CAPITULO II. LOS CARTELES Y ACUERDOS HORIZONTALES VISTOS DESDE EL
3.4 Regla de la Razón y su aplicación al Cartel
Photoacoustic imaging is a hybrid imaging technique based on the photoacoustic (PA) effect. The photoacoustic effect was discovered over a 130 years ago by Alexander Graham Bell during experiments with sunlight51. Bell noticed a distinct sound produced by sudden exposure of materials to sunlight. While the underlying principles are the same, today, photoacoustic imaging typically employs lasers or high power LEDs for illumination and ultrasonic detectors, or transducers, for signal detection.
In general, PAI involves illumination of a turbid imaging target by either a pulsed or modulated light source. Upon illumination, photons are diffusely scattered and absorbed in the target. Volumes inside the target with relatively high absorption coefficients preferentially absorb the light energy and undergo slight but rapid heating, followed by near instantaneous thermo-elastic expansion. The expansion generates a spatially varying three dimensional pressure distribution which is proportional to the locally absorbed energy. Shortly after the illumination is reduced, the pressure is dissipated as an acoustic transient wave in the ultrasound frequency range. To generate images, acoustic transducers of appropriate frequency range, or bandwidth, surround the imaging target and detect the dissipating transient waves. By utilizing an image reconstruction method such as back-projection, the detected signals can be used to solve for the original pressure, and hence absorber, distribution52.
Photoacoustic wave theory:
(Note: Information provided in the remainder of this section is based on “Biomedical Optics: Principles and Imaging”, L.V. Wang and H.I. Wu, 2007)
To gain further understanding of how properties of the PA signal-carrying wave depend on the illumination and resultant pressure distribution, it is instructive to consider a fully symmetrical and homogeneous absorber, such as the dark gray sphere of radius R in Figure 1.5a. It is straight forward to show that in order to generate acoustic waves with a waveform and amplitude that is a good representation of the locally absorbed light energy, two conditions must be met. First, the heat generated
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during the illumination phase must not have sufficient time, referred to as thermal relaxation time τth, to diffuse out of the region of interest before the illumination is reduced. This is referred to as thermal confinement and the thermal relaxation time is given by
τ
th=
𝑑𝑐24𝐷𝑇
1.1
where dc is the size (mm), or characteristic dimension, of the region of interest, and DT is the thermal diffusivity of the material of interest. Therefore, for the sphere of radius R, and a thermal diffusivity similar to soft tissue of DT ~ 0.1 mm2/s, the thermal relaxation time is τth = R
2
/0.4 s.56 This means that for an accurate thermal response profile of a 1 mm sphere, the illumination time must not exceed 1/0.4 = 2.5 s. This can be easily achieved using conventional illumination sources; however, the other condition, namely stress confinement, is more stringent. Stress confinement implies that the thermo-elastically induced pressure wave does not have sufficient time, called stress relaxation time τs, to travel out of the region of interest before the illumination is reduced. The stress relaxation time is given by
τ
s = 𝑑𝑐𝑣𝑠
1.2
where vs is the speed of sound in the material of interest, typically around 1.5 mm/µs for soft tissue. Again, considering the soft tissue sphere with radius R = 1 mm implies that to satisfy stress confinement the illumination time must not exceed τs = 1/1.5 µs, or about 0.7 µs. This can typically be achieved using pulsed lasers and LEDs, as well as some arc lamp sources.
According to fluid mechanics, the fractional volume change dV/V of a region of interest expressed as a function of local change in pressure at constant temperature Δp and local change in temperature at constant pressure ΔT, can be approximated by
𝑑𝑉
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where β is the thermal coefficient of volume expansion and κ is the isothermal compressibility. However, if a very short illumination time τi is implemented then the thermal and stress confinement requirements, described by Equations 1.1 and 1.2, will be significantly exceeded (i.e. τi << τth
and
τi << τs ). This suggests that the fractional volume change during illumination is negligible, or dV/V ~ 0. Therefore, Equation 1.3 can be written as∆𝑝 =
𝛽∆𝑇𝜅
1.4
Equation 1.4 quantifies the local pressure reached as a function of the temperature rise during the illumination pulse. If we further assume that all absorbed light energy was converted into heat, then the temperature rise ΔT can be expressed as a function of the locally absorbed light through
𝛥𝑇 =
𝜇𝑎𝐹𝜌𝐶𝑣
1.5
where µa is the local optical absorption coefficient, F is the local optical fluence, ρ is the local physical density and Cv is the specific heat capacity at constant volume. Using Equation 1.5 the initial local pressure reached can be expressed as a function of the local fluence and absorption coefficient,
∆𝑝 =
𝛽𝜇𝑎𝐹𝜅𝜌𝐶𝑣
= ΓµaF
1.6where Γ = β / κρCv is the dimensionless Grueneisen parameter which parametrizes the material’s inherent photoacoustic energy-conversion efficiency, and is estimated to be around Γ ~ 0.24 for soft tissues.56 The studies undertaken in this work involved soft breast tissues, known to have an optical absorption coefficient near µa ~ 0.06 cm-1. As a result, given the utilized fluence of F ~ 4 mJ/cm2, the expected pressure generated near the surface was ~ 58 Pa.41
Equation 1.6 implies that knowledge of the local fluence F coupled with pressure measurements Δp, can be used to estimate the local absorption coefficient µa. As discussed above, the value of µa is of
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significant importance in the characterization of various tissue types. Indeed Figure 1.4 clearly demonstrates the potential differentiation of tissue constituents based solely on absorption coefficient values.
Nevertheless, Equation 1.6 tells us nothing about how the initial local pressure within some sample can be related to the time-resolved acoustic pressure measurements, which arrive and are collected later in time.
In order to relate the non-local pressure wave measurements to the local initial pressure distribution, the general wave equation describing propagation of pressure waves under thermal confinement is used
( ∇
2–
1 𝑣𝑠 2 𝜕2 𝜕𝑡2) p(r, t) = −
𝛽 𝐶𝑝 𝜕𝐻 𝜕𝑡1.7
where H is the heating function which depends on the temperature, and via Equation 1.5, the optical fluence, and is given by
H( r,t) = ρC
v 𝜕𝑇(𝐫,𝑡)𝜕𝑡 1.8
The right side of Equation 1.7 describes the initial pressure as a function of position and time, and it is clear that in order to generate a wave (i.e. ∂H/∂t ≠ 0) the heating function must not be constant. Once again, this necessary condition is satisfied as long as the optical energy is converted into heat under thermal confinement.
Equation 1.7 is an inhomogeneous differential equation and the solution, typically provided by invoking Green’s function for a δ impulse source, is given by
p(r,t) =
1 4𝜋𝑣𝑠 2 𝜕 𝜕𝑡[
1 𝑡𝑣𝑠∫ drʹ p0(rʹ) δ
(
t -
|𝐫−𝐫ʹ| 𝑣𝑠)]
1.9
Equation 1.9 gives the pressure p at location r and time t, given an initial pressure p0 at location rʹ and
time tʹ= t - |𝐫−𝐫ʹ|
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target, in conjunction with reconstruction algorithms such as back-projection, Equation 1.9 can be used to recover the original pressure and hence absorber distribution. The equation essentially models the propagation of acoustic pressure waves and so can provide an estimate of the expected pressure distribution measured at one location given an initial pressure distribution at a different location.
For example, consider again the dark-gray uniform sphere of radius R, shown in Figure 1.5a. Below the sphere is a plot of the time dependent pressure signal that is representative of homogeneous illumination by the light source L, followed by measurements by the acoustic transducer T. In accordance with the photoacoustic effect and Equation 1.9, the sphere produces an N-shaped signal whose temporal width τ is proportional to R, and whose amplitude is proportional to the optical density, or absorption coefficient, represented by the dark shade of gray. It therefore follows that a sphere of radius 2R and half the optical density, represented by the light shade of gray in Figure 1.5b, will produce a signal with twice the temporal width 2τ and half the amplitude. This example serves to illustrate two critical properties of the photoacoustic effect:
(1) The absorber dimension along the line of acoustic wave detection is proportional to the period, and therefore inversely proportional to the frequency, of the emitted transient wave.
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(a) (b)
Figure 1.5. Representative examples illustrating the expected time-dependent PA waveforms for varying sizes and optical densities of absorbers. Figure (a) shows a dark-shade sphere of radius R along with the expected signal measured at T after homogeneous illumination by L. Figure (b) shows a light-shade sphere of radius 2R. The expected results demonstrate that object size is inversely proportional to PA signal frequency, while object optical density is directly proportional to PA signal amplitude.
While these properties seem relatively simple, they have profound consequences on PAI system design and performance. For example, property (1) implies that complex imaging targets may emit acoustic transients of very wide range of frequencies, while property (2) suggests that such targets may simultaneously emit a wide range of amplitudes. Consequently, designing an effective PAI system requires careful consideration of the intended imaging targets, ensuring that the utilized detector electronics have sufficient dynamic range in both the amplitude and frequency domains. Insufficient range will likely result in images showing a distorted or inaccurate absorber distribution, or may result in images with missing absorbers.
2τ τ
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