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Photoacoustic Tomography

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Photoacoustic Tomography H U A B E I J I A N G Transducer Fixed Mirror Aperture MEMS Mirror Plastic Film J I A N G CRC Press is an imprint of the Taylor the subsequent laser induced transient photoacoustic waves in the range of 1 100 MHz due to the transient thermoelastic expansion of light absorbing components in tissue are detected by wideband unfocused or focused ultrasound transducer s Unique advantages of PAT are that functional or biochemical parame ters such as deoxy hemoglobin HbR oxy hemoglobin HbO2 water H2O lipids and so forth along with vasculature and blood flow can be imaged in high resolution In addition highly specific molecular PAT can be realized through the use of molec ular contrast agents Finally PAT can be made portable for bedside applications is economical and uses non ionization radiation PAT has found its potential clinical applications in several areas In breast imag ing PAT offers the submillimeter resolution ability to quantitatively image the high optical contrast generated through the presence of blood water and lipid which are the predominant transformations associated with malignancy Clinical studies con ducted at multiple institutions and countries have repeatedly shown that 2 to 11 and higher absorption contrasts exist in breast cancers that can be imaged by PAT These studies suggest that PAT has the potential to detect breast tumors at early stages Application of PAT to joint imaging has been recently explored offering an oppor tunity for early detection and monitoring of progressive diseases including osteoar thritis and rheumatoid arthritis In this case the optical contrast is produced through the degraded articular cartilage and the increased water content and turbidity in the synovial cavity Other clinical applications of PAT also are starting to appear including intravascular imaging and intraoperative imaging during cancer treatment While brain imaging and electrophysiology play a central role in neuroscience research and in the evaluation of neurological disorders a single noninvasive modal ity that offers both high spatial and temporal resolution is currently not available PAT may become such a neuroimaging modality at least in animals In Chapter 8 I show an acute epilepsy rat model in which PAT can noninvasively track seizure brain dynamics with both high spatial and temporal resolution and at a depth that is clinically relevant The noninvasive yet whole surface and depth capabilities of PAT actually seeing what is happening during ictogenesis in terms of seizure onset and spread xiiPreface The concept of PAT emerged in the mid 1990s and the field of PAT is now rap idly moving forward Several reviews and edited books on PAT have been published However I have decided to write a book that presents a detailed and comprehensive treatment of PAT It appears timely to produce such a book for the first time in this field The book is essentially a collection of the research work that the author and his colleagues have been pursuing over the past decade This actually presents a unique feature of the book as it allows the principles of PAT and its applications to be treated in a systematic way In addition the collection covers almost every aspect on PAT from mathematics image reconstruction methods instrumentation and phantom animal experiments to clinical applications I believe this book will be particularly useful for graduate students and researchers who wish to enter the field of PAT In Chapter 1 the fundamentals of PAT are presented including the theory on photo acoustic effect various image reconstruction methods and instrumentation Chapter 2 describes various advanced methods for quantitative PAT which allow the recov ery of tissue optical absorption coefficients and or acoustic properties Experimental validations of the reconstruction methods are also presented here The develop ment of several image enhancing schemes including both software and hardware approaches are discussed in Chapter 3 Chapter 4 describes array based PAT systems that are the foundation for the realization of 2D 3D and 4 D PAT Photoacoustic microscopy PAM a variation of PAT is detailed in Chapter 5 The recent devel opment of multimodality methods is presented in Chapter 6 These works describe the combinations of PAT PAM with other imaging methods including DOT fluores cence molecular tomography FMT ultrasound imaging and OCT which represent new directions in the field of PAT Chapter 7 focuses on the discussion of contrast agents based molecular PAT where both nontargeted and cell receptor targeted methods are included Chapter 8 the final chapter describes the clinical applications and animal studies in the areas of breast cancer detection osteoarthritis diagnosis seizure localization intravascular imaging and image guided cancer therapy I wish to thank my previous and current colleagues whose contributions to the works presented have made this book possible I would also like to thank the staff at CRC Press especially associate editor Ashley Gasque and project coordinator Jill Jurgensen who have worked diligently at bringing this book to fruition Finally my wife Yonghong and my family made it all worthwhile Huabei Jiang Gainesville Florida xiii About the Author Huabei Jiang PhD is J Crayton Pruitt Family professor in the Department of Biomedical Engineering at the University of Florida He has published more than 300 peer reviewed scientific articles and patents Dr Jiang is a fellow of the Optical Society of America OSA a fellow of the International Society of Optical Engineering SPIE and of the American Institute of Medical and Biological Engineering AIMBE 1 1 Fundamentals of Photoacoustic Tomography Photoacoustic tomography PAT or optoacoustic tomography OAT is based on the photoacoustic PA effect discovered by Alexander Graham Bell 1880 who observed that audible sound is produced when chopped sunlight is incident on optically absorbing materials Since then techniques based on PA spectros copy and microscopy have been developed and used primarily for the studies of gases and solids Tam 1986 Methods for tomographic PA imaging or PAT did not appear until 1990s first for nonbiological medium Gusev and Karabutov 1993 and then for biological tissue Oraevsky et al 1994 Kruger and Liu 1994 PAT is basically concerned with an inverse problem where a single short pulsed light beam illuminates an object and the photoacoustic waves excited by ther moelastic expansion are measured using wideband ultrasound transducers in multiple locations around the object The geometry of the object and spatial distribution of the optical absorption are obtained from the measured scattered fields using a reconstruction algorithm or formed directly by time resolved pho toacoustic signals In this chapter we present the basic PA theory several image reconstruction methods and essential hardware components needed for imple menting a PAT system 1 1 PHOTOACOUSTIC EFFECT 1 1 1 Empirical DEscription The basic physical principles behind PA effect can be demonstrated using some approximate solutions For generality a 3D model using lateral dimensions x and y and depth z needs to be considered Biological tissue is homogenously irradiated with a short laser pulse of energy E J on an area A m2 and the photon density or radiant exposure J m2 on the surface can be approximately stated as x y 0 E A 1 1 The radiant exposure for a large depth z is estimated by I x y ze z eff 0 1 2 2Photoacoustic Tomography where eff is the effective optical attenuation coefficient which is a function of opti cal absorption coefficient aand reduced scattering coefficient s effaas 3 1 3 It should be noted that the photon density here can be accurately calculated using the photon diffusion equation with any kind of source distribution In PA effect the stress generation depends on the locally absorbed light energy density that is the product of optical absorption coefficient a and photon density Norton and Vo Dinh 2003 a 1 4 With mass density kg m3 and specific heat Cp J kg C the absorbed light energy leads to a temperature increase C T x y zx y zCp 1 5 Additionally due to laser excitation the absorbed light energy also gives rise to a fractional volume expansion dV V dV V p T 1 6 where denotes the isothermal compressibility is the thermal coefficient of vol ume expansion and T and p are the changes in temperature k and pressure Pa respectively The isothermal compressibility can be expressed as Cv C psv 2 1 7 where Cp and Cv are respectively the specific heat capacities at constant pressure and volume and vs is the acoustic velocity If the laser excitation is in both the ther mal and stress confinements the fractional volume expansion is negligible and the local pressure rise can be obtained from Equation 1 6 pT 0 1 8 If we define a set of coefficients as Grueneisen parameter Gusev and Karabutov 1993 Cv C Vsp 2 1 9 Equation 1 8 is rewritten in consideration of Equations 1 4 1 5 and 1 9 pC ava 0 1 10 3Fundamentals of Photoacoustic Tomography For water and diluted aqueous solution can be estimated by the following empirical equation TT 0 00430 0053 00 1 11 in which T0 is the temperature in degrees Celsius p0 is often specified as the initial pressure for a temporal PA wave equation These sets of empirical equations for the calculation of p0 should satisfy the stress confinement and thermal confinement conditions Gusev and Karabutov 1993 Kostli et al 2000 The direct efficient con version to pressure is possible if the acoustic transient does not leave the heated region during the heating process This condition is called stress confinement and is satisfied if the pulse duration tp is smaller than the propagation time of the acoustic transient through the region d with acoustic velocity vs Gusev and Karabutov 1993 Kollkman et al 2007 d v ps 1 12 If the laser pulse width is much shorter than the thermal relaxation time th the excitation is said to be in thermal confinement and heat conduction is negligible dur ing the laser excitation The thermal confinement condition is written d thth 2 1 13 in which th is the thermal diffusivity m2 s and dc eff 1 Example A laser pulse with energy E 30 mJ pulse length 10 ns beam radius r 1 cm and wavelength 550 nm irradiates human blood vessel The radiant exposure is 0 E r2 30 mJ 3 14 cm2 9 55 mJ cm2 Assuming a 25 cm 1 the absorbed light energy density is calculated as 25 9 55 238 75 mJ cm3 2 38 105 J m3 Then we can obtain the temperature increase and initial pressure assuming 1 g cm3 and Cp 4 Jg 1K 1 T x y zx y zCK p 238 75mJ cm 1g cm4 Jg 59 7mk 3311 p0 22 3810 Pa0 22 38 bar0 476bar 0 5 To obtain this pressure stress confinement with a region d 1 a 400 m and sound velocity vs 1 5 m ns needs to be satisfied p 10 ns d vs 266 ns 4Photoacoustic Tomography 1 1 2 rigorous thEory The empirical description presented in Section 1 1 1 is intended to illustrate the PA effect and certainly cannot be used for the purpose of imaging The PA wave genera tion and propagation in biological tissue is rigorously described by the Helmholtz like wave equation To derive the PA wave equation three general elastic mechanical and fluid dynamic equations are responsible for the PA generation and propagation Newton s law of motion equation of continuity and thermal elastic equation r t V r tp r t 1 14 V r t r vrt p r t t T r t s 1 2 1 15 r C t T r tH r t p 1 16 where V is particle velocity and H is the laser excitation source term which can be written as H I t where I t is the temporal illumination function Considering Equations 1 14 1 16 and eliminating V we obtain r r p r t vrt p r t Ct H r t sp 1 1 2 2 2 1 17 If a homogeneous elastic medium is assumed Equation 1 17 is further written p r t vrt p r t Ct H r t sp 1 2 2 2 2 1 18 Equation 1 18 is the general PA wave equation without acoustic attenuation in time domain If a homogeneous acoustic medium is also assumed Equation 1 18 is written as p r t vt p r t Ct H r t p 2 2 0 22 1 19 where v0 is the constant acoustic velocity within the entire problem domain Denoting the following Fourier transform expression for acoustic pressure P rp r ti t dt exp 1 20 5Fundamentals of Photoacoustic Tomography and taking the Fourier transform on Equation 1 18 with respect to variable t we obtain the PA wave equation in frequency domain assuming HI ttt 0 P rk P rik v r C s p 22 1 21 where P is the pressure wave in frequency domain kvs is the wave number described by the angular frequency and the acoustic speed vs The PA wave equation in frequency domain for a homogeneous acoustic medium is accordingly expressed as P rk P rik v r Cp 2 0 2 00 1 22 where kv 00 If both the acoustic velocity and attenuation are heterogeneously distributed the frequency domain PA wave equation is then described as P rkO P rik v r Cp 1 2 0 2 00 1 23 where O is a coefficient that depends on both acoustic velocity and attenuation as follows Ov viAv k v oss 1 22 00 2 1 24 where v0 is the constant acoustic velocity in a reference coupling medium and A is the acoustic attenuation coefficient 1 2 IMAGE RECONSTRUCTION METHODS 1 2 1 DElay anD sum BEam Forming algorithm To form the PA image of an object an image reconstruction algorithm is required A simple yet very effective method is the delay and sum beam forming algorithm that is commonly used in radar signal processing Hoelen and de Mul 2000 The image expression in the case of near field for photoacoustic imaging can be stated as S r t w S t w iii i i i 1 25 where S r t is the image output at a particular spatial point r within the imaging domain Si t is the time resolved acoustic signal from the ith receiver i is the 6Photoacoustic Tomography delay applied to this signal and wi is an amplitude weighting factor which is used to enhance the beam shape to reduce sidelobe effects or to minimize the noise level Note that i Ri r v is just the time needed for the acoustic signal to travel from point r to point Ri where Ri is the spatial location of the ith receiver and v is the assumed constant acoustic velocity The summed signal is typically normalized to make the output independent of the actual set of transducers This delay and sum algorithm has been tested and evaluated using considerable phantom and in vivo experiments Hoelen and de Mul 2000 Manohar et al 2007 1 2 2 itErativE nonlinEar algorithm While the delay and sum algorithm described above is simple to implement it does not account for the diffraction effect that is essential to the PA waves In particular a major assumption has to be made in this and other linear algorithms such as backpro jection or filtered backprojection seen in the literature e g backprojection Kruger et al 1995 Wang et al 2007 that biological tissues are acoustically homogeneous which may not be true in reality This assumption will certainly affect the accuracy in image reconstruction In addition acoustic properties cannot be reconstructed because of this assumption Importantly functional parameters such as deoxy and oxy hemoglobin HbR and HbO2 and water content H2O cannot be obtained with these linear methods see more about quantitative PAT in Chapter 2 To overcome the above mentioned limitations here we describe a nonlinear reconstruction algorithm based on the finite element FE solution to the full PA wave equation without the homogeneous acoustic property assumption made thus far in the literature Finite element method FEM has been a powerful numerical method for solving the Helmholtz type equation because of its computational effi ciency and unrivaled ability to accommodate tissue heterogeneity and geometrical irregularity as well as allow complex boundary conditions and source representa tions This reconstruction approach in PAT is an iterative Newton method with combined Marquardt and Tikhonov regularizations that can provide stable inverse solutions The approach uses the hybrid regularizations based Newton method to update an initial optical property distribution iteratively in order to minimize an object function composed of a weighted sum of the squared difference between computed and measured data Together with the iterative Newton method this nonlinear algorithm is able to precisely solve the Helmholtz wave equation and fulfill reliable inverse computation for an arbitrary measurement configuration The nonlinear algorithm is implemented in both frequency and time domain as follows 1 2 2 1 Frequency Domain FE Based Algorithm The PA wave equation in frequency domain is written if a homogeneous acoustic medium is considered for the case of inhomogeneous acoustic medium see Chapter 2 P rk P rik v r Cp 2 0 2 00 1 26 7Fundamentals of Photoacoustic Tomography Expanding acoustic pressure P as the sum of coefficients multiplied by a set of basis functions j PPj j the finite element discretization of the Helmholtz wave equation 1 26 can be expressed as Pkds ik vC j j N jijij j i Pi 1 0 2 2 2 00 1 27 where the following second order absorbing boundary conditions have been applied Jiang et al 2006 P nP P 2 2 1 28 where ikiki k 3 23 8 1 00 2 0 and iki k 2 1 0 2 0 N is the total number of nodes of the finite element mesh indicates the integration over the problem domain and expresses the integration over the boundary In both the forward and inverse calculations the unknown needs to be separated into real R and imaginary I parts both of which are expanded in a similar fashion to P as a sum of unknown parameters multiplied by a known spatially varying basis function The matrix form of Equation 1 27 is expressed as follows A PB 1 29 where Akds ijjijij j i0 2 2 2 1 30 BIk vCk vC iR kki k pI lli l p00 00 1 31 PP PPN T 12 1 32 To form an image from a presumably uniform initial guess of the optical prop erty distribution we use iterative Newton s method to update R and I from their starting values In this method we Taylor expand P about an assumed R I distribution which is a perturbation away from some other distribution RI such that a discrete set of P values can be expressed as PP PP RIRI R R I I 1 33 where RRR and III If the assumed optical property dis tribution is close to the true profile the left hand side of Equation 1 33 can be 8Photoacoustic Tomography considered as true data observed or measured and the relationship can be truncated to yield oc JPP 1 34 where PPPP PPPP PPPP RR KII L RR KII L MRMR KMIMI L 1 11 1 11 2 12 2 12 1 1 J 1 35 RRR KIII L T 1 2 1 2 1 36 PPP ooo M oT 12 P 1 37 P PP ccc M cT 12 P 1 38 and Pio and Pic are measured and calculated acoustic field data based on the assumed R I distribution data for i 1 2 M boundary locations R k k 1 2 K and I l l 1 2 L are the reconstruction parameters for the optical property profile In order to realize an invertible system of equations for Equation 1 34 is left multiplied by the transpose of J to produce TToc J JIJPP 1 39 where regularization schemes are invoked to stabilize the decomposition of T J J I is the identity matrix and is the regularization parameter determined by com bined Marquardt and Tikhonov regularization schemes We have found that when oc PP trace T J J the reconstruction algorithm generates best results for PAT image reconstruction The process now involves determining the calculated acoustic field data and Jacobian matrix The reconstruction algorithm here uses the hybrid regularization based Newton method to update an initial guess optical prop erty distribution iteratively via the solution of Equations 1 29 and 1 39 so that an object function composed of a weighted sum of the squared difference between com puted and measured acoustic pressures for all acoustic frequencies and optical wave lengths can be minimized 1 2 2 2 Time Domain FE Based Algorithm The time domain PA wave equation 1 19 for an acoustically homogeneous medium can be rewritten as pt v pt tC I t t p 1 2 0 2 2 2 r rr 1 40 where I ttt 0 is typically assumed 9Fundamentals of Photoacoustic Tomography Expanding p as the sum of coefficients multiplied b

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