Download Report

University College London
CoMPLEX
Mini Project 1
Prof. Andrew Taylor
Dr. Adrien Desjardins
Photoacoustic sensing of lipids
January 27, 2016
Christoph Sadée(15084362)
Abstract
A 1210nm laser was used to obtain a signal by using the photoacoustic effect from
CNT-PDMS, fat and muscle tissue from the cow and liver tissue from chicken. The signal
was significantly stronger for the CNT but good alignment showed that a reasonable signal
could also be obtained with muscle tissue. A model was made to attempt describing the
behaviour but further effort has to be made to accurately describe the phenomenon.
CONTENTS
Christioph Sadée
Contents
1 Introduction
3
2 Theory
2.1 EM absorption mechanisms . . . . . . . . . . . . . . . . . . .
2.1.1 Absorption in the microwave domain - thermoacoustic
2.1.2 Absorption in the NIR domain - photoacoustic [14] . .
2.2 Acoustic signal . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Acoustic Signal generation . . . . . . . . . . . . . . . .
2.2.2 Acoustic signal propagation[14] . . . . . . . . . . . . .
2.2.3 Acoustic wave absorption . . . . . . . . . . . . . . . .
2.3 Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Light Diffusion model . . . . . . . . . . . . . . . . . . .
2.4 Acoustic wave absorption . . . . . . . . . . . . . . . . . . . . .
3 Experiment
3.1 Setup . . . . . . . . . . .
3.2 Procedure . . . . . . . .
3.3 Results and Discussion .
3.3.1 Signal . . . . . .
3.3.2 Pulse Compressed
. . . .
. . . .
. . . .
. . . .
Signal
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
5
7
10
11
11
12
13
14
14
15
.
.
.
.
.
16
16
17
19
19
21
4 Model
22
5 Clinical Application
23
6 Conclusion
24
7 Afterword
25
References
25
2
2 THEORY
1
Christioph Sadée
Introduction
Photoacoustic imaging uses an amplitude modulated signal to drive a laser in the near infrared
range. Due to the absorption of light in the medium, the sample is heated. The alteration in
heating causes the tissue to contract and expand and different rates, building up thermal stress.
The thermal stress results in a pressure wave that can be detected with a transducer. The aim
of this report is find such a signal from biological tissues particularly fat and a fibre coating
material.
2
Theory
The opto-/ photo-/ thermoacoustic effect describes the generation of sound in a sample by means
of absorption of an amplitude modulated (AM) electromagnetic source. An electromagnetic
(EM) source in this case can be anything from a Laser diode up to a microwave antenna. When
an electromagnetic wave is incident on a tissue sample, different absorption mechanisms can
take place, depending on the frequency of the incident light. The amount of absorption is
dependent on the chemical composition of the sample and in some cases such as crystals the
angle of incidence[6]. The absorbed energy at the frequencies used is mainly converted to heat
and dissipated in the medium (luminescence is another mechanism that can take place after
absorption). The continuous irradiation of a sample will not lead to the formation of an acoustic
signal only heating. Instead an AM electromagnetic source is required. This represents a change
in amplitude (strength of EM) mostly based on another sinusoidal function. This can be seen in
Figure 2.1, where a source signal with a frequency f1 (frequency of laser) was modulated with
another sinusoidal signal with frequency f2 to obtain an AM signal.
Figure 2.1: Amplitude Modulated Signal[13]
Source and modulation signal from the above figure can both be described by:
Source →
Modulation →
yS = AS sin(2πf1 t)
yM = AM sin(2πf2 t)
The AM Signal is then given by:
3
Christioph Sadée
yAM = [1 + yM ] · yS + k
(2.1)
where k = Offset for continuous output
When an AM Signal is incident on a sample more energy will be deposited at the crests
of the modulation signal as at the troughs of the modulation signal. This leads to different
heating within the sample over time and therefore different thermal expansion. The different
thermal expansion and relaxation in time leads to thermal stresses which generate a pressure
wave. It’s due to this pressure wave propagating outwards that one can obtain an acoustic
signal, generated by absorption of EM radiation. Hence the name photo-/ thermoacoustic effect.
Thinking one step further one should note that since more energy is deposited on the crests
of the modulation signal than on the troughs of the modulation signal, the pressure wave that
will result from this alteration will approximately have the same frequency as the modulated
signal yM with frequency f2 . Hence there will be a pressure wave propagating through the
medium with the same frequency as the modulation signal. If one now alters the frequency of
the modulation then one should also see a change in the pressure wave frequency. One possibility
to continuously alter the frequency of the modulation is by using a linear chirp
fmax − fmin 2
t
(2.2)
yM = Am sin 2π fmin t +
2kf
where fmin = minimum frequency
fmax = maximum frequency
kf = rate of frequency increase
Equation (2.2) increases the modulation frequency from a minimum frequency to a maximum
frequency within a time span of kf . An example of this is given in Figure 2.2, with the following
parameters: fmin = 1MHz, fmax = 25MHz and kf = 5µs.
Figure 2.2: Chirp modulation
One might ask what is the benefit of a chirp modulation as compared to a single frequency
modulation. For this look at the behaviour of an EM wave that gets absorbed in a medium.
Absorption of light in a medium is highly dependent on the frequency of the carrier signal in
response to the medium, for example lipids have an absorption peak at 1210[nm] but have really
low absorption at about 700[nm], the wavelength is related to the frequency via f = c/λ. This is
4
2 THEORY
Christioph Sadée
the same with acoustic waves, some materials will absorb acoustic waves more than others at
certain frequencies. Since the acoustic wave frequency originates from the modulation signal, a
chirp modulation will introduce a frequency range and as it is propagating through a medium
certain frequencies will be more absorbed than others, resulting in an amplitude decrease at
certain frequencies (best seen through a power spectrum). As mentioned the absorption of the
acoustic wave is linked to the frequency response of a medium, hence different media can be
characterised by their different frequency responses. Therefore tissues can be characterised by
1. Their frequency dependent absorption of EM - radiation
2. Their frequency dependent absorption of acoustic waves
Hence the possibility for photoacoustic sensing of lipids. There is also a negative impact of
acoustic frequency dependent absorption in media. Consider an optical fibre which is coated
with a very absorbing material within the NIR range.
Figure 2.3: Coated optical fibre
Now when a chirp modulated light in the NIR range travels through the fibre and is absorbed
in the coating, then the generated acoustic wave will be more absorbed at certain frequency
than others. This results in a non flat power spectrum of the acoustic frequencies. But it would
be of much more use to in this case have a flat power spectrum. In order to prevent this one
can modify Equation (2.2), to obtain a flat power spectrum:
yM
fmax − fmin 2
= Am (f ) sin 2π fmin t +
t
2kf
(2.3)
All that Equation (2.3) is saying is that AM is now a function of frequency as well. This
function has to be found for the material and should allow for an increase in AM at frequencies
where more absorption is taking place, essentially balancing out the effect. AM (f ) can also be
used to cancel out any non linear responses of the laser when driven by an AWG.
2.1
EM absorption mechanisms
As previously mentioned, the absoprtion of EM radiation in tissue is highly dependant on
the frequency. Different mechanisms of absorption take place at different wavelengths, this is
summarised in Figure 2.4 and in relation to wavelength and frequency in Figure 2.5.
5
2.1 EM absorption mechanisms
Figure
2.4:
Mechanisms[10]
Christioph Sadée
Absorption
Figure 2.5: EM Spectrum[10]
1) Molecular rotation and torsion [8]: This type of absorption mainly takes place at microwave frequencies which are in the range of 300MHz up to 300GHz and a wavelength range
from 100cm up to 0.1cm. Due to molecular rotation and torsion EM energy is converted to
kinetic energy which causes heating within the sample. This absorption can be described with
Maxwell’s Equation for dispersive media, incorporating a Debye or a more accurate Cole media
model. This type of absorption is part of the broadly used term thermoacoustics.
2) Molecular vibration[8]: This type of absorption mainly takes place at infrared frequencies
which are in the range of 300GHz up to 430THz and a wavelength range from 1mm up to
750nm. This also includes the near infrared region (NIR) which has a frequency range of 214THz
up to 400THz and a wavelength range from 750nm to 1400nm and will be the main focus
later in this report. EM energy is converted via molecular vibration to kinetic energy and
then dissipated as heat. Absorption of NIR radiation in Biological tissue is weak and hence
considerable penetration can be achieved, allowing deep imaging, this is called the Near Infrared
Window. The main limiting factor is scattering, when light is deflected and a change in direction
occurs, which increases the path travelled and therefore the likelihood of absorption. Due to
scattering, a diffusion based model can describe the bulk interaction. This type of absorption is
used in photoacoustics a sub group of thermoacoustic.
3)Electron level changes[8]: This type of absorption mainly takes place in the visible region
which are in the frequency range of 430THz up to 770THz and a wavelength range from 700nm
up to 430nm. Light is absorbed via excitation of an atom or molecule (no ionisation). The
energy is then either released by motion of the molecules leading to heating within the sample
or by radiative relaxation such as luminescence where the energy is re-emitted by relaxation of
an electron from the excited state to the ground state. Luminescence does not cause heating
of the sample, but the re-emitted photons can be absorbed again. In Biological tissue a great
amount of scattering is taking place as well, and the penetration depth is very shallow and
therefor not often used in Medical imaging.
4)Photoionisation and higher[8]: UV light and X-rays have enough energy to cause ionisation,
which can lead to breaking chemical bonds and free radicals. In biological tissue this type of
radiation is mainly damaging but still used for imaging due to the great penetration depth.
Even higher energies lead to Compton scattering and Pair production, used in radiotherapy and
PET scans.
6
2 THEORY
2.1.1
Christioph Sadée
Absorption in the microwave domain - thermoacoustic
Microwaves mainly loose energy due to molecular rotation and torsion in biological tissue and
are not noticeably scattered. Therefore Maxwell’s equations can be used to describe the problem
[4]
∇ · E = ρ/0
∂B
∇×E=−
∂t
∇·B=0
(2.4)
(2.5)
∇ × B = µ0 0 J + µ0 0
(2.6)
∂E
∂t
(2.7)
where E = lectric field
B = magnetic field
J = current density
ρ = charge density
0 = electric permittivity of vacuum
µ0 = magnetic permeability of vacuum
Faraday’s law Equation (2.5) and Ampére’s law Equation (2.13) describe the propagation of
an electromagnetic wave, combining Equation (2.6) to (2.13) gives the wave equation for either
field, i.e.:
∂2
2 2
c ∇ − 2 E=0
∂t
|
{z
}
(2.8)
d’Alembert operator
where c = √
1
0 µ0
So far the expressions would only describe the propagation of an EM wave through vacuum,
they would not take into account the different electric permittivity values of tissue and it’s
absorbing nature. Hence a more complex version of Ampere’s law and Faraday’s law is required.
A medium like tissue is a dielectric and can therefore be polarised when an EM wave passes
through it, it’s polarisation is highly dependent on the frequency. Therefor the medium is said to
be dispersive. The dispersion arises from the fact that the polarization and the current density
cannot follow the rapid change of the electromagnetic field at high frequencies, which then leads
to absorption of energy in the medium. This is represented with the electric displacement field
in the frequency domain being out of face with the electric field Ê, like a phasor (”hat” denotes
the frequency domain):
D̂(r, ω) = 0 ∞ Ê(r, ω) + P̂(r, ω) = 0 [∞ + χ̂e (ω)] Ê(r, ω)
|
{z
}
c (ω)
7
(2.9)
2.1 EM absorption mechanisms
Christioph Sadée
where D̂ = dielectric displacement field
P̂ = 0 χ̂e (ω)Ê = electric polarization
∞ = relative permittivity at upper end of frequency band
c = complex permittivity
χ̂e = electric susceptibility
Note that there is an equivalent version of Equation (2.9) for magnetic dispersion, but this
is negligible for biological tissue, hence the magnetic field changes to B = µ0 µr H, where µr is
the relative permeability of the medium considered to be constant over the applied frequency
range. The electric susceptibility in Equation (2.9) describes the degree of polarisation of the
medium in response to an applied electric field, it is a complex quantity. Since Equation (2.9) is
in the frequency domain one must either convert Maxwell’s equations to the frequency domain
or transform Equation (2.9) into the time domain. Considering the latter, one has to realise
that the product of two frequency dependent quantities in Equation (2.9) leads to a convolution
integral in the time domain [6]
D(r, t) = 0 [∞ + χe (r, t)] ∗E(r, t)
|
{z
}
(2.10)
(r,t)
t
Z
E[r, (t − τ )]χe (r, τ )dτ
D(r, t) = 0 ∞ E(r, t) + 0
(2.11)
0
Applying this to Ampere’s and Faraday’s law, gives their dispersive media equivalent
∂H
(2.12)
∂t
∂E
(2.13)
∇ × H = (r, t) ∗
∂t
Next an analytic expression for χe (r, t) is required, describing the material behaviour. P.
Debye [3] was the first to propose an exponential decay, allowing for the lagging of the polarisation
behind the applied field. Those media are called Debye media.
∇ × E = −µ0 µr
χe (t) = χe (0)e−t/τ0 u(t)
(2.14)
where u(t) = Heaviside step function
τ0 = relaxation time specific to media
To find χe (0) sub Equation (2.14) into the complex permittivity expression in Equation (2.9)
by using a Fourier transform:
Z
c (ω) = 0 ∞ +
∞
−jωt
χe (t)e
|
{z
χ̂e
χe (0)τ0
c (ω) = 0 ∞ +
1 + jωτ0
dt
0
8
}
(2.15)
2 THEORY
Christioph Sadée
when ω = 0
c (0) = dc 0 = 0 inf + 0 τ0 χe (0)
where dc denotes the relative permittivity at the lower frequency band when ω = 0,
rearranging yields:
dc − ∞
τ0
χe (0) =
(2.16)
Plugging into Equation (2.14) gives the required expression for the time domain susceptibility
χe (t) =
dc − ∞ − t/τ0
e
H(t)
τ0
(2.17)
And the complex permittivity in the frequency domain is given by:
c (ω) = 0
dc − ∞
∞ +
= 0 [0r − j00r ]
1 + jωτ0
(2.18)
where 0r and 00r represent the real and imaginary relative permittivity respectively. The
most absorption for this medium will take place when
τ0 =
1
1
=
ω
2πf
or
0r = 00r
For a medium like water this corresponds to a microwave frequency of about 10GHz at 0◦ C,
which then corresponds to τ0 ≈ 3.18 · 10−10 , dc − ∞ is a scaling factor, stating how much is
absorbed. Unfortunately Equation (2.18) is not enough to model the behaviour of tissue which
can have multiple absorption peaks within the microwave domain. Therefore multiple Debye
poles should be considered, introducing several peak absorptions at different frequencies, in
addition an electric conductivity term can be added[6]:
(ω) = 0
M
X
∆m
∞ +
1 + jωτ0m
m=1
!
+
σ
jω0
(2.19)
where M = Debye poles
∆m = dc,m − ∞
One further complication is required to describe biological tissue accurately [11], proposed
by Cole [1], the introduction of an exponential parameter α
(ω) = 0
M
X
∆m
σ
∞ +
+
1−α
m
1 + (jωτ0m )
jω0
m=1
!
(2.20)
α takes a value between 0 and 1 and essentially stretches the relaxation, it is an empirical
formula which can be fitted to describe the dielectric behaviour of tissue. Several tables are
available spanning the behaviour of various tissues from 10Hz up to 100GHz. For example
http://niremf.ifac.cnr.it/docs/DIELECTRIC/AppendixC.html obtained the following fit
for fat:
9
2.1 EM absorption mechanisms
Christioph Sadée
Figure 2.6: Cole model fit to experimental data for fat, from 10Hz up to 100GHz[5]
2.1.2
Absorption in the NIR domain - photoacoustic [14]
Scattering of light plays a major role in the NIR domain in turbit media (i.e. human tissue),
which doesn’t allow the modelling of light using Maxwell’s equation, instead light behaviour can
be described as a diffusive process in a bulk medium. But there are important considerations
for this model. First consider that the total attenuation coefficient is given by∗ :
µt = µa + µs
(2.21)
where µa = absorption coefficient
µs = scattering coefficient
Those parameters are related to the mean free paths of total attenuation 1/µt , absorption
and scattering 1/µs . The mean free paths refer to how long a photon can travel before it
is attenuated, absorbed or scattered. It is important for the validity of the scheme that the
mean free path of total attenuation is much smaller than the dimensions of the problem and
that the photons are scattered many times before being absorbed (or leaving the medium). The
dimensions of the problem is the size of the medium to be analysed, i.e. distance from one
boundary to another, which must have a certain size above 1/µt since the diffusion only applies
to the bulk but doesn’t describe processes at the molecular level. The sufficient scattering is
required since otherwise no diffusion of light would take place. The diffusion of light is then
given by the diffusion equation
1/µa
1 ∂φ(r, t)
= ∇ · [D∇φ(r, t)] − µa (r)φ(r, t) + S(r, t)
|
{z
} | {z }
c ∂t
Absorption
∗
Source
Please note that prior notation such as µ0 = magnetic permeability doesn’t apply in this chapter
10
(2.22)
2 THEORY
Christioph Sadée
where c = Speed of light in the medium
φ = fluence rate
1
D = diffusion coefficient =
3µtr
µtr = transport attenuation coefficient = µa + µs (1 − g)
g = dimensionless anisotropy
S = source term
The fluence rate φ is defined as Ref[page 34 book]: ”The radiant power incident on a small
sphere, divided by the cross-sectional area of that sphere.” Note that in the case of uniform
irradiation of the surface and a non time varying source, Equation (2.22) simplifies to an ODE
and a 1D steady state problem, who’s solution exhibits the behaviour of the Beer-Lambert Law,
essentially exponential decay [8]:
φ(r) =
φ(r0 ) −µef f r
e
2πDr
where µef f = effective optical absorption =
(2.23)
p
3µa µtr
The absorption term in Equation (2.22) is the energy that is absorbed by the tissue, this is
then either converted to heat or re-emitted by the process of luminescence. The source term in
Equation (2.22) is the laser light incident on the sample.
2.2
2.2.1
Acoustic signal
Acoustic Signal generation
When a short laser pulse passes through a medium then the fractional volume expansion caused
by it at position r is given by [14]
dV
= −κp(r) + βT (r)
V
(2.24)
where κ = compressibility = 1/ρvs2
ρ = density
vs2 = speed of sound in medium
β = thermal coefficient of volume expansion
(2.25)
If the length scale of the heated volume is large compared to the acoustic confinement time
then an acoustic impulse response to the heating can be assumed and the fractional volume
expansion can be neglected:
p0 (r) =
βT (r)
κ
11
(2.26)
2.2 Acoustic signal
Christioph Sadée
Ref[] states that approximately a 1mK increase in temperature in tissue gives rise to an
about 800Pa pressure rise. Next the temperature rise T (r) is related to the local absorption
given by the absorption term in Equation (2.22).
Z +∞
1
1
Wa (r)
=
µa (r)φ(r, t)dt =
µa (r)ψ(r)
(2.27)
T (r) =
ρCv
ρCv −∞
ρCv
where Wa = specific optical absorption
ψ = optical fluence
Cv = specific heat capacity at constant volume
Combining Equation (2.27) and (2.26), gives
po (r) =
β
Wa (r) = ΓWa (r)
κρCv
(2.28)
Γ is the so called Grüneisen parameter, it describes the efficiency with which heat energy is
converted to pressure in a given medium, it is a dimensionless quantity and more often expressed
with the specific heat capacity at constant temperature
Γ=
2.2.2
βvs2
Cp
(2.29)
Acoustic signal propagation[14]
An acoustic signal can in it’s most simple form be described as a pressure wave propagating
within an inviscid medium (zero viscosity). The behaviour is given by the d’Alembert operator
as already used in Equation (2.8):
∂2
2 2
c ∇ − 2 p(r, t) = 0
(2.30)
∂t
For the photoacoustic effect a source term must be added which is generated within the
domain of the wave equation and not externally†
∂2
β ∂ 2T
2 2
c ∇ − 2 p(r, t) = − 2 2
(2.31)
∂t
κvs ∂t
| {z }
pressure source
∂2T
Next an expression for
/∂t2 must be found. Using the heat equation but neglecting the
heat-diffusion part since the laser pulse is very short:
∂T
1
) +
∇
· (D∇T
=
µa (r)φ(r, t)
∂t
ρCp
(2.32)
Denote µa (r)φ(r, t) = H(r, t), then differentiating in time and subbing into Equation (2.31)
gives:
∂2
β ∂H(r, t)
2 2
c ∇ − 2 p(r, t) = −
(2.33)
∂t
Cp
∂t
†
for ultrasound generated externally one can impose a boundary condition to simulate the source
12
2 THEORY
Christioph Sadée
An analytic solution exists for Equation (2.33) using Green’s function:
Z
|r − r0 |
β ∂
1
0
0
H r ,t −
p(r, t) =
dr
4πCp ∂t
|r − r0 |
vs
2.2.3
(2.34)
Acoustic wave absorption
Equation (2.30) is a very idealised version of an acoustic wave propagating even in an inviscid
medium. Acoustic waves get attenuated, either by absorption or scattering. The attenuation is
a function of frequency and increases with increasing frequency. It is often described as a power
law[12]:
P (x, ω) = P0 e−α0 ω
ηx
(2.35)
where P = Pressure
α = attenuation coefficient
η = attenuation exponent coefficient
Suggested models consider fractional wave equations and are difficult to solve. A different
approach is given in [15], which considers acoustic wave propagation in dispersive media, the
formulation has close ties to the propagation of electromagnetic waves in dispersive medias as
previously presented in Section 2.1.1. Consider that Equation (2.30) can be broken down into
two couples PDE’s for pressure and velocity:
∂
u(x, t)
∂t
∂
∇ · u(x, t) = −κ p(x, t)
∂t
∇p(x, t) = −ρ
(2.36)
(2.37)
Next consider that the medium compressibility κ is a function of frequency, [9] proposed the
following model:
M
X
iω
1
κl
κ(ω) = 2 +
cρ
−iω +
l
1
τl
(2.38)
Equation (2.38) can be brought into a form equivalent to the Debye multi pole expression in
Equation (2.20). Therefore the same behaviour holds for the relaxation time τl , absorption will
be greatest when τl = 1/2πf . Denote
κ0 =
1
c2 ρ
M
κ∞
X
1
= 2 +
Kl
cρ
l
Fourier transforming Equation (2.38) into the time domain yields:
κ(x, t) = κ∞ (x)δ(t) +
M
X
κl (x)
l=1
13
τl (x)
e−t/τl u(t)
(2.39)
2.3 Numerical solutions
Christioph Sadée
In the frequency domain the pressure term of Equation (2.37) is multiplied with the frequency
dependent compressibility Equation (2.38). In the time domain this has to be represented as
the convolution as seen below
∂
p(x, t)
(2.40)
∂t
∗ denotes the convolution of κ(x, t) with p(x, t). Therefore dispersive acoustic media are
described by Equation (2.36) and (2.40).
∇ · u(x, t) = −κ(x, t) ∗
2.3
Numerical solutions
2.3.1
Light Diffusion model
The light diffusion equation as given by Equation (2.22) and stated again below for convenience,
can be numerically solved using the finite difference crank-nicolson method [2].
1 ∂φ(r, t)
= ∇ · [D∇φ(r, t)] − µa (r)φ(r, t) + S(r, t)
{z
} | {z }
|
c ∂t
Absorption
Source
A 2D model was chosen because of less computational power and fewer complexity. The
discretisation is performed as follows:
n+1
φn+1
i,j − φi,j
= −ci,j µa |i,j φni,j
∆t
1
n+1
n+1
n+1
n+1
+ ci,j
D
φ
−
φ
−
D
φ
−
φ
i+1/2,j
i−1/2,j
i+1,j
i,j
i,j
i−1,j
2(∆x)2
+ Di+1/2,j φni+1,j − φni,j − Di−1/2,j φni,j − φni−1,j
1
n+1
n+1
Di,j+1/2 φn+1
+ ci,j
− Di,j−1/2 φn+1
i,j+1 − φi,j
i,j − φi,j−1
2
2(∆y)
n
n
n
n
n
+ Di,j+1/2 φi,j+1 − φi,j − Di,j−1/2 φi,j − φi,j−1 + ci,j Si,j
(2.41)
Rearranging n + 1 terms on the left and n terms on the right and grouping coefficients
together yields
n+1
n+1
n+1
n+1
αi,j φn+1
i−1,j + βi,j φi,j + γi,j φi+1,j + δi,j φi,j−1 + ζi,j φi,j+1 =
n
− αi,j φni−1,j + (−βi,j + 2 − ∆tci,j µa |i,j )φni,j − γi,j φni+1,j − δi,j φni,j−1 − ζi,j φni,j+1 + ci,j Si,j
(2.42)
All the coefficients of Equation (2.42) are then filled into a matrix A along its diagonals.
Another φ vector is constructed containing all the φn+1
i,j vectors, another ”b” column vector is
constructed containing all the known values, those are all values on the RHS of Equation (2.42),
This results in the following linear system:
A φn+1 = b
(2.43)
φn+1 = A\b
(2.44)
It can be solved by inversion:
14
2 THEORY
2.4
Christioph Sadée
Acoustic wave absorption
Equation (2.36) and (2.40) were analysed using the FDTD scheme, but first the convolution in
Equation (2.40) had to be re-expressed since otherwise storage of each time step would have
been necessary, since the convolution integral requires all previous time steps which would make
the scheme computationally highly intensive [15]. The convolution in Equation (2.40) is fully
expressed like:
!
M
X
∂
∂
κl (x) −t/τl
∂
u(t)
(2.45)
p(x, t) ∗
e
κ(x, t) ∗ p(x, t) = κ∞ (x) p(x, t) +
∂t
∂t
∂t
τ
l (x)
l=1
Replacing the term in brackets with state variable Sl :
Sl (x, t) = p(x, t) ∗
κl (x) −t/τl
u(t)
e
τl (x)
(2.46)
Differentiating the state variable to introduce an intermediate step:
∂
1
κl (x)
Si (x, t) = − (x)Sl (x, t) +
∂t
τl
τl (x)
(2.47)
Placing Equation (2.46) into Equation (2.45) and substituting into Equation (2.37) yields:
M
M
X1
X κl (x)
∂
(x)Sl (x, t) −
= ∇ · u(x, t)
− κ∞ (x) p(x, t) +
∂t
τ
τ
l
l (x)
l=1
l=1
(2.48)
Therefore the following order is used for computing u and p:
First Equation (2.36)
∂
∇p(x, t) = −ρ u(x, t)
∂t
Second Equation (2.47)
∂
1
κl (x)
Si (x, t) = − (x)Sl (x, t) +
∂t
τl
τl (x)
Thrid Equation (2.48)
M
M
X1
X κl (x)
∂
− κ∞ (x) p(x, t) +
(x)Sl (x, t) −
= ∇ · u(x, t)
∂t
τ
τ
l
l (x)
l=1
l=1
Note how no convolution has to computed anymore, the initial step for S is assumed to
be 0. The FDTD scheme was constructed in such a way that the p nodes were integer values
(not as seen in [REF]), in order to match the φ nodes from the light diffusion scheme. The
discretisation in 2D is as follows
First Equation (2.36)
n+1/2
dt
P |ni+1,j − P |ni,j
∆xρi+1/2,j
dt
n−1/2
= ux |i,j+1/2 −
P |ni,j+1 − P |ni,j
∆xρi,j+1/2
n−1/2
ux |i+1/2,j = ux |i+1/2,j −
n+1/2
uy |i,j+1/2
15
(2.49)
(2.50)
Christioph Sadée
Second Equation (2.47)
n−1/2
S|n+1/2 = e−dt/τi ,j S|i,j
+
κ|i,j ∆t −dt/2τ |i,j
e
P |i,j
τ |i,j
(2.51)
Thrid Equation (2.48)
P |n+1
i,j
−∆t κ
=e
κ|i,j
∞ |i,j τ |i,j
P |n−1
i,j
κ|i,j
∆t − ∆t
2 κ∞ |i,j τ |i,j
+
e
κ∞ |i,j
n+1/2
"
−
∆x
n+1/2
−
n+1/2
ux |i+1/2,j − ux |i−1/2,j
n+1/2
uy |i,j+1/2 − uy |i,j+1/2
∆y
+
1
n+1/2
τ |i,j S|i,j
#
(2.52)
Not that there was a mistake in Equation (2.52) in [15], the exponent in the second exponential
had a minus missing. Instead of using Berengers boundaries as suggested, 2nd oder absorbing
boundary condition from [6] were used.
3
Experiment
Experimental work was conducted to find acoustical signals in different media with the photoacoustic effect. Media included a highly absorbing carbon nanotube composite, used to cover
optical fibre to produce ultrasound, fat and muscle tissue from the cow and liver tissue from
chicken.
3.1
Setup
The Setup was changed several times in the course of the work. This was especially due
to the availability of new equipment that wasn’t accessible at the start. Figure 3.1 shows a
representation of the lab.
Figure 3.1: Set Up
16
3 EXPERIMENT
Christioph Sadée
Matlab was used to send a chirp modulation waveform to the AWG which then generated
the signal to drive the laser diode. The laser diode had an output at 1215nm close to the best
absorption wavelength of fat. An optical fibre connected to the laser diode was placed into a
water tank. To hold it in place it was first held by a chuck and 3 translation stages, later the
chuck was replaced with a syringe hypodermic needle due to its smaller size and better ability
to fix the flexible fibre, both set-ups can be seen in Figures 3.2 and 3.3. The samples were held
on a thin test cover slice which was held in place by another stand. The transducer was placed
opposite the sample and connected to an amplifier.
Figure 3.2: Set Up 1
Figure 3.3: Set Up 2
The signal was then detected with a National Instrument Digital Acquisition Card (NI-DAQ)
and visualised on a computer using a Labview code.
3.2
Procedure
Fist a chirp modulation waveform was coded in Matlab. The waveform can be seen in Figure 3.4,
it is padded with zero values after termination of the chirp at 5[µs], this is necessary to
obtain ultrasound signals in pulses. In order to have continuous output during the chrip, the
offset k from Equation (2.1) was kept larger than the chirp modulation amplitude AM from
Equation (2.2). The total offset and modulation amplitude was never raised about 1.2V in order
to not damage the diode, imaging was performed below 1V. The waveform repetition rate was
set to be 3500 times in one second, although higher repetition rates were also tried.
17
3.2 Procedure
Christioph Sadée
Figure 3.4: Chirp waveform
Next at the start of each imaging day, a large tank was filled with deionised water. The
optical fibre and transducer were placed into the tank facing each other, the distance between
them corresponded to about the focal length of the transducer. It is of great importance that
the transducer is focused on the fibre tip and NOT on the sample which will later be placed as
close as possible to the fibre tip. Reason for this is that the optical fibre is not fully straight
as previously mentioned, which makes it very hard to align it coaxially with the transducer
axis. Hence only signal that will be generated at the tip of the fibre will most likely be picked
up. If focusing the transducer on the sample, then not even that is sure and finding a signal
is almost impossible. A hypodermic needle was used to straighten the fibre further for better
imaging. A pulse echo was used to exactly focus the transducer on the sample. This method
uses the Amplifier as seen in Figure 3.1 in a different setting to send out acoustic pulses and
record the reflections. The 3 translation stages attached to the optical fibre were used to move
the tip to a point of greatest reflection. Following that 4 different meat samples were prepared
as see in Figure 3.5 and placed on cover slides. An additional Carbon nanotube (CNT) in
Polydimethylsiloxane (PDMS) sample was used as seen in Figure 3.6, from now on referred to
CNT-PDMS sample. The sample is made of the same material as the covered optical fibre and
highly absorbing and hence produces a large acoustic signal.
18
3 EXPERIMENT
Christioph Sadée
Figure 3.6: Carbon Nanotubes+PDMS Sample
Figure 3.5: Meat Samples
The samples were placed as close to the optical fibre as possible using another stand as seen
in Figure 3.3. Next the laser was turned on and the waveform was send to the AWG. For safety
purposes, safety googles have to be worn when ever the laser is turned on.
3.3
3.3.1
Results and Discussion
Signal
Once the best Set-up and aligned was figured out, it was easy to obtain a signal from the carbon
sample but still rather hard from the meat samples. A considerable amount of averaging must
be employed to obtain an acoustic signal from meat. Averaging overlays several measurements
and divides by the total amount of measurements overplayed and therefore decreases the noise
and amplifies the signal.
CNT-PDMS samples
Figure 3.7: CNT-PDMS 1000 averages
Figure 3.8: CNT-PDMS 1000 averages
19
3.3 Results and Discussion
Christioph Sadée
Figure 3.9: CNT-PDMS 1000 averages, thin sample
All 3 figures show large acoustic signals relative to the noise level after 1000 averages. Note
how the lower frequencies are present where as the higher frequencies are attenuated or maybe
not generated. Figure 3.9 was a thiner slice than the other two samples, it is clearly visible
that there is a general decrease in amplitude at higher frequencies which could be due to signal
attenuation by the media, particularly since the thinner slice has more of the high frequencies
as compared to the thicker slices in Figure 3.7 and Figure 3.8. But there are 2 large indents
which are more likely to correspond to a non linear output out the laser diode with increasing
frequency, a photodetector should be used to detect such irregularities.
Beef Muscle samples
Figure 3.10: Beef muscle 20000 averages
Figure 3.11: Beef muscle 20000 averages
Figure 3.10 and Figure 3.11 are the only two signals that have been found for meat with
this setup with only 20000 averages, emphasising how import proper alignment is. The signal is
not as clearly visible as with the CNT-PDMS sample but one can notice the waveform at 15µs.
20
3 EXPERIMENT
Christioph Sadée
The large output from 0 to 5µs was an unidentifiable noise which does not correspond to the
signal since it occurs far too early. Other signals were obtained at higher averaging for tissue
but even then pulse compression was required to make them visible.
3.3.2
Pulse Compressed Signal
Pulse compression was performed on each measurement to make some signals visible, the code
is in the appendix. Pulse compression requires the modulation chirp to be recoded to fit the
sampling rate of the NI-DAQ. The received transducer signal is then cross correlated with the
chirp to give the output as seen in Figure 3.12.
Figure 3.12: Pulse Compression Procedure
Figure 3.13: Pulse compression liver 100000 av-Figure 3.14: Pulse compression liver 100000 averages
erages
21
Christioph Sadée
Figure 3.15: Pulse compression liver 100000 averages
Note the small peaks after 15µs in each of the lower pictures. These correspond to pulse
compressed signal, better visible after cross correlation.
4
Model
Figure 4.1: Acoustic wave Propagation with Figure 4.2: Acoustic wave propagation with 8
one source
sources
22
5 CLINICAL APPLICATION
Christioph Sadée
Figure 4.3: Acoustic wave propagation with 8 sources later stage
The above scheme shows the propagation of Ricker wavelets to test the FDTD algorithm for
dispersive acoustic signal propagation. Note how Figure 4.2 has lost in amplitude significantly
as compared to Figure 4.2, this is partially due to the relaxation which was set to be τ0 = 10−6 .
The FDTD scheme has to be further complemented by fitting the appropriate values to the
material to obtain a realistic behaviour.
5
Clinical Application
There is a broad spectrum of clinical applications for Photoacoustic Tomography (PAT), but it
would be particularly useful for detecting plaques, stenoses and calcification of arteries . Due to
the small size of the optical fibre, it could fit into most arteries, with the smallest devices about
1.25mm as seen in Figure 5.1, b), this is so called intra vascular photoacoustics (IVPA).
23
Christioph Sadée
Figure 5.1: a) Set up for Ex vivo imaging [7], b) Catheter tip with optical
fibre and transducer on 10cent coin
Figure 5.2: [7] Imaging of left anterior descending
artery ex vivo
The catheter with the imaging tip could be inserted during key-hole surgeries, for example
to detect atherosclerosis in arteries, the different plaques can be detected by tuning the laser
frequency to the corresponding highest absorption of the material, so for example fatty plaques
are best detected at 1210nm. Imaging of a left anterior descending artery ex vivo can be seen
in Figure 5.2, note the calcification in a), also c) and d) illustrate the difference of using light
at different frequencies. c) was imaged with a laser wavelength of 1210nm which is ideal for
lipid detection hence one can see the plaque within the artery where ascompared to d) far less
is seen, which is due to the change in wavelength to about 1230nm which is not particularly
absorbing for fat. The tuning up and down of the laser results in different contrasts hence giving
a functional image, associating absorption at a particular wavelength to a particular tissue. So
far the uses discussed were mainly for ultrasound generated within tissue, but there is also a
use for ultrasound generated within the coating of an optical fibre. Instead of the absorption in
tissue, light is absorbed in a strongly absorbing medium, covering the fibre with a thin sheet.
The CNT-PDMS material tested in this lab report is such a material. The fibre can then be
used like a normal transducer to send out signals but it’s side is much smaller than the smallest
medical transducer with a piezoelectric crystal. Again intravenous ultrasound imaging could be
a possible application.
6
Conclusion
Detection of a signal in CNT-PDMS was clearly shown also signal from tissue was detect. To
detect fat tissue seemed to be harder even though the laser was at the required frequency of
1210nm. One should consider placing the Transducer on the same side as the optical fibre, in
that way no acoustic attenuation due to the sample will take place or only minimally and a
better Signal should be detected. The code was a side project and has to be further developed.
Obtaining Signal from the samples presented was really hard and the first 4 weeks were spend
24
REFERENCES
Christioph Sadée
looking for a signal. Also there was a big problem with setting up everything and installing
drivers as a new computer arrived just half through the project.
7
Afterword
References
[1] Kenneth S Cole and Robert H Cole. Dispersion and absorption in dielectrics i. alternating
current characteristics. The Journal of Chemical Physics, 9(4):341–351, 1941.
[2] John Crank and Phyllis Nicolson. A practical method for numerical evaluation of solutions
of partial differential equations of the heat-conduction type. In Mathematical Proceedings
of the Cambridge Philosophical Society, volume 43, pages 50–67. Cambridge Univ Press,
1947.
[3] Peter JW Debye et al. collected papers of peter jw debye. 1954.
[4] Richard Phillips Feynman, Robert B Leighton, and Matthew Sands. [Lectures on physics];
The Feynman lectures on physics. 2 (1969). Mainly electromagnetism and matter. AddisonWesley, 1969.
[5] S Gabriel. Radiation absorption mechanism.
DIELECTRIC/AppendixC.html", November 1997.
"http://niremf.ifac.cnr.it/docs/
[6] Umran S Inan and Robert A Marshall. Numerical electromagnetics: the FDTD method.
Cambridge University Press, 2011.
[7] Krista Jansen, Antonius FW Van Der Steen, Heleen MM van Beusekom, J Wolter Oosterhuis,
and Gijs van Soest. Intravascular photoacoustic imaging of human coronary atherosclerosis.
Optics letters, 36(5):597–599, 2011.
[8] Stephan Kellnberger. Thermoacoustic Imaging in time and frequency domain. Theory and
experiments. PhD thesis, München, Technische Universität München, Diss., 2013, 2013.
[9] Adrian I Nachman, James F Smith III, and Robert C Waag. An equation for acoustic
propagation in inhomogeneous media with relaxation losses. The Journal of the Acoustical
Society of America, 88(3):1584–1595, 1990.
[10] R Nave. Radiation absorption mechanism. "http://hyperphysics.phy-astr.gsu.edu/
hbase/mod3.html", Unkonw Unkown.
[11] JW Schuster and RJ Luebbers. An fdtd algorithm for transient propagation in biological
tissue with a cole-cole dispersion relation. In Antennas and Propagation Society International
Symposium, 1998. IEEE, volume 4, pages 1988–1991. IEEE, 1998.
[12] Thomas L Szabo. Time domain wave equations for lossy media obeying a frequency power
law. The Journal of the Acoustical Society of America, 96(1):491–500, 1994.
[13] Unkown.
Amplitude modulated signal.
"http://astarmathsandphysics.com/
igcse-physics-notes/372-amplitude-modulation-am-and-frequency-modulation-fm.
html", Unkonw Unkown.
25
REFERENCES
Christioph Sadée
[14] Ashley J Welch and Martin JC Van Gemert. Optical-thermal response of laser-irradiated
tissue, volume 1. Springer, 1995.
[15] Xiaojuen Yuan, David Borup, James Wiskin, Michael Berggren, Steven Johnson, et al.
Simulation of acoustic wave propagation in dispersive media with relaxation losses by using
fdtd method with pml absorbing boundary condition. Ultrasonics, Ferroelectrics, and
Frequency Control, IEEE Transactions on, 46(1):14–23, 1999.
26