Inverse Transport Problems with Internal Data and Applications · Kui Ren (UT Austin) Inversion Transport with Internal Data June 15, 2015 4 / 47. PAT: Transport of Photons Thescattering

Inverse Transport Problems with Internal Data andApplications

Kui Ren

Department of Mathematics & ICESUniversity of Texas at Austin

Banff Workshop on Hybrid Methods in Imaging

June 15, 2015

Kui Ren (UT Austin) Inversion Transport with Internal Data June 15, 2015 1 / 47

Presentation Outline

1 Introduction to Photoacoustic Tomography

2 Non-Scattering Regime

3 Single Scattering Regime

4 Two General Stability Results

5 Generalization to Quantitative Fluorescence PAT


Photoacoustic Tomography (PAT)

Ultrasound Transducer

Ultrasound Transducer

NIR

Photons

Figure 1: Photoacoustic Tomography: To recover scattering, absorption andphotoacoustic efficiency properties of tissues from boundary measurement ofacoustic signal. Two processes: propagation of radiation and propagation ofultrasound. Time scale separation between the two processes.


PAT: Transport of Photons

Let Ω ⊂ Rd be the medium, and Sd−1 be the unit sphere in Rd . Thenthe time-integrated density of photons at x traveling in directionv ∈ Sd−1, u(x,v), solves the radiative transport equation:

v · ∇u(x,v) + (σa(x) + σs(x))u(x,v) = σs(x)KΘ(u)(x,v), in Xu(x,v) = g(x,v), on Γ−

where

X = Ω× Sd−1, and Γ− ≡ (x,v) ∈ ∂Ω× Sd−1 s.t. v · ν(x) < 0.

The functions σa(x), σs(x) are respectively the absorption andscattering coefficients of the medium.


PAT: Transport of Photons

The scattering operator KΘ usually takes the form of

KΘ(u)(x,v) =

∫Sd−1

Θ(v,v′)u(x,v′)dv′

with the kernel Θ(v,v′) being symmetric and normalized in appropriatesense (such as the Henyey-Greenstein scattering function thatdepends only on v · v′).


PAT: Photoacoustic Effect

The initial pressure field generated though photoacoustic effect atx ∈ Ω is given by:

H(x) = γ(x)σa(x)

∫Sd−1

u(x,v)dv ≡ γ(x)σa(x)KI(x)

The Grüneisen coefficient γ measures the efficiency of thephotoacoustic effect.


PAT: Acoustic Field

The acoustic wave equation for the pressure field:

1c2(x)

∂2p∂t2 −∆p = 0, in R+ × Rd

p(0,x) = H ≡ γ(x)σa(x)KI(u)(x), in Rd

∂p∂t

(0,x) = 0, in Rd

It turns out that change of optical properties has very small impacton the ultrasound speed c(x). Therefore the coupling between theradiative transport and acoustic process is only through the initialpressure field.


PAT: Measurements

In PAT, acoustic signals are measured for sufficiently large time Ton a surface Σ ⊂ ∂Ω:

z(t ,x) = p(t ,x)|(0,T )×Σ

The objective is to recover information on γ(x), σa(x) and σs(x)

from measured data.

The reconstructions are often performed in two steps:

p(t ,x)|(0,T )×Σ −→ H(x) −→ (γ, σa, σs)


PAT: Acoustic Reconstructions

The case c = 1 has been studied by many authors: Agranovsky,Ambartsoumian, Ammari, Anastasio, Arridge, Finch, Haltmeier,Kuchment, Kunyansky, Nguyen, Patch, Quinto, Rakesh, Scherzer,Wang, and many more.

The case of c = c(x) is much more complicated, and has beenstudied in: Acosta-Montalto IP 15, Hristova-Kuchment-Nguyen IP08, Hristova IP 09, Qian-Stefanov-Uhlmann-Zhao SIAM 11,Stefanov-Uhlmann IP 09, Stefanov-Uhlmann TAMS 12,Stefanov-Yang IP 15, Tittelfitz IP12 (elastic media), and more.

Acoustic attenuation effects can also be considered:Ammari-Bretin-Jugnon-Wahab-LNM11, Haltmeier etal SPIE07,Kowar-Scherzer-LNM12, La Riviére-Zhang-Anastasio OL06,Patch-Greenleaf 06, Treeby-Zhang-Cox IP10.


Inverse Transport in QPAT

We focus on the quantitative step of PAT: To recover one or more of thecoefficients among γ, σa and σs from the radiative transport equation:

v · ∇u(x,v) + (σa + σs)u(x,v) = σs(x)KΘ(u)(x,v), in Xu(x,v) = g(x,v), on Γ−

with the internal data of the form:

H(x) = γ(x)σa(x)KI(u)



There are many computational results recently on inversetransport with internal data for applications in QPAT. Very littletheoretical results exist in the literature.

The Bal-Jollivet-Jugnon, IP 10 result is the first theoretical resulton inverse transport with internal data.

Theorem (Bal-Jollivet-Jugnon, IP 10)Let γ be known. Then the data

Λ : g(x,v) 7→ H(x)

uniquely and stably determines σa(x), σs(x) and some information onΘ.



Bal-Jollivet-Jugnon, IP 10 can be easily generalized to show that:

Theorem (Mamonov-R., CMS 14)

(a) Λ = Λ =⇒ (σa(x), σs(x)) = (σa(x), σs(x)) if γ = γ; (b) Λ = Λ =⇒(γ, σa) = (γ, σa) if σs(x) = σs(x); and (c) Λ = Λ =⇒ (γ, σs) = (γ, σs) ifσa(x) = σa(x).

When the medium is diffusive (meaning mainly that σa is small while σs

is large compared to size of the domain), Bal-R., IP 11 showed thatonly two of coefficients can be reconstructed even with full albedo data.









Inverse Transport in Non-Scattering Regime

Transport of photons (σs = 0):

v · ∇u(x,v) + σa(x)u(x,v), = 0 in Xu(x,v) = g(x,v), on Γ−

Interior data from acoustic inversion:

H(x) = γ(x)σa(x)KI(u)(x)

Objective: to reconstruct γ and σa(x) from two internal data sets.

This problem has applications in sectional photoacoustic imaging:Zhu & Sevick-Muraca, OE 2011, Elbau-Scherzer-Schulze, IP2012.Kui Ren (UT Austin) Inversion Transport with Internal Data June 15, 2015 14 / 47

Inverse Transport in Non-scattering Regime

Let x ∈ Ω, we define τ±(x,v) = infs ∈ R+|x± sv /∈ Ω as the distanceit takes for a photon at x traveling in direction ±v to exit the domain.

Ω

),( vx+τ

),( vx−τ

),( vx



Then the interior data H(x) generated with source g andcoefficients (γ, σa) can be written as

H(x) =

γ(x)σa(x)

∫Sd−1

g(x− τ−(x,v)v,v)e−∫ τ−(x,v)

0 σa(x−τ−(x,v)v+sv)dsdv

The main idea will be to take special forms of the illuminationsource g(x,v) such that the integration over directions can besimplified.



Theorem (Mamonov-R., CMS 14)Let (γ, σa) and (γ, σa) be two sets of coefficients and H = (H1,H2) andH = (H1, H2) the corresponding interior data. Then there exists g1 andg2 such that:(a) H1 = H1 implies σa(x) = σa(x) if γ = γ with

c‖H1 − H1‖L∞(Ω) ≤ ‖σa − σa‖L∞(Ω) ≤ C‖H1 − H1‖L∞(Ω);

(b) H = H implies (γ, σa) = (γ, σa) with

‖γ(x)σa(x)〈e−∫ τ−(x,v′)

0 σa(x−τ−(x,v′)v′+sv′)ds〉v′

−γ(x)σa(x)〈e−∫ τ−(x,v′)

0 σa(x−τ−(x,v′)v′+sv′)ds〉v′‖L∞(Ω) ≤ c‖H−H‖(L∞(Ω))2 .


Inverse Transport with Collimated Sources

If we have two collimated sources of the form:

g1(x,v) = g(x)δ(v− v′), x ∈ Σ−(v′).

g2(x,v) = g(x− τ+(x,v′)v′)δ(v + v′), x ∈ Σ+(v′).

We take two data sets in the following setting:

v′

Σ−

-v′x′

Σ+Ω

x

Figure 2: Two collimated sources supported on Σ− and Σ+ respectivelybut pointing to opposite directions.Kui Ren (UT Austin) Inversion Transport with Internal Data June 15, 2015 18 / 47


Then the data sets can be written as

H1(x′ + τ−(x,v′)v′)g(x′)γ(x′ + τ−(x,v′)v′)

= σa(x′ + τ−(x,v′)v′)e−∫ τ−(x,v′)

0 σa(x′+sv′)ds

and

H2(x′ + τ−(x,v′)v′)g(x′)γ(x′ + τ−(x,v′)v′)

= σa(x′+τ−(x,v′)v′)e−∫ τ+(x,v′)

0 σa(x′+τ(x,v′)v ′−sv′)ds



We now take the logarithm of the ratio of H1 and H2 and use therelation τ(x,v) = τ+(x,v) + τ−(x,v) to obtain

lnH2

H1(x′ + τ−(x,v′)v′) =

−∫ τ+(x′,v′)

0σa(x′ + sv′)ds + 2

∫ τ−(x,v′)

0σa(x′ + sv′)ds.

Differentiation of this result with respect to τ−(x,v′) (equivalent tothe directional differentiation v′ · ∇x) will allow us to reconstructthe quantity 2σa(x′ + τ−(x,v′)v′) = 2σa(x) a.e.

The coefficient γ can then be reconstructed by solving thetransport equation with the reconstructed σa and computingγ = H1/(σa〈u1〉v) (or γ = H2/(σa〈u2〉v)).


Inverse Transport with Point Sources

Assume that we have collimated sources of the form:

g1(x,v) = g(v)δ(x− x′)

g2(x,v) = g(v)δ(x− x′′)

We take two data sets in the following setting:

Ω

v

v′′v′

x

x′′x′

Figure 3: Two point sources located at x′ and x′′ respectively.



Then the data sets can be written as

H1(x′ + τ−(x,v′)v′)g(v′)γ(x′ + τ−(x,v′)v′)

= σa(x′ + τ−(x,v′)v′)e−∫ τ−(x,v′)

0 σa(x′+sv′)ds

and

H2(x′′ + τ−(x,v′′)v′′)g(v′′)γ(x′′ + τ−(x,v′′)v′′)

= σa(x′′ + τ−(x,v′′)v′′)e−∫ τ−(x,v′′)

0 σa(x′′+sv′′)ds



We now take the logarithm of the ratio of H1 and H2 and use therelation τ−(x,v′)v′ · v + τ−(x,v′′)v′′ · (−v) = d(x′,x′′) to obtain

ln(

H2(x′′ + τ−(x,v′′)v′′)g(v′)γ(x′ + τ−(x,v′)v′)H1(x′ + τ−(x,v′)v′)g(v′′)γ(x′′ + τ−(x,v′′)v′′)

)=∫ τ−(x,v′)

0σa(x′ + sv′)ds −

∫ τ−(x,v′′)

0σa(x′′ + sv′′)ds

Differentiation of this result with respect to τ−(x,v′) (equivalent tothe directional differentiation v′ · ∇x) will allow us to reconstructthe quantity 2σa(x′ + τ−(x,v′)v′) = 2σa(x) a.e..

The coefficient γ can then be reconstructed by solving thetransport equation with the reconstructed σa and computingγ = H1/(σa〈u1〉v) (or γ = H2/(σa〈u2〉v)).


Numerical Reconstructions in Non-scattering Media

Figure 4: Reconstructions of a piecewise constant absorption coefficient witha collimated source. Left to right: true absorption coefficient σa, interior dataH, σa reconstructed with noiseless data and σa reconstructed with noisy data(noise level 5%).


Numerical Reconstructions in Non-scattering Media

Figure 5: Reconstructions of σa (top row) and γ (bottom row) using a pair ofcollimated sources. Left to right: true coefficients, coefficients reconstructedwith noiseless data and coefficients reconstructed with noisy data. The noisydata contains 5% random noise.









Inverse Transport in Single-Scattering Regime

The same idea works for the single-scattering model.

Let u0 be the solution of the free transport equation

v · ∇u0(x,v) + (σa + σs)u0(x,v) = 0, in Xu0(x,v) = g(x,v), on Γ−

Let u1 be the solution of the transport equation

v · ∇u1(x,v) + (σa + σs)u1(x,v) = σs(x)KΘ(u0)(x,v), in Xu1(x,v) = g(x,v), on Γ−

The internal datum for the single scattering model is:

H(x) = γ(x)σa(x)KI(u1)(x)


Inverse Transport in Single-scattering Regime

We have the following uniqueness result with two data sets.

TheoremLet σs be known. Let H = (H1,H2) and H = (H1, H2) be data setsgenerated with coefficients (γ, σa) and (γ, σa) respectively usingsource (g1,g2). Assume that H = H. Then (a)(σa(x), σs(x)) = (σa(x), σs(x)) if γ = γ; (b) (γ, σa) = (γ, σa) ifσs(x) = σs(x); and (c) (γ, σs) = (γ, σs) if σa(x) = σa(x).


Numerical Reconstructions in Single-scattering Media

Figure 6: Reconstructions of σa (top row) and γ (bottom row) using a pair ofcollimated sources and the single scattering model. Left to right: truecoefficients, coefficients reconstructed with noiseless data and coefficientsreconstructed with noisy data. The noisy data contains 5% random noise.









Stability for Recovery of σa

Let us come back to the QPAT problem:

v · ∇u + (σa + σs)u = σsKΘ(u), in Xu(x,v) = g(x,v), on Γ−

with internal datum H(x) = γσaKI(u).

TheoremLet γ and σs be known. Let H and H be two data sets generated withcoefficients σa and σa respectively. There there exists g such thatH = H implies σa = σa. Moreover,

c1‖H − H‖L∞(Ω) ≤ ‖σa − σa‖L∞(Ω) ≤ c2‖H − H‖L∞(Ω)

with c1, c2 constants that depend on Ω, γ and σs.


Stability for Recovery of σa: Idea of Proof

The proof of this result relys on our ability to control the solution of thetransport equation in the following way: selecting g such that thesolution to the transport equation


satisfies the constraint

u(x,v) = KI(u)(x).


Stability for Recovery of σa: Idea of Proof

We verify that the difference w = u − u solves:

v · ∇w + (σa + σs)w = σsKΘ(w)− (σa − σa)u, in Xw(x,v) = 0, on Γ−

The difference of the data gives:

(H − H)/γ = σaKI(w) + (σa − σa)KI(u)

Therefore w = u − u solves:

v · ∇w + (σa + σs)w = σsKΘ(w) + σauKI(u)

KI(w)− H−HγKI(u)

, in Xw(x,v) = 0, on Γ−


Stability for Recovery of (γ, σa)

Similar idea works for the two coefficients QPAT problem:


with internal datum H(x) = γσaKI(u).

TheoremLet σs be known. Let H = (H1,H2) and H = (H1, H2) be the data setsgenerated with coefficients (γ, σa) and (γ, σa) respectively. There thereexists g1 and g2 such that H = H implies γ = γ and σa = σa. Moreover,

‖σa − σa‖L2(Ω) ≤ c1‖H1/H2 − H1/H2‖L2(Ω)

‖γ − γ‖L2(Ω) ≤ c2‖H1 − H1‖L2(Ω).









Fluorescence Photoacoustic Tomography (fPAT)

NIR

Photons

xλ

Photons at

Photons at

Photons at mλ

mλPhotons at

mλ

mλ

Figure 7: Fluorescence Generation: (i) fluorescent biochemical markers areinjected into biological objects; (ii) markers then accumulate on target (forinstance cancer) tissues; (iii) upon excitation by external sources (atwavelength λx ), the markers emit fluorescent light (at wavelength λm).

Fluorescence PAT (fPAT) aims at reconstructing the fluorescenceconcentration and its quantum efficiency from measured acousticsignal generated by both excitation and emission photons.


fPAT: The Mathematical Model

Let us denote by ux and um the photon densities at λx and λm

respectively, then

v · ∇ux + (σa,xi + σa,xf )ux = σs,xQ(ux ), in Xv · ∇um + σa,mum = σs,mQ(um) + ησa,xf KI(ux ), in X

ux (x,v) = gx (x,v), um(x,v) = 0 on Γ−.

The operator Q = KΘ − I.

The fluorescence absorption coefficient σa,xf (x) is proportional tothe concentration ρ(x) and the extinction coefficient ε(x) of thefluorophores, i.e. σa,xf = ε(x)ρ(x).


fPAT: Internal Data from Photoacoustic Effect

In the propagation process, both the excitation light and thefluorescence light get absorbed by the tissue. The absorbedenergy then generates a pressure field following the photoacousticeffect:

H(x) = γ(x)[(σa,xi + (1− η)σa,xf )KI(ux ) + σa,mKI(um)

].

The aim is mainly to reconstruct η and σa,xf from datum H.

The problem in diffusive regime has been analyzed in R.-ZhaoSIIMS 13.


Theory of QfPAT: Reconstructing η

If we are only interested in reconstructing η assuming σa,xf isknown:

v · ∇ux + (σa,xi + σa,xf )ux − σs,xQ(ux ) = 0, in Xv · ∇um + σa,mum − σs,mQ(um) = ησa,xf KI(ux ), in X

ux (x,v) = gx (x,v), um(x,v) = 0 on Γ−

from the internal data

H(x) = γ[(σa,xi + (1− η)σa,xf

)KI(ux ) + σa,mKI(um)

]


Theory of QfPAT: η

Theorem

Let p ∈ [1,∞] and the source gx ∈ Lp(Γ−) be such that the transportsolution ux satisfies KI(ux ) ≥ c > 0. Let H and H be two data setsgenerated with coefficients (η, σa,xf ) and (η, σa,xf ) respectively. ThenH = H a.e. implies η = η a.e.. Moreover, the following stabilityestimate holds,

c‖H − H‖Lp(Ω) ≤ ‖(η − η)σa,xf KI(ux )‖Lp(Ω) ≤ C‖H − H‖Lp(Ω)

where the constants c and C depend on Ω and the coefficients σa,xi ,σa,m, σs,x , σs,m, and Ξ.

Moreover, there is an explicit reconstruction procedure.


Theory of QfPAT: η

Here is the reconstruction procedure:

S1. Given σa,xf , solve the first transport equation in the system (withthe boundary condition) for ux ;

S2. Subtract S(x) = (σa,xi + σa,xf )KI(ux ) from H/γ to getσa,mKI(um)− ησa,xf KI((ux );

S3. Solve for um from transport equation

v · ∇um + σa,m(um − KI(um))− σs,mQ(um) = S(x)− Hγ, in X

um(x,v) = 0,on Γ−.

S4. Reconstruct η = 1−(Hγ−σa,xiKI(ux )−σa,mKI(um)

)/(σa,xf KI(ux )).


Theory of QfPAT: Reconstructing σx ,f

Let us now assume that we know η and attempt to reconstructσa,xf :

v · ∇ux + (σa,xi + σa,xf )ux − σs,xQ(ux ) = 0, in Xv · ∇um + σa,mum − σs,mQ(um) = ησa,xf KI(ux ), in X

ux (x,v) = gx (x,v), um(x,v) = 0 on Γ−

from the internal data

H(x) = γ[(σa,xi + (1− η)σa,xf

)KI(ux ) + σa,mKI(um)

]


Theory of QfPAT: σx ,f

Theorem

Let gx ∈ Lp(Γ−) (p ∈ [1,∞]) be such that the transport solution ux

satisfies ux = KI(ux ) ≥ c > 0. Let H and H be data sets generatedwith coefficient pairs (η, σa,xf and (η, σa,xf ) respectively. Then H = Ha.e. implies σa,xf = σa,xf a.e.. Moreover, the following bound holds,

c‖H − H‖Lp(Ω) ≤ ‖(σa,xf − σa,xf )KI(ux )‖Lp(Ω) ≤ C‖H − H‖Lp(Ω),

with c and C depending on Ω, σa,xi , σa,m, σs,x , σs,m, η and Ξ.

This is a nonlinear problem. We do not have a general explicitreconstruction procedure as in the previous case. However, in thelinearized setting we can have a similar explicit reconstructionprocedure.


Theory of QfPAT: Nonuniqueness for SmallCoefficients

If we linearize the problem around the background (η, σa,xf ) = (0,0),we see a problem of nonuniqueness when reconstructing twocoefficients simultaneously. To be precise, the datum simplifies to

1γ

H ′[0,0](δη, δσa,xf ) = δσa,xf KI(ux ) + σa,xiKI(vx ),

with ux the background solution and vx solves

v · ∇vx + (σa,xi + σs,x )vx = σs,xKΘ(vx )− δσa,xf ux , in Xvx (x,v) = 0, on Γ−.

We observe that δη completely disappear from the datum and thetransport equation. Therefore, it can NOT be reconstructed in thissetting.


Theory of QfPAT: Nonuniqueness for SmallCoefficients

We still reconstruct stably the absorption coefficient δσa,xf :

Corollary

Let gx ∈ Lp(Γ−) (p ∈ [1,∞]) be such that the background solutionsatisfies ux = KI(ux ) ≥ c > 0. Denote by H ′[0,0] and H ′[0,0] theperturbed data sets generated with perturbed coefficients (δη, δσa,xf )and (δη, δσa,xf ) respectively. Then H ′[0,0] = H ′[0,0] a.e. impliesδσa,xf = δσa,xf a.e.. In addition, we have,

c‖H ′[0,0]−H ′[0,0]‖Lp(Ω) ≤ ‖(δσx ,f−δσx ,f )KI(ux )‖Lp(Ω) ≤ C‖H ′[0,0]−H ′[0,0]‖Lp(Ω),

with c and C constants that depend on Ω, Ξ, σa,xi and σs,x .


Numerical Reconstruction of η

Figure 8: Reconstruction of the quantum efficiency η. Shown are, from left toright, reconstructions with noise-free data, data containing 2% random noise,data containing 5% random noise, and data containing 10% random noise.


Numerical Reconstruction of σa,xf

Figure 9: Reconstruction of the absorption coefficient σa,xf . Shown are, fromleft to right, reconstructions with noise-free data, data containing 2% randomnoise, data containing 5% random noise, and data containing 10% randomnoise.


Documents

Inverse Transport Problems with Internal Data and Applications · Kui Ren (UT Austin) Inversion Transport with Internal Data June 15, 2015 4 / 47. PAT: Transport of Photons Thescattering