## Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration |

Optics Express, Vol. 19, Issue 23, pp. 22837-22850 (2011)

http://dx.doi.org/10.1364/OE.19.022837

Acrobat PDF (1163 KB)

### Abstract

We demonstrate a method to recover the Young’s modulus (*E*) of a tissue-mimicking phantom from measurements of ultrasound modulated optical tomography (UMOT). The object is insonified by a dual-beam, confocal ultrasound transducer (US) oscillating at frequencies *f*_{0} and *f*_{0} + Δ*f* and the variation of modulation depth (*M*) in the autocorrelation of light traversed through the focal region of the US transducer against Δ*f* is measured. From the dominant peaks observed in the above variation, the natural frequencies of the insonified region associated with the vibration along the US transducer axis are deduced. A consequence of the above resonance is that the speckle fluctuation at the resonance frequency has a higher signal-to-noise to ratio (SNR). From these natural frequencies and the associated eigenspectrum of the oscillating object, Young’s modulus (*E*) of the material in the focal region is recovered. The working of this method is confirmed by recovering *E* in the case of three tissue-mimicking phantoms of different elastic modulus values.

© 2011 OSA

## 1. Introduction

*μ*and

_{a}*μ*respectively) in soft-tissue organs for medical diagnostic imaging [1

_{s}1. C. Kim and L. V. Wang, “Multi-optical-wavelength ultrasound-modulated optical tomography: a phantom study,” Opt. Lett. **32**, 2285–2287 (2007). [CrossRef] [PubMed]

2. L. V. Wang, “Ultrasound-mediated biophotonic imaging: A review of acousto-optical tomography and photo-acoustic tomography,” Dis. Markers **19**, 123–138 (2004). [PubMed]

3. S. Leveque, A. C. Boccara, M. Lebec, and H. S. Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. **24**, 181–183 (1999). [CrossRef]

4. A P Gibson, J C Hebden, and S R Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2005). [CrossRef] [PubMed]

*μ*. However, to extract such information accurate mathematical models describing the interaction of US wave with tissue, and the propagation of light through the insonified ROI are required. In the past, considerable effort has been expended to arrive at and numerically study such interaction models [5

_{a}5. S. Sakadzic and L. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: An analytical model for aniostropically scattering media,” Phys. Rev. E **66**, 026603 (2002). [CrossRef]

8. W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B **204**, 14–19 (1995). [CrossRef]

9. C. Kim, R. J. Zemp, and L. V. Wang, “Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography,” Opt. Lett. **31**, 2423–2425 (2006). [CrossRef] [PubMed]

12. E. Bossy, A. Funke, K. Daoudi, A. Boccara, M. Tanter, and M. Fink, “Transient optoelastography in optically diffusive media,” Appl. Phys. Lett. **90**174111 (2007). [CrossRef]

*M*to

*μ*,

_{a}*μ*,

_{s}*n*and <

*a*

^{2}> (in the ROI) opening up the possibility of recovery of these parameters from the set of measurements {

*M*} over many ‘views’. A partial differential equation (PDE) describing the propagation of the amplitude correlation of light (

*G*) through a turbid medium insonified by an US beam is derived in [7

7. S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. **96**, 163902 (2006). [CrossRef] [PubMed]

*G*. Alternatively, the above non-linear parameter estimation problem can be recast as a linear source recovery problem of reconstructing the ultrasound-induced sources of perturbation in

*G*(which are

*a*(

*r*) and Δ

*n*(

*r*)) from the boundary measurements of this perturbation (

*G*).

^{δ}*M*.vs.frequency of US forcing, in particular the frequency at which

*M*has a dominant peak. To facilitate measurement of

*M*at different US forcing frequencies we employ (as detailed further in Section 2) a confocal two-region transducer, one region resonating at a frequency (

*f*) and the other at (

*f*+ Δ

*f*). The mixing of US pressure amplitudes produces a forcing at the beat frequency Δ

*f*[15

15. M. Fatemi, A. Manduca, and J. Greenleaf, “Imaging Elastic Properties of Biological Tissues by Low-Frequency Harmonic Vibration,” Proc. IEEE **91**, 1503–1517 (2003). [CrossRef]

*M*vs frequency curve is indicative of resonance in the VROI. This natural frequency is a function of the Young’s modulus (

*E*) of the material of the ROI, presently assumed to be linear elastic (Hookean) homogeneous and isotropic, which opens the way for a quantitative assessment of

*E*from a measured resonant frequency. We note here that the natural modes of vibration depend not only on elastic properties of the VROI, but also on its shape/geometry and the boundary conditions. Therefore, as further elaborated in Section 3, for the recovery of

*E*one needs to correctly obtain the shape of the VROI followed by an assignment of the correct local boundary conditions.

*M*for various Δ

*f*’s. Apart from this, in this section we also describe a method to arrive at the unknown

*E*from the measured dominant natural frequency using the method of bisection. Section 6 gives our concluding remarks.

## 2. Perturbation from ultrasound: modelling and verification

### 2.1. Ultrasound-induced force distribution in the focal volume

*f*where Δ

*f*can be varied from as low as 10 Hz to 1 kHz. The composite transducer helps us send MHz waves which have reasonable depth of penetration in the object and thus apply a sinusoidal force at the ROI oscillating at Δ

*f*Hz, the beat frequency.

*p*

_{1}(

*f*) and

*p*

_{2}(

*f*+ Δ

*f*) to denote the sinusoidal pressure amplitudes at the ROI from the two regions in the transducer. The average intensity,

*I*(

**X**), of the combined US beam is given by

*ξ*(

**X**) = <

*ξ*̃(

**X**) > is the time average of the energy deposited at

**X**in the ROI, and

*s*the cross-sectional area we are considering in the intersection volume of the two pressure waves. This energy

*ξ*(

**X**) can be shown to be equal to [14

14. E. Konofagou, J. Thierman, and K. Hynynen, “A focused ultrasound method for simultaneous diagnostic and therapeutic applicationsa simulation study,” Phys. Med. Biol. **46**, 2967–2984 (2001). [CrossRef] [PubMed]

*P*

_{0}and

*P*

_{1}are constants dependent on

*T*. The radiation force experienced by the ROI in an area around

**X**is given by

*α*is the sound absorption coefficient and

*c*the velocity of sound in the object. This force can be shown to be equal to [14

14. E. Konofagou, J. Thierman, and K. Hynynen, “A focused ultrasound method for simultaneous diagnostic and therapeutic applicationsa simulation study,” Phys. Med. Biol. **46**, 2967–2984 (2001). [CrossRef] [PubMed]

*p*

_{1}or

*p*

_{2}, at the focal region can be computed by propagating the pressure at the transducer surface. The pressure at the transducer surface can be calculated from the electrical parameters of the transducer, such as the equivalent resistance, capacitance and inductance (

*R*,

*C*and

*L*respectively) of the PZT. The acoustic pressure

*p*

_{0}on the transducer surface is given by [16

16. J. Huang, J.A. Nissen, and Erik Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. **71**(1), 70–75 (1992). [CrossRef]

*ρ*is the material density and

*c*the acoustic wave velocity,

*A*the surface area of the transducer,

*V*the applied voltage and

### 2.2. Ultrasound-induced pressure distribution in the focal volume

*p*

_{0}on the surface of the transducer to its focal volume, where the pressure can be very large, one needs a model that accounts for material nonlinearity apart from absorption and diffraction in the medium. Here we make use of such a model known as the Westervelt equation [13

13. T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am. **107**, 3035–3046 (2000). [CrossRef] [PubMed]

*β*which depends on the average transducer pressure and the density of the medium. The acoustic absorption

*δ*is the sound diffusivity.

13. T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am. **107**, 3035–3046 (2000). [CrossRef] [PubMed]

*α*and

*β*used are

*α*= 1.5

*dBcm*

^{−1}and

*β*= 6.2 to match the values quoted for breast tissue in [17

17. F. A. Duck, “Nonlinear acoustics in diagnostic ultrasound,” Ultrasound Med. Biol. **28**(1), 1–18 (2002). [CrossRef] [PubMed]

*x*–

*z*plane which contains the transducer axis taken to be along the

*z*-direction.

### 2.3. Experimental verification of the shape of the focal volume

*mm*

^{2}which is masked to a smaller area of ∼ 0.38

*mm*

^{2}. The transducer is mounted on an x-y-z stage and the voltages generated are measured at a number of points in the focal region (at many x-y cross-sections). Assuming that the measured voltages are proportional to the force experienced by points in the ROI, cross-sectional plots of the force experienced by the ROI are generated. The measured voltage distributions, giving axial and radial cross-sections, are shown in Figs. 4 and 5 respectively. Their variations are similar to (and compare well with) the computed pressure variation shown in Figs. 1 and 2. Figure 6 is the contour plot of the experimentally measured voltage in the

*x*–

*z*plane.

## 3. Response of the periodically excited focal volume via solution of the momentum-balance equation

^{®}(a commercially available finite element programme) to carry out the eigenanalysis of the vibrating ROI, obtaining as a result the subset of its natural frequencies corresponding to the dominant vibration (due to shear deformation along the transducer axis). Towards this, the initially selected

*∂*Ω (the closed boundary of Ω) obtained experimentally in Section 2.3 is refined during the eigenanalysis. Theoretically verified/corrected

*∂*Ω is obtained by solving the momentum-balance equation over one full cycle of forcing, followed by identifying the boundary nodes (of the VROI) that separate the vibrating region from its non-vibrating complement. (A good initial guess of

*∂*Ω can also be had from the force distribution in the ROI. The boundary nodes may approximately be considered as those where

*F*(

**X**) drops to

*f*= 1 MHz) gives rise to pressure distributions slightly different with an asymmetry compared to the other one (Figs. 2 & 5). This is owing to a defect in the spherical mould on which PZT is coated.

19. S. Sakadzic and L. V. Wang, “High-resolution ultrasound-modulated optical tomography in biological tissues,” Opt. Lett. **29**, 2770–2772 (2004). [CrossRef] [PubMed]

*g*

_{2}(

*τ*) contains an oscillation of the same frequency as that of the scattering centres. The oscillations of the VROI under US forcing are characterized by a set of eigenvalues (natural frequencies) and eigenvectors (mode shapes) which contain both compression and (dilatational) shear modes. The US vibration at the low beat frequency, however excites mainly shear modes in the VROI (along with the dilatational modes excited by the high frequency wave at 2

*f*). The eigenanalysis helps us identify these shear modes corresponding decidedly to the lower end of the eigenspectrum. The shear waves attenuate rather quickly in the surrounding region of the ROI as shown in [12

12. E. Bossy, A. Funke, K. Daoudi, A. Boccara, M. Tanter, and M. Fink, “Transient optoelastography in optically diffusive media,” Appl. Phys. Lett. **90**174111 (2007). [CrossRef]

*ν*= 0.49) jelly-like material used in our experiments and simulations. The forced vibration at the

*k*nodal point of the VROI can be expressed as where Ψ

^{th}*(*

_{l}*t*), a combination of

*sinω*and

_{l}t*cosω*, is the

_{l}t*l*modal basis function,

^{th}*γ*(

_{k}*t*) is the particular solution under US forcing and

*n*is the number of degrees of freedom in the discretized VROI. If the US forcing is at one of the natural frequencies, say

*ω*, then

_{p}*γ*(

_{k}*t*) will be of the form

## 4. Propagation of light through a turbid medium insonified by the US beam

*g*

_{2}(

*τ*)). To such an object a focused US beam brings in an extra forcing, deterministic and localised. Extensive work has gone into the modelling of the effect of US forcing on the object. As pointed out in Section 1, scattering centres in the insonified region undergo periodic oscillations. The region itself, owing to the compression and rarefaction induced by the pressure wave,presents to the light beam a refractive index modulation Δ

*n*. This modulation, owing to the near-incompressibility of the tissue material, should occur with very low amplitudes with very high frequencies. A typical optical path length

*n*} and {

_{i}*l*} are refractive index and path length corresponding to the

_{i}*i*scattering event) gets perturbed to

^{th}*u*is the amplitude of oscillation of the

_{i}*i*scattering centre in the direction of

^{th}*l*) which means that the US-induced path length fluctuation, to a first-order approximation, is Σ(

_{i}*n*+

_{i}u_{i}*l*Δ

_{i}*n*), giving the US-induced phase perturbation as Δ

_{i}*ϕ*= Δ

*ϕ*+ Δ

_{u}*ϕ*=

_{n}*k*

_{0}Σ

*n*+

_{i}u_{i}*k*

_{0}Σ

*l*Δ

_{i}*n*where Δ

_{i}*ϕ*and Δ

_{u}*ϕ*are the perturbations owing to oscillation and Δ

_{n}*n*respectively.

*ϕ*and Δ

_{u}*ϕ*when the medium has either scattering isotropy or anisotropy, which contain optical, acoustic and elastic properties of the medium as parameters [5

_{n}5. S. Sakadzic and L. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: An analytical model for aniostropically scattering media,” Phys. Rev. E **66**, 026603 (2002). [CrossRef]

*G*

_{1}(

*τ*)) of the electric field of light. In the weak scattering limit, assuming that there is no correlation between light arriving through different paths, photons travelling along the same path of length,

*s*, have amplitude correlation given by

*G*

_{1,s}(

*τ*) is given by [5

5. S. Sakadzic and L. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: An analytical model for aniostropically scattering media,” Phys. Rev. E **66**, 026603 (2002). [CrossRef]

*N*– 1) is the number of scattering events along the path of length

*s*. To arrive at the overall amplitude autocorrelation

*G*(

*τ*),

*G*

_{1,s}(

*τ*) is ensemble averaged over all photon paths s: i.e.,

*p*(

*s*) is the probability density function for

*s*. The normalized amplitude autocorrelation

*g*

_{1}(

*τ*) is

*g*

_{2}(

*τ*) is estimated as 1 +

*β*|

*g*

_{1}(

*τ*)|

^{2}where

*β*is a constant dependent on the optics used in the data gathering.

**66**, 026603 (2002). [CrossRef]

*G*

_{1}(

*τ*); further

*p*(

*s*) evaluated through sending photons, using Monte Carlo (MC) simulations or by solving the diffusion equation for light propagation, is used to compute

*G*(

*τ*) and

*g*

_{1}(

*τ*). The effect of periodic phase fluctuation on

*G*(

*τ*) is the appearance of a modulation of the same frequency as that of the US-induced acoustic vibration frequency, Δ

*f*, from which we derive our measurement which is the modulation depth

*M*(Δ

*f*) obtained as the modulus of the Fourier transform of

*g*

_{2}(

*τ*) evaluated at Δ

*f*. The MC simulation carries the advantage that in transporting photons one can compute both

*p*(

*s*) as well as Δ

*ϕ*and their time averages. Δ

_{u}*ϕ*is in fact the perturbation in the accumulated phase

_{u}*ϕ*originating from the oscillations of scattering centres confined only to the insonified ROI.

*E*of the phantom material, and the force at the focal region (evaluated using the pressure distribution in the focal region obtained from the Westervelt equation, in Eqs.(1)–(4) [13

13. T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am. **107**, 3035–3046 (2000). [CrossRef] [PubMed]

*G*(

*τ*) is calculated, from which

*g*

_{1}(

*τ*),

*g*

_{2}(

*τ*) and

*M*are evaluated [19

19. S. Sakadzic and L. V. Wang, “High-resolution ultrasound-modulated optical tomography in biological tissues,” Opt. Lett. **29**, 2770–2772 (2004). [CrossRef] [PubMed]

*f*varying from 40Hz to 130Hz. A typical

*M*vs Δ

*f*plot is shown in Fig. 7 for which

*E*= 11 kPa. It is observed from these simulations that

*M*goes through a prominent peak (for a particular Δ

*f*, say Δ

*f*) and this Δ

_{r}*f*depends on

_{r}*E*of the material. This indicates that the vibration of the ROI induced by the (US) forcing at the beat frequency Δ

*f*goes through a resonant peak at Δ

*f*, which results in a peak in phase modulation of photons traversed through the ROI, observed as a peak in the modulation depth in

_{r}*g*

_{2}(

*τ*). This evidence of a resonant peak and the inferring of the frequency from the

*M*vs Δ

*f*plot are also verified through experiments. As is shown in Section 5 this resonant frequency can be used to recover

*E*of the VROI.

## 5. Experiments

### 5.1. Fabrication and characterization of the tissue-mimicking phantom

*C*for 2 hours with continuous stirring. This solution is allowed to undergo repeated freeze-thaw cycles of freezing at −20°

*C*for 12 hours, followed by thawing at room temperature for 12 hours [21

21. A. Kharine, S. Manohar, R. Seeton, R. G. M. Kolkman, R. A. Bolt, W. Steenbergen, and F. F. M. de Mul, “Poly Vinyl Alcohol gels for use as tissue phantoms in photoacoustic mammography,” Phys. Med. Biol. **48**, 357–370 (2003). [CrossRef] [PubMed]

*E*and

*μ*in the PVA slab can be tailored by varying the number of freeze-thaw cycles. As mentioned in reference [20]

_{s}*E*can be varied from 11 to 97

*kPa*and correspondingly

*μ*vary from 1.4 to 4.5

_{s}*mm*

^{−1}. For further increasing

*μ*we mix polystyrene beads of diameter 0.1

_{s}*μm*whilst forming the PVA slab. The scattering coefficient of the resulting phantom is determined using an indirect method based on the reverse Monte Carlo procedure [20] and

*E*is measured and verified using a dynamic mechanical analyzer (DMA). The

*E*for the three samples used are found to be 11 kPa, 45 kPa and 58kPa. We mixed 0.3

*ml*of polystyrene bead solution in 100

*ml*of stock solution and increased the

*μ*to 8.14

_{s}*mm*

^{−1}.

### 5.2. Measurement of modulation depth

*f*) MHz is driven by an ultra-stable dual-channel function generator with suitable power amplification. To achieve a frequency scan of the ROI starting at a Δ

*f*as low as tens of Hz we drive one of them slightly away from resonance. A slight drop in output because of this deviation is neglected. We note that the ability to generate a stable low frequency acoustic wave critically depends on the frequency stability of the driving function generator. Our ability to select a frequency to match the desired computed resonant modes (see Section 3) is hampered because of the limitation of possible frequencies that can be selected in the function generator. The photons are picked up by a single-mode fibre and delivered to the PC-PMT. For maximizing the signal-to-noise ratio the fibre is aligned carefully to capture a single speckle from the boundary photon fluctuation.

*cm*×6

*cm*×20

*cm*, designed to have a

*μ*equal to that of healthy breast tissue (8

_{s}*mm*

^{−1}) and

*μ*kept at a nominal low value of 0.00025

_{a}*mm*

^{−1}and

*E*= 11 kPa, 45 kPa and 58 kPa are used. Light is gathered over 1200

*s*and the intensity autocorrelation

*g*

_{2}(

*τ*) is measured. For a single

*g*

_{2}(

*τ*), 2000 samples are collected and averaged.

*g*

_{2}(

*τ*) gathered for slabs with

*E*= 11 kPa to 58 kPa are analyzed to arrive at

*M*(experimentally measured modulation depth) which is measured from the area around the peak at Δ

^{e}*f*in the Fourier spectrum of

*g*

_{2}(

*τ*).

*M*vs Δ

*f*is shown in Fig. 7 for a typical slab with

*E*= 11 kPa. The computed

*M*(

*M*) is also shown in the same figure.

^{c}*M*is seen to closely follow

^{c}*M*at the peak, an evidence of the resonance in vibration of the VROI, observed at ∼ 70 Hz.

^{e}*M*with Δ

^{e}*f*for objects with

*E*= 11, 45 and 58 kPa respectively. For comparison, the locations of the resonant modes of the VROI as computed using ANSYS

^{®}(in Section 3) are also shown. These figures prove that the peak resonant modes can be measured using the peak observed in the

*M*.

^{e}*M*and

^{e}*M*only around the resonant frequency. A possible reason for the deviation at other frequencies is the error accrued in the computation of the acoustic radiation force consequent to the uncertainty in the values used to account for coupling loss between transducer and object, losses at the interface between water and phantom and also the variable acoustic absorption in the object. An incorrect magnitude of the forcing used does not however shift the resonant frequency but only its strength. Also, the other errors we may have incurred may have a broad spectrum and therefore hardly contribute to the negligibly small bandwidth around the resonant frequency wherein most of the structural energy is concentrated.

^{c}*M*is found experimentally. This mismatch is owing to the spread of the resonant modes as dictated by the mechanical properties and shape of the vibrating ROI (as computed by ANSYS) and the fact that the set of

^{e}*M*was evaluated at frequencies possible to be selected in the function generator around 1.01 MHz. Limitations on the stability of the function generator used precludes generation of more experimental sample points in these frequency ranges.

^{e}*E*is from the measured resonant frequency data-model mismatch elsewhere has not affected the accuracy of the recovered

*E*. However in an algorithm that tries to reconcile a ‘data-model misfit’, the difference between computed and experimental ‘measurements’ owing to the incorrect input of certain quantities to the algorithm, say the force in the insonified area, can pose serious problems. One possible way to overcome this difficulty is to consider force also as a parameter to be identified in the inversion algorithm.

*f*= 70 Hz,140 Hz and 250 Hz for the three slabs) assuming that the

_{r}*E*’s are not known, we use the method of bisection to arrive at the unknown

*E*. Thus, for a guess of two

*E*values, say

*E*

_{1}and

*E*

_{2}such that

*E*

_{1}<

*E*

_{0}<

*E*

_{2}where

*E*

_{0}is the unknown Young’s modulus to be found, the resonant modes, Δ

*f*

_{1}and Δ

*f*

_{2}, are computed. If the interval between mean

*f*(

_{i}*i*=

*j*with

*j*= 1 or 2) contains Δ

*f*, the guess of

_{r}*E*is modified to

*E*thus reducing the length of the original search interval Δ

_{j}*E*=

*E*

_{2}–

*E*

_{1}. This is continued until the computed resonant frequencies match within a specified tolerance. The

*E*values thus obtained from the measured

*f*’s are 11.4, 44.8 and 58.7 kPa for the three objects investigated.

_{r}## 6. Conclusions

*M*because of resonance will make the usual UMOT based recovery of optical absorption coefficient more accurate, which can be cited as an added benefit resulting from the present work. The fact that the variation of Δ

*f*provides different sets of modulation depth measurements, can be further utilized to do tomographic inversion for

*μ*,

_{a}*n*and

*E*for the ROI, which is left for investigation in the future.

## References and links

1. | C. Kim and L. V. Wang, “Multi-optical-wavelength ultrasound-modulated optical tomography: a phantom study,” Opt. Lett. |

2. | L. V. Wang, “Ultrasound-mediated biophotonic imaging: A review of acousto-optical tomography and photo-acoustic tomography,” Dis. Markers |

3. | S. Leveque, A. C. Boccara, M. Lebec, and H. S. Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. |

4. | A P Gibson, J C Hebden, and S R Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

5. | S. Sakadzic and L. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: An analytical model for aniostropically scattering media,” Phys. Rev. E |

6. | M. Kempe, M. Larionov, D. Zaslarsky, and A. Z. Genack, “Acoustooptic tomography with multiple scattered light,” J. Opt. Soc. Am. A |

7. | S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. |

8. | W. Leutz and G. Maret, “Ultrasonic modulation of multiply scattered light,” Physica B |

9. | C. Kim, R. J. Zemp, and L. V. Wang, “Intense acoustic bursts as a signal-enhancement mechanism in ultrasound-modulated optical tomography,” Opt. Lett. |

10. | R. J. Zemp, C. Kim, and L.V. Wang, “Ultrasound-modulated optical tomography with intense acoustic bursts,” Appl. Opt. |

11. | X. Xu, H. Zhang, P. Hemmer, D. Qing, C. Kim, and L. V. Wang, “Photorefractive detection of tissue optical and mechanical properties by ultrasound modulated optical tomography” Opt. Lett. |

12. | E. Bossy, A. Funke, K. Daoudi, A. Boccara, M. Tanter, and M. Fink, “Transient optoelastography in optically diffusive media,” Appl. Phys. Lett. |

13. | T. Kamakura, T. Ishiwata, and K. Matsuda, “Model equation for strongly focused finite-amplitude sound beams,” J. Acoust. Soc. Am. |

14. | E. Konofagou, J. Thierman, and K. Hynynen, “A focused ultrasound method for simultaneous diagnostic and therapeutic applicationsa simulation study,” Phys. Med. Biol. |

15. | M. Fatemi, A. Manduca, and J. Greenleaf, “Imaging Elastic Properties of Biological Tissues by Low-Frequency Harmonic Vibration,” Proc. IEEE |

16. | J. Huang, J.A. Nissen, and Erik Bodegom, “Diffraction of light by a focused ultrasonic wave,” J. Appl. Phys. |

17. | F. A. Duck, “Nonlinear acoustics in diagnostic ultrasound,” Ultrasound Med. Biol. |

18. | J.E. Marsden and T.J.R. Hughes, |

19. | S. Sakadzic and L. V. Wang, “High-resolution ultrasound-modulated optical tomography in biological tissues,” Opt. Lett. |

20. | C. Usha Devi, R. M. Vasu, and A. K. Sood, “Design, fabrication, and characterization of a tissue-equivalent phantom for optical elastography,” J. Biomed. Opt. |

21. | A. Kharine, S. Manohar, R. Seeton, R. G. M. Kolkman, R. A. Bolt, W. Steenbergen, and F. F. M. de Mul, “Poly Vinyl Alcohol gels for use as tissue phantoms in photoacoustic mammography,” Phys. Med. Biol. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.6150) Imaging systems : Speckle imaging

(110.7170) Imaging systems : Ultrasound

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Medical Optics and Biotechnology

**Virtual Issues**

Vol. 7, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

R. Sriram Chandran, Debasish Roy, Rajan Kanhirodan, Ram Mohan Vasu, and C. Usha Devi, "Ultrasound modulated optical tomography: Young’s modulus of the insonified region from measurement of natural frequency of vibration," Opt. Express **19**, 22837-22850 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22837

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### References

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