Biophotonics_Concepts_to_Applications

the stress induced by the expansion will decrease and can be characterized in terms
of thestress relaxation timeτs. This time is defined by

ss¼

d

vs

ð 10 : 22 Þ

where vsis the speed of sound, which is about 1.48 mm/μs in water and soft tissue. For
proper PAT procedures, the volume expansion of the absorber during the illumination
period should be negligible, which is known asstress confinement. This condition
holds when the laser pulse duration is shorter than the stress relaxation time.

Example 10.11Suppose that in a PAT procedure the heated region in a

sample of soft tissue has a dimension d = 1 mm. What are the thermal

relaxation time and the stress relaxation time?

Solution: Using Eq. (10.21) withαth= 0.13 mm^2 /s yields a thermal relax-

ation time of

sth¼

d^2

ath

¼

ð1mmÞ^2

0 :13 mm^2 =s

¼ 7 :7s

Using Eq. (10.22) with vs= 1.48 mm/μs yields a stress relaxation time of

ss¼

d

vs

¼

1mm

1 :48 mm=ls

¼ 0 : 68 ls

The basic operation of PAT can be illustrated through a simple mathematical
model [ 42 ]. When a short laser pulse excites a tissue sample, the local fractional
volume expansion dV/V can be expressed in terms of the pressure change p (given
in Pa) and the temperature change T (given in K) as

dV

V

¼jpþbT ð 10 : 23 Þ

whereκ is the isothermal compressibility andβis the coefficient of thermal
expansion. Typical values of these parameters areκ≈ 5 × 10 −^10 Pa−^1 for water
and soft tissue andβ≈ 4 × 10 −^4 K−^1 for muscle tissue.
When a laser pulse impinges on a tissue sample, the temperature rise can be
expressed as

T¼

Ae

qCV

¼

laH

qCV

ð 10 : 24 Þ

Hereρis the mass density (1 gm/cm^3 for water and soft tissue), CVis the
specific heat capacity at constant volume [≈4 J/(gm K)], and Ae=μaH is the

314 10 Optical Imaging Procedures