Photo-acoustic Imaging - MIT Mathematicsmath.mit.edu/~vjugnon/pdf/MIT_slides.pdf · 2012-09-07 ·...

Photo-acoustic Imaging

Vincent Jugnon

September 7, 2012

1 Photo-acoustic ImagingImposed boundary conditionsView limitationAcoustic attenuationCluttered sound speedqPAT : an inverse problem for light propagation

2 Wave imaging and topological derivative

The photoacoustic effect

physical phenomenon observed when a medium is exposed to anelectromagnetic wave.the EM wave drops energy in the medium, which heats up, dilatesand emits an (ultra-sonic) acoustic wave.

Mary Evans Picture Library

Photoacoustic imaging

The sound of lightJun 4th 2009From The Economist print edition

Biomedical technology: A novel scanning technique that combines optics with ultrasound couldprovide detailed images at greater depths

IF LIGHT passed through objects, rather than bouncing off them, people might now talk to each other on“photophones”. Alexander Graham Bell demonstrated such a device in 1880, transmitting a conversation on abeam of light. Bell’s invention stemmed from his discovery that exposing certain materials to focused,flickering beams of light caused them to emit sound—a phenomenon now known as the photoacoustic effect.

It was the world’s first wireless audio transmission, and Bell regarded the photophone as his most importantinvention. Sadly its use was impractical before the development of optical fibres, so Bell concentrated insteadon his more successful idea, the telephone. But more than a century later the photoacoustic effect is making acomeback, this time transforming the field of biomedical imaging.

A new technique called photoacoustic (or optoacoustic) tomography, which marries optics with ultrasonicimaging, should in theory be able to provide detailed scans comparable to those produced by magnetic-resonance imaging (MRI) or X-ray computerised tomography (CT), but with the cost and convenience of ahand-held scanner. Since the technology can operate at depths of several centimetres, its champions hopethat within a few years it will be able to help guide biopsy needles deep within tissue, assist withgastrointestinal endoscopies and measure oxygen levels in vascular and lymph nodes, thereby helping todetermine whether tumours are malignant or not. There is even scope to use photoacoustic imaging tomonitor brain activity and gene expression within cells.

To create a photoacoustic image, pulses of laser light are shone onto the tissue being scanned. This heats thetissue by a tiny amount—just a few thousandths of a degree—that is perfectly safe, but is enough to cause thecells to expand and contract in response. As they do so, they emit sound waves in the ultrasonic range. Anarray of sensors placed on the skin picks up these waves, and a computer then uses a process of triangulationto turn the ultrasonic signals into a two- or three-dimensional image of what lies beneath.

The technique works at far greater depths (up to seven centimetres) than other optical-imaging techniquessuch as confocal microscopy or optical-coherence tomography, which penetrate to depths of only about amillimetre. And because the degree to which a particular wavelength of light is absorbed depends on the type

Economist.com http://www.economist.com/science/tq/PrinterFriendly.cfm?story_id=1...

1 of 3 15/06/2009 09:46

Interest for medical imaging

advantages of both ultrasonic imaging and pure optical imagingwithout their drawbacks

fast, cheap and harmless

The considered model

∂2p∂t2 (x , t)− c2

0∆p(x , t) = 0

p(x ,0) =β

CpA(x)

∂p∂t

(x ,0) = 0

where A(x) the ”instantaneous” energy deposition of the EM wave inthe medium, will be modeled later.

The inverse problem

Reconstruct :p(x ,0) = p0(x) =

β

CpA(x)

from boundary measurements∂p∂ν

(y , t) or p(y , t), y ∈ ∂Ω, t ∈ [0 T ].

We assume that we have ideal point-like broadband acoustic receivers,with enough spatial density, so that we can neglect the error in :∫ T

0

∫∂Ω

f (x , t)dσ(x)dt ≈ ∆x∆t∑

i

∑j

f (xi , tj)

The problem has been well studied in the case of :free spacefull view measurementsnon-attenuating mediumhomogeneous sound speed.

We can express the pressure p on the boundary of the medium Ωin terms of spherical means of p0.We can then use a wide range of inversion formulas for thespherical Radon transform to reconstruct p0.

Reconstruction using inverse spherical Radon approachTrue initial condition

The free-space hypothesis implies that ultrasonic waves ”do notsee” the boundary of the observed medium.In some imaging devices, we have to account for acousticboundary conditions :

1c2

0

∂2p∂t2 (x , t)−∆p(x , t) =

1c2

0δ′(t)p0(x)

p(x ,0) =∂p∂t

(x ,0) = 0

p(y , t) = 0 y ∈ ∂Ω

with boundary measurements∂p∂ν

(y , t) on some time interval(0,T ).

we cannot express the measurements in terms of sphericalmeans anymore.

True initial condition Reconstruction using inverse spherical Radon approach

in fact, a duality approach allows the same quality ofreconstruction as in the free space case.

Consider the solutions of the wave equation in the free-space :

1c2

0

∂2v∂t2 (x , t)−∆v(x , t) = 0

with final conditions :

v(x ,T ) =∂v∂t

(x ,T ) = 0 x ∈ Ω

Testing the measures against those ”probe functions”, we get :∫ T

0

∫∂Ω

∂p∂ν

(y , t)v(y , t)dσ(y) =

∫Ω

p0(x)∂v∂t

(x ,0)dx

Consider plane wave test functions of the form :

v(x , t ; θ, s) = δ

(x .θc0

+ s − t)

Compute : ∫ T

0

∫∂Ω

∂p∂ν

(y , t)v(y , t)dσ(y) =∂

∂sR[p0](s, θ)

Now, choosing plane test functions of the form :

v(x , t ; θ, s) = 1− H(

x .θc0

+ s − t)

where H is the Heaviside function. we will have :∫ T

0

∫∂Ω

∂p∂ν

(y , t)v(y , t)dσ(y) = R[p0](s, θ)

In both cases, it is straightforward to obtain p0 using classicalback-projection algorithms.

p0 = R?BR[p0]

We still consider the wave equation :∂2p∂t2 (x , t)− c2

0∆p(x , t) = 0

p(x ,0) = p0(x)∂p∂t

(x ,0) = p1(x)

p(y , t) = 0 y ∈ ∂Ω

and we want to reconstruct p0,p1 from the measurements :∂p∂ν

(x , t) x ∈ Γc .

Applying the previous algorithm to limited-view data fails.

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Once again we are interested in obtaining integral information :

M(v0, v1) =

∫Ω

p0(x)v1(x)− p1(x)v0(x)dx

for example we have :

M(

0, δ(

x .θc0

+ s))

= R[p0](s, θ)

M(

0,eik .x)

= F [p0](k)

and evenM (0, δ(x)) = p0(x)

If we can find gv0,v1 such that v solution of :

1c2

0

∂2v∂t2 (x , t)−∆v(x , t) = 0

v(x ,0) = v0(x),∂v∂t

(x ,0) = v1(x)

v(y , t) = 0 y ∈ ∂Ω\Γcv(y , t) = gv0,v1 (y , t) y ∈ Γc

(1)

vanishes at time T :

v(x ,T ) = 0,∂v∂t

(x ,T ) = 0

Then we get what we want :∫ T

0

∫Γc

gv0,v1 (y , t)∂p∂ν

(y , t)dσ(y)dt = M(v0, v1)

Such a control exists :

Theorem J.L. LionsIf T and Γc satisfy controllability conditions, then for all (v0, v1) inL2(Ω)× H−1(Ω), there exists a control gv0,v1 solution of this controlproblem.

+ constructive proof.

Sharp controllability conditions

Theorem Bardos, Lebeau, RauchThe system is controllable if every ray of geometrical optics, starting at anypoint x ∈ Ω, at time t = 0, hits Γc before time T at a nondiffractive point.

⇒ No “glancing” nor “trapped” rays.

Uncontrollable and controllable geometries

The approach is computationaly expensive : O(n2d+1x ), but highly

parallelizable.

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Theoretically, it is possible to get the same reconstruction as in the fullview case.

We have to consider the attenuated wave equation :

1c2

0

∂2pa

∂t2 (x , t)−∆pa(x , t)−D(t) ∗ pa(x , t) =1c2

0δ′(t)p0(x)

where D takes the form :

D(t) =1√2π

∫R

(K 2(ω)− ω2

c20

)eiωtdω

If not accounted for, attenuation generates a blurring in thereconstruction.

True initial condition

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Reconstruction using inverse spherical Radon on attenuated data

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We propose an algorithm to deal with a power-law attenuation modelof exponent 2 :

K (ω) =ω

c0+ ia|ω|2

which is close to the thermo-visquous model :

K 2(ω) =ω2

c20

11− aiω

at low frequencies.

We link pa(x , t) to p(x , t) (solution with no attenuation) via anintegral operator. Inverting this operator, we get p and using usualalgorithms, we reconstruct p0.The inversion of the operator is achieved using an asymptoticexpansion (w.r.t. to a) given by a stationary phase theorem. Wewill write :

Lφ = Lkφ+ o(ak+1)

and :L−1ψ = L−1

k ψ + o(ak+1)

Going in the Fourier domain, we can write :(∆ + K 2(ω)

)pa(x , ω) =

iωc2

0p0(x)

(∆ +

ω2

c20

)p(x , ω) =

iωc2

0p0(x)

Green’s functions are known for both equations, so we get :

pa(x , t) =1

2π

∫R

ω

c0K (ω)e−iωt

∫ ∞0

p(x , s)eic0K (ω)sdsdω

We want to invert the operator :

Lφ(t) =1

2π

∫R

ω

c0K (ω)e−iωt

∫ ∞0

φ(s)eic0K (ω)sdsdω

We have K (ω) = ωc0

+ iaω2

2 and :

ω

c0K (ω)= 1− i

ac0

2ω +

∞∑k=2

ik (−1)k(ac0ω

2

)k

Lφ(t) =1

2π

∫ ∞0

φ(x , s)

∫R

(1− i

ac0

2ω +

∞∑k=2

(−iω)k(a

2

)k)

eiω(s−t)e−12 c0aω2sdωds

=

(1 +

ac0

2∂

∂t+∞∑

k=2

(−

ac0

2

)k ∂k

∂tk

)(1√

2π

∫ ∞0

φ(x , s)1

√c0as

e− 1

2(s−t)2

c0as ds

)

We have to investigate the operator :

Mφ =1√2π

∫ ∞0

φ(x , s)1√

c0ase−

12

(s−t)2

c0as ds

Stationary phase theoremFor f and ψ sufficiently smooth, if :

Im(f (t0)) = 0 f ′(t0) = 0 f ′′(t0) 6= 0 f ′ 6= 0 in K\t0

then for ε > 0, we have the following approximation :∣∣∣∣∣∣∫

Kψ(t)ei f (t)

ε dt − eif (t0)/ε

(ε−1 f ′′(t0)

2iπ

)− 12 ∑

j<k

εjLjψ

∣∣∣∣∣∣ ≤ Cεk∑α≤2k

sup |ψ(α)(x)|

where the Lj can be explicitly expressed as differential operators.

In our case, we obtain for k = 1:∣∣∣Mφ(t)−(φ(t) +

ac0

2(tφ(t))′′

)∣∣∣ ≤ Ca32∑α≤4

sup |φ(α)(t)|

At the end we have a third order ODE to solve :

pa(x , t) = Lp(x , t) =

(1 +

ac0

2∂

∂t

)(p(x , t) +

ac0

2

(t∂2p∂t2 (x , t) +

∂p∂t

(x , t)))

+ o(a)

for higher order asymptotics, we use :

Mφ(t) =k∑

i=0

(c0a)i

2i i!(t iφ(t))(2i)(t) + o(ak )

In a bounded domain, we adapt the dual approach presentedpreviously.Testing the measurements against appropriate waves, we accessan integral operator of R[p0] that we invert using the stationaryphase theorem.

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In the case of an unknown cluttered sound speed, we look at :1

c(x)2∂2p∂t2 (x , t)−∆p(x , t) = 0

p(x ,0) = p0(x)∂p∂t

(x ,0) = 0

where c(x) randomly varies around a constant :

1c2(x)

=1c2

0

(1 + σµ

(xxc

))

If we apply algorithms for homogeneous sound speed to this case, wemay observe artifacts.

True initial condition Reconstruction using inverse Radon

We combine 2 ideas :Kirchhoff migration can be robustified using space and frequencycorrelations between the dataour reconstruction algorithms for extended targets are based onRadon transforms inversions. The last step of such inversions is aback-projection which is similar to the Kirchhoff migration.

Our methods of inversion read :

R[p0] =W[p]

p0 = R?BR[p0]

where R is the spherical or line Radon transform, R?, its adjoint, is abackprojection :

R?[f ](x) =

∫R

∫∂Ω

f (y , ω)e−iω|x−y |dσ(y)dω

resp.

R?[f ](x) =

∫R

∫Sn−1

f (y , ω)e−iωθ.xdθdω

Kirchhoff migration :

IKM(x) = R?[p](x) =

∫R

∫∂Ω

p(y , ω)e−iω|x−y |dσ(y)dω

The original CINT :

ICINT (x) =

∫R

∫|ω2−ω1|≤Ωd

∫∂Ω

∫|y2−y1|≤Xd

p(y1, ω1)e−iω1|x−y1|

p(y2, ω2)eiω2|x−y2|dσ(y1)dσ(y2)dω1dω2

Our extended target reconstruction :

Irad(x) = R?BW[p](x) =

∫∂Ω

BW[p](y , ω)e−iω|x−y |dσ(y)dω

The proposed CINT algorithm :

ICIR(x) =

∫R

∫R

∫∂Ω

∫∂Ω

e− (ω1−ω2)2

2Ω2d e

− |y1−y2|2

2X2d

BW[p](y1, ω1)e−iω1|x−y1|BW[p](y2, ω2)eiω2|x−y2|dσ(y1)dσ(y2)dω1dω2

Stability Analysis

In a slightly different setting :

finite bandwidth[ω0 −

B2, ω0 +

B2

]with B ω0

xc X0

we can carry out a stability analysis. We have :

SNRradon ∼ e−ω2

0τ2c

2√

1 + Bτc

√1 +

X0

Xc

and :SNRCIR ∼

1√Ω2

dτ2c +

X 2d

X 2c

Reconstruction using inverse Radon

True initial condition Reconstruction using inverse Radon Reconstruction using CINT

A second inverse problem

good reconstruction of the absorbed energy density p0(x) insidethe medium is not the true (relevant) piece of information.it is not intrinsic and depends on both the illumination and onoptical coefficients.need of light propagation model, in terms of intrinsic opticalcoefficients (µa, µs, ...).

The RTE

Consider the radiative transfer equation (RTE) :

s.∇xϕ(x , s) + µt (x , s)ϕ(x , s) =

∫Sn−1

k(x , s′, s)ϕ(x , s′)dσ(s′)

with µt = µa + µs, µs(x , s) =

∫Sn−1

k(x , s, s′)dσ(s′).

In this case we have : p0(x) ∝∫Sn−1

µa(x , s′)ϕ(x , s′)dσ(s′).

A theoretical result for the RTE

We define an Albedo operator A :

L1(∂Ω−) → L1(Ω)

ϕ(x , s) 7→ Aϕ(x) =

∫Sn−1

µa(x , s′)ϕ(x , s′)dσ(s′)

Using the singularities of A, we prove the following stability estimates :

Theorem (G.Bal, A.Jollivet, VJ)

If (µt , µa, k) generates the Albedo A and (µt , µa, k) generates A then :

‖µt − µt‖L∞(Sn−1,W−1,1(X)) +‖µa − µa‖L∞(Sn−1,L1(X))

≤ C‖A− A‖L(L1(∂Ω−),L1(Ω))

If µt , µt ∈ L∞(Sn−1,W r ,p(X ))

‖µt − µt‖L∞(Sn−1,W s,p(X)) ≤ C2‖A− A‖r−s

p(1+r)

L(L1(∂Ω−),L1(Ω))

Assuming a model for k , we can also get stability estimates onanisotropy.

Conclusion

The standard problem of photoacoustics is well-posed, and has beenthoroughly treated.

Consideration of non-idealities may introduce some difficulties and/orill-posedness.

For each non-ideality, we proposed a model, analysed it, developedcorrections based on adapted mathematical tools, and validated themethods on numerical experiments.

Another important issue is to separate intrinsic coefficients fromillumination.

different models can be used to model light.we derived stability estimates for the RTE.still much to be done for this problem (need for theory andalgorithms).

The proposed methods need to be validated on real experimental data.

Ω

∂Ω

ρ=1K=1

KK, ρ

Helmholtz problem with inclusion D = za + δB: ∇.(

1χΩ\D + ρ−1χD(x)∇u)

+ ω2(

1χΩ\D + K−1χD(x))

u = 0∂u∂ν

= g on ∂Ω

background solution : ∆U + ω2U = 0∂U∂ν

= g on ∂Ω

Topological derivative

Choose Ds, ρs, Ks, and compute the quadratic misfit :

E(Ds; ρs,Ks) =

∫∂Ω|(us − umeas)(x)|2dσ(x)

we consider the effect of the addition of asymptotically smallinclusions.the volume derivative of E should be the most negative when Ds isclose to D.

Ω

∂Ω

ρ=1K=1

E(∅ )

K, ρ

Ω

∂Ω

ρ=1K=1

E(D1) ∼ E(∅ )

K’, ρ’

K, ρ

Ω

∂Ω

ρ=1K=1

E(D2) ∼ E(∅ )

K’, ρ’K, ρ

Ω

∂Ω

ρ=1K=1

E(D3) << E(∅ )

K’, ρ’

K, ρ

Classic asymptotic results yield :

Theorem Boundary PerturbationSuppose ωδ 1, then for any x ∈ ∂Ω :

(umeas − U) (x) = −δd(∇U(za)M(ρ−1,B)∇zNω(x , za)

+ω2(K−1 − 1)|B|U(za)Nω(x , za))

+ o(δd )

(us − U) (x) = −δds

(∇U(zs)M(ρ−1

s ,Bs)∇zNω(x , zs)

+ω2(K−1s − 1)|Bs|U(zs)Nω(x , zs)

)+ o(δd

s )

where Nω is a (geometry dependent) Neumann function : ∆Nω(x , z) + ω2Nω(x , z) = −δz∂N∂ν

= 0 on ∂Ω

Volume derivative of E at each point zs :

ITD[U,g](zs) = −∂E [U,g](zs)

∂(δs)d

∣∣∣∣δs=0

Using the asymptotic result :

ITD[U,g](zs) = δdπω4(K−1 − 1)(K−1s − 1)

Re

U(zs)

∫∂Ω

Nω(x , zs)Nω(x , za)dσ(x)U(za)

+δd4π2 (1− ρ)(1− ρs)

(1 + ρ)(1 + ρs)

Re∇U(zs)

∫∂Ω

∇zNω(x , zs)∇zNω(x , za)T dσ(x)∇U(za)

+o(δd )

TheoremThe following identity holds. For any x ∈ ∂Ω, for any z ∈ Ω :(

− I2

+KωΩ)

[Nω(., z)](x) = Γω(x − z)

where Γω is the fundamental solution of the Helmholtz equation in thefree-space.

Define the new misfit function :

E2(Ds; ρs,Ks) =

∫∂Ω

∣∣∣∣(− I2

+KωΩ)

(us − umeas)(x)

∣∣∣∣2 dσ(x)

ITD2[U, g](zs) = C(ω,K ,Ks, δ)Re

U(zs)

∫∂Ω

Γω(x − zs)Γω(x − za)dσ(x)U(za)

+C(ρ, ρs, δ)Re

∇U(zs)

∫∂Ω

∇zΓω(x − zs)∇zΓω(x − za)T dσ(x)∇U(za)

+o(δd )

We have that :∫∂Ω

Γω(x − zs)Γω(x − za)dσ(x) ∼ 1ω

Im Γω(zs − za)

∫∂Ω

∇zΓω(x − zs)∇zΓω(x − za)T dσ(x) ∼ ωIm Γω(zs − za)(

za − zs

|za − zs|

)(za − zs

|za − zs|

)T

Stability Study : Electronic Noise

Situation with 1 inclusion located at za. Computation of themeasurements.Addition of electronic noise

captor position

pote

ntia

l u

captor position

pote

ntia

l u

measure without noisemeasure with electronic noise

Application of the localization algorithms : topological derivative(with or without pre-processing), MUSIC, Kirchhoff migration,backpropagation⇒ estimation zest of za.large number of realizations (300)⇒ estimation of STD(|za− zest |)as a function of the noise level.

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0.3

0.4

0.5

0.6

0.7

electronic noise level sigma (%)

std(

|zes

t−z a|)

topological derivative with −I/2+KΩtopological derivativeMUSICKirchhoff MigrationBackpropagation

Stability Study : Clutter Noise

Situation with 1 inclusion located at za.Realization of one clutter noise, computation of themeasurements.

Example of a realisation of ρ (or K)

captor position

pote

ntia

l u

measure without noisemeasure with clutter noise

Application of the localization algorithms : topological derivative(with or without pre-processing), MUSIC, Kirchhoff migration,backpropagation⇒ estimation zest of za.large number of realizations (300)⇒ estimation of STD(|za− zest |)as a function of the noise level.

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std(

|zes

t−z a|)

topological derivative with −I/2+KΩtopological derivativeMUSICKirchhoff MigrationBackpropagation

Conclusion

Topological derivative algorithm is based on a good heuristic.Use of asymptotic tools to understand and improve it.Approach can be extended to elastic and electromagnetic waves,as well as to the EIT problem.Possibility to use higher-order asymptotics and multi-frequencymeasurements.Solutions needed to deal with limited-view acquisition.

H. Ammari, E. Bossy, VJ and H. KangMathematical models in photo-acoustic imagingSIAM Review (Vol.52, No.4).

H. Ammari, M. Asch, L. Guadarrama Bustos, VJ and H. KangTransient imaging with limited view data(under revision) to appear in SIAM Journal on Imaging Sciences.

H. Ammari, E. Bretin, VJ and A. WahabPhotoacoustic imaging for attenuating acoustic mediaMathematical Modeling in Biomedical Imaging II, Lecture Notes in Mathematics, to appear.

H. Ammari, E. Bretin, J. Garnier and VJCoherent interferometric algorithms for photoacoustic imagingsubmitted to SIAM Journal on Imaging Sciences.

G. Bal, A. Jollivet and VJInverse transport theory of photoacousticsInverse Problems, 26 025011.

H. Ammari, E.Bossy, VJ and H. KangQuantitative photoacoustic imaging of small absorbers(under revision) to appear in SIAM Journal on Applied Mathematics.

H. Ammari, J. Garnier, VJ and H. KangStability and resolution analysis for a topological derivative based functionsubmitted to SIAM Journal on Imaging Sciences.

Photo-acoustic Imaging - MIT Mathematicsmath.mit.edu/~vjugnon/pdf/MIT_slides.pdf · 2012-09-07 ·...

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Transcript of Photo-acoustic Imaging - MIT Mathematicsmath.mit.edu/~vjugnon/pdf/MIT_slides.pdf · 2012-09-07 ·...