the algorithm unstable. The FADI-FDTD method is unconditionally stable, and therefore we can see that CFLN beyond 1 still yield considerably accurate results. Nevertheless, the accuracy of the FADI-FDTD method deteriorates as the CFLN increases, especially towards higher frequency region. This is because as the frequency increases, higher CFLN (and thus larger time step) fails to resolve these higher frequency components sufficiently. Therefore, larger chosen time step increases the simulation speed, but at the expense of sacrificing a certain degree of accuracy. Based on these results, we can see that the chosen time step in FADI-FDTD is always a compromise between efficiency and accuracy.

Fig. 4 Magnitude of (a) transmission and (b) reflection coefficients versus frequency for various CFLNs.

4.2. Specific absorption rate (SAR)

Since we have successfully modeled human blood across a range of optical frequency based on complex conjugate pole-residue pairs along with its implementation using the FADI-FDTD method, the power loss density formula in Eq. (30) can be modified to better suit our scenario. For a dispersive medium which has a complex permittivity of ε = ε , where ε and ε are the real and negative imaginary (dissipative) parts, respectively, the power loss density can be found using [26

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

, 27

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

]
sl=12ωɛ|E˜|2.
(31)
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.

At steady state, the SAR values at different locations of the capillary are computed using Eq. (29) and Eq. (31) and the density, ρ of the human blood is 1060 kg/m3 [25

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. , 43, 58–62 (2007).

]. 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 (log10 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 S=12Re(E˜×H˜*) of 1 mW/m2 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]

]. Thus, the SAR calculation reveals that blood clot has a higher absorption compared to normal blood, which could be useful for medical applications and diagnostics.

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 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 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.

We have also performed similar numerical example using the explicit FDTD method. Tables 3 and 4 show the maximum and average SAR of the capillary computed using the FADI-FDTD method and explicit FDTD method at hemoglobin concentrations of 15 g/dL and 33 g/dL. It can be seen that the calculated SAR values are quite close to each other. This shows that using the FADI-FDTD method at CFLN=4 does not suffer too much from accuracy deficiency. It is also found that the simulation time incurred by the FADI-FDTD method at CFLN=4 in Matlab codes is around 3 times faster than that of the explicit FDTD method. These further show that while maintaining accuracy, the FADI-FDTD method is promising in achieving higher efficiency for modeling biological media.

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

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5. Conclusion

This paper has presented 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 has been proposed to model the complex permittivity of hemoglobin at optical frequency. Two hemoglobin concentrations at 15 g/dL and 33 g/dL have been 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 has validated the accuracy of our model and method. The specific absorption rate (SAR) distribution of human capillary at optical frequency has also been shown. While maintaining accuracy, the unconditionally stable FADI-FDTD method has exhibited high efficiency in modeling hemoglobin.

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. 37, 1633–1668 (1991).

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. 107B, 161–166 (1994). [CrossRef]

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

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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]

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]

8.

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).

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, 064019 (2005). [CrossRef]

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]

12.

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

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]

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]

19.

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

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. 57, 2683–2690 (2009). [CrossRef]

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). [CrossRef] [PubMed]

23.

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).

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. , 43, 58–62 (2007).

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). [CrossRef]

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

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  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]
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  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. 57, 2683–2690 (2009). [CrossRef]
  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). [CrossRef] [PubMed]
  23. 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).
  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. , 43, 58–62 (2007).
  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). [CrossRef]

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