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 Markus Haltmeier Resolution of Photoacoustic Tomography 1/ 22
Photoacoustic tomography (PAT) PAT with integrating line detectors Photoacoustic tomography (PAT) 1 Mathematical model Classical approach PAT with integrating line detectors 2 Basic setup Factors influencing resolution Resolution analysis Markus Haltmeier Resolution of Photoacoustic Tomography 2/ 22
Photoacoustic tomography (PAT) Mathematical model PAT with integrating line detectors 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). detector absorbers optical illumination Applications: Cancer diagnostics, imaging of small animals Markus Haltmeier Resolution of Photoacoustic Tomography 3/ 22
Photoacoustic tomography (PAT) Mathematical model PAT with integrating line detectors 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: in R 3 × (0 , ∞ ) , ∂ 2 t p ( x , t ) = ∆ p ( x , t ) in R 3 , p ( x , 0) = I ( x ) · µ abs ( x ) =: f ( x ) in R 3 . ∂ t p ( x , 0) = 0 Markus Haltmeier Resolution of Photoacoustic Tomography 4/ 22
Photoacoustic tomography (PAT) Mathematical model PAT with integrating line detectors Classical approach Inverse problem in PAT Denote by W 3 D f := p the solution of the 3D wave equation. 1 Measure W 3 D f outside of region B including support of f . 2 Reconstruction function f ( x ) inside B from those values. detector absorbers optical illumination Math. Problem depends on type of measurements. Markus Haltmeier Resolution of Photoacoustic Tomography 5/ 22
Photoacoustic tomography (PAT) Mathematical model PAT with integrating line detectors Classical approach Classical approach: Ideal point detectors 1 Assume point-wise data on ∂ B Data = ( W 3 D f )( z , t ) , for ( z , t ) ∈ ∂ B × (0 , ∞ ) . Function f can be reconstructed uniquely and stably. 2 Exact inversion formula in case of ball B R : f ( x ) = 1 � ( ∂ t W 3 D f )( z , | x − z | ) , for x ∈ B R . 2 R ∂ B R Derived in [Finch-Patch-Rakesh ’04]. Markus Haltmeier Resolution of Photoacoustic Tomography 6/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Alternative: PAT with integrating line detectors 3D object θ ⊥ Line detectors: Measure integrals of W 3 D f over lines in di- rection θ projection θ Proposed in [Burgholzer-Hofer-Paltauf-MH-Scherzer ’05] . Markus Haltmeier Resolution of Photoacoustic Tomography 7/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Data from ideal line detectors X-ray transform in direction θ ∈ S 1 × { 0 } : � for y ∈ θ ⊥ . ( X h )( θ, y ) := ( X θ h )( y ) := h ( s θ + y ) ds R 1 Line detectors measure restriction of X -ray transform: �� X W 3 D f � L f := S 1 × ∂ D × (0 , ∞ ) . � � 2 Function f can be reconstructed uniquely and stably from L f . Markus Haltmeier Resolution of Photoacoustic Tomography 8/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Two step reconstruction f = 3D object θ ⊥ Commutation relation: X θ W 3 D f = W 2 D X θ f . X θ f = 2D object θ Two step algorithm: 1 For fixed θ : Recover initial data X θ f of 2D wave equation from solution W 2 D X θ f on curve ∂ D . 2 Recover 3D image f from projection images X θ f by applying inverse X-ray transform (2D Radon transform). Markus Haltmeier Resolution of Photoacoustic Tomography 9/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis First step: Inverse problem for 2d wave equation in R 2 × (0 , ∞ ) detector ∂ 2 t p − △ p = 0 , in R 2 p ( y , 0) = F ( y ) , object in R 2 ( ∂ t p )( y , 0) = 0 , Given restriction W 2 D F = p | ∂ D × (0 , ∞ ) → Reconstruct F Markus Haltmeier Resolution of Photoacoustic Tomography 10/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Equivalence to circular mean Radon transform �� � � ( M 2 D F )( y , r ) = S 1 F ( y + r σ ) ds ( σ ) 2 π 1 Solution formula + analytic inversion � t r ( M 2 D F )( y , r ) ( W 2 D F )( y , t ) = ∂ t √ dr t 2 − r 2 0 � r ( W 2 D F )( y , t ) ( M 2 D F )( y , r ) = 2 √ dt r 2 − t 2 π 0 Inversion of Wave Eq. W 2 D ⇄ Inversion of spherical M 2 D Markus Haltmeier Resolution of Photoacoustic Tomography 11/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis FBP type inversion formulas In case D = D R inversion formulas for wave equation (and circular means) have been found recently: [Kunyansky ’07] 1 [Finch-MH-Rakesh ’07] 2 �� 2 R � � ( ∂ r r ∂ r M 2 D F )( y 0 , r ) log | r 2 − ρ 2 F ( y ) = 0 | dr ds ( y 0 ) ∂ D R 0 Here ρ 0 = | y − y 0 | . Markus Haltmeier Resolution of Photoacoustic Tomography 12/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis 2D Example with PAT scanner in Innsbruck Figure: Setup in Innsbruck, kidney, reconstructed projection. Pictures provided by Markus Holotta and Harald Grossauer. Markus Haltmeier Resolution of Photoacoustic Tomography 13/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Many factors influence the resolution 1 Non-constant sound speed laser beam 2 Attenuation of US waves electromagnetic pulse 3 Limited view/angle/data 4 Detectors are not perfect lines e 3 B R 5 Finite bandwidth of e 2 detection system e 1 Markus Haltmeier Resolution of Photoacoustic Tomography 14/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors 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: �� ϕ ∗ t w ∗ z ( X W 3 D f ) � L ϕ,ψ f = S 1 × ∂ D × (0 , ∞ ) . � � Here ψ ( r ) = radial profile of the laser beam. ϕ ( t ) = impulse response function. Markus Haltmeier Resolution of Photoacoustic Tomography 15/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Approximate line detectors II Properties: L ϕ,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 L ϕ,ψ f )( x ) = blurred reconstruction . Our aim: Find point spread function (PSF, blurring kernel), i.e. find analytic expression for L − 1 L ϕ,ψ f . Markus Haltmeier Resolution of Photoacoustic Tomography 16/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors 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 L ϕ,ψ f ∈ ran( L ) and L − 1 L ϕ,ψ f � � ( x ) = (Φ band ∗ x Ψ line ∗ x f ) ( x ) , with the blurring kernels x ∈ R 3 , Φ band ( x ) := − πϕ ′ ( | x | ) / (2 | x | ) , � ∞ Ψ line ( x ) := − 1 ∂ ξ ψ ( ξ ) x ∈ R 3 . ξ 2 − | x | 2 d ξ , π � | x | Markus Haltmeier Resolution of Photoacoustic Tomography 17/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors 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. Markus Haltmeier Resolution of Photoacoustic Tomography 18/ 22
Basic setup Photoacoustic tomography (PAT) Factors influencing resolution PAT with integrating line detectors Resolution analysis Idea of proof (detector PSF) II 1 In mathematical terms: ( X θ Ψ line ) ∗ z ( W 2 D X θ f ) = W 2 D � � ( X θ Ψ line ) ∗ z ( X θ f ) . 2 Some manipulations: ( X θ Ψ line ) ∗ z ( X θ W 3 D f ) = X θ W 3 D � � Ψ line ∗ x f . Adjusting X θ Ψ line = ψ (inverse Abel transform) shows the theorem. Markus Haltmeier Resolution of Photoacoustic Tomography 19/ 22
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