Effect of Photoacoustic Radar Chirp Parameters on Profilometric Information

Transcription

1 Effect of Photoacoutic Radar Chirp Parameter on Profilometric Information A thei ubmitted to the Faculty of Engineering in partial fulfillment of the requirement for the degree of Mater of Applied Science in Mechanical Engineering by Zuwen Sun Ottawa-Carleton Intitute for Mechanical and Aeropace Engineering Univerity of Ottawa Ottawa, Ontario, Canada, K1N 6N5 January 18 Zuwen Sun, Ottawa, Canada, 18

2 Abtract Photoacoutic imaging for biomedical application ha attracted much reearch in recent year. To date, mot of the work ha focued on puled photoacoutic. Recent development have een the implementation of a radar pule compreion methodology into continuou wave photoacoutic modality, however very little theory ha been developed in upport of thi approach. In thi thei, the one-dimenional theory of radar photoacoutic for pule compreed linear frequency modulated continuou inuoidal laer photoacoutic i developed. The effect of the chirp parameter on the correponding photoacoutic ignal i invetigated, and guideline for chooing the chirp parameter for aborber profilometric detection are given baed on the developed theory and imulation. Simulated reult are alo compared to available experimental reult and how a good agreement. Key word: Photoacoutic, Pule compreion, Autocorrelation, photoacoutic radar. ii

3 Acknowledgement My deep appreciation i preented to Dr. Natalie Baddour, who i my upervior, for her patient help and profeional advice during my tudy, a well a her financial upport which made my life in Canada eaier. I would alo like to appreciate Dr. Liang Ming, who firt gave me the opportunity to tudy at the Univerity of Ottawa. At lat, my grateful thank to my parent, parent in law, and my uncle and aunt for their endle help and upport. iii

4 Table of Content Abtract... ii Acknowledgement... iii Table of Content... iv Lit of Figure... vii Nomenclature... ix 1 Introduction Background: Introduction to Photoacoutic Imaging Motivation Objective Contribution of the thei Outline of the thei... 5 Literature Review A Brief Hitory of Photoacoutic Breat Cancer Detection Method Puled Laer Excitation Photoacoutic Continuou Wave Laer Excitation Photoacoutic Pule Compreion Photoacoutic Preure Repone Fourier Tranform and Autocorrelation Convention Sytem Layout Equation for Preure Tranfer Function and Impule Repone Square Function Aborber in Space Finite Length Step Exponential Aborber in Space The Chirp Autocorrelation of a Finite-Duration Chirp Real Laer Chirp Waveform iv

5 3.1 Preure repone autocorrelated chirp and quare aborber Antiderivative approach Piece by Piece integration Autocorrelated coine chirp preure repone Preure repone autocorrelated chirp with an exponential decay finite aborber Antiderivative Piece by Piece Integration Reolution Signal to Noie Ratio Bandlimited Aborber Cae 1 chirp bandwidth equal to aborber bandwidth Cae - chirp bandwidth maller than aborber bandwidth Cae 3 - chirp bandwidth bigger than aborber bandwidth The Effect of Chirp Bandwidth Square Function Aborber in Space Output of the Receiver-Filter - Reolution and SNR Effect of chirp parameter effect on aborber profilometric information and SNR Effect of Chirp duration T Effect of Frequency Concluion Signal to Noie Ratio Trend Bandlimited Aborber Square Aborber Concluion Comparion of Theory to Experimental Reult Summary and Concluion Thei overview Contribution of the thei v

6 7.3 Future work Reference Appendix - Shifted Chirp Autocorrelation... 1 vi

7 Lit of Figure Figure 1 Photoacoutic tomography... 1 Figure Aborbing ytem model... 4 Figure 3 Model of aborber and urrounding Figure 4 Block diagram of photoacoutic model Figure 5 Square aborber model... Figure 6 Finite exponential decay aborber model... 5 Figure 7 Form of L(t)... 8 Figure 8 Another poible form of L(t)... 8 Figure 9 Real part of a complex chirp Figure 1 Truncated chirp waveform approx Figure 11 plot of R ( t) and R ( t ) II II Figure 1 Shifted coine chirp Figure 13 Autocorrelation of hifted coine chirp Figure 14 Autocorrelation of unhifted coine chirp Figure 15 Comparion between unhifted chirp autocorrelation and manipulated hifted chirp autocorrelation Figure 16 Photoacoutic Signal from Envelope of Autocorrelated Chirp... 4 Figure 17 Photoacoutic Signal from Envelope of Autocorrelated Chirp... 4 Figure 18 Photoacoutic Signal from Envelope of Autocorrelated Chirp Figure 19 Photoacoutic ignal from the autocorrelated coine chirp Figure Photoacoutic Signal from Autocorrelated co Chirp Figure 1 Ideal preure repone for quare aborber Figure Ideal preure repone for exponential decay aborber Figure 3 Different chirp weeping condition forδ=δ a Figure 4 Different chirp weeping condition forδ<δ a Figure 5 Different chirp weeping conditionδ>δ a vii

8 Figure 6 Aborber frequency content remain contant Figure 7 Frequency content of quare aborber Figure 8 SNR of bandlimited aborber with chirp weeping different frequency 5 range A) 3 1 Hz a, chirp center frequency moving from Hz to 6 1 Hz ; B) 3 1 Hz, 1 1 Hz, chirp 5 5 center frequency moving from 6 1 Hz to 6 1 Hz Hz Hz 5 5 a a ; C ) 31, 5 1, chirp center frequency moving from Hz to 6 1 Hz Figure 9 SNR v. Chirp Bandwidth Figure 3 Square aborber in frequency domain Figure 31 Energy pectrum of quare aborber Figure 3 SNR change with chirp center frequency Figure 33 SNR v. Chirp Bandwidth for quare aborber Figure 34 Experimental Preure Repone (Line 1) from [19]... 9 Figure 35 Simulation reult of preure repone... 9 viii

9 l Thickne of aborber. Nomenclature c Speed of ound. a Light decay factor. a Aborber optical aborption coefficient. Aborber thermo-expanion coefficient. p Initial preure of acoutic wave. T Input waveform duration. Chirp bandwidth. f Chirp center frequency. It Input waveform in time domain. I Input waveform in frequency domain. rt Receiver-filter impule repone in time domain. R Receiver-filter tranfer function in frequency domain. g z,t Aborber impule repone in time domain. G z, Aborber tranfer function in frequency domain. R t Autocorrelation of a function f t ff S ff Energy pectral denity of a ignal. p( z, t ) Preure repone. nt Noie. A z Aborber hape in pace. Aˆ Spatial Fourier tranform of A z z Qt Integral of It, t Q t I d. ix

10 RII t Envelope of the autocorrelation of a finite-duration chirp. R t Approximation of approx II RII _ co R t. t Autocorrelated coine chirp waveform. approx R t Approximation of R t II _ co II II _ co x

11 1 Introduction 1.1 Background: Introduction to Photoacoutic Imaging Photoacoutic phenomenon (alo called optoacoutic phenomenon) wa dicovered more than a century ago [1]. However, the application of thi phenomenon to biomedical imaging tarted only around everal decade ago [], [3]. Preently, the intended purpoe of uing thi imaging method in biomedical engineering i mainly for early cancer detection. X-ray radiography (mammography), magnetic reonance imaging (MRI), and ultraound are the three prevailing imaging technique that are currently available for breat cancer detection. All thee technique have their own advantage of detecting cancer, but they alo have limitation. The main advantage of photoacoutic imaging (alo called photoacoutic tomography, PAT) i that it combine the advantage of both optical and acoutical method: enitive optical aborption contrat and low acoutic cattering in oft tiue. Uing afe illumination ource, the photoacoutic effect can be applied to biological tiue. In addition, optical aborption i highly related to molecular contitution and formation. Thu, photoacoutic ignal contain functional and molecular information [4]. The governing phyical principle behind PAT are traight forward. Figure 1 Photoacoutic tomography 1

12 A hown in Figure 1, the aborbing material i irradiated by a modulated electromagnetic wave. Some of the energy of the incident wave will be aborbed by the aborber which then caue a heating effect. In turn, thi heating will caue a thermoelatic expanion which lead to the generation of an ultraonic wave. The ultraonic wave i then detected by a tranducer and analyzed by different ignal proceing method. Since the ultraonic wave i generated becaue of the aborption of the incident electromagnetic wave, the PAT methodology i actually detecting difference between tiue aborption coefficient. The time delayed profile of the ultraonic wave detected by the tranducer thu carrie tiue aborption coefficient profilometric information. A there i a large difference in optical aborption between blood and urrounding tiue, the ultraound wave induced by the electromagnetic irradiation carrie information about the optical aborption property of the tiue and can be ued for imaging of the microvacular ytem or tiue characterization. There are preently two different approache of excitation, or in other word, two different light ource that can be ent to the target tiue. The mot common excitation ource ued in pat decade ha been a puled laer ource, which utilize an extremely hort duration and high intenity laer pule a the excitation ource. The other ource that ha alo recently attracted reearcher attention i a longer duration continuou wave laer beam that ha a lower intenity. 1. Motivation Much reearch ha been done to advance the cience of photoacoutic imaging. However, mot of the work to date ha focued on puled laer excitation. Thi approach ue powerful nanoecond electromagnetic wave pule to generate the acoutic tranient, a een in the work of Kruger [5], [6] and Wang [7] [9]. Thi modality allow one to find the ditribution of heat ource directly from the hape of the photoacoutic ignal [1]. However, the puled laer modality ha diadvantage

13 mainly due to the hard-to-control depth localization of the aborber, a well a the fact that the incident energy level i limited by afety tandard for in-vivo tiue imaging [11]. Furthermore, the laer ource in thi modality i expenive and larger in ize compared to a continuou wave diode laer ource. An alternative excitation ource mut be developed in order to avoid the diadvantage of puled laer photoacoutic. The alternative continuou wave excitation modality which ha recently attracted reearcher attention ue a linear frequency modulated inuoidal laer waveform rather than a imple hort pule. The intenity of the incident laer beam i modulated into a linearly modulated inuoidal wave form and pule compreion technique from radar are implemented. Thi kind of incident waveform i uually referred to a a Chirp in traditional radar technology. Mandeli and hi team originally developed thi modality [11], [13] [18], which they refer to a photoacoutic radar. The idea i that pule compreion technique are implemented in the imaging modality in order to achieve a better ignal to noie ratio (SNR) and better reolution [18]. However, the effect of the laer chirp parameter on the correponding PA ignal are till not well undertood. The received ignal doe not appear to carry ufficient aborber profilometric information. In order to obtain acceptable aborber profilometric information, a theory of photoacoutic radar need to be developed and the effect of the parameter of the chirp on the reulting photoacoutic ignal need to be invetigated. 1.3 Objective The goal of thi thei i to develop a one-dimenional theory of linear frequency modulated inuoidal laer chirp photoacoutic imaging, and to invetigate the chirp parameter effect on correponding photoacoutic ignal through imulation. Subequently, the goal i to develop guideline for chooing the chirp parameter to obtain a photoacoutic ignal which carrie acceptable aborber profilometric 3

14 information. The effect of the chirp parameter on the photoacoutic ignal to noie ratio i alo invetigated. In thi thei, the photoacoutic radar theory will be developed in one dimenion. The model repreenting the aborbing and urrounding tiue will be conidered a a three layer model a hown in Figure. Figure Aborbing ytem model Two different aborber profile will be conidered. The firt one i where the aborber i modelled a a quare function in pace, which implie that the intenity of the incident laer light doe not decay a it pae through the aborber. The econd cae conidered i where the laer intenity will decay in the aborber and the decay i modelled a being exponential. The mathematical expreion of the photoacoutic ignal will be developed for thee two different cae and then imulated with ymbolic computer algebra oftware (Maple). Subequently, the effect of the different chirp parameter will be invetigated via imulation. 1.4 Contribution of the thei The contribution of thi thei are hown a follow: 1. Two different aborber profile have been propoed and their correponding mathematical model have been developed in cloed form. 4

15 . Mathematical expreion of the pule-compreed photoacoutic ignal repone of two different type of aborber under illumination by a linear frequency modulated laer chirp have been developed. 3. Mathematical expreion of the Signal-to-Noie Ratio (SNR) for a bandlimited aborber and quare aborber have been developed. 4. The effect of the chirp parameter on the correponding photoacoutic ignal are invetigated. 5. Guideline on how to chooe the parameter of the chirp in order to obtain a photoacoutic ignal with acceptable aborber profilometric information are propoed. 6. The effect of the different chirp parameter on the photoacoutic SNR i determined through development of cloed-form expreion and imulation. 7. Simulation are performed uing the chirp parameter ued in available experimental reult [19], and compared with the experimental reult. The comparion how that the guideline developed in thi thei for chooing chirp parameter predict a imilar reult to that obtained in the experiment. 1.5 Outline of the thei The thei i organized a follow. Chapter preent the literature review, which give an overview of hitorical and current reearch. In Chapter 3, cloed-form photoacoutic ignal repone for a quare aborber and exponential decay aborber are developed and illutrated. In Chapter 4, the cloed form expreion of SNR for bandlimited and quare aborber are developed and invetigated. Several imulation 5

16 are performed in Chapter 5 to illutrate the chirp parameter effect on the photoacoutic ignal, and guideline on how to chooe the parameter to receive acceptable aborber profilometric information are preented. In Chapter 6, a comparion between the imulation with parameter ued in experimental reult and the obtained experimental reult carried out. Concluion are drawn in Chapter 6, in which future reearch direction are alo uggeted. 6

17 Literature Review.1 A Brief Hitory of Photoacoutic In 188, Alexander Graham Bell and Sumner Tainter firt reported the photoacoutic phenomenon []. A year later, Bell found that onoroune, under the influence of intermittent light, i a property common to all matter [1]. However, interet in the photoacoutic field did not lat long. Interet wa revived later in 1938 when Veingerov followed by Pfund and Luft ued the photoacoutic effect for nondiperive infrared ga analyi [1]. In 197, Kreuzer firt introduced the laer-excited photoacoutic effect when analyzing ga concentration []. The utilization of a laer a the light ource increaed the enitivity by everal order of magnitude. In 1975, the photoacoutic effect wa firt introduced in the biomedical field by Maugh [3].. Breat Cancer Detection Method X-ray radiography (mammography), magnetic reonance imaging (MRI), and ultraound are the three commonly ued imaging technique for breat cancer detection []. All thee technique have their own advantage but alo limitation. The main limitation of X-ray mammography i it incapability to detect leion in radiologically dene breat. The minimal ize of tumor detectable by mammography i about 5-1 mm if a tumor doe not contain calcification [4]. Another diadvantage of X-ray mammography i it utilize harmful ionizing radiation. MRI i known for it good reolution. However, the enitivity of MRI in many intance i not very good, and thi technology i alo very expenive [5]. Ultraound i another modality that ha been ued for imaging and for detection of 7

18 tumor. The main advantage of ultraound i that it provide a good reolution and larger depth penetration. However, ultraound imaging i incapable of producing high contrat image [6]. The detection of tumor in many cae i difficult becaue of the low contrat in acoutic propertie between the tumor and normal tiue. Photoacoutic imaging combine the advantage of the optical and acoutical imaging method. It ha the advantage of both enitive optical aborption contrat [7] and low acoutic cattering in oft tiue [6]. Uing afe non-ionizing illumination ource, the photoacoutic effect can be applied to biological tiue. In addition, optical aborption i highly related to molecular contitution and formation. Thu, photoacoutic ignal contain functional and molecular information [4]. Photoacoutic imaging ha drawn attention during the pat everal decade. Much progre in many different technique ha been made to apply the photoacoutic phenomenon to imaging application [], [3], [14], [15], [17]. One way to categorize the different technique i by their excitation ource. The two mot general categorie of photoacoutic ue puled laer excitation ource and continuou wave laer excitation ource..3 Puled Laer Excitation Photoacoutic The mot common excitation ource for photoacoutic application ha been a puled electromagnetic wave. Thi method ue powerful nanoecond electromagnetic wave pule to generate the acoutic tranient. Kruger [5], [6], Wang [4], [7], [9], [9] [33] worked in the puled laer excitation modality. The excitation wave mot commonly ued are radio frequency electromagnetic wave pule, microwave pule and infra-red laer pule. The main reaon for uing puled laer light i that it allow one to find the 8

19 ditribution of heat ource directly from the hape of the photoacoutic ignal [1]. However, the puled laer modality ha diadvantage mainly due to the hard-to-control depth localization of the aborber, a well a the limitation on the incident energy level becaue of afety tandard for in-vivo tiue imaging [11]. A pule with extremely high intenity cannot be ued a it may caue damage to the tiue. Much progre ha been made in trying to find other method that do not need the high intenity laer pule but alo do not loe the high ignal amplitude generated by a high intenity pule. Liu et al. [8] found that intead of uing one ingle high intenity hort pule, uing multiple low-energy picoecond pule excitation may olve thi problem. They alo found the ignal to noie ratio increaed when the number of pule increaed. From one ingle high intenity pule to multiple low intenity pule wa an improvement in the approach. Thi uggeted another approach, what if many more pule with lower energy in the ame time interval were added, until they became a continuou wave?.4 Continuou Wave Laer Excitation Photoacoutic The above lead to the alternative excitation modality, continuou wave laer excitation. Much reearch ha been done to apply a continuou wave to photoacoutic tomography [13], [15], [17] [19]. Thi modality commonly ue a continuou laer wave intead of hort duration pule a the incident light ource. Continuou wave photoacoutic wa firt introduced by Mandeli reearch group uing a frequency weep (chirp) laer ource with heterodyne modulation [11], [17]. The reult howed thi method can be more preciely controlled in depth imaging 9

20 comparing to puled laer modality. However, the photoacoutic ignal peak i broadened due to the extended expoure time. Therefore, the major merit of puled laer modality, which allow one to find the ditribution of heat ource directly from the hape of the photoacoutic ignal, i gone. Another ignal proceing method i matched-filtering. The received photoacoutic ignal i cro-correlated with a modulated incident wave form by the matched-filter. The merit of uing a matched-filter are detailed in many radar or ignal proceing book [33], [34]..5 Pule Compreion The main reaon that linear frequency modulated inuoidal chirp are ued with a matched-filter i the pule compreion phenomenon, which i commonly ued in traditional radar technique [4], [33], [35], [36]. The matched filter wa hown to be the complex conjugate of the tranmitted ignal [4], [37]. When a received photoacoutic ignal i cro-correlated with a tranmitted wave form, the energy of the output ignal will be concentrated in a much narrower time interval (the ignal ha been compreed) [35], [36]. Telenkov, Lahkari and Mandeli performed many invetigation on continuou wave photoacoutic radar [1] [16], [19], [38], [39], which they refer to a Frequency Domain Photoacoutic (FD-PA). The firt attempt at uing cro-correlation ignal proceing method in FD-PA wa introduced in 6 [4]. In thi tudy, the algorithm of FD-PA wa introduced and an experimental device wa built. Teting reult for turbid phantom and ex-vivo chicken breat pecimen howed clear peak which carrie the inhomogeneity (aborber) information. Further tudy [41] howed that although acoutic preure wave induced by an 1

21 intenity modulated continuou wave laer have much lower amplitude than thoe generated by high peak-power nanoecond pule, the ignal to noie ratio (SNR) can be increaed dramatically via pecific modulation coding and coherent ignal proceing. A year later, invetigation on SNR, axial reolution and detection depth wa performed [1], [18]. The experimental reult howed FD-PA can provide comparable or better depth enitivity with millimeter-cale axial reolution due to uperior SNR. Thee feature imply that FD-PA i a competitive imaging method comparing to puled photoacoutic technique [1], [18]. Lahkari and Mandeli further reearch on FD-PA give detailed theoretical and experimental comparion between variou key parameter of the puled photoacoutic modality and FD-PA imaging modality. The experimental reult howed much maller SNR difference between puled and FD photoacoutic [15]. The reult alo howed that increaing the chirp frequency bandwidth dratically reduce the SNR [15]. Hence, it wa attempted to find the optimal chirp bandwidth which generate the bet SNR [14]. It wa hown that the optimal bandwidth generating the highet SNR for a low-frequency tranducer i roughly centered at the peak frequency of the tranducer, and for high-frequency tranducer, the optimal bandwidth i centered on the lower frequencie [14]. However, the theory behind the SNR optimization i till not clear. Further tudy on the theory of FD-PA howed a more detailed view of chirp parameter effect on SNR [38]. The reult explained that the SNR of the filtered photoacoutic ignal i proportional to the chirp duration. However, chirp duration i not the only parameter that control SNR. Further tudy need to be done in order to fully undertand the controlling chirp parameter on SNR. Furthermore, the experimental photoacoutic ignal generated by linear frequency modulated chirp after cro-correlation ignal proceing [13], [19], [4], [41] did not give a much aborber profilometric information a obtained with the puled laer photoacoutic technique. Thi lead to the main motivation of thi work, which i to recontruct the approach for continuou wave linear frequency modulated laer 11

22 photoacoutic and to give guideline on how to chooe the parameter of the chirp to obtain ufficient aborber depth profilometric information and optimal SNR. 1

23 3 Photoacoutic Preure Repone A dicued earlier, matched-filtering i implemented in continuou wave photoacoutic tomography. A linear frequency modulated inuoidal waveform commonly ued in radar technique ha a property of pule compreion after matched-filtering. However, the theory behind thi technology in photoacoutic i till not clear. In thi chapter, the 1-D theory for two pecific aborber model and their correponding photoacoutic ignal after receiver-filter are developed. The incident laer waveform i limited to be a linear frequency modulated coine chirp. The aborber model conidered are a quare aborber and an exponential decay aborber which will be explained in detail later. 3.1 Fourier Tranform and Autocorrelation Convention Fourier tranform and autocorrelation have everal different convention which can be found in many ource [33], [4], [43]. In thi thei, the choen convention for the temporal Fourier tranform i the angular frequency, non-unitary convention for forward and invere tranform given by it 1 it f ( f ( t) e dt f t f e d (3.1) where a tilde ( ) over the variable ha been ued to denote a temporal Fourier tranform, and auming uitability of the function f for Fourier tranformation. The patial Fourier tranform from regular pace with patial variable z to patial frequency pace with patial frequency variable z i defined a in (3.1), with z replacing t and z replacing in (3.1). The patial Fourier tranform i denoted with an over hat over the variable, o that ˆ tranform pair with repect to the patial variable. f z f z denote a Fourier 13

24 The energy of a ignal i defined a [4] 1 E f f ( t) dt f ( ) d (3.) where the lat equality in equation (3.) follow from Pareval theorem [33]. Since the integral on the right-hand ide i the energy of the ignal, the integrand f ( ) can be interpreted a an energy denity function [44] decribing the energy per unit frequency contained in the ignal at frequency. In light of thi, the energy pectral denity of a ignal f t i defined a it S f ( ) f ( ) f ( ) f ( t) e dt (3.3) ff where the overbar indicate a complex conjugate. The energy pectral denity i mot uitable for tranient that i, pule-like ignal having a finite total energy. Now conider the invere Fourier tranform of the energy pectral denity: it i i ' it S ff e d f ( ) e d f ( ') e d ' e d i ' it i f ( ) f ( ') e e e d d d ' t ' (3.4) Changing the order of integration a hown in (3.4) and uing the definition of the invere Fourier tranform of the delta function give 1 it S e d f ( ) f ( ') t ' d d ' ff f ( t ') f ( ') d ' The lat integral in equation (3.5) i exactly the auto-correlation of the function traditionally defined a ( ') ( ') ' (3.5) f t, Rff t f t f d (3.6) Hence, it follow that the autocorrelation of a function and it energy pectral denity are a Fourier tranform pair R t S ff. ff 14

25 The autocorrelation of a periodic function f t i defined a an average power over one waveform a 1 t T Rff t f f t d T, (3.7) where t i any arbitrary value, and T i the period of the function. t 3. Sytem Layout For a one-dimenional ytem, the aborbing material urrounded with the cattering medium i conidered to have the imple layout hown in Figure 3. Figure 3 Model of aborber and urrounding In Figure 3, the origin of the z-axi i placed at the center of aborber. The aborber i conidered a an infinite heet with a thicknel. The urrounding cattering medium on negative ide of z axi i conidered a an infinite layer from to l. The urrounding cattering medium on poitive ide of z axi i conidered a an infinite 15

26 layer from l to.the peed of ound c i conidered to be the ame in all three layer. C p i the pecific heat, a i the optical aborption coefficient of the chromophore aborber that ha been heated by an optical pule with fluence F. The incident laer waveform It i tranmitted from the negative ide of z axi, outide of the aborber. The aborber aborb energy from the incident laer and generate a ound wave (or in other word, preure repone) denoted by p z, t. The preure repone i then detected by a tranducer and be ent to a receiver-filter. The output of the receiver-filter yz, t i the final photoacoutic ignal. 3.3 Equation for Preure Diebold [45] give a concie explanation of the governing equation for the preure that reult from launching a photoacoutic wave. When light pule are delivered to biological tiue, the tiue aborb the light and convert it to heat, generating an initial preure rie due to the thermoelatic expanion. The initial preure give rie to the acoutic wave, which i then detected by a tranducer. The governing equation i given by:, p r t c t C p t 1 H ( r, t) (3.8) where i the thermal expanion coefficient, c i the peed of ound, C p i the pecific heat, H( r, t ) i the energy per unit volume and time depoited by the optical radiation beam, and p( r, t ) i the preure of the acoutic wave, a function of pace and time. In thi work, it i aumed that the heating function H( r, t ), reult from a 16

27 chromophore aborber that i optically thin with an optical aborption coefficient a that i heated by an optical pule with a fluence of F. We aume that H( r, t ) i a eparable function of pace and time o that the preceding aumption imply that H( r, t) a FAr I t. Ar i a function of pace that decribe the geometry of the aborber and It i a function that decribe the time dependence of the incident optical wave, a hown chematically in Figure 3. Taking the temporal Fourier tranform of equation (3.8) and auming one dimenional Carteian coordinate o that aborber geometry i a function of the depth variable z only, r z, then d ikcaf p z, k p z, I A z (3.9) dz C p where k i the wavenumber. We now take a patial Fourier tranform / c (denoted with an overhat) in the patial variable z, and then equation (3.9) become ˆ ˆ ikc F p k p I Aˆ a,, z z z z C p a I pˆ, z I pˆ, z ikc F C ikp c p Aˆ z z k Aˆ z z k (3.1) where p i the initial preure, 17 a i the aborption coefficient of tiue, and F i the local light fluence. Here we can introduce the Grüneien parameter, which i a contitutive parameter ued in photoacoutic. The Grüneien parameter Γ of tiue, relate the initial preure p to the light aborption by the following expreion [9]: p F (3.11) a In general, it i neceary to directly meaure the Grüneien parameter of tiue. However, it can alo be etimated in term of the iobaric volume expanion

28 coefficient β, the pecific heat C p, and the peed of ound c in tiue uing c (3.1) C The Grüneien parameter varie between different type of tiue. For example, the Grüneien parameter of fat tiue i etimated to be between.7 and.9. A more accurate value hould be obtained by direct meaurement. p 3.4 Tranfer Function and Impule Repone When viewing the ytem hown in Figure 3 a an input/output problem, a hown in Figure 4, the tranfer function concept i very ueful. Figure 4 Block diagram of photoacoutic model The incident laer waveform It i aborbed by the aborber which ha the impule repone denoted by G z, t. The energy in the incident wave i then converted to acoutic energy and generate an acoutic preure wave p z, t. The acoutic preure wave received by a tranducer i then tranmitted to a ha an impule repone receiver-filter which rt. The ignal received at the receiver-filter alo contain noie nt. In thi thei, we aume the noie to be a zero-mean additive Gauian white noie [4] which ha a double-ided power pectral denity of Snn. Furthermore, the noie i aumed to be tatitically independent of both the 18

29 tranmitted input waveform It and the aborber impule repone, G z t. The output of receiver-filter yz, t contain the photoacoutic ignal y, y z t. noie ignal, n z t and the Baed on thi concept, we can write the equation (3.1) a ˆ p ik ˆ p I A G k I, ˆ, z z z c z k (3.13) where ˆ p ik ˆ c F ik G k A Aˆ a, z z z c z k C p z k (3.14) i the tranfer function. The notation G k ˆ z, with the overhat and tilde, i ued a a reminder that thi function i operating in both the patial and temporal Fourier domain. Furthermore, it i clear from equation (3.14) that G k ˆ i only a z function of and k, along with the aumed contant parameter c, C,, F. z p a Taking the invere patial Fourier tranform of the tranfer function G k ˆ, 1 ˆ pik 1 z G z, k G, k e d Aˆ e d i z izz z z z z c z k (3.15) Theorem 5 from [46] tate that the following reult hold true: or i ikz 1 ( I e d ikz k e k e z 4k ikz k e k e z iz 4k (3.16) k i ikz z I i ikz k e z k i ikz k e z k (3.17) or 19

30 1 k e ik I 1 ikz k e ik ikz z z (3.18) Here, i an analytic function defined on the poitive real line that remain bounded a x goe to infinity (ha no pole). The different olution in (3.16) to (3.18) repreent inward, outward propagating wave or tanding wave, therefore the phyic of the problem govern the choice between equation (3.16) to (3.18). Given the definition of the Fourier tranform that i being currently ued, the preented reult in (3.18) atifie the Sommerfeld radiation condition, enuring an outwardly propagating wave. Given thi reult, and auming ˆ that A ha no pole and remain bounded, it follow z pik 1 ˆ iz z G zk Aze dz c k z 1 ˆ ikz Ak e z pik ik c 1 ˆ ikz Ak e z ik ˆ ikz p A k e z ˆ ikz c Ak e z (3.19) where k i the wave number. Temporal invere Fourier tranformation of c equation (3.19) give the ytem impule repone in the time and pace domain a

31 1 it G z, t G zk e d p 4 c p c z ˆ i c it A e e d z c z i ˆ A e e d z c c i t A z ct z A z ct z (3.) Equation (3.) how that the impule repone of the aborber ha exactly the ame patial hape of the inhomogeneity, although it i a function in time wherea the hape of the aborber i a function of pace. The peed of ound i the converting factor that relate ditance in pace to duration in time. Now, the field of puled photoacoutic involve the meaurement and analyi of photoacoutic repone to input pule that are ufficiently hort o a to be modelled a Dirac-delta function. Hence, puled photoacoutic can be conidered the branch of photoacoutic where the primary goal i the direct meaurement of the ytem impule repone and equation (3.) make it mathematically clear that in doing o the aborber profile i directly obtained. Furthermore, the time for acoutic wave to travel can be ued to etimate the location of the aborber. 3.5 Square Function Aborber in Space Now conider the pecific aborber which i conidered a quare function in pace. Light propagation in thi aborber i aumed to have no decay. That i, aumed to be given by A z i where l l Az A u z u z (3.1) uz i a Heaviide unit tep function, and A i an arbitrary contant caling factor. A plot of (3.1) with l.5 m, A 1 i hown in Figure 5. The 1

32 thickne l of the aborber i alway taken a.5m in other experimental reearche [18], [19]. The patial Fourier tranform of Figure 5 Square aborber model A z i given by ˆ in 1 ilz ilz lz A z e e z iz (3.) Subtituting equation (3.) into equation (3.13) for the preure in patial and temporal Fourier domain give ˆ p ik p z I A c k, ˆ z ilz ilz p ik 1 I e e c z k iz z (3.3) There i a pole at zero here, o Theorem 5 in [46] cannot be ued immediately. However, partial fraction can be ued to eparate the pole from the ret of the expreion. Therefore, uing partial fraction, equation (3.3) can be written a ilz ilz ˆ p 1 z pz I e e ck z k z (3.4) Now taking the invere patial Fourier tranform give ilz ilz ilz ilz p z 1 izz izz p z I e e e d z e e e dz ck z k z p i / / 1 z zl iz zl iz zl/ iz zl/ z I e e d e e dz z ck z k z (3.5) Uing theorem 5, equation (3.18), on the firt term and tandard integral table on the econd term give

33 ik ( zl/) ke z l/ p ik p z I ik ( zl/) ck ke zl/ ik p ke ik ik ( zl/) I ik ( zl/) ck ke ik I iu z iu z zl/ zl/ p l l ck (3.6) Cleaning up equation (3.6) give Putting it all together give ik ( zl/) p e z l / p z I ik ( zl/) cik e z l / ik ( zl/) p / e z l I ik ( zl/) cik e z l / p l l cik I u z u z ik ( zl/) ik ( zl/) e e z l / p p z I e e l z l cik ik ( zl/) ik ( zl/) e e z 1/ ik ( zl/) ik ( zl/) / / Invere (temporal) Fourier tranforming equation (3.8) via (3.7) (3.8) 1 it p z, t pz, e d (3.9) Then, z l z l I i t i t c c c c e e d z l / i p z t e e e d l z l 4 i z l z l i t i t p I it c c c c, / / z l z l I i t i t c c c c e e d z 1/ i (3.3) 3

34 Now it i a known property of Fourier tranform that dividing by i in the frequency domain i the ame a integrating in time in the time domain o that where we ue (3.3) can be written a 1 I i t it e d I d Q t (3.31) Qt to denote the integral of z l / z l / It. Uing equation (3.31), equation Qt Qt z l / c c p z l / z l / p z t Qt Qt Qt l z l c c z l / z l / Qt Qt z 1/ c c, / / (3.3) Equation (3.3) i derived by invere temporal Fourier tranform p z, hown by equation (3.8). However, the problem can eaily be olved in the time domain. The preure repone p z, t will be the convolution of the input waveform the aborber impule repone G z, t, which will be given by,, It and p z t I G z t d (3.33) For a quare aborber hown by equation (3.1), equation (3.33) become p z, t p z l t c c z l t c c z l t c c z l t c c I d z I d z (3.34) A verification of our prior reult, if the input time function i a delta function, t, then it integral i the Heaviide tep function Qt u t I t, and for 4

35 z z l, equation (3.3) indicate a tep function that turn on at time / l/ z l and turn off at / c, or in other word a tep that lat for an amount of time lc econd. Thi i in keeping with our prior reult that aid that in repone to a very hort incident pule in time, the preure function ha the ame hape a the patial hape of the inhomogeneity. The reult here i more general a it i valid for any temporal function of input It and quare wave aborber. Furthermore, the olution in (3.3) ha three region of validity. The middle region l / z l / i actually inide the inhomogeneity and in practice a tranducer would never be placed there. c 3.6 Finite Length Step Exponential Aborber in Space We now conider the repone to the aborber patial function az az A z e u z e u z l (3.35) where a i poitive contant and uz i the Heaviide tep function. Thi aborber denote a light decay when propagating inide the aborber. A plot of (3.35) with a1, l.5 i hown in Figure 6. Figure 6 Finite exponential decay aborber model The patial Fourier tranform of A z i given by 5

36 Aˆ z l ai z 1 e a i a i z z (3.36) The preure repone in both patial and temporal Fourier domain i given by ˆ p ik p z I A c k ˆ z z l ai z p ik 1 p ik e I I c z k a iz c z k a iz Thi can be written via partial fraction a pˆ I ia ia e l ai z l ai z p z z 1 e z c a k ia z k k z ia z z (3.37) (3.38) We conider the patial invere Fourier tranform term by term. The firt term in equation (3.38) i p 1 c a k Second term in equation (3.38) p ikz p I a ik e z z ikz a ik e z la ik zl p I a ik e e z l z la ik zl c a k a ik e e z l Lat two term in equation (3.38) give p ki az az p3,4 z ie u z ie u z l c a k p I ik az az c a k e u z e u z l (3.39) (3.4) (3.41) Putting the term all together p I ikz la ikzl ikz la a ik e a ik e e z az ikz la ik zl p z ike a ik e a ik e e z l c a k ik z l a ik e a ik e e z l (3.4) We note that the particular olution, a term proportional to az e only appear in the area z l (which i inide the aborber itelf). Outide of thi area, the repone 6

37 conit of term proportional to e e e multiplication by i in the frequency domain). The term ikz la ik z l plu it time derivative (a ikz la ik zl e e e i two propagating wave, with the econd being attenuated by al e compared to the firt. If we et a in equation (3.4), it yield the ame reult a equation (3.8), which i for quare function aborber. Rearrange equation (3.4) give pi ikz la ik zl ikz la ikzl az ikz la ik zl ikz la ik zl ikz la ikzl ikz la ikzl ca e e e i e e e z p z i e c a e e e i e e e z l ica ca e e e i e e e z l (3.43) Although equation (3.43) look complicated, we can make ome interpretation. Firt conider the invere temporal Fourier tranform of the leading term and we define thi function to be above in (3.18) a Ltand can be implified uing Theorem 5 from [47], given i cat I caie t p I c ai it p L t : e d 4 ic i cat a I caie t cai cat p I cai e t cat 4ca I cai e t (3.44) Equation (3.44) are decaying exponential in time with amplitude controlled by I c ai. The general hape of (3.44) will be a hown in Figure 7 or Figure 8. 7

38 Figure 7 Form of L(t) Figure 8 The form of the time derivative of Another poible form of L(t) Lt i given by cat p I cai e t L' t : cat 4 I cai e t (3.45) Clearly, from Figure 7, the derivative i not defined at, although the function itelf i continuou at time zero. Having found cloed-form expreion for Lt and it firt time derivative, equation (3.43) can now be interpreted in the time domain a z la ( z l) z la ( z l) calt cae Lt L ' t e L ' t z c c c c az z la ( z l) z la ( z l) pz t e L ' t calt cae Lt L ' t e L ' t z l c c c c z la ( z l) z la ( z l) calt cae Lt L ' t e L ' t z l c c c c (3.46) Similar to the quare aborber, the preure repone can be olved in time domain only uing equation (3.33). The only difference i the aborber impule repone G z, t. A for the exponential decay aborber hown by equation (3.35). The expreion for preure repone become 8

39 p z, t p z l t c c z l t c c z l t c c z l t c c ac t I e d z ac t I e d z (3.46) 3.7 The Chirp In prior ection, the expreion for the preure repone p z, t for two different aborber with any incident wave form It have been derived. The expreion of the ignal paed through receiver-filter y, z t with an incident liner frequency modulated inuoidal chirp will be demontrated in the following ection. The chirp i a commonly ued waveform in many application. For convenience in calculation, it i common to ue the complex form of the chirp, but the actual ignal i the real part of the complex chirp I t i Kt e (3.47) where K i the chirp parameter, a poitive contant. It ha the dimenion of (Hz) o that Kt i dimenionle. The real verion of the ignal i given a I ( ) co re t Kt (3.48) The plot of equation (3.48) i hown in Figure 9. 9

40 Figure 9 Real part of a complex chirp. The chirp can be regarded a a ignal with an intantaneou frequency defined a d Kt It ha been hown that the temporal Fourier tranform of dt I Kt rad/ Kt Hz (3.49) i 4 K It i given by [44] i 1 4 e e (3.5) K Further, it ha been hown that I () t ha the Fourier tranform given by re I re 1 co 4 K 4 K (3.51) In practice, the chirp i truncated to be a finite duration pule with a center frequency, f, which i often non-zero. The truncated complex chirp i given by Kt t i ft i t t t IT t rect e rect e T T (3.5) The ubcript T indicate the duration of the chirp and may be omitted if the meaning i clear. In equation (3.5), f i the tandard converion from Hz to radian per econd and K. The function rect z i the rectangular function, introduced by Woodward [48], and defined a rect z 1 1 z 1 z (3.53) 3

41 Figure 1 how how the truncated chirp look like in time domain. Figure 1 Truncated chirp waveform A before, the complex chirp can be regarded a a ignal with intantaneou frequency defined a t dt t rad/ f Kt Hz (3.54) dt Thu, during the T econd interval of the pule, the intantaneou frequency change linearly from f KT / Hz to f KT / Hz. The frequency weep, KT i then the difference in thee two value. The product of duration and frequency weep T i then given by KT and i called the time-bandwidth product of the chirp, D T KT. The time-bandwidth product (dimenionle) i alo referred to a the Diperion Factor in older paper, for example [49]. The function I t a defined in equation (3.5) i an even function, therefore it T Fourier tranform i alo even. In particular, I will be real and even o that T the autocorrelation of a complex chirp IT t i alo real and even. 3.8 Autocorrelation of a Finite-Duration Chirp The auto-correlation of a finite duration chirp may be computed from 31

42 ( ) ( ) R t I t I d (3.55) II T T which ha been hown to be given by [49] 1 ift R in II t e t Kt Kt (3.56) The envelope of the auto-correlation of a finite-duration chirp i therefore given by in t Kt T T t T R t t (3.57) II otherwie Thi can be written in an alternate form a t in t 1 t T T 1 T t T RII t T t t 1 T otherwie (3.58) For mall value of time (epecially for large time-bandwidth product), thi i approximately given by the inc function in t approx RII t RII t T t T t (3.59) Therefore, we can define the approximation of approx R t a R II II t, given by the right hand ide of equation (3.59). The approx R t and II R t are envelope of the autocorrelated chirp. A full II expreion of autocorrelated real coine chirp i given by The approximation of in t Kt T co ft T t T RII _ co t t otherwie RII _ co t i then given by (3.6) 3

43 The plot of approx II _ co in t T co f t T t T R t t approx R t and R II (3.61) II otherwie t are hown in Figure 11. The chirp parameter ued to draw the plot are a follow: T , 3 1 Hz, f. approx Figure 11 plot of R ( t) and R ( t ) II II For the plot in Figure 11, the firt zero croing of equation (3.59) i at t 1/ and the duration of the main lobe of the inc function i / / KT, which will be referred to a the effective pule duration. The important point i that the energy of the autocorrelation of the chirp i concentrated in a much narrower duration of time than the original chirp IT t, which had a duration of T. The pule compreion ratio i then defined a the ratio of the duration of the original pule IT t to the duration of the autocorrelation of the pule, which i given by 33

44 t duration of IT t T T CR duration of R / II (3.6) Thu, we ee that the pule compreion ratio i directly proportional to the time-bandwidth product, T. From Figure 11, ince the firt zero croing of approx II R t i at t 1/.333, 66 point have been ampled from.5 μ to.33 μ for calculation convenience, and found the maximum error i 3.38% uing the equation Error max approx RII ( ti ) RII ( ti ) max, i (3.63) R ( t ) II i A hown in Figure 11 and the correponding maximum error, the difference between approx R t and R II II t could be ignored. 3.9 Real Laer Chirp Waveform In previou ection, the chirp a hown in Figure 9 wa being ued. The magnitude of thi chirp contain negative value, which doe not make phyical ene if the quantity being repreented i an intenity value a in the cae of laer intenity. The real laer chirp mut ocillate above zero intenity. The hifted chirp (hifted up) ha the wave form co I t I Kt f t (3.64) where I i an arbitrary contant hift factor to enure that the function alway remain poitive. A plot of equation (3.64) with Figure 1. f 1 Hz, I 1 i hown in 34

45 Figure 1 Shifted coine chirp The autocorrelation equation for the hifted chirp become extremely long and take everal page, Therefore, it i hown in the appendix A. A plot of the autocorrelation with parameter f T m 6 Hz, 1,.8 1 i hown in Figure 13. Figure 13 Autocorrelation of hifted coine chirp The plot of the unhifted coine chirp hown in equation (3.61) with the ame parameter except I i hown in Figure 14. Figure 14 Autocorrelation of unhifted coine chirp 35

46 Comparion between Figure 13 and Figure 14 how that by hifting the chirp up, the autocorrelation of the chirp i alo hifted by a factor. The hape doe not change much, epecially for the main lobe. Manipulation on hifted chirp autocorrelation have been made to invetigate the difference. Figure 15 Comparion between unhifted chirp autocorrelation and manipulated hifted chirp autocorrelation The manipulated autocorrelation of hifted chirp and the unhifted chirp autocorrelation are hown in Figure 15. The manipulation i done by multiplying the hifted chirp autocorrelation by a factor of and ubtracting.6. A hown in Figure 15, the difference i mall. In order to quantify the error in the main lobe, 66 point in the time interval.5 μ to.33 μ with contant pacing have been ampled for unhifted unhifted chirp autocorrelation R t and manipulated hifted chirp II autocorrelation hifted RII t. The maximum error in thi time interval wa calculated a unhifted hifted RII ti RII ti Errormax max i 1.. N (3.65) unhifted R II t where N i the number of point, in thi cae 66. hifted R t and R unhifted II II i The maximum error between t in Figure 15 i.97%. With thi reult, it i jutifiable to ue the impler equation, I t cokt ft in imulation and obtain reult that are imilar to thoe obtained with a hifted chirp, without unneceary mathematical complication. 36

47 3.1 Preure repone autocorrelated chirp and quare aborber In order to invetigate the chirp parameter effect on correponding photoacoutic ignal after the receiver-filter y z, t, the mathematical expreion of y, a quare aborber are derived in thi ection. z t for Antiderivative approach Previouly, in equation (3.8), the temporal Fourier tranform of the preure repone for a quare function aborber wa hown to be given by ik ( zl/) ik ( zl/) e e z l / p p z I e e l z l i ik ( zl/) ik ( zl/) e e z l / ik ( zl/) ik ( zl/) / / (3.66) where k c i the wavenumber. Equation (3.66) then can be written in input/output form a,, p z G z I (3.67) where Gz, i the tranfer function of the ytem at ome meaurement point z, and i given by G z l l i z l i z l c c p l l i i z l i z l c c e e z l / (reflection) e e z l / (tranmiion) (3.68) It i noted that Gz, poee lightly different form for meaurement made in reflection (tranducer on irradiated ide of the ample) or tranmiion (tranducer on the other ide of the ample). Temporal invere Fourier tranforming equation (3.68) give the ytem impule repone a 37

48 z l z l u t u t z l / (reflection) p c c c c G zt z l z l u t u t z l / (tranmiion) c c c c In equation (3.69), (3.69) ut i the Heaviide tep function o that the hape of the impule repone i a quare rectangle function in time at any fixed obervation point z. It wa further demontrated that invere temporal Fourier tranformation of (3.66) lead to z l z l Qt Qt z l / c c c c p z l z l p z, t Qt Qt Qt l / z l / c c c c z l z l Qt Qt z l / c c c c where the function (3.7) Qt i the integral of the time function given by t 1 I it Qt e d I d i (3.71) In order to obtain the photoacoutic ignal after the receiver-filter where the received filter i a matched filter to the input chip that i the receiver filter i the complex conjugate of the input chirp, we need to replace approx RII I in equation (3.7) with (the autocorrelation of the input ignal) in equation (3.71) o that approx t approx ( ) Via Maple, equation (3.7) can be found a Q t R d (3.7) II 38