Frequency Domain Photoacoustics: Specifics Of Signal .
Fields-MITACSMathematics of Medical ImagingJune 21, 2011Frequency Domain Photoacoustics:Specifics of Signal Processing andImage ReconstructionSergey TelenkovUniversity of TorontoCentre for Advanced Diffusion WaveTechnologies (CADIFT), Toronto, Canada
Photoacoustic Tomography: Objectives and MethodsImaging Objectives:1. Positions and dimensions of photoacoustic sources.2. Characteristics of PA sources: absorption coefficient,chemical composition, blood flow rate etc.Standard Methods:1. Short (nanosecond) laser irradiation and broadband detection.2. Photoacoustic microscopy with high frequency ( 30 MHz) sources.3. Photoacoustic spectroscopy with narrow band tunable sources.
Photoacoustic Imaging with Intensity ModulatedCW Laser Source (Frequency Domain PA)OpticalContrast µaLaserSource TModulation waveforms(t)I(t)AcousticPressure2 i ω βq̃ ( ⃗r , ω)Cpq̃ (⃗r , ω) μ a ( ⃗r ) I ( ⃗r ) F̃ (ω)2 p̃ ( ⃗r , ω) k ̃p ( ⃗r ,ω) Confinement conditions in FD:1/ 2DLT T μ a 1ωω ωa μ a c a( )Difficulties of FD photoacoustics:1. Low optical power (0.1 – 1 W) Low SNR2. Long pulse duration ( 1 ms) Poor spatial resolutionRaw signal after 1000 averages
Spatially-Resolved PA Imaging with ChirpedWaveformsf2Train of frequency-modulated pulsesCorrelation function of chirped PA signalf0f1TchTch 1msCorrelation Processor 1B( τ) r(t τ) s(t) dt R̃* (ω) S̃ (ω) ei ω τ d ω 2 π Correlation image of optical contrastin scattering mediumA2 T chB( τ) 2sinPeakAmplitude[ ( )]πmττ1 T chT chπ m τ /T chSide lobes1cmcos (ω 0 τ)Harmoniccarrierm Tch f
Analytical Model of PA Generation1-D Model with Acoustic ImpedanceDiscontinuityRef. Chirp1 – 5 MHzL μ eff zI ( z) I 0 eDzρfcfρscsρfcf2p̃ zz (z , ω) k ̃p ( z , ω) q̃ ( z , ω) μ a I 0 e μ a z i ωβq̃ ( z , ω)CpF̃ (ω),PA TransferFunctionF̃ (ω) - Spectrum of the lasermodulation waveformMethod of transfer functions:̃ PA (ω) I 0 F̃ (ω) Φ( ⃗r )p̃ ( ⃗r , ω) H( ζ k f i μ a ) cos( k s L) (i k s ζ μ a c s /c f ) sin( k s L) ( ζ k f i μ a ) e μH PA (ω) C p (μ 2a c 2s ω2 )i(1/c 2s ζ 2 /c 2f ) sin( k s L) (2 ζ /c s c f ) cos (k s L) iβμ a c s1-D solution for exponential source: i k f zp̃ ( z ,ω) H̃ PA (ω) I 0 F̃ (ω) e,k f ω /c faLPA PressureSpectrum
Correlation Processing of Chirped PA SignalsNoise-free correlation functionSimulation Results for a 1-D layer:Layer thickness: 5 mmρscs 1.54 MRyalsρfcf 1.48 MRyalsAbsorption: 4 cm-1Optical Modulation:Sine chirp: 1 – 5 MHzChirp duration: 1 msCorrelation function of noisy signalZero-mean Gaussian noise:Input SNR - 40 dBCoherent averaging of 1000 chirpsSNR Improvement 56 dBAxial Resolution: ca/ f 1 mm
Signal-to-Noise of Frequency Domain PAMeasurementsMatched Filter (Correlation):1iω τ̃B( τ) R(ω)S̃ (ω)e d ω2π B(τ) ℜ B(τ) ℑ B(τ)22SNR of Matched Filter (Single Chirp):2B (0) E s f sSNR MF 〈 P NB 〉 0.43 σ 2Multiple Chirps: Coherent vsIncoherent Averaging of Np Chirps1) Coherent Averaging (Phase retained):σ2〈 P N 〉 ;NpGaussian noise PSD:〈 P N 〉 2 σ2N 0 f s/2fsE [ A] SNR A22exp( A /2 σ A )2σA Esσπ2σ A ,σ A 2fsVar 0.43 σ A2 π Esσ2 f sNp- Noise BackgroundE s σ2Var [ B N ] 0.43f sNpNoise of matched filter(Rayleigh distribution):PDF E [ B N ] 2- Noise Variance(E s E [ B N ])2 f s N p20.43σ E s2) Incoherent Averaging (Post processing):Np1B av ( τ) B(τ) B (τ) B(τ ) n BN p i 1 NE [ n B ] π Esσ;2 fsSNR - Independent on NP2( )π EsE s σ2 fs20.43 σ E sf sNp
SNR of Coherent vs Incoherent AveragingTwo methods of signal detection with different level of the input noise(zero-mean Gaussian noise with std deviation σ)Coherent Averaging of 100 chirpsIncoherent Averaging of 100 chirpsσ 100SNR -23 dBCoherent Averaging of 100 chirpsIncoherent Averaging of 100 chirpsσ 150SNR -25 dB
SNR and Laser Safety LimitMaximum Permissible Exposure (1064nm, 10-7 – 10s):E MPE 5.5 T1/ 42[ J /cm ]E2SNR of Matched Filter Processing:2E1SNR MF E s A s T ch5.5t 0.25Signal Amplitude:A s I L - Laser Irradiance [W/cm2]Assuming:Then: 3/ 4I L I MPE 5.5 T chSNR I2 T ch TMPEE CW I L t L 1/ 2chFor I IMPE shorter chirp duration isexpected to give higher SNRP 1.76 W;P 0.8 WLaser safety curve(diam 5 mm)
Amplitude and Phase of Correlation ProcessingCorrelation function of a linearfrequency modulated chirpA2 T chB( τ) 2sin[ ( )]πmττ1 T chT chπ m τ /T chcos (ω 0 τ)Heterodyne mixing (Stretch Processor):ReceivedChirpLow itudeLock-InPhaseFrequency Scan2Reference: r (t) r 0 exp [i(2 π f 1 t πb t )]2Delayed chirp: s(t) s 0 exp[i (2 π f 1 (t τ) π b(t τ) )]Downshifted signal:b 2V (t ) s(t) r (t) r 0 s 0 exp [i(2 πb τ t 2 π( f 1 τ τ ))]2Sine wave frequency: F bτbPhase: Θ 2π( f 1 τ τ 2)2b 4 x 109τ 30 µsF 120 kHz
Phased Array Correlation ImagingCorrelation Phased Array – multichannel matchedfilter processing and beamforming in frequency domainYImage plane1) Array acquisition and FFT of signal matrix̃ (ω) R̃ * (ω) S̃ i (ω) - matrix Ne x Nt, Nt 100kB̃ i (ω) WBeamrArray2) Digital beamforming, i.e. spatial filtering by creatingdirectional beams and beam steeringθNeXŨ (ω ,θ) w n B̃ n (ω) exp( i ω t n (θ))n 1k ω22 k x k ycat n (θ) Ũ (k ,θ) - Spatial spectrum3) Backprojection:u(x , y) FFT 1 [ Ũ ]Optical inclusion inscattering phantomSystem PSF, SNR -34dB1cm Bilinear interpolation1cmDiscrete chromophoresin Intralipid1cmynsin(θ) t fca
ConclusionsDepth-resolved PA imaging with CW laser sources is feasible using chirped opticalexcitation and correlation signal processing.Correlation processing of coded PA response can significantly increase SNR( 50 dB) and provide axial resolution 1 mm.High repetition rates ( 1 kHz) can easily implemented using inexpensive laser diodes.To achieve maximum SNR performance multiple chirps must be averagedcoherently in pre-processing and chirp duration should be set according to MPE.Phase information can be potentially utilized for PA imaging using heterodyne mixingtechnique.Phased array PA correlation imaging was demonstrated using conventional ultrasoundarray and frequency domain reconstruction algorithm.
AcknowledgementsProf. Mandelis for support and Rudolf Alwi for experimental assistance.The CADIFT research group at the University of Toronto.
Centre for Advanced Diffusion Wave Technologies (CADIFT), Toronto, Canada Frequency Domain Photoacoustics: Specifics of Signal Processing and Image Reconstruction Fields-MITACS Mathematics of Medical Imaging June 21, 2011. Imaging Objectives: 1. Positions and dimensions of photoacoustic sources. . Photoacoustic microscopy with high frequency .
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