Analysis of Spatial Resolution in Thermo- and Photoacoustic - - PowerPoint PPT Presentation

Photoacoustic tomography (PAT) PAT with integrating line detectors

Analysis of Spatial Resolution in Thermo- and Photoacoustic Tomography

Markus Haltmeier, Gerhard Zangerl and Otmar Scherzer

Infmath Imaging Group, University Innsbruck (→ Vienna) Austria

AIP Conference – July 2009

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Photoacoustic tomography (PAT) PAT with integrating line detectors

1

Photoacoustic tomography (PAT) Mathematical model Classical approach

2

PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

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Photoacoustic tomography (PAT) PAT with integrating line detectors Mathematical model Classical approach

Photoacoustic tomography (PAT)

Hybrid imaging technique:

1 Convert optical illumination into acoustic wave. 2 Detect acoustic (pressure) waves. 3 Reconstruct initial pressure (related to structure of object).

• ptical illumination

absorbers detector

Applications: Cancer diagnostics, imaging of small animals

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Photoacoustic tomography (PAT) PAT with integrating line detectors Mathematical model Classical approach

Forward problem: Wave equation in R 3

Assumptions:

1 Pulsed Illumination: Intensity = I(x)δ(t) 2 Sound speed constant 3 No ultrasound attenuation

IVP for 3D wave equation: ∂2

t p(x, t) = ∆p(x, t)

in R 3 × (0, ∞) , p(x, 0) = I(x) · µabs(x) =: f (x) in R 3 , ∂tp(x, 0) = 0 in R 3 .

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Photoacoustic tomography (PAT) PAT with integrating line detectors Mathematical model Classical approach

Inverse problem in PAT

Denote by W3D f := p the solution of the 3D wave equation.

1 Measure W3D f outside of region B including support of f . 2 Reconstruction function f (x) inside B from those values.

• ptical illumination

absorbers detector

• Math. Problem depends on type of measurements.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Mathematical model Classical approach

Classical approach: Ideal point detectors

1 Assume point-wise data on ∂B

Data = (W3D f )(z, t) , for (z, t) ∈ ∂B × (0, ∞) . Function f can be reconstructed uniquely and stably.

2 Exact inversion formula in case of ball BR:

f (x) = 1 2R

• ∂BR

(∂t W3D f )(z, |x − z|) , for x ∈ BR . Derived in [Finch-Patch-Rakesh ’04].

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Alternative: PAT with integrating line detectors

Line detectors: Measure integrals

• f

W3D f over lines in di- rection θ

3D object projection θ⊥ θ

Proposed in [Burgholzer-Hofer-Paltauf-MH-Scherzer ’05].

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Data from ideal line detectors

X-ray transform in direction θ ∈ S1 × {0}: (X h)(θ, y) := (Xθ h)(y) :=

• R

h(sθ + y)ds for y ∈ θ⊥ .

1 Line detectors measure restriction of X-ray transform:

L f :=

• X W3D f
• S1×∂D×(0,∞) .

2 Function f can be reconstructed uniquely and stably from L f . Markus Haltmeier Resolution of Photoacoustic Tomography 8/ 22

Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Two step reconstruction

Commutation relation: Xθ W3D f = W2D Xθ f .

f = 3D object Xθ f = 2D object θ⊥ θ

Two step algorithm:

1 For fixed θ: Recover initial data Xθ f of 2D wave equation

from solution W2D Xθ f on curve ∂D.

2 Recover 3D image f from projection images Xθ f by applying

inverse X-ray transform (2D Radon transform).

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

First step: Inverse problem for 2d wave equation

∂2

t p − △p = 0 ,

in R 2 × (0, ∞) p(y, 0) = F(y) , in R 2 (∂tp)(y, 0) = 0 , in R 2 detector

• bject

Given restriction W2D F = p|∂D×(0,∞) → Reconstruct F

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Equivalence to circular mean Radon transform

(M2D F)(y, r) =

• S1 F(y + rσ)ds(σ)

2π

1 Solution formula + analytic inversion

(W2D F)(y, t) = ∂t t r (M2D F)(y, r) √ t2 − r2 dr (M2D F)(y, r) = 2 π r (W2D F)(y, t) √ r2 − t2 dt Inversion of Wave Eq. W2D ⇄ Inversion of spherical M2D

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

FBP type inversion formulas

In case D = DR inversion formulas for wave equation (and circular means) have been found recently:

1

[Kunyansky ’07]

2

[Finch-MH-Rakesh ’07] F(y) =

• ∂DR

2R (∂rr∂r M2D F)(y0, r) log |r2 − ρ2

0| dr

• ds(y0)

Here ρ0 = |y − y0|.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

2D Example with PAT scanner in Innsbruck

Figure: Setup in Innsbruck, kidney, reconstructed projection. Pictures provided by Markus Holotta and Harald Grossauer.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Many factors influence the resolution

1 Non-constant sound speed 2 Attenuation of US waves 3 Limited view/angle/data 4 Detectors are not perfect

lines

5 Finite bandwidth of

detection system

BR laser beam electromagnetic pulse e1 e2 e3 Markus Haltmeier Resolution of Photoacoustic Tomography 14/ 22

Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Approximate line detectors

Include following practical constraints: Detection system has a finite bandwidth. Laser beam integrates pressure over cylindrical volume (with radial weight). Measured data: Lϕ,ψ f =

• ϕ ∗t w ∗z (X W3D f )
• S1×∂D×(0,∞) .

Here ψ(r) = radial profile of the laser beam. ϕ(t) = impulse response function.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Approximate line detectors II

Properties: Lϕ,ψ f is blurred version of line data L f . De-blurring severely Ill-posed (unstable). Inexact knowledge of ψ and ϕ. Common practise: Apply L−1 to blurred data: (L−1 Lϕ,ψ f )(x) = blurred reconstruction . Our aim: Find point spread function (PSF, blurring kernel), i.e. find analytic expression for L−1 Lϕ,ψ f .

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Blurring for approximate line detectors

Assume supp(ϕ ∗t ψ) ⊂ [−τ, τ], where τ := dist

• supp(f ), ∂B
• .

Theorem (MH, Scherzer, Zangerl ’09) We have Lϕ,ψ f ∈ ran(L) and

• L−1 Lϕ,ψ f
• (x) = (Φband ∗x Ψline ∗x f ) (x) ,

with the blurring kernels Φband(x) := −πϕ′(|x|)/(2|x|) , x ∈ R 3 , Ψline(x) := − 1 π ∞

|x|

∂ξψ(ξ)

• ξ2 − |x|2 dξ ,

x ∈ R 3 .

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Idea of proof (detector PSF)

1 Translational invariance of 2D wave equation: Blurring of

solution is equivalent to burring of initial data.

2 Convolution theorem for the X-ray transform: Blurring of 2D

projection is equivalent to blurring of 3D object.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Idea of proof (detector PSF) II

1 In mathematical terms:

(Xθ Ψline) ∗z (W2D Xθ f ) = W2D (Xθ Ψline) ∗z (Xθ f )

• .

2 Some manipulations:

(Xθ Ψline) ∗z (Xθ W3D f ) = Xθ W3D Ψline ∗x f

• .

Adjusting Xθ Ψline = ψ (inverse Abel transform) shows the theorem.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Estimate spatial resolution

Full with half maximum of the PSF is typical parameter for spatial resolution.

1 Approximate line detectors:

alaser = width of the detecting laser beam .

2 Approximate point detectors (similar analysis)

atransducer = size of ultrasound detector . Typical values atransducer = 1 cm and alaser = 0.1 cm show improved spatial resolution of PAT with line detectors.

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Photoacoustic tomography (PAT) PAT with integrating line detectors Basic setup Factors influencing resolution Resolution analysis

Some relevant references

[Burgholzer/Hofer/Paltauf/MH/Scherzer] Thermoacoustic tomography with integrating area and line detectors. IEEE Ultrason. and Freq. Contr. (2005) [Finch/MH/Rakesh] Inversion of spherical means and the wave equation in even dimensions. SIAM Journal of applied mathematics (2007) [Xu/Wang] Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic

• r photoacoustic reconstruction. Phys. Rev. E (2003)

[MH/Scherzer/Zangerl] Influence of bandwidth and detector size to the spatial resolution of photo-acoustic tomography. In preparation (2009) Photoacoustic Imaging in Biology and Medicine: http://pai.uibk.ac.at/ Markus Haltmeier Resolution of Photoacoustic Tomography 21/ 22

Photoacoustic tomography (PAT) PAT with integrating line detectors

Conclusion

Photoacoustic Tomography . . .

1 Emerging hybrid imaging technique. 2 Acoustic waves induced by optical illumination.

Resolution . . .

3 Depends on bandwidth and detector size. 4 Analytical expressions of blurring kernels. 5 Resolution spatially invariant. 6 Line detectors show improved resolution.

Thank you for your attention!

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