Theory of Frequency Domain Photoacoustic Tomography

Theory and analysis of frequency-domain photoacoustictomography

Natalie Baddoura�

Department of Mechanical Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa Ontario,Canada K1N 6N5

�Received 8 August 2007; revised 2 January 2008; accepted 24 February 2008�

A new frequency-domain approach to photoacoustic tomography has recently been proposed,promising to overcome some of the shortcomings associated with the pulsed photoacousticapproach. This approach offers many of the benefits of pulsed photoacoustics but requires a differentset of equations for modeling of the forward and inverse problems due to the longer time scalesinvolved in the optical input signal. The theory of photoacoustic tomography with an optical inputthat is not necessarily a short pulse is considered in this paper. The full optical, thermal, and acousticgoverning equations are derived. A transfer function approach is taken for the solution and analysisof this problem. The results and implications are compared with those of pulsed photoacoustics andtraditional ultrasonic diffraction tomography. A Fourier diffraction theorem is also presented, whichcould be used as a basis for the development of tomographic imaging algorithms.© 2008 Acoustical Society of America. �DOI: 10.1121/1.2897132�

PACS number�s�: 43.35.Ud, 43.20.Bi, 43.60.Pt �LLT� Pages: 2577–2590

I. INTRODUCTION

In recent years, noninvasive laser-based diagnostic andimaging techniques have been proposed and developed. Pho-toacoustic signal generation is a new technique, which hasdemonstrated great potential for visualization of the internalstructures and function of soft tissue. It has particularlyshown great potential for small animal imaging.1,2 With thistechnique, a short-pulsed laser source is used to irradiate thesample. The energy absorbed produces a small temperaturerise, which induces a pressure inside the sample throughthermal expansion. This pressure acts as an acoustic sourceand generates further acoustic waves, which can be detectedby ultrasound transducers positioned outside the sample. Asthere is a large difference in optical absorption betweenblood and surrounding tissue, the ultrasound wave inducedby the laser irradiation carries information about the opticalabsorption property of the tissue. This approach is thus suit-able for the imaging of the microvascular system or for tis-sue characterization.

This imaging technique has contrast similar to that ofpure optical imaging and spatial resolution similar to that ofpure ultrasonic imaging. It therefore combines the advan-tages of two imaging modalities in a single modality. Theissue of the strong scattering of light in media such as bio-logical tissue is overcome and the ability of acoustic wavesto travel long distances without significant distortion or at-tenuation is also exploited. Photoacoustic detection hasshown concrete promise of imaging in turbid media at depthsgreater than the full thickness of skin.3,4

The potential for high contrast is the most potent advan-tage of the technique. The large variation in the optical ab-sorption and scattering properties of different tissue constitu-ents can be exploited. Sources of naturally occurring

a�
Electronic mail: [email protected]
J. Acoust. Soc. Am. 123 �5�, May 2008 0001-4966/2008/123�5

absorption contrast include chromophores—features that se-lectively absorb light at certain wavelengths—such as bloodvessels, tumors, hemoglobin �and its various oxygenatedstates�, melanin, beta-carotene, and lipids. Of all of these,hemoglobin is perhaps the most significant. It offers strongoptical contrast at optical wavelengths giving the techniquethe potential to image blood vessels for directly assessingarterial disease or mapping the vasculature. It can also beexploited to detect abnormal tissue morphologies such ascancerous lesions and vascular lesions that are accompaniedby changes in the surrounding vasculature and tissue oxy-genation status.5,6 For three-dimensional �3D� imaging, theachievable resolution depends not only on the experimentalapproach but also on the choice of image reconstructionalgorithm.7–14 Careful design and verification of tomographicalgorithms are required to ensure stable, rapid, and artifact-free imaging. A good review of photoacoustic imaging isgiven in Ref. 12.

The general field of photoacoustic tomography has sofar been based entirely on pulsed laser excitation and time-of-flight measurements of acoustic transients to determinethe position and optical properties of subsurfacechromophores.4,7,8,15–17 The field has recently experiencedrapid development due to promising results for subsurfacemeasurements and imaging of turbid media.2,18

A novel Fourier-domain photothermoacoustic �FD-PTA�imaging methodology has recently emerged.5,19 For FD-PTA,the acoustic wave is generated by periodic modulation of alaser. In general, frequency-domain approaches in place oftime-domain approaches have been shown to yield highersignal-to-noise ratios in other imaging modalities.20 In thispaper, the term photothermoacoustic imaging will be used todenote photoacoustic imaging where a short pulse is not nec-essarily used. This may imply a frequency-domain approachwith narrow-band lock-in amplifiers or a time-domain ap-
proach where the pulse used is not necessarily short enough
© 2008 Acoustical Society of America 2577�/2577/14/$23.00

to fulfil the assumptions upon which photoacoustic imagingis based. Thus, the data can be acquired either as a steady-state complex intensity with measurable amplitude and phaseor as time-varying intensities as the response to an inputpulse. With the restriction of a short pulse removed, there isa greater degree of flexibility in the choice of input opticalexcitation. With this higher degree of flexibility and controlavailable over the distribution in space and time of the illu-minating sources, the potential to design an optimal acquisi-tion system in a realistic fashion presents itself. Such a de-sign must be based on a proper theoretical analysis. Thispaper aims to consider the theoretical development of thenecessary background upon which to base the tomographyalgorithms and/or experiment design.

The layout of the paper is as follows: in Sec. II, thegoverning equations are derived. Sections III–V present signconventions, geometry, and some background mathematics.Sections VI and VII present the solution to the presentedproblem, based on the use of the first Born approximation.Section VIII includes necessary mathematical simplifica-tions, whereas Sec. IX considers the Green function theoremfor a general illumination function. Section X presents ananalysis of the implications of the mathematical results. Sec-tions XI–XIII present the Fourier diffraction theorem for thisproblem, both for a generalized illumination function and fora planar illumination function and consider the implicationsof this theorem on the achievable resolution of this approachto imaging. Section XIV concludes the paper.

II. GOVERNING EQUATIONS

Governing equations for the coupled photothermoacous-tic problem are presented here and are culled from,5,21,22

where the photonic, thermal, or acoustic phenemena areseparately discussed. The equations presented here are as-sembled to represent a complete model consisting of the fullcoupled photonic, thermal, and acoustic phenomena and arepresented in infinite space in order to allow a diffractiontomographic approach. The first equation is the equation forthe diffuse photon density waves �DPDW� that describes thepropagation of photons in turbid media. The second equationuses the DPDW wave as a source term to the heat equationand describes the thermal expansion of the turbid media.Finally, the thermal term is used as an inhomogeneous sourceterm to the acoustic wave equation where the variable ofinterest is the acoustic displacement potential.

A. Light propagation in scattering media

The gold standard for describing light propagation inturbid media is the radiative transport equation. The radiativetransport equation governs light propagation in random me-dia such as clouds, fog, and biological tissues. It takes intoaccount scattering and absorption due to inhomogeneities inthe propagating medium. Understanding the optical responsedue to inhomogeneities inside a uniform absorbing and scat-tering medium is important for imaging purposes. A com-monly used approximation to the transport theory is the dif-fusion approximation, which describes time and frequency
dependent photon diffusion. This approximation is particu-
2578 J. Acoust. Soc. Am., Vol. 123, No. 5, May 2008 Nata

larly appealing due to its inherent mathematical tractability.This approximation is valid in the regime of multiple scat-tering such that the wavelength of light is much smaller thanthe thickness of the medium.6 In this approximation, the mul-tiply scattered light intensity is described by the diffusionequation. The equation for the DPDW, which describes thephoton density �d�r , t� in the solid due to incident energyintensity S�r , t� �optical source function� is given in the timedomain by

D�2�d�r,t� −1

c

�

�t�d�r,t� − �a�d�r,t� = − S�r,t� . �1�

In the preceding equation, �a is the optical absorption coef-ficient �m−1�, c is the speed of light in the turbid medium�m/s�, and D is the optical diffusion coefficient �m�. Theoptical diffusion coefficient is defined as

D =1

3��a + �s��, �2�

where �s� is the reduced scattering coefficient �m−1�. Thereduced scattering coefficient �s� is related to the scatteringcoefficient �s such that �s�= �1−g��s where g is the meancosine of the scattering function p�� of the photon over allspatial directions described by the solid angle �. It can beconsidered to be an assymetry parameter for anisotropic mul-tiple scattering and can be defined as

g =�4�p��cos � d�

�4�p��d�. �3�

Note that −1�g�1. Isotropic scattering is given by g=0.Net backward scattering implies negative values of g whilenet forward scattering implies positive values of g. Clearlythe reduced scattering coefficient �s� becomes equal to thescattering coefficient �s only in the case of isotropic scatter-ing when g=0.

The diffusion approximation to the radiative transportequation, as given in Eq. �1� is valid when the absorptioncoefficient �a is much smaller than the reduced scatteringcoefficient �s�, or �a��s�.

23 It should be noted that evenwhen this condition is satisfied, the diffusion approximationbreaks down when the point-scatterer source is near theboundaries of the turbid medium and in general may not beapplicable near the surface of a turbid medium.6 This topic isaddressed in Ref. 6, where it is found that for a turbid me-dium such as skin, the near-surface ballistic “skin layer” ison the order of 7–50 �m thick. A concise review of theliterature on this subject is also included in that reference. Itis noted that for the work considered here, an infinite domainis used and thus boundary layer effects are not considered.

Taking the Fourier transform of the DPDW equation, weobtain the Helmholtz pseudowave equation in the frequencydomain:

�2�d�r,�� + kp2�d�r,�� = −

S�r,��D

, �4�

where kp2 =−�a /D− i� /cD and �d�r ,��=F��d�r , t� ; t→�� is

the Fourier transform of �d�r , t�.

lie Baddour: Theory of frequency-domain photoacoustic tomography

B. Thermal propagation equation

The equation describing the temperature rise in the ma-terial that follows as a result of the generation of the photo-nic diffuse density wave is given by the heat equation:

�2T�r,t� −1

�

�tT�r,t� = −

�a

s�d�r,t� . �5�

In the frequency domain, this equation becomes anotherHelmholtz pseudowave equation given by

�2T�r,�� + kt2T�r,�� = − �a

��r,��s

, �6�

where kt2=−i� / and �=�d+�c. In the previous equation,

�=�d+�c is the total DPDW wave and is the sum of thediffuse and coherent photon density waves. The variable isthe thermal diffusivity of the material and s is the thermalconductivity of the material.

C. Acoustic propagation equation

Finally, the acoustic displacement potential is describedby the wave equation, leading to the Helmholtz equation inthe frequency domain. The wave equation can be dealt withby introducing a displacement potential �s�r , t�, which is re-lated to the displacement vector by24

u�r,t� = ��s�r,t� . �7�

Only longitudinal waves are assumed to propagate and as aresult ��u�r , t�=0 and the displacement potential satisfiesthe wave equation

�2�s�r,t� −1

cs2

�2

�t2�s�r,�� =s s

�scs2 T�r,t� . �8�

Taking the Fourier transform of this equation leads to aHelmholtz equation in the frequency domain with the ther-mal term acting as a source term, yielding

�2�s�r,�� + ks2�s�r,�� =

s s

�scs2 T�r,�� , �9�

where ks2=�2 /cs

2. The thermoelastic pressure in the solid isthen obtained from the displacement potential by P�r , t�=−�s��2 /�t2��s�r , t�, which in the frequency domain be-comes P�r ,��=�s�

2�s�r ,��. Here cs is the speed of soundin the medium, �s is the material density, and s is the coef-ficient of thermal expansion of the material. Although thepartial differential equation describing the variations of pres-sure is not an additional equation as the same information iscontained in the equation for the displacement potential, it isnevertheless instructive to examine the equation for pressure:

�2P�r,t� −1

cs2

�2

�t2 P�r,t� = −s s

cs2

�2

�t2T�r,t� . �10�

It can immediately be seen that the acoustic pressure is gov-erned by the second time derivative �“acceleration”� of thetemperature dependence. It is noted that acoustic inhomoge-neities and acoustic absorption are not modeled in this for-
mulation.
J. Acoust. Soc. Am., Vol. 123, No. 5, May 2008 Natalie Bad

In the preceding equations k refers to the wave numberand the subscript indicates which phenomena �photonic, ther-mal or acoustic� the wave number is referring to. Specifi-cally, kp refers to the wave number for the DPDW �photonic�equation, kt refers to the thermal wave number, and ks refersto the acoustic wave number. It should be noted that only theacoustic wave number is real, with the thermal and photonicwave numbers being complex quantities, hence the use of theterm “pseudowave” equation to describe the resulting Helm-holtz equation in order not to avoid confusion with theproper Helmholtz equation with, the real wave number.

In this paper, our main interest lies in reconstructing theinhomogeneous distributions of optical absorption. Further, itis assumed in this formulation that the only inhomogeneitiespresent are in the optical absorption coefficient.

III. NOTATION AND SIGN CONVENTIONS

From the preceding section, it can be seen that the pho-tonic, thermal, and acoustic wave numbers are defined by thesquares of their quantities and arise as a result of taking theFourier transform of the corresponding equation. In particu-lar,

kp2 = −

�a

D−

i�

cD, kt

2 = −i�

, ks

2 =�2

cs2 �11�

In the rest of the paper, we will have great use of the wavenumbers themselves, namely, kp, ks, and kt, which are thesquare roots of the equations in Eq. �11�. Each k will bewritten as the sum of a real and an imaginary part, so thatkp=kpr+ ikpi with kpr denoting the real part of kp and kpi

denoting the imaginary part of kp. Similar notations will ap-ply to the other k’s. As there are two square roots in theprevious equations, we will use the convention that a particu-lar �photonic, thermal, acoustic� k is the square root of thecorresponding k2 such that the imaginary part of k is posi-tive. Hence kp is the square root of kp

2 such that kpi�0 and soforth. This convention will be important in derivations inorder to ensure boundedness of solutions.

It is noted that the variable � is the Fourier frequencyvariable corresponding to time. The �temporal� Fourier trans-form is defined in the conventional way as

F̂�� = �−�

�

f�t�e−i�tdt , �12�

with the inverse transform defined by

f�t� =1

2��

−�

�

F̂��ei�td� . �13�

The two-dimensional �2D� and 3D spatial Fourier transformof functions will be required and is defined as

F̂3D��x,�y,�z� = �−�

� �−�

� �−�

�

f�x,y,z�

�e−i�xxe−i�yye−i�zzdx dy dz . �14�

In this case ��x ,�y ,�z� are the spatial frequency variables in

dour: Theory of frequency-domain photoacoustic tomography 2579

the Fourier domain. We will also require use of a variable ofthe form

�2 = �x2 + �y

2 − k2. �15�

We will adopt the convention that the subscript on � indi-cates the subscript of the corresponding k. That is, we define�p

2 =�x2+�y

2−kp2, �t

2=�x2+�y

2−kt2, and �s

2=�x2+�y

2−ks2. Simi-

larly, the use of � will be more frequent than �2 and thus aconvention is required to indicate which square root is beingused. We will write �=�r+ i�i where this is defined as thesquare root of �2 such that �r�0. Similar to the conventionfor ks the subscripts for � get carried over to real and imagi-nary parts so that, e.g., �s=�sr+ i�si can be written.

IV. GEOMETRY

As for standard acoustic diffraction tomography theory,we assume a background medium infinite in extent and aninhomogeneity structure of finite extent. The previouslygiven equations for the scattered field are valid for all pointsoutside the inhomogeneity and for arbitrary source–detectorconfigurations. The assumption of an infinite domain is thesimplest case for physical insight and can later be modifiedfor different geometries. As it is also the assumption typi-cally made for standard acoustic diffraction tomography, aswell as for diffuse photonic wave tomography, this assump-tion will allow for straightforward comparisons.

We further specialize our formulation to the case wherethe acoustic wave is measured by a plane of detectors so thatthe acoustic wave is measured in the z=zd plane. We are thusinterested in the Fourier transform of the wave measured inthe z=zd plane.

V. FIRST BORN APPROXIMATION

For tomographic imaging purposes, our main interestlies in reconstructing the inhomogeneous distributions of op-tical absorption. We therefore consider the Helmholtz equa-tion for the photon density wave:

��2 + kp2�r��d�r�,�� = 0. �16�

We now write

kp2�r�� = kp0

2 n2�r��, n2�r�� =�a0

�a�r��, �17�

where kp0

2 =−�a0/D− i� /cD and �a0

is the optical absorptioncoefficient of the assumed homogeneous medium surround-ing the object region Q and n�r�� is a measure of the variationof the values of the optical absorption coefficient in the ob-ject from that of the surrounding homogeneous �reference�region. In this development, it is assumed that the only in-homogeneities are in the optical absorption coefficient. It isalso assumed that the object has finite size. This leads to

��2 + kp0

2 ��d�r�,�� = −1

Doa�r��d�r�,�� , �18�

where the object function is given by


oa�r�� = kp0

2 �n2�r�� − 1�, r� � Q

0, r� � Q .�19�

It is noted that the preceding exactly follows the develop-ment of the diffraction tomography problem with the waveequation, with the exception of the complex nature of thephotonic wave vector. The effect of the inhomogeneities ofthe object region appears as a source term on the right-handside of the Helmholtz equation �18�, with oa�r�� being theobject function representing the inhomogeneities of the scat-tering object region Q. The object function is zero outsidethe object region and its nonzero value within the objectregion represents the ratio of optical absorption coefficients.

The DPDW field �d�r� ,�� shall be considered to be thesum of two components, �d0�r� ,�� and �ds�r� ,��, namely,

�d�r�,�� = �d0�r�,�� + �ds�r�,�� . �20�

The component �d0 �r�� is known as the incident field �orequivalently the illumination function� and is the fieldpresent without any inhomogeneities. It is given by the solu-tion to

��2 + kp0

2 �r��d0�r�,�� = 0. �21�

The component �ds�r� ,��, known as the secondary field, willbe that part of the total field that can be attributed solely tothe inhomogeneities. Note that this term is often referred toas the scattered field, however, the term “secondary” will beused here to avoid confusion with the true scattered field inthe optical sense. The secondary component of the field willnecessarily have to satisfy

��2 + kp0

2 ��ds�r�,�� = −1

Doa�r��d�r�,�� . �22�

It should be noted that �d�r� ,�� in the right-hand side of Eq.�22� is unknown. However, for small enough inhomogene-ities, �d�r� ,�� can be replaced with its free-space value�d0�r� ,��, which itself can be found from Eq. �21�. Replac-ing �d�r� ,�� with �d0�r� ,�� on the right-hand side of Eq.�21� is known as the first Born approximation. Hence, underthe first Born approximation, the secondary component ofthe DPDW field must satisfy

��2 + kp0

2 ��dB�r�,�� = −1

Doa�r��d0�r�,�� , �23�

where �dB�r� ,�� is used to denote the first Born approxima-tion to the secondary DPDW field. Thus, �d�r� ,��=�d0�r� ,��+�dB�r� ,��, with �d0�r� ,�� and �dB�r� ,�� givenby Eqs. �21� and �23�, respectively. In the rest of what fol-lows, we will use kp to denote kp0

for ease of notation, as noexplicit reference to kp0

is needed once the first Born ap-proximation has been made.

VI. SOLUTION TO THE FORWARD PROBLEM

The complete problem formulation starts with Eq. �21�for the illumination function �which is a DPDW� along withEq. �23�, the Born solution to the inhomogeneous DPDW
problem. The solution to this is subsequently used as source

dzp.

terms in Eq. �6�, the solution of which is then further used inEq. �9�. Hence the full problem consists of Eq. �21� alongwith Eqs. �6�, �9�, and �23�.

To begin, a general illumination function is considered.That is, the sequentially coupled equations, Eqs. �6�, �9�, and�23�, are solved without �yet� specifying a form for the illu-mination function. Next, we proceed by taking the spatialFourier transform of both sides of the equations, howeveronly with respect to the x and y variables leaving the z vari-able untransformed. This gives

d2

dz2�dB��x,�y,z� − �p2�dB��x,�y,z,��

= −1

DOa��x,�y,z��d0��x,�y,z,�� , �24�

where �p2 =�x

2+�y2−kp

2. Denote Oa�d0��x ,�y ,z�ªOa��x ,�y ,z��d0��x ,�y ,z ,�� for brevity. Similarly, thethermal and acoustic equations become

d2

dz2T��x,�y,z,�� − �t2T��x,�y,z,��

= −�a

s�dB��x,�y,z,�� , �25�

d2

dz2�s��x,�y,z,�� − �s2�s��x,�y,z,��

= −s s

�scs2 T��x,�y,z,�� , �26�

where �t2=�x

2+�y2−kt

2 and �s2=�x

2+�y2−ks

2. Note that Eqs.�24�–�26� can be solved using the one-dimensional �1D�Green function with the appropriate � as the wave vector.Hence,

�dB��x,�y,z,�� = �−�

� 1

2�pe−�pz−zp

� 1

DOa�d0��x,�y,zp,��dzp. �27�

The same Green function is also used to solve thermal equa-tion �25� to give

T��x,�y,z,�� = �−�

� 1

2�te−�tz−zt

�a

s�dB��x,�y,zt,��dzt.

�28�

Using the expression in Eq. �27�, this can be simplified to

T��x,�y,z,�� =�a

4sD�t�p�

−�

� �−�

�

e−�tz−zte−�pzt−zpOa

��d0��x,�y,zp,��dzpdzt. �29�

Finally, the 1D Green expansion can also be used to solve
acoustic equation �26� to give

�s��x,�y,z,�� = �−�

� 1

2�se−�sz−zs

s s

�scs2 T��x,�y,zs,��dzs.

�30�

The expression for T from Eq. �29� is substituted into Eq.�30� and the order of integration is switched, yielding

�s��x,�y,z,�� =− �a s

8�p�t�sD�scs2�

−�

� �−�

� �−�

�

e−�pzt−zp

�e−�tzs−zte−�sz−zsOa�d0��x,�y,zp,��dzs dzt

�31�

Recalling that the pressure is given by P�r� ,��=�2�s�s�r� ,��, the Fourier transform of the pressure detectedin a plane is obtained

P��x,�y,z,�� =− �a sks

2

D�

−�

� �−�

� �−�

� e−�pzt−zp

2�p

�e−�tzs−zt

2�t

e−�sz−zs

2�s

�Oa�d0��x,�y,zp,��dzs dzt dzp

=− �a sks

2

D�

−�

� �−�

� �−�

�

�gp�zt − zp;�p�gt�zs − zt;�t�gs�z

− zs;�s�Oa�d0��x,�y,zp,��dzs dzt dzp. �32�

In the integrand, the photonic, thermal and acoustic Greenfunctions have been identified and labeled with the appropri-ate subscript, in keeping with the conventions of this paper.That is, for i= p, t, and s representing photonic, thermal andacoustic phenomena:

gi�z;�i� =e−�iz

2�i. �33�

Equation �32� provides a good interpretation of the physicsof the problem. In the Born approximation, the product ofthe object function and illumination functionOa�x ,y ,z��d0�x ,y ,z , t� can be considered as the input, withthe only time dependence being in the incident diffuse pho-ton density wave function. Let us refer to this product ofobject and illumination functions as the heterogeneity func-tion. The system acts on this heterogeneity function inputthrough the three Green functions. The system can be con-sidered to be acting on the input with the photonic, followedby thermal, followed by acoustic Green function with thetotal system Green function being a product of the three.Essentially, we have

P��x,�y,z,��

=− �a sks

2

D�

−�

� �−�

� �−�

�

�gpts�z,zp,zt,zs;�p,�t,�s�Oa

��d0��x,�y,zp,��dzs dzt dzp, �34�


where gpts�z ,zp ,zt ,zs ;�p ,�t ,�s�=gp�zt−zp ;�p�gt�zs−zt ;�t��gs�z−zs ;�s�.

This is a convenient way of thinking of the effect of thesystem as any simplifications or alternative modeling as-sumptions can be directly reflected in the individual Greenfunctions and it is known that the total system Green func-tion, given by gpts, will be a product of the three individualones and thus any changes can be analyzed directly. Notethat the total system Green function, gpts, is a four-dimensional �4D� function of �z ,zp ,zt ,zs� and when inte-grated can be considered to be the sequential convolution ofthe Green functions for the three subsystems. The integra-tions are performed over zp, zt, and zs to ultimately yield afunction of z. The �’s are considered to be parameters in theGreen function�s�. Simplifying this expression for the totalsystem Green function will now be considered.

VII. SIMPLIFYING THE INTEGRALS

The expression for the pressure as given in Eq. �32� canbe simplified. Using Lemma 1 �derived in the Appendix�, thefollowing results are important.

Result 1

�−�

� e−�tzs−zt

2�t

e−�sz−zs

2�sdzs

= �−�

�

gt�zs − zt;�t�gs�z − zs;�s�dzs =gs�z − zt;�s�

�ks2 − kt

2�

+gt�z − zt;�t�

�kt2 − ks

2�. �35�

Proof: This result follows directly from the applicationof Lemma 1, which is given in the Appendix.

Result 2

�−�

� �−�

� e−�pzt−zp

2�p

e−�tzs−zt

2�t

e−�sz−zs

2�sdzs dzt

= �−�

� �−�

�

gp�zt − zp;�p�gt�zs − zt;�t�gs�z − zs;�s�dzs dzt

=gs�z − zp;�s�

�ks2 − kt

2��ks2 − kp

2�+

gt�z − zp;�t��kt

2 − ks2��kt

2 − kp2�

+gp�z − zp;�p�

�kp2 − ks

2��kp2 − kt

2��36�

Proof: We write

�−�

� �−�

�

gp�zt − zp;�p�gt�zs − zt;�t�gs�z − zs;�s�dzs dzt

= �−�

�

gp�zt − zp;�p��−�

�

gt�zs − zt;�t�gs�z

− zs;�s�dzs dzt �37�

and then invoke Result 1.


VIII. GREEN FUNCTION THEOREM FOR A GENERALILLUMINATION FUNCTION

Results 1 and 2 can be used to simplify Eq. �32� to give

P��x,�y,z,�� =− �a sks

2

D�

−�

�

Oa�d0��x,�y,zp,��

�� gs�z − zp;�s��ks

2 − kt2��ks

2 − kp2�

+gt�z − zp;�t�

�kt2 − ks

2��kt2 − kp

2�

+gp�z − zp;�p�

�kp2 − ks

2��kp2 − kt

2� dzp. �38�

Let us further define the following variables:

As ª�a

2ks2

�ks2 − kt

2��ks2 − kp

2�, �39�

At ª�a

2ks2

�kt2 − ks

2��kt2 − kp

2�, �40�

Ap ª�a

2ks2

�kp2 − ks

2��kp2 − kt

2�. �41�

With these new variables, Eq. �38� can be written as

P��x,�y,z,�� =− s

D�a�

−�

�

Oa�d0��x,�y,zp,��gpts

��z − zp;�s,�p,�t�dzp, �42�

where gpts�z ;�s ,�p ,�t�=Apgp�z ;�p�+Atgt�z ;�t�+Asgs�z ;�s�It can be seen that this is now a one dimensional �1D�

problem with the pressure being given by a convolution ofthe heterogeneity function with the system Green function.Further, it is also clear that the system Green function, whichwas formerly 4D and a product of the subsystem Green func-tions, is now 1D and a sum of the subsystem Green func-tions.

A. Interpretation as a convolution

Equation �42� can be interpreted as a convolution whereit can be written as

P��x,�y,z,�� =− s

D�a�Oa�d0��x,�y,z,�� * gpts

��z;�s,�p,�t��z� . �43�

Taking the Fourier transform with respect to z of both sidesturns products into convolutions and convolutions into prod-ucts, leading to

P��x,�y,�z,�� =− s

D�aOa * �d0��x,�y,�z,�� · Gpts

��x,�y,�z,�� , �44�

where, Gpts��x ,�y ,�z ,��=ApGp��x ,�y ,�z ,��+At

�Gt��x ,�y ,�z ,��+AsGs��x ,�y ,�z ,��


The capital G is used to denote the full Fourier trans-form of each corresponding g function, for each of the pho-tonic, thermal, and acoustic Green function. The form ofeach of these transfer functions is given by

G��x,�y,�z,�� = F� e−��z

2��

=2��

2��

1

��2 + �z

2 =1

�x2 + �y

2 + �z2 − k2 ,

�45�

which is precisely the three-dimensional �3D� wave equationtransfer function. In this form, it is easier to visualize thephysical implications of the photonic, thermal, and acoustictransfer functions separately, as the combined transfer func-tion is a sum of the three subsystem transfer functions. Themagnitude of each transfer function is shown in Figs. 1–3,respectively. The transfer functions are 4D hypersurfaces infrequency space but have been plotted in Figs. 1–3 with �y

=0 for clarity. In Figs. 1–3 each wave number in the denomi-

FIG. 1. �Color online� Photonic transfer function.

FIG. 2. �Color online� Thermal transfer function.


nator of the transfer function of the form of Eq. �45� has beennormalized so that k =1. Thus, the spatial frequencies dis-played in the plots are normalized frequencies rather thantrue frequencies. To obtain the true frequency, each spatialfrequency would have to be multiplied by the k2 corre-sponding to that particular transfer function. The thermaltransfer function dies off much more quickly in spatial fre-quency space than the photonic transfer function as the decayof both is primarily controlled by the exponent in the decay-ing exponential function. As the thermal wave number issmaller, the corresponding gamma exponent is larger for thethermal transfer function and thus the thermal transfer func-tion goes to zero far more quickly than the photonic transferfunction. As both the photonic and thermal wave numbersare complex, the corresponding transfer function reaches itsmaximum at �x=�y =�z=0 and then goes to zero for in-creasing spatial frequencies. Thus, the photonic and thermaltransfer functions can be considered to be low-pass filters.However, the situation for the acoustic transfer function isnot the same. The acoustic wave number is real, thus itstransfer function, as given by Eq. �45� reaches a maximumwhen the denominator is zero. This would occur for �x

2

+�y2+�z

2=ks2. As illustrated in Fig. 3, in two dimensions this

would be �x2+�z

2=ks2, a circle in 2D spatial frequency space

�and a sphere in 3D space�. The variation in the heights ofthe spikes in Fig. 3 is an artifact of the discretization theFourier space and not an inherent property of the transferfunction. The acoustic transfer function can be considered tobe a band-pass filter, with frequencies satisfying �x

2+�y2

+�z2=ks

2 passed on for detection.

B. Block diagram and comparison with ultrasonicimaging

From Eq. �44�, it can be seen that the detected pressurein Fourier frequency space �spatial and temporal� can be con-sidered as a sequence of operations on the object �inhomo-geneity� function. The object function is first convolved withthe illumination function. This resulting function �termed theheterogeneity� function is then multiplied by the system

FIG. 3. �Color online� Acoustic transfer function.

transfer function. For the photothermoacoustics problem, this


system transfer function is actually a sum of the three sub-system transfer functions. This process yields the pressureeverywhere. This is shown schematically in Fig. 4. Clearly,the act of detecting the pressure on some surface is yet an-other transfer function, which acts on the resulting pressurefunction.

Viewing the photothermoacoustic process in the blockdiagram of Fig. 4 allows immediate comparison with theusual ultrasonic imaging modality. For traditional ultrasonicimaging, the inhomogeneity being imaged �i.e., the objectfunction� represents inhomogeneities in the speed of soundin the object. In comparison, with PTA imaging, it is theinhomogeneities in the absorption coefficient of the tissuethat are being imaged. The large variation in the optical ab-sorption of different tissue constituents is thus expoited asthese are the high-contrast inhomogeneities that are beingimaged. For ultrasonic imaging, the illumination function istypically a plane-wave acoustic wave. This plane wave is adelta function in the spatial frequency domain and its convo-lution with the object function yields a shift of the objectfunction in spatial Fourier space. In PTA imaging, the illu-mination function is a DPDW. Finally, the system transferfunction for ultrasonic imaging is the acoustic transfer func-tion alone, similar to the acoustic subsystem transfer functionderived here. For PTA imaging, the system transfer functionconsists of the sum of the acoustic, photonic, and thermaltransfer functions with the effect of each to be discussedshortly.

C. Comparison with pulsed photoacoustic imaging

In pulsed photoacoustic tomography, the pulse durationis so short that the thermal conduction time is greater thanthe thermoacoustic transit time and the effect of thermal con-duction can be ignored.25 The equation describing the result-ing thermoacoustic pressure wave propagation is givenby12–14,25,26

�2p�r�,t� −1

cs2

�2

�t2 p�r�,t� = − scs

Cp

�

�tH�r�,t� , �46�

where Cp is the specific heat, H is the heating function de-fined as the thermal energy deposited by the energy sourceper time and volume, s is the coefficient of thermal volumeexpansion, and cs is the speed of sound. The heating functioncan be written as the product of a spatial absorption function

Objectfunction Convolution

IlluminationFunction

multiply

SystemTransferFunction

Pressure

FIG. 4. Schematic of PTA process in the spatial and temporal frequencydomain.

and a temporal illumination function of the rf source,


H�r�,t� = I0oa�r��t� , �47�

where I0 is a scaling factor proportional to the incident ra-diation intensity and oa�r�� describes the absorption propertiesof the medium—essentially the inhomogeneity whose imageis sought. The function ��t� describes the shape of the irra-diating pulse and is a nonnegative function whose integrationover time equals the pulse energy. Taking the Fourier trans-form with respect to time and the 3D Fourier transform withrespect to space of Eq. �46� yields

P��x,�y,�z,�� = I0i�

Cp��

�Oa��x,�y,�z�Gs��x,�y,�z,�� , �48�

where

Gs��x,�y,�z,�� =1

�x2 + �y

2 + �z2 − ks

2 �49�

is the acoustic transfer function. This can be directly com-pared with Eq. �43� for the full photo-thermo-acoustic prob-lem. We see that there are several differences. First, thetransfer functions are not the same. The pulsed photoacousticproblem yields the purely acoustic transfer function whereasthe full photothermoacoustic problem has a transfer function,which is a sum of the acoustic, thermal, and photonic trans-fer functions. Second, for the pulsed photoacoustic problem,the system transfer function is directly multiplied with theobject function. However, for the photothermoacoustic prob-lem, the object function is first convolved with the illumina-tion function before being multiplied by the system transferfunction. As the choice of illumination function is somewhatunder the control of the designer, it should be expected thatthe photothermoacoustic approach might yield greater abilityto “probe” the object function and thus better imaging re-sults.

IX. DIMENSIONAL ANALYSIS

The relative importance of acoustic, thermal, and photo-nic effects will be governed by three separate characteristicfrequencies. To analyze the effects of acoustic, thermal, andphotonic effects, let us define these characteristic frequenciesas

�s ª �acs ¬1

�s, �t ª �a

2 ¬

1

�t, �p ª �ac ¬

1

�p.

�50�

The subscripts have been used along with this paper’s con-vention of s , t, and p referring to acoustic, thermal, and pho-tonic phenomena, respectively. Each characteristic frequencyalso defines a characteristic time, with this time defined asthe inverse of the corresponding characteristic frequency.The time �s is the transit time of sound through the depth oflight penetration. This is the time taken for stress to traversethe heated region. For frequencies ��s, or equivalently fortimes t��s, this condition is known as stress confinement.12

For frequencies ��t, then heat conduction is considered to25
be negligible as the thermal diffusion length is shorter than

the heating zone. Thus the time �t can be interpreted as acharacteristic time scale for the heat dissipation of the ab-sorbed photonic energy by thermal conduction.12 It is a mea-sure of the relative sizes of the depth of penetration of thediffuse photon density wave �heating zone� and the thermaldiffusion length. This condition has been referred to as ther-mal confinement.12 It follows that �s, �t, and �p can be inter-preted as the times taken for acoustic, thermal, and photonicwaves to traverse the depth of light penetration �the heatingzone�. Thermal and stress confinements are usually discussedin terms of time scales rather than frequencies. Generally,�p��s��t.

In keeping with the work of Gusev and Karabutov,25wealso define

� =cs

2

=

�s2

�t. �51�

Gusev and Karabutov note that �Ref. 25, p. 16� the wavevector of the acoustic wave is much smaller than the wavevector of the thermal wave across the entire frequency range.This implies

�

cs��

→

�

cs2 � 1 → � �

�s2

�t= �. �52�

This is simply the statement that the acoustic wavelength ismuch longer than the thermal wavelength, expressed in termsof input frequency and characteristic frequencies. It followsimmediately from this statement that

ks2

kt2 =

�i

cs2 =

��t

�s2 i � 1i , �53�

where i=�−1, so that the magnitude of the ratio of thesquared acoustic to thermal wave number is very small.Similarly,

kp2

kt2 = −

�ai

D�+

cD=

1

�aD� �t

�p−

�t

�i �

− 1

�aD

�t

�i , �54�

and can thus be considered as a measure of the degree ofthermal confinement. If the input frequency ��t, then themagnitude of the ratios of the squared photonic to thermalwave number is very small �thermal confinement�. Finally,we can also write

kp2

ks2 = −

�acs2

D�2 −ics

2

cD�=

− �s2

�2 �1 +�

�pi 1

�aD. �55�

This last ratio of squared wave numbers can be considered asa measure of the degree of stress confinement and in generalcannot be considered to be a small quantity. In summary, wehave that thermal confinement implies that kp

2 /kt2 is a small

quantity and that stress confinement implies that kp2 /ks

2 is asmall quantity. It is noted that both conditions can be ex-pressed in terms of ratios of the appropriate wave numberswith thermal confinement involving the ratio of the photonicto thermal wave numbers and stress confinement involvingthe ratio of photonic to acoustic wave number.

Some typical examples of parameters are now given.
From Ref. 12, it can be obtained that the absorption coeffi-

cients of the electric field in fat �low water content� andmuscle �high water content� are about 0.1 and 0.9 cm−1, re-spectively, at 3 GHz and about 0.03 and 0.25 cm−1, respec-tively, at 300 MHz. Typical thermal diffusivity for most softtissues is 1.4�10−3 cm2 /s. The speed of sound is1.5 mm /�s. Telenkov et al. gives the thermal diffusivity ofsoft tissue as 0.11 mm2 /s.27 Background optical propertiesare given as �a=0.02 cm−1 and �s�=8.0 cm−1.28 The speed oflight in tissue, c, is on the order of 2�108 m /s.29 Giventhese typical parameters, then �p=2�108 s−1, �s=3�103 s−1, and �t=5.6�10−7 s−1. Given these numbers, it iseasy to see that, in general, ��t will be easily achievedwhereas it is not necessarily true that ��s. Thus, in gen-eral, it follows that the magnitude of the ratio of the photonicand thermal wave numbers is very small and that the mag-nitude of the ratio of the acoustic and thermal wave numberis also small. However, the ratio of photonic to acousticwave number will depend on the frequency of the illumina-tion function.

Based on the preceding analysis, the values of As, At,and Ap can be simplified. Using �a

2 /ks2=�s

2 /�2 and �a2 /kt

2

= i�t /�, along with Eqs. �53� and �54�, these can be rewrittenas

At ª− �t

2/�s2

�1 − ks2/kt

2��1 − kp2/kt

2� � −�t

2

�s2 � 1, �56�

As ª�ti/�

�ks2/kt

2 − 1��1 − kp2/ks

2� � −�ti

�

1

�1 − kp2/ks

2� , �57�

Ap ª�ti/�

�kp2/ks

2 − 1��kp2/kt

2 − 1� � −�ti

�

1

�kp2/ks

2 − 1� = − As.

�58�

Thus, the system Green function can now be written as

gpts�z;�s,�p,�t� = Apgp�z;�p� + Atgt�z;�t� + Asgs�z;�s�

� As�gs�z;�s� − gp�z;�p�� , �59�

and it follows that the thermal subsystem Green function hasvery little effect on the final output.

Stress confinement: As previously discussed, under thecondition of stress confinement, kp

2 /ks2 becomes a small quan-

tity. It then further follows that under stress confinement Ap

and As simplify to

As = −�ti

�= − Ap. �60�

As previously discussed, At is considered small for most fre-quencies. The system Green function still has the same formas given by Eq. �59�.

Near field and far field regions: It can be seen fromEq. �59� that the system Green function is a sum of photonicand acoustic Green functions. Recall that these functions are
given by by the form,

g�z� =e−��z

2��

, �61�

where �� takes on the appropriate form for photonic andacoustic wave numbers and is an “effective” wave number.For acoustic waves, �� is purely real for values of ks

2��x2

+�y2 and thus decays exponentially. The Green function will

approach zero for values of z greater than about5 /��r—recalling that it is the real component of �� thatcauses the decay in the Green function. As �s acts as aneffective wave number, the inverse of �s behaves as an ef-fective wavelength. For ks

2��x2+�y

2, then �s is purely imagi-nary and thus does not decay. This is the “low-pass” versionof the Green function as essentially only these lower spatialfrequencies are propagated and the others are attenuated.More specifically, the acoustic Green function is really aband-pass function as spatial frequencies such that �s=0,that is �x

2+�y2=ks

2, are the ones that are most highly selected,causing the denominator of the Green function to go to zero.The corresponding �� for the DPDW is always complex andthus always has a decaying component. Hence, “far enough”away from a given inhomogeneity, only the acoustic contri-bution will remain as the photonic contribution to the sys-tem, as given in Eq. �59�, will have completely decayed. Asincreasing spatial frequencies only serve to enhance the pos-sible decay of the exponential term, the largest allowable z


�before the Green function decays to zero� occurs at �x

=�y =0. It is the real part of kp that dictates the rate of decayof the Green function. Recall that

kp2 = −

�a

D−

i�

cD. �62�

As the speed of light, c, is large, then for all but the highestof frequencies we can approximately write kp

2 =−�a /D andthus kp= i��a /D. For the typical values previously men-tioned, ��a /D�70 m−1 and it follows that the decay con-stant is roughly about 1.4 cm. If the detector is roughlywithin this order of magnitude or closer to the inhomogene-ity then both the acoustic and photonic contributions to thefinal pressure should be considered. However, for distancesabout 4 to 5 times this far, then the photonic contributionbecomes negligible and only the acoustic Green functionplays a role. This distinction is of importance and shouldhave an effect on any inversion algorithms employed.

X. GENERALIZED FOURIER DIFFRACTION THEOREMFOR GENERAL ILLUMINATION FUNCTION

Given the exponential functional form of the subsystemGreen functions, Eq. �42� can be interpreted in terms of theFourier transform. For a general Green function of the formg�z−zp�=e−��z−zp / �2��, we can write for any function h of
space and temporal frequency:
�−�

�

h��x,�y,zp,��e−��z−zp

2��

dzp = �−�

�

h��x,�y,zp,��e−��r+i��i�z−zp

2��

dzp

= �e−��r+i��i�z

2��

−�

�

h��x,�y,zp,��e��rzpei��izpdzp, z � zp

e��r+i��i�z

2��

−�

�

h��x,�y,zp,��e−��rzpe−i��izpdzp, z � zp

= �e−��r+i��i�z

2��

F3D�h�r�,��e��rz��z=−��i, z � zp �transmission�

e��r+i��i�z

2��

F3D�h�r�,��e−��rz��z=��i, z � zp �reflection� .

�63�

Here, F3D denotes the 3D Fourier transform and it can beobserved that each subsystem Green function has the effectof returning the 3D Fourier transform of the product of theheterogeneity function and a �decaying� exponential, evalu-ated on �z= ��i, weighted by appropriate factors. Thepositive sign is chosen for measurements in reflection andthe negative sign for measurements in transmission. Theweighting factor e��r+i��i�z /2�� is a constant for a fixedreceiver plane and its form is very much reminiscent of theform of each subsystem Green function. In fact, as withoutloss of generality the inhomogeneity can be taken to be lo-

cated at the origin, then the previous equation can be morecompactly written as

�−�

�

h��x,�y,zp,��e−��z−zp

2��

dzp

= g��z�F3D�h�r�,��e��rz��z=−��i sgn�z�, �64�

where g��z� is the appropriate �thermal, photonic or acoustic�Green function. Positive z’s are chosen for transmission mea-surements and negative z’s are chosen for reflection measure-ments. With respect to the convergence of the integrals in Eq.
�63�, it should be noted that the object function has finite

support, otherwise the basic premise of the Born approxima-tion is no longer valid. With finite support, the integrals willconverge, despite the presence of the exponential terms inthe integrand.

Result: Applying the result of Eq. �63� on Eq. �42�, weobtain the generalized Fourier diffraction theorem for thecoupled PTA problem:

P��x,�y,z,�� =− s

D�a�

−�

�

Oa�d0��x,�y,zp,��gpts�z

− zp;�s,�p,�t�dzp =− s

D�a�

�=p,t,sA�g��z�F3D

��Oa�d0��x,�y,z,��e��rz��z=−��i sgn�z�.

�65�

This Fourier diffraction theorem tells us that the spatial 2DFourier transform of the measured response on a plane isproportional to the sum of three 3D Fourier transforms of theheterogeneity function �weighted with an exponential�,evaluated at �z= ��i. The positive z is chosen for measure-ments in reflection and the negative z is chosen for measure-ments in transmission. The three separate 3D Fourier trans-forms are due to each of photonic, thermal, and acousticgeneralized wave number, ��. The � has been defined as thesquare root of �2 such that the real part is positive, so that theexponential functions in front of the 3D Fourier transformare always well behaved at infinity.

XI. FOURIER DIFFRACTION THEOREM FOR PLANE-WAVE ILLUMINATION FUNCTION

We now specialize the Fourier diffraction theorem asderived earlier to the case of plane-wave illumination. Spe-cifically, it is assumed that

�d0��x,�y,z,�� = �d0�z,�� = eikpz. �66�

For plane-wave illumination, Eq. �65� becomes

P��x,�y,z,��

=− s

D�a�

−�

�

Oa��x,�y,z�eikpzgpts�z − zp;�s,�p,�t�dzp

=− s

D�a�

�=p,t,sA�g��z�F3D

��oa�r��e��r−kpi�z��z=�−kpr−��i�sgn�z�. �67�

As �p/t/s2 =�x

2+�y2−kp/t/s

2 , then the terms g��z� in Eq. �67� al-ways have a positive decaying component with the exceptionof �s=��x

2+�y2−ks

2 when �x2+�y

2�ks2. Hence for “large

enough” z, the acoustic terms for which �x2+�y

2�ks2 are

propagated and the other terms are attenuated, effectively alow-pass filter. In this case, �si=�ks

2−�x2−�y

2, and the Fou-rier region of the 3D transform of oa�r��e��r−kpi�z that is de-tected is where �z= � �kpr+�si� or ��z�kpr�2=�si

2 =ks2−�x

2

−�y2, implying ��z�kpr�2+�x

2+�y2=ks

2. This last expressionis for a sphere in 3D spatial frequency space, centered at
�0,0 , �kpr� and of radius ks and is the photothermoacoustic

equivalent of the Fourier diffraction theorem as used in tra-ditional ultrasonic detection.

XII. FOURIER PLANE COVERAGE

As mentioned in the previous section, it can be seen thatin the case of plane-wave illumination, the Fourier region isa hemisphere �depending on whether the detection is intransmission or reflection� centered at �0,0 , �kpr� and ofradius ks. It should be clear that the resolution with which theobject can be imaged depends on how much of its Fouriertransform can be measured. The Fourier plane coverage—theportion of the full 3D Fourier transform of the objectionfunction that is obtained via measurement—is thus a goodindicator of what it is theoretically possible to image. Fur-ther, as similar Fourier plane coverage results can be ob-tained or derived for both traditional ultrasonic imaging andpulsed photoacoustic imaging, different imaging modalitiescan be compared by comparing their Fourier plane coverage.

In particular, let us compare the Fourier plane coveragefor the case of fixed frequency and multiple views. In otherwords, we compare the coverage of the object function ifmeasurements are made at a specific frequency but thetransmission/detection is made at various angles, eventuallyobtaining full 360° coverage of the subject.

For graphical simplicity, circles in the 2D Fourier planerather than spheres in the 3D Fourier plane can be consid-ered. For the ultrasonic case, this analysis is performed inRef. 30. It can be observed that for the frequency domainphotoacoustic modality, at fixed frequency and multipleviews, the Fourier plane coverage is controlled by the loca-tion of the center of the circle, kpr and the radius of the circle,ks. By normalizing with ks, the relevant quantities becomethe location of the center of the circle and the circle itself �ofradius 1�. As the view angle is changed, new information isobtained at a different location in the Fourier plane. TheFourier plane coverage for fixed frequency and multipleviews is shown in Figs. 5–9. Various circle centers areshown. For Fig. 5, we see the Fourier coverage for a circle atcenter 0.5 and radius 1. It does not matter if the center is at

FIG. 5. �Color online� Rotated semicircle of radius 1, center at 0.5.

�0,0.5� or �0.5, 0�, etc. as these theoretical measurements are


made through a full 360° rotation about the object. We seethat this is essentially a band-pass filter for the object func-tion as only information in an annulus type region of theFourier plane is obtained. Figure 6 shows the Fourier cover-age for a circle at center 1 and radius 1. This case corre-sponds to the ultrasonic case.30 This case is a low-pass filter,in comparison with the band-pass filter of the previous case.The difference between the ultrasonic case and the photo-thermoacoustic case is that in the ultrasonic case, both thecenter of the circle and radius of the circle are controlled byks, whereas in the photothermoacoustic case, the center of thecircle is controlled by kpr, whereas the radius of the circle iscontrolled by ks—two separate, although related, quantities.Figure 7 shows the case for a circle of radius 1, centered at 0.It can be shown that this corresponds to the pulsed photoa-coustic case. In a way, this is an unreasonable comparison asin the case of pulse excitation more than one frequencywould be present and thus it could be argued that the single-frequency-multiple-view scenario is unrealistic. However,for the sake of analysis we proceed with this case, in particu-lar as this would also apply when kpr is close to zero. We seethat in this case, the band-pass filter narrows and essentiallypasses only those frequencies that are on the fixed circle. TheFourier coverage is minimal and in fact, no new informationis gained by making measurements at various views. To gainmore information about the rest of the Fourier plane, mea-surements would have to be made at additional frequencies,

FIG. 6. �Color online� Rotated semicircle, radius 1, center at 1.

FIG. 7. �Color online� Rotated semicircles, radius 1, center at 0.


thus changing the radius of the circle. In Fig. 8, the case withcenter at 0.2 �a small center compared to radius� is demon-strated and it can be seen that this corresponds to a narrowband-pass filter. Clearly, the case where the center is at orclose to zero is the limiting band-pass case. In Fig. 9, thecase where the center is at 4, and thus greater than the radiusof the circle �which is 1� is shown. This is again anotherband-pass case, although the location of the information onthe Fourier plane is different. It can be seen from Fig. 9 thatas the center of the circle starts to move away from theorigin, additional measurements �more views� need to bemade in order to “fill in” the information in the Fourier planedisk. In constrast, for the very narrow band-pass filters suchthose of Figs. 7 and 8, fewer views �measurements� need tobe taken before all possible information at that frequency hasbeen obtained. Hence, the higher kpr, the more views need tobe taken to fill in the Fourier disk and vice versa.

All of these simulations where done for the case of 40views to achieve a full 360° rotation about the object. Thesmaller the center of the circle, as controlled by kpr, thenarrower the annulus of the band-pass filters. For small kpr,less and less information is gained by making measurementsat multiple views and the same frequency. In the limit ofkpr=0, additional views yield no new information, and mea-surements at different frequencies must be obtained in orderto gain more information about the 3D Fourier transform of

FIG. 8. �Color online� Rotated semicircle, radius 1, center at 0.2.

FIG. 9. �Color online� Rotated semicircle, radius 1, center at 4.


the object. The case where center and radius of the circle areequal corresponds to the traditional ultrasonic case androughly speaking achieves best coverage of the Fourier planeat a fixed frequency. This case becomes a low-pass filter casein comparison with the band-pass filter of all the other cases.The power behind the photothermoacoustic modality is thatthe center and radius of the circle control the band-pass na-ture of the detection modality and these quantities are con-trolled by separate but related quantities, kpr and ks. Properexperiment design would be a matter of controlling thechoice of frequencies and views in order to obtain coverageof the 3D Fourier transform of the object that is adequate forthe intended application.

XIII. CONCLUSIONS

In conclusion, the governing equations for photoacousticimaging were derived where a short input pulse is not as-sumed. The full coupled problem with diffuse photon densitywaves, thermal and acoustic waves was presented. It wasshown that the forward problem could be interpreted as aconvolution between the system transfer function and theheterogeneity function where the heterogeneity function isdefined as the product of the object and illumination func-tions. Although truly a product of the three subsystem trans-fer function, the total system transfer function was eventuallyshown to be a sum of the three subsystem transfer function.Analysis of relative sizes of each showed that in general thethermal Green function can be neglected, whereas the pho-tonic one may be negligible under certain circumstances. Fi-nally, the preceding results led to a Fourier diffraction theo-rem for PTA imaging where the observed resulting pressuremeasured on a plane is shown to be a spherical slice of the3D Fourier transforms of the object function. Fourier planecoverage was considered. It is expected that these findingswill be useful in the design of experiments to ensure properFourier plane coverage and also as the basis of reconstruc-tion algorithms for PTA imaging. Future work will extendthese results to more general geometries and consider incor-porating additional effects into the model.

APPENDIX

Define the following function:

f�z,zt;�t,�s� ª �−�

�

e−�tzs−zte−�sz−zsdzs. �A1�

Note that zs is a �dummy� integration variable and that��t ,�s� are parameters. We have used �t ,�s in Eq. �A1� andwe shall make use of their specific definition in what follows.

Lemma 1:

f�z,zt;�t,�s� ª �−�

�

e−�tzs−zte−�sz−zsdzs

=2�te

−�sz−zt

ks2 − kt

2 −2�se

−�tz−zt

ks2 − kt

2 . �A2�

Proof: As f is essentially a function of z and zt, two
possible cases need to be considered to evaluate the integral.

In the first possible case, z�zt, whereas in the second pos-sible case, z�zt. In the first case, if z�zt, then f as given inEq. �A1� can be written as

f�z,zt;�t,�s� = �−�

z

e−�t�zt−zs�e−�s�z−zs�dzs + �z

zt

e−�t�zt−zs�

�e−�s�zs−z�dzs + �zt

�

e−�t�zs−zt�e−�s�zs−z�dzs.

�A3�

This sum of integrals can be easily evaluated to give

f�z,zt;�t,�s� =e−�t�zt−z� + e−�s�zt−z�

�t + �s+

e−�s�zt−z� − e−�t�zt−z�

�t − �s.

�A4�

In the second case, if z�zt, then f as given in Eq. �A1� canbe written as

f�z,zt;�t,�s� = �−�

zt

e−�t�zt−zs�e−�s�z−zs�dzs + �zt

z

e−�t�zs−zt�

�e−�s�z−zs�dzs + �z

�

e−�t�zs−zt�e−�s�zs−z�dzs.

�A5�

This sum of integrals can similarly be easily evaluated togive

f�z,zt;�t,�s� =e−�t�z−zt� + e−�s�z−zt�

�t + �s+

e−�s�z−zt� − e−�t�z−zt�

�t − �s.

�A6�

Clearly, the preceding two cases can be summarized by useof the absolute value notation so that

f�z,zt;�t,�s� =e−�tz−zt + e−�sz−zt

�t + �s+

e−�sz−zt − e−�tz−zt

�t − �s

=2�te

−�sz−zt

�t2 − �s

2 −2�se

−�tz−zt

�t2 − �s

2

=2�te

−�sz−zt

ks2 − kt

2 −2�se

−�tz−zt

ks2 − kt

2 , �A7�

where we have made use of the definitions of � so that �t2

−�s2=ks

2−kt2. The proof is complete.

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Theory of Frequency Domain Photoacoustic Tomography