A Transfer Function Approach to Photoacoustic Signal Analysis

A Transfer Function Approach to Photoacoustic Signal Analysis Natalie Baddour, Mechanical Engineering Department, University of Ottawa 770 King Edward Av., Ottawa, ON, K1N 6N5 Abstract: Photoacoustic signal generation has shown potential for medical tomography. With this technique, a short-pulsed laser source is used to irradiate a sample, resulting in an induced acoustic pressure field which can be subsequently detected by ultrasound transducers positioned outside the sample. This paper aims to present a consistent and unified approach to the mathematical modelling of the photoacoustic problem. Temporal and three-dimensional Fourier transforms are used to develop a consistent transfer function approach. A generalized version of the Fourier slice theorem is presented and proved. Closed form solutions to special cases are given in spherical polar coordinates and also discussed.

progress towards achieving these imaging goals [2-6].

However, a number of different approaches have been developed by each group. This paper aims to present a unifying framework for the mathematical theory of photoacoustic imaging, using a transfer function/Green's function approach. A Photoacoustic Fourier theorem is presented and proved. This result has appeared in an indirect way in [4, 7, 8], and this paper aims to present the theorem in a clear and general way, with no assumption regarding the detection geometry. This theorem is extremely important as it forms the basis for most reconstruction algorithms. The representations in this paper are embedded in timedependent, spatially three dimensional descriptions.

Keywords: modelling, photoacoustic, transfer function, Fourier, imaging.

2. GOVERNING EQUATIONS

1. INTRODUCTION Photoacoustic signal generation is a new technique which has demonstrated great potential for non-invasive medical tomography. With this technique, a short-pulsed laser source is used to irradiate the sample. The energy absorbed produces a small temperature rise which induces a pressure inside the sample through thermal expansion. This pressure acts as an acoustic source and generates further acoustic waves which can be detected by ultrasound transducers positioned outside the sample. Since there is a large difference in optical absorption between blood and surrounding tissue, the laser irradiation induces an ultrasound wave on the inhomogeneities within the investigated volume.

Hence, the acquired photothermoacoustic signals carry information about the optical absorption property of the tissue. This approach is suitable for the imaging of the microvascular system or for tissue characterization, with contrast similar to that of pure optical imaging and spatial resolution similar to that of pure ultrasonic imaging. It therefore combines the advantages of two imaging modalities in a single modality. The issue of the strong scattering of light in media like biological tissue is overcome and the ability of acoustic waves to travel long distances without significant distortion or attenuation is also exploited. Photoacoustic detection has shown concrete promise of imaging in turbid media at depths potentially through the full thickness of skin [1, 2].

Several groups of researchers have made significant

The physical principle behind this imaging modality is the photoacoustic effect. This entails the generation of an acoustic wave as a result of the absorption of light pulse.

While optical energy can be converted to mechanical energy through various pathways, it is often the case that thermal expansion is the dominant mechanism. In pulsed photoacoustic tomography, the pulse duration is so short that the thermal conduction time is greater than the thermoacoustic transit time and the effect of thermal conduction can be ignored [1]. The equation describing the thermoacoustic wave propagation with a thermal expansion source term is given by [1, 2, 6, 7, 9]

β ∂ 1 ∂2 G G G H ( r , t ) (1) ∇ p (r ,t ) − 2 2 p (r ,t ) = − s C p ∂t cs ∂t 2

where p is the pressure of the acoustic wave, Cp is the specific heat, H is the heating function defined as the thermal energy deposited by the energy source per unit time and volume, β s is the coefficient of thermal volume expansion and cs is the speed of sound. The light incident on the sample must be pulsed or modulated in order for photoacoustic waves to be generated. The heating function can be written as the product of a spatial absorption function and a temporal illumination function of the RF source

G G H ( r , t ) = I oφ ( r )η ( t )

(2)

where Io is a scaling factor proportional to the incident radiation intensity and φ describes the absorption properties

of the medium. This is essentially the inhomogeneity whose image is sought and the aim of photoacoustic imaging is to reconstruct φ from pressure measurements made at the surface. This will be referred to as the inheterogeneity function since it represents the absorbed energy and is thus a function of the actual inhomogeneity of the medium and of the illuminating beam. The function η(t) describes the shape of the irradiating pulse and is a non-negative function whose integration over time equals the pulse energy.

As for standard diffraction tomography theory, a background medium infinite in extent is assumed along with an inhomogeneity structure of finite extent. The previouslygiven equation for the pressure field is valid for all points outside the inhomogeneity and for arbitrary source-detector configurations. This is the same geometry assumption made for standard acoustic diffraction tomography, as well as for diffuse photonic wave tomography, thus allowing for straightforward comparisons.

3. GREEN'S FUNCTION AND TRANSFER FUNCTIONS

The spatial Green’s function may be defined as

G G G G G (r | x ) = G (r − x ) =

(

C p ω x2

β s I oiω G φ (ω )η (ω ) 2 2 2 + ω y + ωz − ks

(3)

)

G

β I iωη (ω ) ∞ G G G G G p(r , ω ) = s o ∫−∞ φ ( x )G(r − x )dx Cp

spatial frequency variable corresponding to position x and so forth. Additionally, k s = ω/cs . Although the same variables are used to denote a function and its Fourier transform, it should be clear from the arguments which is being indicated. In cases where it may not be clear, a tilde (~) will be used to denote the function in spatial Fourier space. For shorthand notation, let ωk = ω x + ω y + ω z , so 2

that

2

2

ωk is the length of the spatial Fourier vector.

Then by

G GG β s I oiωη (ω ) ∞ φ (ω ) G iω ⋅r G p(r , ω ) = e dω ∫ ω2 − k2 (2π )3 C p −∞ k s

)

(4)

β I iω G G G p (ω , ω ) = s o G (ω )φ (ω )η (ω ) , Cp

G GG β s I oiωη (ω ) ∞ φ (ω ) G iω ⋅r G p(r , ω ) = e dω ∫ 3 (2π ) C p −∞ ωk2 − k s2 GG

β I iωη (ω ) ∞ eiω ⋅r = s o 3 ∫ ω2 − k 2 (2π ) C p −∞ k s

(

)

∞

G

φ ( x )e ∫ ) −∞

G G − iω ⋅ x

(8)

where the Fourier transform of the spatial Green's function is the spatial transfer function:

G G (ω ) =

(

1

ωk2

− k s2

)

.

(9)

Another way of thinking about equation (8) is to write it as

βI G G G p (ω, ω ) = s o iωG (ω ) {φ (ω )η (ω )} Cp G G = Gtotal (ω , ω ) {φ (ω )η (ω )}

(10)

βI G G Gtotal (ω , ω ) = s o iωG (ω ) Cp

(11)

where

is the 'total' system transfer function acting on the product of the input pulse and the heterogeneity function.

G G dxdω

4. PHOTOACOUTICS FOURIER THEOREM We return to the main statement of the solution, the equation for the spatial pressure distribution

G GG β s I oiωη (ω ) ∞ φ (ω ) G iω ⋅r G p(r , ω ) = e dω ∫ ω2 − k2 (2π )3 C p −∞ k s

(

)

(12)

Using the coordinate transformation from cartesian to spherical polar coordinates as given, the integral in (12) will be rewritten. The subscript 'ω' is dropped for brevity.

Without loss of generality, it can be assumed that the vector G r is in the positive z direction so that

Using the definition of the Fourier transform, the above equation can be re-written as

(

(7)

2

spatial inverse transformation of equation (3), the pressure function becomes

(

(6)

The frequency domain equivalent to equation (7) is equation (3), which can now be interpreted in terms of the spatial transfer function as

where the shorthand notation (ω ) = (ω x , ω y , ω z ) has been used to denote a point in 3D spatial Fourier frequency space.

Here, ω is the temporal frequency variable and ω x is the

eiω ⋅( r − x ) G ∫ ωk2 − ks2 dω (2π )3 −∞

This is the Green's function for the Helmholtz equation. It then follows that equation (5) can be rewritten such that it can be clearly interpreted as a convolution in space

Taking the full spatial and temporal Fourier transforms of the governing equation and rearranging yields

G p (ω , ω ) =

G G G

∞

1

(5)

G G G r = (0, 0, r ) = rkˆ and ω ⋅ r = ρ r cos(ψ ) = μρ r , where G G r = r . For the case where r is in the positive z direction,

the spatial inverse Fourier transform integral in equation (12) can be written as

∫ (2π )3 −∞ =

G

∞

1

(

)

2π

1 (2π )

1 ∞

∫ ∫

3

function, evaluated on the sphere ρ

φ (ω ) iωG ⋅rG G e dω ωk2 − ks2

−1

0

φ ( ρ , μ ,θ ) iμρ r 2 ∫ ρ 2 − ks2 e ρ d ρ d μ dθ 0

(13)

By the first mean-value theorem for integration, there must exist some value of ( μ * , θ * ) for −1 ≤ μ ∗ ≤ 1, 0 ≤ θ ∗ ≤ 2π such that 2π

1

0

(14)

φ ( ρ , μ ∗ ,θ ∗ ) = φ (− ρ , μ ∗ ,θ ∗ ) , which implies that the

Fourier transform of the heterogeneity function is defined to have an even extension in the radial dimension. Using Cauchy's integral formula, the last integral can be simplified to yield

β I iωη (ω ) G p(r , ω ) = s o ⋅ 8π rC p

(15)

⎡φ ( ks , μ ∗ , θ ∗ )eiks r + φ ( ks , μ ∗ , θ ∗ )e−iks r ⎤ ⎣ ⎦ The integral in equation (15) must converge since the inhomogeneity must have compact support, and must thus eventually become zero. This expression contains both

eiks r and e −iks r spherical waves. The incoming eiks r waves are disregarded based on causality and the strength of the remaining wave is doubled to keep the energy balance correct. Therefore, it follows that

β I iω e−iks r G p (r , ω ) = s o η (ω )φ k s , μ * , θ * Cp 4π r β s I oiω Cp

(

)

(

)

η (ω )φ ks , μ * , θ * G (r )

(

= Gtotal (r , ω )η (ω )φ k s , μ * , θ *

Cp

G G G p (ω , ω ) = Gtotal (ω , ω )φ (ω )η (ω )

(17)

to writing in the physical spatial domain

In the last line, we have made use of the fact that

G p (r , ω ) = Gtotal (r , ω )η (ω )φ ( ks , μ * ,θ * ) .

(18) G The function Gtotal (ω , ω ) is a transfer function in the allfrequency (time and space) domain. Equation (18) implies that Gtotal ( r , ω ) also behaves as a transfer function in the spatial domain. The statement as given in equation (18) is particularly important for several reasons. First, it is immediately obvious how a change in Green's function or its description can be incorporated. Furthermore, equation (18) states that the pressure function at any location is directly * * * * proportional to φ ( k s , μ , θ ) where φ ( k s , μ , θ ) is the

3D spatial Fourier transform of the heterogeneity function evaluated on the sphere ρ = ks = ω cs and at some angular values ( μ * ,θ * ) such that −1 ≤ μ ∗ ≤ 1, 0 ≤ θ ∗ ≤ 2π . Lastly, the Fourier rotation theorem can be used to gather more information about the spatial Fourier transform of the heterogeneity function on that sphere. The reader is G reminded that this property says that if F (ω ) is the multi-

G G f (r ) for any vector r , G G then the Fourier transform of f ( Rr ) is given by F ( Rω ) , dimensional Fourier transform of

where R is a rotation matrix. From symmetry, the Green's functions G (r ) and Gtotal ( r , ω ) are unaffected by rotation,

(16)

)

Gtotal (r , ω ) is defined in equation (11) as

β s I oiω

relationship between input and output should be one of convolution in physical space, not multiplication.

The implication of this last statement is that it is possible to go from writing in the spatial Fourier domain

∞ φ ( ρ , μ ∗ ,θ ∗ ) iρ r 1 = ∫ ρ 2 − ks2 e ρ d ρ (2π )2 ir −∞

where

the input and heterogeneity functions to yield the resulting output pressure in physical space. This is interesting because Gtotal (r , ω ) is actually a Green's function, implying the the

−1

−i ρ r iρ r φ ( ρ , μ ∗ , θ ∗ ) ⎡⎣e − e ⎤⎦ 2 = ρ dρ iρ r (2π )2 ∫0 ρ 2 − ks2

=

Gtotal (r , ω ) acts as a transfer function – that is it multiplies

a. Statement of the Photoacoustics Fourier Theorem

φ ( ρ , μ , θ ) iμρ r 2 ∫ ρ 2 − ks2 e ρ d ρ d μ dθ 0

∞

1

space. This is a sphere in Fourier space centred at zero and of radius k s . It should also be noted that in equation (16),

1 ∞

∫ ∫

(2π )3

= ks in spatial frequency

G (r ) . Equation (16) makes the important

G statement that the resulting pressure field at r is proportional to the Fourier transform of the heterogeneity

as is clearly the time-dependent η (ω ) . Therefore, the effect of rotation is to locate a different set of angles R( μ * ,θ * ) = μ R∗ ,θ R∗ , obtained via rotation, on the same

(

)

ρ = ks = ω cs sphere in spatial Fourier space such that G p ( Rr , ω ) = Gtotal (r , ω )η (ω )φ ( k s , μ R∗ , θ R∗ ) . (19) Similarly, making a measurement at the same angular position but at a different value of r will only produce a change in the proportionality constant given by Gtotal ( r , ω )

and yield the same spatial location on the heterogeneity Fourier transform sphere φ ( ks , μ * , θ * ) .

heterogeneity function will also be a bandlimited version.

Some special cases will now be individually considered. a. Special Case: Point Source at the Origin

5. SPHERICALLY SYMMETRIC OBJECT FUNCTIONS We consider a special case of equation (12) where the G Fourier transform φ (ω ) is spherically symmetric so that it is a function of ρ only and does not depend on the angles μ or θ . Under this assumption, the angular integration can be performed. In the special case of spherical symmetry of the heterogeneity function, then the expression in the spatial Fourier domain

G G G p (ω , ω ) = Gtotal (ω , ω )φ (ω )η (ω )

(20)

(21)

where φ ( k s ) is the 3D spatial Fourier transform of the heterogeneity function evaluated at

ρ = k s = ω cs .

Furthermore, from equation (11) it can be seen that the net effect of Gtotal in the time domain is a time delay from the

e −iks r term and a derivative from the iω term. Hence, from equation (21), the temporal pressure response will be a timedelayed version of the derivative of the temporal convolution of the input pulse with the heterogeneity function. One of the huge benefits of choosing the input pulse to be a diracdelta function is that convolution with the dirac-delta function returns the function itself. Therefore, if the input pulse is chosen to be a dirac-delta function, the pressure response observation will be almost a direct observation of the heterogeneity function itself.

Since in the special case of spherical symmetry there is no angular dependence of pressure, then the pressure need only be measured at a single point. For detection at a radius of r = rd , the pressure is given by

⎛ω ⎞ p (rd , ω ) = Gtotal ( rd , ω )η (ω )φ ⎜ ⎟ ⎝ cs ⎠

G

φ ( r ) = δ 3 (r ) =

1 δ (r) 4π r 2

(23)

is the dirac-delta function at the origin, written in spherical polar coordinates. More importantly, the Fourier transform of φ is 1 and equation (22) becomes

p (rd , ω ) = Gtotal ( rd , ω )η (ω ) .

(24)

Inverting equation (24) gives

becomes in the physical spatial domain

p (r , ω ) = Gtotal (r , ω )η (ω )φ ( k s ) ,

Consider the special case where there is a point source at the origin. In that case

(22)

By measuring the pressure at a single point, information about the Fourier transform of the heterogeneity function has been obtained at the point ρ = k s = ω cs . Therefore,

measurements at several different frequencies, ω , will need to be made in order to obtained sufficient information about the heterogeneity function in order to permit inverse Fourier inversion. Thus, the frequency content of the input pulse η (ω ) determines the Fourier plane coverage of the heterogeneity function and the resolution possible. If the input pulse is band-limited, then the reconstruction of the

p(rd , t ) =

⎛ r ⎞ βs Io η′⎜ t − d ⎟ .

8π rd C p ⎝ cs ⎠

(25)

In this case, the pressure response is a time-shifted version of the derivative of the input pulse that is detected at an arbitrary point. b. Special Case: Point Source at the Origin and Input DiracDelta Pulse Consider the special case where there is a point source at the origin and the input pulse is also a a dirac-delta function occuring at t0 so that η (t ) = δ (t − t0 ) → η (ω ) = e

− iωt0

.

In this case, the measured pressure is simply the Green's function itself

p(rd , ω ) = Gtotal ( rd )η (ω ) =

β s I oiω e−iks rd −iωt0 . e C p 4π rd

(26)

This can be easily inverse Fourier transformed to the timedomain to yield

p (rd , t ) = −

β s Io 4π rd C p ⎛

1

⎛

δ ⎜ t − t0 −

rd ⎞ ⎝ ⎜ t − t0 − ⎟ cs ⎠ ⎝

rd ⎞ ⎟ (27) cs ⎠

where the result δ ′( x) = − δ ( x) x has been used. Equation (27) is a scaled, time-shifted delta-dirac pulse, with the magnitude of the time-delay given by t0 + rd / cs . c. Special Case: Gaussian Heterogeneity Function It is assumed that the heterogeneity function is a (spherically symmetric) Gaussian, so that

φ (r ) =

1

e− ( r / a )

a π

2

(28)

for some positive a. The 3D Fourier transform of this can be found to be

φ ( ρ ) = e

−

a2ρ 2 4

⎛ω ⇒ φ ⎜ ⎝ cs

⎞ − ⎟=e ⎠

a 2ω 2 4 cs2

.

(29)

It then follows that equation (22) becomes

proportional to a is a pulse of width

⎛ω ⎞ p (rd , ω ) = Gtotal ( rd , ω )η (ω )φ ⎜ ⎟ ⎝ cs ⎠ =

β s I oiω e Cp

− ird

ω cs

4π rd

−

e

(30)

a 2ω 2 4 cs2

η (ω )

The temporal pressure response will be a time-shifted version of the derivative of the convolution of the input temporal pulse with the guassian heterogeneity function.

If it is also assumed that the input pulse is a Dirac-delta dunction so that η (ω ) = 1 , then equation (30) can be inverse transformed back into the time domain to give

β I c 2 ( c t − r ) −⎜ t − c ⎟ p(rd , t ) = − s o3 / s2 s 3 d e ⎝ s ⎠ π rd C p a

a

2

(31)

This is a time-shifted version of the first derivative of the input guassian. For an assumed heterogeneity with a width related to a, the time-response has a width related to a / cs .

This is not an unreasonable result and it is emphasized that this assumes a dirac-delta temporal input. d. Special Case: Gaussian Heterogeneity Function and Guassian Input Pulse

η (t ) =

b π

e

− b 2t 2

⇒ η (ω ) = e

−

b 2ω 2 4

.

(32)

Substituting into equation (30) and inverse Fourier transforming back into the time domain yields

( cst − rd ) cs ⋅ βs Io p (rd , t ) = 3/ 2 C pπ rd c 2b 2 + a 2 3 / 2 s 2

(

⎛ ( c t − r )2 ⎞ ⋅ exp ⎜ − 2s 2 d 2 ⎟ ⎜ cs b + a ⎟ ⎝ ⎠

)

comparison to the case where the input pulse is a dirac-delta function, the pressure response pulse has been widened by the guassian nature of the input pulse, in direct proportion to the width of the input pulse. However, any uncertainty in the precise width of the input pulse would imply uncertainty into the size of the heterogeneity being detected. If b a / cs so that b is negligible compared to a / cs , the effect of the width of the input pulse will have very little effect on the pressure response and can essentially be ignored. It is noted that this condition is the same as that for the often-mentioned requirement of photoacoustic stress confinement.

The Fourier transform of the Heterogeneity function can be isolated to give

⎛ω ⎝ cs

φ ⎜

⎞ p (rd , ω ) C p 4π rd + iks rd . e ⎟= η (ω ) β s I oiω ⎠

Or

1 ⎛ ω φ⎜ cs3 ⎝ cs

⎞ p (rd , ω ) C p 4π rd e ⎟= 3 ( ) η ω β ω c I i ⎠ s s o

ird ω cs

(34)

(35)

The left hand side of equation (35) is written to make it clear that the scale properties of the (3D) Fourier transform will be used. Taking the inverse Fourier transform of both sides of equation (35) and using the scale properties of the 3D Fourier transform gives

It is now assumed that the input pulse is not a true delta function but is also a guassian pulse so that

1

a2 b + 2 . In cs 2

6. SOLVING FOR THE HETEROGENEITY FUNCTION

2

rd ⎞ cs2

⎛

equation (31), which is reasonable since a delta pulse can be approximated with a narrow enough gaussian pulse.

However, the potential problem with equation (33) is that if the width of the input temporal pulse is not known very well, then it will be difficult or impossible to extract the exact width of the heterogeneity gaussian, which is typically the quantity being sought in medical imaging applications. From equation (33), it can also be seen that the response to an input pulse of width proportional to b and a heterogeneity of width

.(33)

The functional form in equation (33) is very similar to that in

⎧⎪ p(r , ω ) ird ω ⎫⎪ d φ ( cs r ) = 3 e cs ⎬ ⎨ η ω i ω ( ) cs β s I o ⎪⎩ ⎪⎭ (36) ird ω ⎧ ⎫ 2C p rd −1 ⎪ p(rd , ω ) cs ⎪ F1D ⎨ =− 3 e ;ω → r ⎬ cs β s I o r ⎪⎩ η (ω ) ⎪⎭ C p 4π rd

F3−D1

This can be written as

φ (r ) = −

⎛ r + rd Q ⎜ cs2 β s I o r ⎝ cs

2C p rd

⎞ ⎟. ⎠

(37)

where

⎧ p(r , ω ) ⎫ Q( r ) = F1−D1 ⎨ d ;ω → r ⎬ . ⎩ η (ω ) ⎭

presented and proved. This theorem is important in that it forms the basis for most reconstruction algorithms.

(38) ACKNOWLEDGMENT

Note that if the input pulse is a delta function so that η (t ) = δ (t ) → η (ω ) = 1 , then

Q( r ) = F1−D1 { p (rd , ω ); ω → r} = p (r )

(39)

In other words, for a delta function input pulse, the pressure pulse as measured as a function of time, is directly the shape of the inhomogeneity function after appropriate scaling by 1 r.

7. DISADVANTAGE OF THE DIREC-DELTA PULSE It was noted in the discussion on the example with the guassian heterogeneity function that if the input pulse is also a gaussian, it may be difficult to discern the size of the heterogeneity function if there is any uncertainty in the width of the input pulse. It has also been observed that if the inputpulse is a Dirac-Delta pulse that observations made are almost directly those of the heterogeneity function itself. So, in a certain sense, the choice of input pulse as a Dirac-Delta pulse is optimal. However, perfect delta functions are difficult to produce in real-world conditions and it is more likely that the input pulse begins to resemble a gaussian function as the input pulse acquires a finite width. If the heterogeneity function is also roughly assumed to be spherical with a gaussian-like radial profile, which is a reasonable approximation to a heterogeneity , then it becomes apparent from the previous discussion that any lack of information regarding the length of the input pulse will become uncertainty in the true size of the heterogeneity function. Hence, it is noted that there is an inherent conflict between the system's ability to resolve inhomogeneities and the system's ability to detect weak signals from deep inhomogeneities. The reason that is shorter pulses enable an almost direct observation of the heterogeneity function with little uncertainty and also the ability to discriminate between closely spaced heterogeneities. However, longer pulses will allow the system to absorb energy over a longer period of time and thereby distinguish between the photoacoustic signal from the heterogeneity and other interfering energy sources. That is, longer pulses tend to increase the signal-tonoise ratio.

8. SUMMARY In this paper, a consistent and unified approach to the photoacoustic problem has been presented using a Green's function/transfer function approach. Spherical polar coordinate systems were considered and the solutions for special cases of each were presented. Most importantly, a generalized version of the Fourier slice theorem was

This work was financially supported by the Natural Science and Engineering Research Council of Canada (NSERC).

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