## 2-D PSTD Simulation of the time-reversed ultrasound-encoded deep-tissue imaging technique |

Biomedical Optics Express, Vol. 5, Issue 3, pp. 882-894 (2014)

http://dx.doi.org/10.1364/BOE.5.000882

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### Abstract

We present a robust simulation technique to model the time-reversed ultrasonically encoded (TRUE) technique for deep-tissue imaging. The pseudospectral time-domain (PSTD) algorithm is employed to rigorously model the electromagnetic wave interaction of light propagating through a macroscopic scattering medium. Based upon numerical solutions of Maxwell’s equations, the amplitude and phase are accurately accounted for to analyze factors that affect the TRUE propagation of light through scattering media. More generally, we demonstrate the feasibility of modeling light propagation through a virtual tissue model of *macroscopic dimensions* with numerical solutions of Maxwell’s equations.

© 2014 Optical Society of America

## 1. Introduction

*through*biological tissues, Xu

*et al.*proposed a technique, time-reversal of ultrasonically encoded light (TRUE) [2

2. X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics **5**(3), 154–157 (2011). [CrossRef] [PubMed]

*virtual light source*that is frequency-shifted by the acousto-optic effect. This

*virtual light source*plays the role of a guide star: after phase-conjugation, light propagates and scatters in reversed directions through the scattering medium towards the virtual light source. The combination of ultrasound-modulation and optical phase conjugation enables directing light through scattering medium to a specific position. As shown by [1, 3

3. K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics **6**(10), 657–661 (2012). [CrossRef] [PubMed]

## 2. Methods

4. J. L. Hollmann, R. Horstmeyer, C. Yang, and C. A. DiMarzio, “Analysis and modeling of an ultrasound-modulated guide star to increase the depth of focusing in a turbid medium,” J. Biomed. Opt. **18**(2), 025004 (2013). [CrossRef] [PubMed]

5. S. H. Tseng, “PSTD Simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express **17**(7), 5490–5495 (2009). [CrossRef] [PubMed]

6. S. H. Tseng, “Investigating the Optical Phase Conjugation Reconstruction Phenomenon of Light Multiply Scattered by a Random Medium,” IEEE Photon. J. **2**(4), 636–641 (2010). [CrossRef]

5. S. H. Tseng, “PSTD Simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express **17**(7), 5490–5495 (2009). [CrossRef] [PubMed]

6. S. H. Tseng, “Investigating the Optical Phase Conjugation Reconstruction Phenomenon of Light Multiply Scattered by a Random Medium,” IEEE Photon. J. **2**(4), 636–641 (2010). [CrossRef]

7. Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. **37**(2), 917–926 (1999). [CrossRef]

**is the electric field,**

*E**k*is the wave number, and

_{x}**F**represents the Fourier transform. Since computer calculations are based on discrete numbers, discretization of the continuous electromagnetic fields is necessary. According to the Nyquist sampling theorem, the spatial derivatives calculated in Eq. (1), with a coarse grid of 2 spatial samples per wavelength, allows the PSTD technique to achieve similar accuracy as the FDTD technique, which requires 20 spatial samples per wavelength. The coarse spatial grid points makes it possible to simulate large-scale optical phenomena with economic computational memory. In order to model a light scattering system isolated in free space, an anisotropic perfectly matched layer (APML) absorbing boundary condition [8

8. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. **44**(12), 1630–1639 (1996). [CrossRef]

2. X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics **5**(3), 154–157 (2011). [CrossRef] [PubMed]

*finite-width,*CW light source in a PSTD simulation that was not possible before. By modeling the overlapping region of the ultrasound pulse and incident light beam as a localized light source embedded within the scattering medium, we can trace and analyze the light from the virtual source and simulate the phase-conjugated light focusing at the target position.

*virtual light source*is surrounded by randomly positioned dielectric cylinders to model light propagation through scattering medium. We use a simulation grid with a uniform spatial resolution of 0.33 μm. The scattering medium is modeled by a rectangular region inside vacuum (

*n*= 1.0) packed with randomly positioned,

*r*-μm-diameter dielectric cylinders with a refractive index of

*n*= 1.2. A standard APML absorbing boundary condition [8

8. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. **44**(12), 1630–1639 (1996). [CrossRef]

*piecewise continuous*electromagnetic field (

*E**) generated from a localized region [9] by adding into the simulated space a finite-width, sinusoidal CW plane-wave at each time-step:where (x*

_{i}_{0}, y

_{0}) is the center of the light source;

*k*and

*ω*is the wavenumber and angular frequency, respectively. Edges of the added electromagnetic field are smoothened by a Gaussian profile in the y-direction to minimize numerical artifacts. A CW light of arbitrary wavelength can be generated from this rectangular region to model light from the

*virtual light source.*To relate with experiments, a realistic wavelength such as 532 nm, 1 μm can be modeled. The optical field is added into space as a “soft source” [10] to prevent numerical artifacts from the

*source region*. This allows modeling the TRUE virtual light source as a finite-width, CW light source where the incident light beam and ultrasound overlap in the scattering medium.

*w*) is modeled:

*w*= 0.67 μm, 2 μm, 5, μm, 10 μm, and 20 μm. The back-propagation of light is simulated at various wavelengths: 532 nm, 694 nm, 1064 nm, up to 3 μm (frequency ranging from 563 THz to 100 THz). The intensity profile of the back-propagated light reaching the virtual light source is compared to the light intensity profile of just emitted from the virtual light source; the root-mean-square (RMS) error is calculated and shown in Fig. 4. Simulation results show that the back-propagation of light through a scattering medium is insensitive to the dimensions of the virtual light source—the TRUE virtual light source can be modeled by a coherent finite-width plane wave or a speckle-like virtual light source with no significant difference in performance.

## 3. Analyzing the density of scattering media

*within*a scattering medium is simulated; factors upon which the effectiveness of TRUE light delivery depends are analyzed. As shown in Fig. 5, light propagating through a scattering medium is randomized by the irregular geometry. To analyze this problem, we simulate a localized light source embedded inside a 200-μm-by-300-μm scattering medium consisting of

*N*randomly positioned, 5-μm-diameter, dielectric (

*n*= 1.2) cylinders. Scattering characteristic is proportional to the difference of the refractive index between the scatterer and the surrounding medium; for biological structures, the refractive index difference is typically less than 1.2. The scattering medium is modeled by an aggregate of dielectric (

*n*= 1.2) cylinders positioned in vacuum (

*n*= 1); the reduced scattering coefficient

*μs’*= 0.0113 (1/ μm). Arbitrary wavelength of light can be modeled with the reported simulation technique; here we model an incident wavelength of 1 μm. By phase-conjugating the outgoing light recorded in the

*OPC region*at the right edge of the simulation grid (

*x*= 200 μm,

*y*= 0 to 300 μm), light back-propagates towards the virtual light source. For denser scattering media, scattered light spreads into a wider range of directions. Intensity pattern of light propagating through the scattering medium of various densities are shown in Fig. 5(a), 5(b), 5(c).

*N*= 0, 100, 200, 600, and 676 randomly-positioned, 5-μm-diameter, dielectric (

*n*= 1.2) cylinders. For the case of no scatterers (

*N*= 0), the amplitude profile exhibits a high-frequency wriggle resulting from near field diffraction of light just emerging from the virtual light source. With scatterers randomly positioned in space (

*N*≠ 0), this high-frequency wriggle is dominated by the interference of light from the irregular geometry and becomes less pronounced. As the scattering medium becomes denser with more cylinders, light is scattered into a wider range of directions. Furthermore, an increased percentage of light is backscattered away, resulting in less light delivered into the scattering medium [6

6. S. H. Tseng, “Investigating the Optical Phase Conjugation Reconstruction Phenomenon of Light Multiply Scattered by a Random Medium,” IEEE Photon. J. **2**(4), 636–641 (2010). [CrossRef]

## 4. Radius of constituent scatterers

*r*-μm-radius dielectric cylinders (refractive index

*n*= 1.2). Cross-sectional intensity profiles at the target position are compared (Fig. 7. inset-figure); the root-mean-square error is calculated and compared for various radius

*r*. Simulation results indicate that the effectiveness of the delivery of light through a scattering medium is insensitive to the size constituent scatterers.

## 5. Fraction of phase-conjugated light

*OPC region*(as depicted in Fig. 1). With a larger

*OPC region*, more light is phase-conjugated and propagates towards the target position. We investigate the effect of different cross-sectional widths of the

*OPC region*. As shown in Fig. 8 from bottom to top, each profile corresponds to different cross-sectional width of the

*OPC region*: 50 μm, 100 μm, 200 μm, and 300 μm; as more light is phase-conjugated, the focal point width doesn’t change but the signal-to-background ratio increases.

## 6. Modeling TRUE delivery of light through a virtual tissue model

11. W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods **4**(9), 717–719 (2007). [CrossRef] [PubMed]

*OPC region*at the right edge of the simulation is phase-conjugated and back-propagates towards the virtual light source. Maximum amplitude of the phase-conjugated light emerges at the center of the tissue model. A zoomed-in view of the center of the tissue model is shown in the inset-figure where light converges at the virtual light source and continues to propagate outwards. As shown in Fig. 9, via multiple scattering, the phase-conjugated light propagates through the virtual tissue model forming a pattern of optical paths similar to a river network where smaller creeks flow into larger rivers. Figure 9 demonstrates that the TRUE phenomenon for biological tissue structure can be analyzed with the reported simulation technique. Furthermore, arbitrary irregular geometry, such as a denser or sparser distribution of biological cells can be readily modeled. In addition, the reported simulation enables placing the CW virtual light source at

*arbitrary*position (inside or outside) the scattering medium that was not possible before [4

4. J. L. Hollmann, R. Horstmeyer, C. Yang, and C. A. DiMarzio, “Analysis and modeling of an ultrasound-modulated guide star to increase the depth of focusing in a turbid medium,” J. Biomed. Opt. **18**(2), 025004 (2013). [CrossRef] [PubMed]

*macroscopic*scale without heuristic approximations.

## 7. Summary

*arbitrary*position (

*inside or outside*) the scattering medium. Our simulation is based upon numerical solutions of Maxwell’s equations; by accounting for the phase and amplitude, we accurately simulate the light propagation and interference of the TRUE technique. Specifically, we analyze possible factors that affect light delivery of the TRUE technique, including the virtual light source, the density of the scattering media, the size of the constituent scatterers, and the fraction of phase-conjugated light. Simulation results show that the effectiveness of TRUE light delivery is insensitive to the morphology of the virtual light source (Fig. 3), the density of scattering media (Fig. 6) or the size of the constituent scatterers (Fig. 7). However, the amount of phase-conjugated light plays a critical role of light delivery—with increased phase-conjugated light, the signal-to-background ratio of the TRUE technique increases.

*mm-thick*biological tissue and focus at the target position (Fig. 9). More generally, our results show that such macroscopic light scattering problem can be rigorously analyzed based upon numerical solutions of Maxwell’s equations.

## Acknowledgments

## References and links

1. | Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. H. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun. |

2. | X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics |

3. | K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics |

4. | J. L. Hollmann, R. Horstmeyer, C. Yang, and C. A. DiMarzio, “Analysis and modeling of an ultrasound-modulated guide star to increase the depth of focusing in a turbid medium,” J. Biomed. Opt. |

5. | S. H. Tseng, “PSTD Simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express |

6. | S. H. Tseng, “Investigating the Optical Phase Conjugation Reconstruction Phenomenon of Light Multiply Scattered by a Random Medium,” IEEE Photon. J. |

7. | Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens. |

8. | S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. |

9. | Y. Huang, C. Tsai, W. Ting, and S. H. Tseng, “PSTD simulation of the continuous-wave optical phase conjugation phenomenon,” In Review. |

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

11. | W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods |

**OCIS Codes**

(290.4210) Scattering : Multiple scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Optics of Tissue and Turbid Media

**History**

Original Manuscript: February 11, 2014

Manuscript Accepted: February 20, 2014

Published: February 25, 2014

**Citation**

Snow H. Tseng, Wei-Lun Ting, and Shiang-Jiu Wang, "2-D PSTD Simulation of the time-reversed ultrasound-encoded deep-tissue imaging technique," Biomed. Opt. Express **5**, 882-894 (2014)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-5-3-882

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### References

- Y. M. Wang, B. Judkewitz, C. A. DiMarzio, and C. H. Yang, “Deep-tissue focal fluorescence imaging with digitally time-reversed ultrasound-encoded light,” Nat. Commun.3, 928 (2012).
- X. Xu, H. Liu, and L. V. Wang, “Time-reversed ultrasonically encoded optical focusing into scattering media,” Nat. Photonics5(3), 154–157 (2011). [CrossRef] [PubMed]
- K. Si, R. Fiolka, and M. Cui, “Fluorescence imaging beyond the ballistic regime by ultrasound pulse guided digital phase conjugation,” Nat. Photonics6(10), 657–661 (2012). [CrossRef] [PubMed]
- J. L. Hollmann, R. Horstmeyer, C. Yang, and C. A. DiMarzio, “Analysis and modeling of an ultrasound-modulated guide star to increase the depth of focusing in a turbid medium,” J. Biomed. Opt.18(2), 025004 (2013). [CrossRef] [PubMed]
- S. H. Tseng, “PSTD Simulation of optical phase conjugation of light propagating long optical paths,” Opt. Express17(7), 5490–5495 (2009). [CrossRef] [PubMed]
- S. H. Tseng, “Investigating the Optical Phase Conjugation Reconstruction Phenomenon of Light Multiply Scattered by a Random Medium,” IEEE Photon. J.2(4), 636–641 (2010). [CrossRef]
- Q. H. Liu, “Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm,” IEEE Trans. Geosci. Rem. Sens.37(2), 917–926 (1999). [CrossRef]
- S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag.44(12), 1630–1639 (1996). [CrossRef]
- Y. Huang, C. Tsai, W. Ting, and S. H. Tseng, “PSTD simulation of the continuous-wave optical phase conjugation phenomenon,” In Review.
- A. Taflove and S. C. Hagness, Computational Electrodynamics: the finite-difference time-domain method (Artech House, 2000).
- W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods4(9), 717–719 (2007). [CrossRef] [PubMed]

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