Abstract

We address a certain inverse problem in ultrasound-modulated optical tomography: the recovery of the amplitude of vibration of scatterers [p(r)] in the ultrasound focal volume in a diffusive object from boundary measurement of the modulation depth (M) of the amplitude autocorrelation of light [ϕ(r,τ)] traversing through it. Since M is dependent on the stiffness of the material, this is the precursor to elasticity imaging. The propagation of ϕ(r,τ) is described by a diffusion equation from which we have derived a nonlinear perturbation equation connecting p(r) and refractive index modulation [Δn(r)] in the region of interest to M measured on the boundary. The nonlinear perturbation equation and its approximate linear counterpart are solved for the recovery of p(r). The numerical results reveal regions of different stiffness, proving that the present method recovers p(r) with reasonable quantitative accuracy and spatial resolution.

© 2011 Optical Society of America

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  1. L. H. V. Wang, S. L. Jacques, and X. M. Zhao, “Continuous wave ultrasonic modulation of scattered laser light to image objects in turbid media,” Opt. Lett. 20, 629–631 (1995).
    [CrossRef] [PubMed]
  2. M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light,” J. Opt. Soc. Am. A 14, 1151–1158 (1997).
    [CrossRef]
  3. L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
    [CrossRef] [PubMed]
  4. G. Yao and L. V. Wang, “Theoretical and experimental studies of ultrasound-modulated optical tomography in biological tissue,” Appl. Opt. 39, 659–664 (2000).
    [CrossRef]
  5. S. Leveque, A. C. Boccara, M. Lebec, and H. Saint-Jalmes, “Ultrasonic tagging of photon paths in scattering media: parallel speckle modulation processing,” Opt. Lett. 24, 181–183 (1999).
    [CrossRef]
  6. S. Sakadzic and L. V. Wang, “Ultrasonic modulation of multiply scattered coherent light: An analytical model for anisotropically scattering media,” Phys. Rev. E 66, 026603 (2002).
    [CrossRef]
  7. L. V. Wang, “Mechanisms of ultrasonic modulation of ultiply scattered light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
    [CrossRef] [PubMed]
  8. D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
    [CrossRef]
  9. J. L. Harden and V. Viasnoff, “Recent advances in DWS-based micro-rheology,” Curr. Opin. Colloid Interface Sci. 6, 438–445(2001).
    [CrossRef]
  10. D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
    [CrossRef]
  11. D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
    [CrossRef] [PubMed]
  12. S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
    [CrossRef] [PubMed]
  13. S. Sakadzic and L. V. Wang, “Correlation transfer equation for ultrasound-modulated multiply scattered light.,” Phys. Rev. E 74, 036618 (2006).
    [CrossRef]
  14. C. U. Devi, R. M. Vasu, and A. K. Sood, “Application of ultrasound-tagged photons for measurement of amplitude of vibration of tissue caused by ultrasound: theory, simulation, and experiments,” J. Biomed. Opt. 11, 049802(2006).
    [CrossRef]
  15. This perturbation approach is commonly used to solve the inverse paramater recovery problem. See, for example, H. M. Varma, B. Banerjee, D. Roy, A. K. Nandakumaran, and R. M. Vasu, “Convergence analysis of the Newton algorithm and a pseudo-time marching scheme for diffuse correlation tomography,” J. Opt. Soc. Am. A 27, 259–267 (2010).
    [CrossRef]
  16. A. Björck, Numerical Methods for Least Squares Problems (SIAM, 1989).
  17. S. Sakadzic and L. V. Wang, “Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse,” J. Opt. Soc. Am. A 24, 2797–2806 (2007).
    [CrossRef]
  18. In general the measurement F(p,r,ωa)=|F˜(p,r,ωa)|. However, from the numerical simulations, for the objects we have chosen, we have noticed that the imaginary part of F˜(p,r,ωa) is negligible compared to its real part. Therefore, we have taken the measurement as F(p,r,ωa)≈F˜(p,r,ωa).
  19. J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.
  20. H. M. Varma, A. K. Nandakumaran, and R. M. Vasu, “Study of turbid media with light: recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light,” J. Opt. Soc. Am. A 26, 1472–1483(2009).
    [CrossRef]
  21. H. M. Varma, B. Banerjee, D. Roy, A. K. Nandakumaran, and R. M. Vasu, “Convergence analysis of the Newton algorithm and a pseudo-time marching scheme for diffuse correlation tomography,” J. Opt. Soc. Am. A 27, 259–267(2010).
    [CrossRef]

2010

2009

2007

2006

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

S. Sakadzic and L. V. Wang, “Correlation transfer equation for ultrasound-modulated multiply scattered light.,” Phys. Rev. E 74, 036618 (2006).
[CrossRef]

C. U. Devi, R. M. Vasu, and A. K. Sood, “Application of ultrasound-tagged photons for measurement of amplitude of vibration of tissue caused by ultrasound: theory, simulation, and experiments,” J. Biomed. Opt. 11, 049802(2006).
[CrossRef]

2002

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

2001

L. V. Wang, “Mechanisms of ultrasonic modulation of ultiply scattered light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

J. L. Harden and V. Viasnoff, “Recent advances in DWS-based micro-rheology,” Curr. Opin. Colloid Interface Sci. 6, 438–445(2001).
[CrossRef]

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

2000

1999

1997

1995

L. H. V. Wang, S. L. Jacques, and X. M. Zhao, “Continuous wave ultrasonic modulation of scattered laser light to image objects in turbid media,” Opt. Lett. 20, 629–631 (1995).
[CrossRef] [PubMed]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

1993

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

1989

A. Björck, Numerical Methods for Least Squares Problems (SIAM, 1989).

1972

J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.

Banerjee, B.

Björck, A.

A. Björck, Numerical Methods for Least Squares Problems (SIAM, 1989).

Boas, D. A.

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Boccara, A. C.

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Devi, C. U.

C. U. Devi, R. M. Vasu, and A. K. Sood, “Application of ultrasound-tagged photons for measurement of amplitude of vibration of tissue caused by ultrasound: theory, simulation, and experiments,” J. Biomed. Opt. 11, 049802(2006).
[CrossRef]

Durian, D. J.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Gang, H.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Genack, A. Z.

Harden, J. L.

J. L. Harden and V. Viasnoff, “Recent advances in DWS-based micro-rheology,” Curr. Opin. Colloid Interface Sci. 6, 438–445(2001).
[CrossRef]

Jacques, S. L.

Kempe, M.

Kenneth, P.

J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.

Larionov, M.

Lebec, M.

Leveque, S.

Lions, J. L.

J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.

Magenes, E.

J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.

Nandakumaran, A. K.

Pine, D. J.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Roy, D.

Saint-Jalmes, H.

Sakadzic, S.

S. Sakadzic and L. V. Wang, “Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse,” J. Opt. Soc. Am. A 24, 2797–2806 (2007).
[CrossRef]

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

S. Sakadzic and L. V. Wang, “Correlation transfer equation for ultrasound-modulated multiply scattered light.,” Phys. Rev. E 74, 036618 (2006).
[CrossRef]

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

Sood, A. K.

C. U. Devi, R. M. Vasu, and A. K. Sood, “Application of ultrasound-tagged photons for measurement of amplitude of vibration of tissue caused by ultrasound: theory, simulation, and experiments,” J. Biomed. Opt. 11, 049802(2006).
[CrossRef]

Varma, H. M.

Vasu, R. M.

Viasnoff, V.

J. L. Harden and V. Viasnoff, “Recent advances in DWS-based micro-rheology,” Curr. Opin. Colloid Interface Sci. 6, 438–445(2001).
[CrossRef]

Wang, L. H. V.

Wang, L. V.

S. Sakadzic and L. V. Wang, “Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse,” J. Opt. Soc. Am. A 24, 2797–2806 (2007).
[CrossRef]

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

S. Sakadzic and L. V. Wang, “Correlation transfer equation for ultrasound-modulated multiply scattered light.,” Phys. Rev. E 74, 036618 (2006).
[CrossRef]

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

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

L. V. Wang, “Mechanisms of ultrasonic modulation of ultiply scattered light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

G. Yao and L. V. Wang, “Theoretical and experimental studies of ultrasound-modulated optical tomography in biological tissue,” Appl. Opt. 39, 659–664 (2000).
[CrossRef]

Weitz, D. A.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Yao, G.

Yodh, A. G.

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192–215 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

Zaslavsky, D.

Zhao, X. M.

Zhu, J. X.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Appl. Opt.

Curr. Opin. Colloid Interface Sci.

J. L. Harden and V. Viasnoff, “Recent advances in DWS-based micro-rheology,” Curr. Opin. Colloid Interface Sci. 6, 438–445(2001).
[CrossRef]

J. Biomed. Opt.

C. U. Devi, R. M. Vasu, and A. K. Sood, “Application of ultrasound-tagged photons for measurement of amplitude of vibration of tissue caused by ultrasound: theory, simulation, and experiments,” J. Biomed. Opt. 11, 049802(2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. E

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

S. Sakadzic and L. V. Wang, “Correlation transfer equation for ultrasound-modulated multiply scattered light.,” Phys. Rev. E 74, 036618 (2006).
[CrossRef]

Phys. Rev. Lett.

L. V. Wang, “Mechanisms of ultrasonic modulation of multiply scattered coherent light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

L. V. Wang, “Mechanisms of ultrasonic modulation of ultiply scattered light: an analytic model,” Phys. Rev. Lett. 87, 043903 (2001).
[CrossRef] [PubMed]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855–1858 (1995).
[CrossRef] [PubMed]

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

Phys. Scr.

D. A. Weitz, J. X. Zhu, D. J. Durian, H. Gang, and D. J. Pine, “Diffusing-wave spectroscopy: the technique and some applications,” Phys. Scr. T49, 610–621 (1993).
[CrossRef]

Other

In general the measurement F(p,r,ωa)=|F˜(p,r,ωa)|. However, from the numerical simulations, for the objects we have chosen, we have noticed that the imaginary part of F˜(p,r,ωa) is negligible compared to its real part. Therefore, we have taken the measurement as F(p,r,ωa)≈F˜(p,r,ωa).

J. L. Lions, E. Magenes, and P. Kenneth, Non-Homogeneous Boundary Value Problems and Applications (Springer-Verlag, 1972), Vol. 2.

A. Björck, Numerical Methods for Least Squares Problems (SIAM, 1989).

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Figures (10)

Fig. 1
Fig. 1

(a) Gray-level image of the object used in the simulations given in terms of p ( r ) where the insonified region has one inhomogeneous inclusion, (b) gray-level image of the object used in the simulations given in terms of p ( r ) where the insonified region has two inhomogeneous inclusions.

Fig. 2
Fig. 2

(a) Recovered image obtained from the object of Fig. 1a using the nonlinear-perturbation-equation-based algorithm. Data for the algorithm were generated taking refractive index modulation into account as well. (b) Cross-sectional plot through the center of the inhomogeneity in Fig. 2a compared with a similar plot from the original in Fig. 1a.

Fig. 3
Fig. 3

(a) Same Fig. 2a, but obtained from the object in Fig. 1b, (b) same Fig. 2b, but obtained from the reconstructions of Fig. 3a.

Fig. 4
Fig. 4

(a) Same as Fig. 2a, except that data used are generated without taking refractive index modulation into account, (b) cross-sectional plot through the center of the inhomogeneity of the reconstruction in Fig. 4a.

Fig. 5
Fig. 5

(a) Same as Fig. 4a, but obtained from the object shown in Fig. 1b, (b) cross-sectional plot through the center of the inhomogeneities of the reconstruction in Fig. 5a.

Fig. 6
Fig. 6

(a) Same as Fig. 4a, except that the reconstruction algorithm uses the linear perturbation equation, (b) cross-sectional plot through the center of the inhomogeneity of the reconstruction of Fig. 6a.

Fig. 7
Fig. 7

(a) Same as Fig. 6a, but for the object shown in Fig. 1b, (b) cross-sectional plot through the center of the inhomogeneity of the reconstruction of Fig. 7a.

Fig. 8
Fig. 8

(a) Plot of the data domain error Θ i versus iteration number, i, for the object given in Fig. 1b with refractive index modulation included in the simulation. The reconstruction uses the nonlinear perturbation equation. (b) Plot of the object domain error ϵ i = p original p recovered 2 2 versus iteration number, i, for the object given in Fig. 1b with refractive index modulation included in the simulation. The reconstruction algorithm uses the nonlinear perturbation equation.

Fig. 9
Fig. 9

(a) Same as Fig. 8a, except that data used are generated without taking refractive index modulation into account, (b) same as the one in Fig. 8b, except that data used are generated without taking refractive index modulation into account.

Fig. 10
Fig. 10

(a) Same as Fig. 9a, but with the linear- perturbation-based-equation used for inversion, (b) same as Fig. 9b, but with the linear-perturbation-based-equation used for inversion.

Equations (35)

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φ a , b o ( t ) = k ^ s · ( d b ( t ) d a ( t ) )
φ a , b n ( t ) = η ρ v a 2 r a r b A ( r , t ) d r ,
φ a , b = φ a , b o + φ a , b n .
d a = P a k a ρ v a 2 sin ( ω a t k a · r a + φ in ) .
exp ( j Δ φ ) 1 | r b r a | l tr 1 Δ φ 2 l tr 2 ,
Δ φ 2 l tr = | P a | 2 sin 2 ( ω a τ 2 ) ( l tr k a · k ^ s ) 2 1 + ( l tr k a · k ^ s ) 2 [ S a 2 ( k ^ s · k ^ a ) 2 + η 2 ( k ^ s · k ^ a ) 2 2 η S a ] .
Δ I = I ( r a , k ^ s , τ ) exp ( μ t | r b r a | ) [ 1 | r b r a | μ t Δ φ 2 l tr 2 ] ,
k ^ s · I ( r , k ^ s , τ ) = ( μ a + μ s ) I ( r , k ^ s , τ ) + S ( r , k ^ s ) + μ s 4 π p ( k ^ s , k ^ z ) ( 1 1 2 Δ φ 2 l tr ) I ( r , k ^ s , τ ) d k ^ s .
· D ϕ ( r , τ ) + ( μ a + μ s φ ^ ( τ ) ) ϕ ( r , τ ) = S 0 ( r 0 ) ,
φ ^ ( τ ) = 1 2 | P a | 2 sin 2 ( ω a τ 2 ) [ η 2 ( k a l tr ) tan 1 ( k a l tr ) + S a 2 3 2 η S a ] .
ϕ ( r , τ ) + D ϕ ( r , τ ) n = 0 , r Ω .
· D ϕ ( r , τ ) + ( μ a + μ s φ ^ ( τ ) + B ( r , τ ) ) ϕ ( r , τ ) = S 0 ( r 0 ) ,
· D ϕ ( r , τ ) + ( μ a + B ( r , τ ) ) ϕ ( r , τ ) = S 0 ( r 0 ) ,
ϕ + D ϕ n = 0 .
· D ( ϕ + ϕ δ ) ( r , τ ) + ( μ a + B ( r , τ ) + A ( τ ) χ I p ( r , τ ) ) ( ϕ + ϕ δ ) ( r , τ ) = S 0 ( r 0 ) ,
( ϕ + ϕ δ ) ( r , τ ) + D ( ϕ + ϕ δ ) ( r , τ ) n = 0 , r Ω .
M ( p , r , ω ) = 0 ( ϕ + ϕ δ ) ( r , τ ) e j ω τ d τ , r Ω .
F ˜ ( p , r , ω a ) | r Ω = 0 ϕ δ ( r , τ ) e j ω a τ d τ .
· D ϕ δ ( r , τ ) + ( μ a + B ( r , τ ) + A ( τ ) χ I p ( r , τ ) ) ϕ δ ( r , τ ) = A ( τ ) χ I p ϕ ,
ϕ δ ( r , τ ) + D ϕ δ ( r , τ ) n = 0 , r Ω .
minimize p L ( Ω ) Θ ( p ) = 1 2 F ( p ) F e L 2 ( Ω ) 2 + β 2 p L 2 ( I ) 2 .
· D ϕ 2 δ + ( μ a + B ( r , τ ) + A ( τ ) χ I p ) ϕ 2 δ = A ( τ ) χ I p δ ( ϕ + ϕ δ ) ,
ϕ 2 δ ( r , τ ) + D ϕ 2 δ ( r , τ ) n = 0 , r Ω .
· D ψ + ( μ a + B ( r , τ ) + A ( τ ) χ I p ) ψ = 0 ,
ψ + D ψ n = q + ,
Ω q + ϕ 2 δ = Ω A ( τ ) χ I p δ ( ϕ + ϕ δ ) ψ .
( F p j ) i = 0 [ A ( τ ) χ I ( ϕ + ϕ δ ) G R ψ e j ω a τ ] r = r j d τ .
· D ϕ δ ( r , τ ) ( μ a + B ( r , τ ) ) ϕ δ ( r , τ ) = A ( τ ) χ I p ϕ ( r , τ ) ,
· D ψ ( r , τ ) ( μ a + B ( r , τ ) ) ψ ( r , τ ) = 0 ,
ψ + D ψ n = q + .
Ω q + ϕ δ = Ω A ( τ ) χ I ψ p ϕ .
( F p j ) i = 0 [ A ( τ ) χ I ϕ G R ψ e j ω τ ] r = r j d τ ,
· D Φ δ ( ω a ) + ( μ a + C 0 2 χ I p ) Φ δ ( ω a ) = C 0 2 χ I p π .
Δ p = ( J T J + λ I ) 1 J T Δ F ,
100 ( Θ i Θ i 3 Θ i 3 ) < 0.1.

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