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Quantitative Photoacoustics Using the Transport Equation Simon - - PowerPoint PPT Presentation

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Quantitative Photoacoustics Using the Transport Equation Simon Arridge 1 Ben Cox 3 Teedah Saratoon 3 Joint work with: Tanja Tarvainen 1 , 2 1 Department of Computer Science, University College London, UK 2 Department of Physics and Mathematics,

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SLIDE 1

Quantitative Photoacoustics Using the Transport Equation

Simon Arridge1 Joint work with: Ben Cox3 Teedah Saratoon3 Tanja Tarvainen1,2

1Department of Computer Science, University College London, UK 2Department of Physics and Mathematics, University of Eastern Finland, Finland 3Department of Medical Physics, University College London, UK

Colóquio Brasileiro de Matemática, July 29-August 2, 2013

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 1 / 54

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SLIDE 2

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 2 / 54

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SLIDE 3

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 3 / 54

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SLIDE 4

Introduction

Outline

Photoacoustic Imaging

  • utline of photoacoustic imaging

Photoacoustic image reconstruction Spectroscopic photoacoustic imaging Artefacts in photoacoustic imaging Quantitative Photoacoustic Imaging Models of light transport Multispectral reconstructions Unknown scattering: diffusion-based inversions Unknown scattering: using radiative transfer equation

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 4 / 54

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SLIDE 5

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 5 / 54

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SLIDE 6

PhotoAcoustic Tomography

PhotoAcoustic Signal Generation

Spatially varying chromophore concentrations (naturally occuring or contrast agents) give rise to optical absorption in the medium. The absorption and scattering coefficients µa and µ′

s determine the

fluence distribution Φ, and thence the absorbed energy distribution H. This energy generates a pressure distribution p0 via thermalisation, which because of the elastic nature

  • f tissue, then propagates as an

acoustic pulse. The pulse is detected by a sensor resulting in the measured PA time series p(t).

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 6 / 54

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SLIDE 7

PhotoAcoustic Tomography

PhotoAcoustic Imaging : 3 Modes

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 7 / 54

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SLIDE 8

PhotoAcoustic Tomography

PhotoAcoustic Signal Generation

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 8 / 54

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SLIDE 9

PhotoAcoustic Tomography

PhotoAcoustic Spherical BackProjection

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 9 / 54

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SLIDE 10

PhotoAcoustic Tomography

Optical part of the direct problem

Optical part of the direct problem H(r) = µa(r)Φ(r) absorbed absorption light energy density coefficient fluence = heat per unit volume

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 10 / 54

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SLIDE 11

PhotoAcoustic Tomography

Acoustic part of the direct problem

Acoustic part of the direct problem p(r)|t=0 = Γ(r)H(r) = Γ(r)µa(r)Φ(r) Grüneisen parameter

  • c2∇2 − ∂2

∂t2

  • p

=

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 11 / 54

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SLIDE 12

PhotoAcoustic Tomography

Fabrey-Perot Detector

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 12 / 54

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SLIDE 13

PhotoAcoustic Tomography

3D PhotoAcoustic Scanner

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 13 / 54

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SLIDE 14

PhotoAcoustic Tomography

PAT Acoustic Inversion (Image Reconstruction)

Initial value Problem

  • c2∇2 − ∂2

∂t2

  • p

= p|t=0 = ΓµaΦ ∂p ∂t

  • t=0

=

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 14 / 54

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SLIDE 15

PhotoAcoustic Tomography

PAT Acoustic Inversion (Image Reconstruction)

Initial value Problem

  • c2∇2 − ∂2

∂t2

  • p

= p|t=0 = ΓµaΦ ∂p ∂t

  • t=0

= Boundary value Problem (t running backwards from T to 0)

  • c2∇2 − ∂2

∂t2

  • p

= p(r, t)|t=T = p(r, t)|∂Ω = pobs(rs, t)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 14 / 54

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SLIDE 16

PhotoAcoustic Tomography

PAT Acoustic Inversion (Image Reconstruction)

Initial value Problem

  • c2∇2 − ∂2

∂t2

  • p

= p|t=0 = ΓµaΦ ∂p ∂t

  • t=0

= Boundary value Problem (t running backwards from T to 0)

  • c2∇2 − ∂2

∂t2

  • p

= p(r, t)|t=T = p(r, t)|∂Ω = pobs(rs, t)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 14 / 54

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SLIDE 17

PhotoAcoustic Tomography

Heterogeneous Sound Speed

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 15 / 54

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PhotoAcoustic Tomography

Spectroscopic PAT

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 16 / 54

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SLIDE 19

PhotoAcoustic Tomography

Spectroscopic PAT

absorption at different wavelengths gives spectral images

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 17 / 54

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SLIDE 20

PhotoAcoustic Tomography

Spectroscopic PAT

absorption at different wavelengths gives spectral images but fluence is also different at different wavelengths

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 17 / 54

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PhotoAcoustic Tomography

Spectroscopic PAT

absorption at different wavelengths gives spectral images but fluence is also different at different wavelengths tumour type LS174T

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 17 / 54

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SLIDE 22

PhotoAcoustic Tomography

Spectral Distortion

Spectral Distortion Spectrum corrupted by wavelength dependence of fluence

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 18 / 54

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SLIDE 23

PhotoAcoustic Tomography

Structural Distortion

Structural Distortion Structural distortion due to non-uniform internal light fluence Structural distortion at each wavelength = spectral distortion at each point

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 19 / 54

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SLIDE 24

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 20 / 54

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SLIDE 25

PhotoAcoustic Forward Model

Second order wave equation for homogeneous media

  • ∇2 − 1

c2 ∂2 ∂t2

  • p(r, t) = 0,

p(r, 0) = p0(r), ∂ ∂t p(r, 0) = 0 (1) p(r, t) = 1 c2 g(r, t|r0, t0)∂p0(r) ∂t0 − p0(r)∂g(r, t|r0, t0) ∂t0

  • dr0

(2) g(r, t|r0, t0) = c2 (2π)3 sin(c0kt) cok eik·(r−r0)dk k = |k| (3) p(r, t) = 1 (2π)3 p0(r) cos(c0kt)eik·(r−r0)dkdr0 (4) = F−1 {F {p0(r)} cos(c0kt)} (5) Simple numerical algorithm using wave propagator cos(c0kt) (Cox and Beard 2005)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 21 / 54

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SLIDE 26

PhotoAcoustic Forward Model

Linear Lossless Acoustic Equations

p(r, t) = F−1 {F {p0(r)} cos(c0kt)} (6) Cannot input a time-varying pressure, so no use for time-reversal imaging FFT ⇒ periodic boundary conditions (wave wrapping) Instead: solve equivalent first-order system ∂u ∂t = − 1 ρ0 ∇p linearised momentum conservation (7) ∂ρ ∂t = −ρ0∇ · u linearised mass conservation (8) p = c2

Pρ

linearised equation of state (9) p acoustic pressure, u particle velocity, ρ acoustic density

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 22 / 54

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SLIDE 27

PhotoAcoustic Forward Model

k-Space Acoustic Propagation Model

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 23 / 54

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SLIDE 28

PhotoAcoustic Forward Model

Modelling Acoustic Absorption

Photoacoustic waves may contain frequencies much higher than conventional ultrasound imaging (tens of MHz) Acoustic absorption in soft tissue over ranges of interest (1-50 MHz or so) typically takes the form α = α0ωy, 1 ≤ y ≤ 1.5 What wave equations account for absorption like this? Can they be time-reversed to correct for absorption during image reconstruction?

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 24 / 54

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PhotoAcoustic Forward Model

Modelling Absorption & Dispersion

Using Kramers-Kronig relations to find a purely dispersive term. Incorporating these into the acoustic equation of state gives p = c2     1

  • adiabatic

+ τ ∂ ∂t (−∇2)y/2−1

  • absorption only

+ η(−∇2)(y+1)/2−1

  • dispersion only

    ρ Separate dispersion and absorption terms: time reversal is then possible p = c2

  • 1−τ ∂

∂t (−∇2)y/2−1 + η(−∇2)(y+1)/2−1

  • ρ

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 25 / 54

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SLIDE 30

PhotoAcoustic Forward Model

Absorption & Dispersion in Time Reversal Imaging

Absorbing wave equation: no time symmetry Absorption → amplification, dispersion unchanged

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 26 / 54

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SLIDE 31

PhotoAcoustic Forward Model

Time Reversal PAT with Absorption Correction

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 27 / 54

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SLIDE 32

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 28 / 54

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SLIDE 33

PhotoAcoustic Tomography

Quantitative PhotoAcoustic Tomography

[slide courtesy of Ben Cox]

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 29 / 54

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PhotoAcoustic Tomography

Quantitative PhotoAcoustic Tomography

Aim: to extract distributions of chromophores from multiwavelength PAT images How are chromophores and PAT images related? [slide courtesy of Ben Cox]

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 29 / 54

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Quantitative PhotoAcoustic Tomography

Outline

The Optical Inverse Problem Fluence Models Parameter Estimation

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 30 / 54

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SLIDE 36

Modelling in Optical Tomography

Radiative Transfer Equation (RTE)

The radiative transfer equation is an integro-differential equation expressing the conservation of energy which takes the following time-independent form as required in PAT (ˆ s · ∇ + µa(r) + µs(r)) φ(r, ˆ s) = µs

  • Sn−1 Θ(ˆ

s, ˆ s′)φ(r, ˆ s′)dˆ s′ + q(r, ˆ s) (10) where φ(r, ˆ s, t) is the radiance, Θ(ˆ s, ˆ s′) is the scattering phase function, Wave effects, polarisation, radiative processes, inelastic scattering, and reactions (such as ionisation) are all neglected in this model. By writing a variational form of equation(10) it can be discretised using the finite element method (Tarvainen2005). When the radiance φ, source q or phase function Θ depend strongly on the direction ˆ s it is necessary to discretise finely in angle ˆ s, and the model can become computationally intensive.

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 31 / 54

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SLIDE 37

Modelling in Optical Tomography

Physical Models of Light Propagation

The Radiative Transfer Equation (RTE) is a natural description of light considered as photons. It represents a balance equation where photons in a constant refractive index medium, in the absence of scattering, are propagated along rays l := r0 + lˆ s ˆ s · ∇U + µaU = 0 ≡ TµaU = 0 (11) whose solution U = U0 exp

  • −
  • l

µa(r0 + lˆ s)dl

  • (12)

is the basis for the definition of the Ray Transform gˆ

s(p) := − ln

U U0

  • =

∞

−∞

µa(pˆ s⊥ + lˆ s)dl ≡ gˆ

s = Rˆ sµa

(13)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 32 / 54

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SLIDE 38

Modelling in Optical Tomography

The Radiative Transfer Equation

In the presence of scattering, and with source terms q, eq.(11) becomes [Tµtr − µsS] U = q (14) where µtr = µs + µa is the attenuation coefficent and S is the scattering

  • perator, which is local (non propagating).

A series solution for eq.(14) can be formally written as U =

  • T −1

µtr + T −1 µtr µsST −1 µtr + . . .

  • T −1

µtr µsS

k T −1

µtr . . .

  • q

(15) This is the method of successive approximation (Sobolev 1963). The first term may be found from the Ray Transform, giving an alternative equation for the collided flux [Tµtr − µsS] Ucollided = µsS T −1

µtr q uncollided

(16)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 33 / 54

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SLIDE 39

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 40

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 41

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 42

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 43

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 44

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 45

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 46

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 47

Modelling in Optical Tomography

RTE solutions

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 34 / 54

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SLIDE 48

Modelling in Optical Tomography

Diffusion Approximation

In the Diffusion pproximation (DA), the radiance is approximated by first order spherical harmonics only (ˆ s ≡ [Y1,−1, Y1,0, Y1,1]), giving φ(r, ˆ s) ≈ 1 4πΦ(r) + 3 4π ˆ s · J(r) (17) where Φ(r) and J(r) are the photon density and current defined as Φ(r) =

  • Sn−1 φ(r, ˆ

s)dˆ s (18) J(r) =

  • Sn−1 ˆ

sφ(r, ˆ s)dˆ s. (19) Inserting the approximation (17) into equation (10) results in a second

  • rder PDE in the photon density

− ∇ · D∇Φ(r) + µaΦ(r) = q0(r) ≡ DΦ = q0 , (20) with D =

1 µa+(1−g)µs). Equation(20) and its associated frequency and

time domain versions, including the Telegraph Equation, are the most commonly used in DOI.

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 35 / 54

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SLIDE 49

Quantitative PhotoAcoustic Tomography

Overview

The inverse problems in QPAT. Solid lines indicate linear operators and dot-dash lines those that are inherently nonlinear. The acoustic pressure time series p(t) are the measured data and the chromophore concentrations Ck are the unknowns. The concentrations may be obtained step by step : linear acoustic inversion A−1, thermoelastic scaling ˆ Γ−1 nonlinear optical inversion for the

  • ptical coefficients T −1 and finally

L−1

λ

a linear spectroscopic inversion

  • f the absorption spectra to recover

the chromophore concentrations.

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 36 / 54

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SLIDE 50

Quantitative PhotoAcoustic Tomography

PAT images and chromophores

PAT images ∝ absorbed energy distribution h(r, λ) p0(r, λ) is related to absorption coefficient µa(r, λ) via the fluence, Φ(r, λ) and the Grüneisen parameter: po(r, λ) = ΓH(r, λ) = Γµa(r, λ)Φ(r, λ) µa is related to chromphores concentrations C(k) via specific absorption coefficients ǫk: µa(r, λ) =

K

  • k=1

ǫk(λ)C(k)(r) Inverse problem is non-linear but well-posed. Solve using diffusion or transport methods

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 37 / 54

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SLIDE 51

Quantitative PhotoAcoustic Tomography

MultiSpectral QPAT

chromophore concentration C(x) scattering parameter a(x) C range: 5-15 g/l HbO2 µ′

s = aa0λ(nm)−b mm−1, a0 = 500, b = 1.3, a range: 5-10

Wavelength Dependence

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 38 / 54

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SLIDE 52

Quantitative PhotoAcoustic Tomography

MultiSpectral QPAT reconstructions

Reconstructed Concentration Reconstructed Scattering ’a’

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 39 / 54

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SLIDE 53

Quantitative PhotoAcoustic Tomography

Inverse Problem

Find the absorption and scattering coefficients µa, µs given the absorbed energy density image h(r) = p0(r) Γ = µa(r)Φ(µa(r), µs(r)) when the fluence Φ is unknown. Strategy used here : fit a model of light transport to the reconstructed data { ˆ µa, ˆ µs} = arg min

µa,µs

  • E := ||hobs − F(µa, µs)||2 + R(µa, µs)
  • where F(µa, µ′

s) = µaΦ((µa, µ′ s) is the forward model of optical energy

absorption, and R is a regularisation term. In this talk, the regularisation term is Total Variation.

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 40 / 54

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SLIDE 54

Quantitative PhotoAcoustic Tomography

Linearisation

Discretise parameters into a suitable basis µa(r) =

  • j

µajuj(r), µs(r) =

  • j

µsjuj(r) The functional gradient vectors are given by ga = ∂E ∂µaj = −

  • m,k

(hm

k − µakΦm k )JAm kj + ∂R

∂µaj gs = ∂E ∂µsj = −

  • m,k

(hm

k − µakΦm k )JSm kj + ∂R

∂µsj with the absorption and scattering Jacobians respectively JAm

kj = Φm k δkj + µak

∂Φm

k

∂µaj , JSm

kj = µak

∂Φm

k

∂µsj . (21)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 41 / 54

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SLIDE 55

Quantitative PhotoAcoustic Tomography

Gauss-Newton Approach

By combining the absorption and scattering Jacobians for every illumination into a single Jacobian matrix, J ∈ RMK×2K J =    JA1 JS1 . . . JAM JSM    , the Hessian, H ∈ R2K×2K, may be approximated by H ≈ JTJ. The update to the absorption and scattering coefficients, [δµak, δµsk]T, can then be calculated by a Newton step according to δµak δµsk

  • = −H−1

ga gs

  • .

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 42 / 54

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SLIDE 56

Quantitative PhotoAcoustic Tomography

Construction of Jacobians

For the Radiative Transfer Equation the following equations were used to directly calculate the Jacobians JA and JS column by column (ˆ s · ∇ + µak + µsk) ∂φm

k (ˆ

s) ∂µaj − µsk

  • Sn−1

Θ(ˆ s, ˆ s′)∂φm

k (ˆ

s′) ∂µaj dˆ s′ = −δkjφm

k (ˆ

s) (ˆ s · ∇ + µak + µsk) ∂φm

k (ˆ

s) ∂µsj − µsk

  • Sn−1

Θ(ˆ s, ˆ s′)∂φm

k (ˆ

s′) ∂µsj dˆ s′ = δkj

  • Sn−1

Θ(ˆ s, ˆ s′)φm

k (ˆ

s′)dˆ s′ − δkjφm

k (ˆ

s)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 43 / 54

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SLIDE 57

Quantitative PhotoAcoustic Tomography

Construction of Jacobians

For the diffusion approximation (µak − ∇ · Dk∇) ∂Φm

k

∂µaj = −δkjΦm

k

(µak − ∇ · Dk∇) ∂Φm

k

∂Dj = ∇ ·

  • δkj∇Φm

k

  • ∂Φm

k /∂µsj is then obtained from ∂Φm k /∂Dj using the relation

∂D/∂µs = −3D2(1 − g).

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 44 / 54

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SLIDE 58

Quantitative PhotoAcoustic Tomography

RTE-based Inversions

Fixed-point iteration, known scattering (Yao, Sun, Jiang 2009) Use separated unscattered, singly-scattered and multiply scattered components (Bal, Jollivet, Jugnon 2010): can’t do in practice. Gauss-Newton inversions with TV regularization (C., Tarvainen, Arridge, 2011; Tarvainen, Cox, Kaipio, A. 2012) Gradient-based inversions with Tikhonov reg. (Saratoon, Tarvainen, Cox, A., submitted)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 45 / 54

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SLIDE 59

Quantitative PhotoAcoustic Tomography

RTE-based Inversions (Gauss-Newton)

Using 4 images from 4 illumination directions (Tarvainen, Cox., Kaipio, A. 2012)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 46 / 54

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SLIDE 60

Quantitative PhotoAcoustic Tomography

SVD comparison

SVD of Hessian reveals different information content

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 47 / 54

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SLIDE 61

Quantitative PhotoAcoustic Tomography

Matrix Free method

Explicit construction of Jacobians is too expensive ⇒ use matrix free method based on adjoint fields Limited memory BFGS optimisation Using 4 images from 4 illumination directions, Tikhonov regularisation (Saratoon, Tarvainen, Cox, A., submitted)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 48 / 54

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SLIDE 62

Quantitative PhotoAcoustic Tomography

Gauss-Newton vs. Gradient-Based

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 49 / 54

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SLIDE 63

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

6

Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 50 / 54

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SLIDE 64

Summary

Photoacoustic imaging: great potential as biomedical imaging method Importance of spectroscopic aspect of photoacoustics sometimes

  • verlooked (as it is not present in thermoacoustics?)

Much progress made in quantitative photoacoustics recently Linearized approaches probably not sufficient in practice Complete 3D problem is of large scale Accuracy of light models in ballistic regime (close to surface) Questions about Grüneisen parameter remain. (Under DA not possible to recover all of Γ, µa, D with one wavelength, but with multiple wavelengths it could be (Bal, Ren 2011, Ren, Bal 2011).)

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 51 / 54

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Quantitative PhotoAcoustic Tomography

Discussion

Multi-Wavelength vs Single Wavelength Inversions Nonlinearity PA Generation Efficiency

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 52 / 54

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SLIDE 66

Outline

1

Introduction

2

PhotoAcoustics

3

PhotoAcoustic Forward Model

4

Quantitative PhotoAcoustic Tomography

5

Summary

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Acknowledgements

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 53 / 54

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SLIDE 67

Acknowledgements

Collaborators :

UCL: P .Beard, M.Betcke, T.Betcke, Y.Kurylev, B.Cox, J.Laufer, T.Saratoon, B.Treeby, Kuopio: J. Kaipio, V. Kolehmainan, T. Tarvainen, M. Vaukhonen,

Funding

This work was supported by EPSRC grant EP/E034950/1 and by the Academy of Finland (projects 122499, 119270 and 213476) Other funding : MRC, Wellcome Trust, CEC Framework, Royal Society

S.Arridge (University College London) QPAT using RTE IMPA, Rio de Janiero 27-07-13 54 / 54