Anuj Kaushik, Deepak Sonker, and Ratan K. Saha, "Study on angular distribution of differential photoacoustic cross-section and its implication in source size determination," J. Opt. Soc. Am. A 36, 387-396 (2019)

Angular distribution of a differential photoacoustic cross-section (DPACS) has been examined for various nonspherical axisymmetric particles. The DPACS as a function of measurement angle has been computed for spheroidal particles with varying aspect ratios and fitted with a tri-axes ellipsoid form factor model to extract shape parameters. Similar study has been carried out for normal and pathological red blood cells, and fitting has been performed with the tri-axes ellipsoid and finite cylinder form factor models to evaluate cellular morphology. It is found that an enhancement of the DPACS occurs as the surface area of the photoacoustic source normal to the direction of measurement is increased. It decreases as the thickness of the source along the same direction increases. For example, the DPACS for normal erythrocyte along the direction of symmetry is nearly 20 times greater than a pathological cell. Further, the first minimum appears slightly later ($\approx 4\xb0$) for a healthy cell compared with that of a diseased cell. Shape information of spheroids can be precisely estimated by the first model. Both models provide accurate estimates of shape parameters for normal red blood cells (errors within 4%). It may be possible to assess cellular morphology from an angular profile of the DPACS using form factor models.

Ting Feng, Qiaochu Li, Cheng Zhang, Guan Xu, L. Jay Guo, Jie Yuan, and Xueding Wang Opt. Express 24(17) 19853-19862 (2016)

References

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Comparative Study between the Nominal Values of the Morphological Parameters of the Spheroids Considered in This Study and the Estimated Values of the Semi-axes along the $x$, $y$, and $z$ Directions, Respectively^{a}

PA Source Spheroid

Nominal Values (μm)

Angular Range for Fitting

Estimated Values (μm)

Tri-axes Ellipsoid

$\varphi =0\xb0$

$\varphi =90\xb0$

Fitting Error (%)

$\mathrm{AR}=1\text{:}2$

$a=6.93$

0–43°

${\varrho}_{1}=6.93$

5.83

0.48

$a=6.93$

${\varrho}_{2}=5.59$

6.93

$b=3.46$

${\varrho}_{3}=3.46$

3.46

$\mathrm{AR}=1\text{:}4$

$a=8.72$

0–35°

${\varrho}_{1}=8.74$

7.70

0.23

$a=8.72$

${\varrho}_{2}=8.90$

8.74

$b=2.18$

${\varrho}_{3}=2.14$

2.14

$\mathrm{AR}=1\text{:}8$

$a=10.96$

0–28°

${\varrho}_{1}=10.95$

10.41

0.08

$a=10.96$

${\varrho}_{2}=10.19$

10.95

$b=1.37$

${\varrho}_{3}=1.39$

1.39

$\mathrm{AR}=2\text{:}1$

$a=4.36$

90–114°

${\varrho}_{1}=4.38$

3.84

0.26

$a=4.36$

${\varrho}_{2}=5.27$

4.38

$b=8.72$

${\varrho}_{3}=8.67$

8.67

$\mathrm{AR}=4\text{:}1$

$a=3.46$

90–108°

${\varrho}_{1}=3.46$

2.85

0.10

$a=3.46$

${\varrho}_{2}=4.32$

3.46

$b=13.86$

${\varrho}_{3}=13.86$

13.86

$\mathrm{AR}=8\text{:}1$

$a=2.75$

90–102°

${\varrho}_{1}=2.75$

2.75

0.10

$a=2.75$

${\varrho}_{2}=1.89$

2.75

$b=22.01$

${\varrho}_{3}=22.01$

22.01

Note that the curve fitting error is defined as fitting $\text{error}=\frac{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}]-10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{{\mathrm{FF}}^{2}(k,\theta )}{{\mathrm{FF}}^{2}(k,{\theta}_{m})}]\Vert}{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}]\Vert}\times 100\%$; $\Vert \text{\hspace{0.17em}}\Vert $ refers to the Euclidean norm and $\theta $ varied from ${\theta}_{m}$ (where DPACS became maximum) to the first minimum in the increasing $\theta $ direction. Fitting error was found to be the same in both the directions ($\varphi =0$ and 90°).

Table 4.

Comparison between the Actual and the Evaluated Values of the Morphological Parameters for Normal and Pathological RBCs^{a}

PA Source

Nominal Values (μm)

Angular Range for Fitting

Estimated Values (μm)

Tri-axes Ellipsoid

Finite Cylinder

$\varphi =0\xb0$

$\varphi =90\xb0$

Fitting Error (%)

Estimates

Fitting Error (%)

Normal RBC

${R}_{e}=3.82$

0–38°

${\varrho}_{1}=3.65$

2.83

0.13

$\mathrm{\Gamma}=3.74$

1.0

$t/2=0.7$

${\varrho}_{2}=4.10$

3.65

$L=2.19$

$h/2=1.42$

${\varrho}_{3}=1.88$

1.88

Stomatocyte1

${R}_{e}=3.18$

0–34°

${\varrho}_{1}=3.76$

4.61

2.0

$\mathrm{\Gamma}=3.81$

1.8

$t/2=1.36$

${\varrho}_{2}=2.43$

3.76

$L=2.11$

$h/2=1.47$

${\varrho}_{3}=1.82$

1.82

Stomatocyte2

${R}_{e}=3.27$

0–33°

${\varrho}_{1}=3.67$

4.87

6.6

$\mathrm{\Gamma}=3.24$

7.4

$t/2=0.21$

${\varrho}_{2}=2.07$

3.67

$L=\mathrm{Not}$

$h/2=1.16$

${\varrho}_{3}=\mathrm{Not}$

Not

converging

converging

converging

Note that the curve fitting error is defined as fitting $\text{error}=\frac{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}\right]-10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{{\mathrm{FF}}^{2}(k,\theta )}{{\mathrm{FF}}^{2}(k,{\theta}_{m})}\right]\Vert}{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}\right]\Vert}\times 100\%$; $\Vert \text{\hspace{0.17em}}\Vert $ refers to the Euclidean norm and $\theta $ varied from ${\theta}_{m}$ (where DPACS became maximum) to the first minimum in the increasing $\theta $ direction. Fitting error was found to be the same in both the directions ($\varphi =0$ and 90°).

Tables (4)

Table 1.

Numerical Values for Physical Parameters Used in Computation

Comparative Study between the Nominal Values of the Morphological Parameters of the Spheroids Considered in This Study and the Estimated Values of the Semi-axes along the $x$, $y$, and $z$ Directions, Respectively^{a}

PA Source Spheroid

Nominal Values (μm)

Angular Range for Fitting

Estimated Values (μm)

Tri-axes Ellipsoid

$\varphi =0\xb0$

$\varphi =90\xb0$

Fitting Error (%)

$\mathrm{AR}=1\text{:}2$

$a=6.93$

0–43°

${\varrho}_{1}=6.93$

5.83

0.48

$a=6.93$

${\varrho}_{2}=5.59$

6.93

$b=3.46$

${\varrho}_{3}=3.46$

3.46

$\mathrm{AR}=1\text{:}4$

$a=8.72$

0–35°

${\varrho}_{1}=8.74$

7.70

0.23

$a=8.72$

${\varrho}_{2}=8.90$

8.74

$b=2.18$

${\varrho}_{3}=2.14$

2.14

$\mathrm{AR}=1\text{:}8$

$a=10.96$

0–28°

${\varrho}_{1}=10.95$

10.41

0.08

$a=10.96$

${\varrho}_{2}=10.19$

10.95

$b=1.37$

${\varrho}_{3}=1.39$

1.39

$\mathrm{AR}=2\text{:}1$

$a=4.36$

90–114°

${\varrho}_{1}=4.38$

3.84

0.26

$a=4.36$

${\varrho}_{2}=5.27$

4.38

$b=8.72$

${\varrho}_{3}=8.67$

8.67

$\mathrm{AR}=4\text{:}1$

$a=3.46$

90–108°

${\varrho}_{1}=3.46$

2.85

0.10

$a=3.46$

${\varrho}_{2}=4.32$

3.46

$b=13.86$

${\varrho}_{3}=13.86$

13.86

$\mathrm{AR}=8\text{:}1$

$a=2.75$

90–102°

${\varrho}_{1}=2.75$

2.75

0.10

$a=2.75$

${\varrho}_{2}=1.89$

2.75

$b=22.01$

${\varrho}_{3}=22.01$

22.01

Note that the curve fitting error is defined as fitting $\text{error}=\frac{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}]-10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{{\mathrm{FF}}^{2}(k,\theta )}{{\mathrm{FF}}^{2}(k,{\theta}_{m})}]\Vert}{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}]\Vert}\times 100\%$; $\Vert \text{\hspace{0.17em}}\Vert $ refers to the Euclidean norm and $\theta $ varied from ${\theta}_{m}$ (where DPACS became maximum) to the first minimum in the increasing $\theta $ direction. Fitting error was found to be the same in both the directions ($\varphi =0$ and 90°).

Table 4.

Comparison between the Actual and the Evaluated Values of the Morphological Parameters for Normal and Pathological RBCs^{a}

PA Source

Nominal Values (μm)

Angular Range for Fitting

Estimated Values (μm)

Tri-axes Ellipsoid

Finite Cylinder

$\varphi =0\xb0$

$\varphi =90\xb0$

Fitting Error (%)

Estimates

Fitting Error (%)

Normal RBC

${R}_{e}=3.82$

0–38°

${\varrho}_{1}=3.65$

2.83

0.13

$\mathrm{\Gamma}=3.74$

1.0

$t/2=0.7$

${\varrho}_{2}=4.10$

3.65

$L=2.19$

$h/2=1.42$

${\varrho}_{3}=1.88$

1.88

Stomatocyte1

${R}_{e}=3.18$

0–34°

${\varrho}_{1}=3.76$

4.61

2.0

$\mathrm{\Gamma}=3.81$

1.8

$t/2=1.36$

${\varrho}_{2}=2.43$

3.76

$L=2.11$

$h/2=1.47$

${\varrho}_{3}=1.82$

1.82

Stomatocyte2

${R}_{e}=3.27$

0–33°

${\varrho}_{1}=3.67$

4.87

6.6

$\mathrm{\Gamma}=3.24$

7.4

$t/2=0.21$

${\varrho}_{2}=2.07$

3.67

$L=\mathrm{Not}$

$h/2=1.16$

${\varrho}_{3}=\mathrm{Not}$

Not

converging

converging

converging

Note that the curve fitting error is defined as fitting $\text{error}=\frac{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}\right]-10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{{\mathrm{FF}}^{2}(k,\theta )}{{\mathrm{FF}}^{2}(k,{\theta}_{m})}\right]\Vert}{\Vert 10\text{\hspace{0.17em}}{\mathrm{log}}_{10}\left[\frac{\sigma (k,\theta )}{\sigma (k,{\theta}_{m})}\right]\Vert}\times 100\%$; $\Vert \text{\hspace{0.17em}}\Vert $ refers to the Euclidean norm and $\theta $ varied from ${\theta}_{m}$ (where DPACS became maximum) to the first minimum in the increasing $\theta $ direction. Fitting error was found to be the same in both the directions ($\varphi =0$ and 90°).