6. Maxwell relations and Euler’s cycle
Where U(S, V) : the internal energy
H(S, P) : Enthalpy
F(T, V) : Helmholtz free energy
G(T, P) : Gibbs free energy
S : entropy
V : volume
P : pressure
T : temperature
Photoacousticeffect
compressibility
Coefficient of
thermal
expansion
7. Specific heat capacities and Compressibilities
The following relations are drove from
Energy conservation ( 𝐸𝑠𝑦𝑠 = 𝑄 − 𝑊) with
internal energy(U) and enthalpy(H)
𝜿 𝒔 =
𝑪 𝑽
𝑪 𝑷
𝜿 𝑻
𝜿 𝒂𝒅 = −
𝟏
𝑽
(
𝝏𝑽
𝝏𝑷
) 𝒂𝒅 = −
𝟏
𝑽
(
𝝏𝑽
𝝏𝑷
) 𝒔 = 𝜿 𝒔(𝒊𝒔𝒆𝒏𝒕𝒓𝒐𝒑𝒊𝒄)
When the system satisfy the following conditions
1. 𝜹𝑸 = 𝟎
2. 𝑹𝒆𝒗𝒆𝒓𝒔𝒊𝒃𝒍𝒆 𝒔𝒚𝒔𝒕𝒆𝒎
Because 𝑑𝑆 ≡ (
𝜹𝑸
𝑻
) 𝒓𝒆𝒗
𝜿 𝑻 = −
𝟏
𝑽
(
𝝏𝑽
𝝏𝑷
) 𝑻
𝜹𝑸 = 𝝏𝑼 + 𝑷𝝏𝑽
𝑪 𝑽 = (
𝜹𝑸
𝝏𝑻
) 𝑽 = (
𝝏𝑼
𝝏𝑻
) 𝑽
𝜹𝑸 = 𝝏𝑯 − 𝑽𝝏P
𝑪 𝑷 = (
𝜹𝑸
𝝏𝑻
) 𝑷 = (
𝝏𝑯
𝝏𝑻
) 𝑷
Photoacousticeffect
8. Speed of sound Volume expansion
The correct formula for the speed of sound
𝑽 𝒔𝒐𝒖𝒏𝒅 =
𝟏
𝝆𝑲 𝒔
=
𝟏
𝝆
𝑪 𝑽
𝑪 𝑷
𝜿 𝑻
𝜿 𝑻 =
𝑪 𝑷
𝝆𝒗 𝒔
𝟐
𝑪 𝑽
Where 𝝆 : the mass density
(~ 1000 𝑘𝑔 𝑚3
for water and soft tissue)
𝑪 𝑷: the specific heat capacity at constant pressure
𝑪 𝑽: the specific heat capacity at constant volume
(~4000 𝐽 𝑘𝑔 𝐾 for muscle)
𝒅𝑽
𝑽
=
1
𝑉
[(
𝜕𝑉
𝜕𝑃
) 𝑇 𝑑𝑃 + (
𝜕𝑉
𝜕𝑇
) 𝑃 𝑑𝑇]
= −𝜿𝒑 + 𝜷𝑻
where 𝜿 : the isothermal compressibility (= 𝜿 𝑻)
(~5 × 10−10
𝑃𝑎−1
for water or soft tissue)
𝜷 : the thermal coefficient of volume expansion
(~ 4 × 10−4
𝐾−1
for muscle) (isobaric)
𝒑 : the changes in pressure (Pa)
𝑻 : the changes in temperature (K)
Photoacousticeffect
9. Local pressure rising (the initial pressure)
𝑷 𝟎 =
𝜷 𝑻
𝜿
=
𝜷
𝜿𝝆𝑪 𝑽
𝜼𝒕𝒉 𝑨 𝒆
where 𝑨 𝒆: the specific optical absorption (𝐽/𝑚3
)
𝜼𝒕𝒉: the percentage that is converted into heat
Photoacousticeffect
𝚪 =
𝜷
𝜿𝝆𝑪 𝑽
=
𝜷 𝒗 𝒔
𝟐
𝑪 𝑷
Grueneisen parameter (dimensionless)
𝑷 𝟎 = 𝚪𝜼𝒕𝒉 𝑨 𝒆 = 𝚪𝜼𝒕𝒉 𝝁 𝒂 𝑭
Where 𝝁 𝒂 : the optical absorption coefficient (1/𝑚)
𝑭 : the optical fluence (𝐽/𝑚2
)
10. Grueneisen parameter (dimensionless)
𝚪 =
𝜷
𝜿𝝆𝑪 𝑽
=
𝜷 𝒗 𝒔
𝟐
𝑪 𝑷
𝜦 𝝀 =
𝑨
𝑭
= 𝜶𝜞𝝁 𝒂(𝝀)
Photoacoustic measurement of the Gruneisen parameter of tissue, Da-Kan Yao, Lihong V. Wang, et al. , 2014, DOI: 10.1117/1, JBO
Temperature mapping using photoacoustic and thermoacoustic tomography, Haixin Ke, Lihong V. Wang, et al, 2012, DOI: 10.1117/12.909000, Proc. SPIE 8223
Grueneisen parameter can be measured from the signal amplitude (𝛬 𝜆 )
where 𝑨 : the peak-to-peak voltage amplitude
𝑭 : the light fluency
𝜶 : calibration factor,
which depends on the central frequency
of the transducer 𝜔0
22℃
Lipid 0.69±0.02
Fat tissue 0.81±0.05
Serum 0.132±0.002
Blood 0.124+0.000333𝐶 𝐻𝑏02
Beef muscle 0.15 (22℃)
0.21 (37℃)
Photoacousticeffect
20. General solution of the PA equation
.
.
SolutionofPhotoacousticequation
Initial condition
Source
Boundary condition
21. Green’s function approach
SolutionofPhotoacousticequation
This integration has discontinuities at 𝑘 = ±𝜔/𝑣𝑠,
but it can be evaluated by Cauchy’s contour integration
and it also has to be analyzed in spherical coordinate
since the wave is the spherical wave.
25. Limitations of the solution
• We cannot say that photoacoustic equation is an adjoint problem
• Tissue is heterogeneous and has viscosity, elasticity and so on
• Actually, Green’s function has the physical meaning as the response of a
point absorber to step heating, rather than impulse heating
Next works..
• Understand the solution in case of array imaging system
• Transform the solution using Numerical analysis with several examples
