# Skull’s Photoacoustic Attenuation and Dispersion Modeling with Deterministic Ray-Tracing: Towards Real-Time Aberration Correction

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## Abstract

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## 1. Introduction

## 2. Materials and Method

#### 2.1. Theoretical Background

#### 2.1.1. Mathematical Modeling of the PA Wave Propagation in a Single-Layer Heterogeneous Medium

#### 2.1.2. PA Wave Propagation at Interfaces (Interaction with Skull Tissue)

#### 2.2. Simulation Framework

#### 2.2.1. Ultrasound Ray Vector Space

#### 2.2.2. Modeling of Impulse Response

#### 2.2.3. Convolution

## 3. Validation

#### 3.1. Numerical Validation

#### 3.2. Analytical Validation

#### 3.3. Experimental Validation

#### 3.3.1. Experimental Setup

#### 3.3.2. Phantom Preparation

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of the PA wave transmission through the skull. The interfaces are assumed to be parallel with the Cartesian coordinates, where z is oriented orthogonal to the interfaces and x runs along the interface plane. Here, $\mathbf{i}$ denotes the incident ray direction vector, $\mathbf{r}$ denotes the reflected ray direction vector and $\mathbf{t}$ denotes the transmitted ray direction vector (the subscripts L and S refer to the longitudinal and shear rays, respectively). Also ${\mathbf{t}}_{L-L}$ and ${\mathbf{t}}_{L-S}$ are the longitudinal transmitted ray direction vectors generated by the incident longitudinal and shear rays in the skull, respectively. ${\theta}_{i}$, ${\theta}_{L}$ and ${\theta}_{S}$ are the angle of incidence, longitudinal transmittance, and shear transmittance, respectively. $\mathbf{n}$ is the normal vector orthogonal to the interface.

**Figure 2.**2-D illustration of a simple model of skull used in our simulations. The model consists of a skull bone with a thickness of h located above the brain tissue. The spherical PA imaging target with the radius of ${r}_{0}$ is located at a depth of d below the skull layer and a flat ultrasound transducer with an element diameter of $2{r}_{d}$ is remained in contact with the outer-skull surface and the coupling gel.

**Figure 3.**Schematic of the experimental setup. DAQ: Data acquisition system, AMP: Amplifier, OPO: Optical parametric oscillator.

**Figure 4.**Photographs of (

**a**) mouse skull bone, (

**b**) rat skull bone, (

**c**) dog frontal skull bone, (

**d**) dog parietal skull bone.

**Figure 5.**Numerical validation of the proposed method versus the k-space algorithm. Absolute PA signal profiles simulated by k-space algorithm (dotted line) and our simulation method (solid line). Dashed rectangles show the main bipolar pulse of the simulated PA signal. In these simulations, the setup in Figure 2 with the parameters, h = 7 mm, k = 10 mm, ${r}_{0}$ = 1 mm, ${r}_{d}$ = 0.025 inch, and T = 5 cm is used. (

**a**) Minimum error of 0.11% obtained for d = 1.7 cm. (

**b**) Maximum error of 0.25% obtained for d = 2.7 cm.

**Figure 6.**Simulation results. (

**a**) PA signals simulated by our simulation method in the presence and absence of a skull tissue. The setup in Figure 2 with the parameters, h = 7 mm, d = 2.7 cm, k = 10 mm, ${r}_{0}$ = 1 mm, ${r}_{d}$ = 0.25 inch, and T = 5 cm is used. (

**b**) Corresponding frequency spectrum of the simulated PA signals.

**Figure 7.**Experimental results. (

**a**) PA signal amplitude obtained with and without dog frontal skull bone. The skull bone was placed in the detection path. (

**b**) Corresponding frequency spectrum of the experimental PA signals.

**Figure 8.**Simulation results. (

**a**) PA signals simulated by our simulation method in the absence of the skull tissue for two different diameters of spherical absorber 2 mm (dotted line) and 0.2 mm (solid line). The setup in Figure 2 with the parameters, h = 0 mm (i.e., without skull), d = 2.7 cm, k = 10 mm, ${r}_{0}$ = 1 mm or 0.1 mm, ${r}_{d}$ = 0.025 inch, and T = 5 cm is used. (

**b**) Corresponding normalized magnitude of the frequency spectrum of the simulated PA signals.

**Figure 9.**PA signal amplitude attenuation versus skull thickness. Simulation results (solid line) and experimental phantom results (dotted line). In simulation, the setup in Figure 2 with the parameters, d = 1.7 cm, k = 10 mm, ${r}_{0}$ = 1 mm, ${r}_{d}$ = 0.25 inch, and T = 5 cm is used. The thicknesses of the skull are as follow; 0.5 mm, representing mouse skull thickness, 1 mm, representing rat skull thickness, 1.5 mm, representing neonatal skull thickness, 4.5 mm, representing dog frontal skull thickness, 5.98 mm, and 7.68 mm, representing human frontal skull thickness, 9 mm, representing dog parietal skull thickness, and 9.61 mm, representing human frontal skull thickness.

**Figure 10.**Simulation results of the effect of skull thickness on the signal broadening (solid line) and time shift (dotted line) of transcranially recorded signal relative to the undisturbed signal. In these simulations, the setup in Figure 2 with the parameters, d = 1.7 cm, k = 10 mm, ${r}_{0}$ = 1 mm, ${r}_{d}$ = 0.25 inch, and T = 5 cm is used. The thicknesses of the skull are as follow; 0.5 mm, representing mouse skull thickness, 1 mm, representing rat skull thickness, 1.5 mm, representing neonatal skull thickness, 4.5 mm, representing dog frontal skull thickness, 5.98 mm, and 7.68 mm, representing human frontal skull thickness, 9 mm, representing dog parietal skull thickness, and 9.61 mm, representing human frontal skull thickness.

**Figure 11.**Simulation results of the effect of different target locations on the PA peak signal amplitude attenuation (solid line) and signal broadening (dotted line). In these simulations, the setup in Figure 2 with the parameters, h = 7 mm, k = 10 mm, ${r}_{0}$ = 1 mm, ${r}_{d}$ = 0.25 inch, and T = 5 cm is used. The imaging target is moved away from the inner-skull surface from 1 cm to 3 cm.

Symbol (Unit) | Soft Tissue | Skull | Water (Coupling Medium) |
---|---|---|---|

$\rho $ (kg/m${}^{3}$) | 1000 [39] | 1800 [27] | 1000 [65] |

${c}_{L}$ (m/s) | 1500 [66] | 2900 [27] | 1486 [65] |

${c}_{S}$ (m/s) | — | 1444 [27] | — |

${\alpha}_{0L}$ (Np/cm) | 0.05 [66] | 1.70 [39] | 0.00 [65] |

${y}_{L}$ | 1.18 [66] | 0.93 [39] | — |

${\alpha}_{0S}$ (Np/cm) | — | 3.41 [39] | — |

${y}_{S}$ | — | 0.93 [39] | — |

**Table 2.**Normalized standard deviation (NSD) errors between the k-Wave and the proposed method for the simulation of different skull thicknesses with constant target depth, d = 1.7 cm.

Skull Thickness (mm) | NSD (%) |
---|---|

1 | 0.19 |

4 | 0.14 |

7 | 0.11 |

**Table 3.**Normalized standard deviation (NSD) errors between the k-Wave and the proposed method for the simulation of different target depths with a constant skull thickness, h = 7 mm.

Target Depth (cm) | NSD (%) |
---|---|

1.7 | 0.11 |

2.7 | 0.25 |

3.7 | 0.21 |

**Table 4.**Estimated errors (${E}_{L}$ and ${E}_{S}$) between the analytical and numerical calculations of the longitudinal and shear intensity transmission coefficients, ${I}_{L}$ and ${I}_{S}$, for three different incidence angles.

${\mathit{\theta}}_{\mathit{i}}$ (${}^{\circ}$) | ${\mathit{I}}_{\mathit{L}\left(\mathit{anl}\right)}$ | ${\mathit{I}}_{\mathit{S}\left(\mathit{anl}\right)}$ | ${\mathit{I}}_{\mathit{L}\left(\mathit{num}\right)}$ | ${\mathit{I}}_{\mathit{S}\left(\mathit{num}\right)}$ | ${\mathit{E}}_{\mathit{L}}$ (%) | ${\mathit{E}}_{\mathit{S}}$ (%) |
---|---|---|---|---|---|---|

0 | 0.414 | 0.000 | 0.421 | 0.000 | 1.69 | 0.00 |

15 | 0.192 | 0.026 | 0.185 | 0.025 | −3.65 | −3.85 |

30 | 0.133 | 0.048 | 0.139 | 0.046 | 4.51 | −4.17 |

**Table 5.**Running time of k-Wave and the proposed method for different maximum frequency supported by grid size in k-Wave and two different number of frequencies modeled by our method.

${\mathit{f}}_{\mathit{max}}$ (MHz) | $\mathit{dx}$ ${}^{\mathit{a}}$ (mm) | N${}^{\mathit{b}}$ | ${\mathit{t}}_{1}$ ${}^{\mathit{c}}$ (s) | ${\mathit{t}}_{2}$ ${}^{\mathit{d}}$ (s) | |
---|---|---|---|---|---|

64 Frequency | 32 Frequency | ||||

1 | 0.75 | 86${}^{3}$ | 849.45 | 35.06 | 19.74 |

1.5 | 0.5 | 128${}^{3}$ | 2571.89 | 40.20 | 20.16 |

3 | 0.25 | 256${}^{3}$ | >18,000 | 42.66 | 22.14 |

5 | 0.15 | 427${}^{3}$ | — ^{e} | 48.69 | 25.12 |

7.5 | 0.1 | 640${}^{3}$ | — ^{e} | 59.14 | 27.92 |

^{e}—: Data is not available.

**Table 6.**Running time of k-Wave and the proposed method for the simulation of different skull thicknesses and two different target depths.

h (mm) | Target Depth = 1.7 cm | Target Depth = 3.7 cm | ||||||
---|---|---|---|---|---|---|---|---|

N${}^{\mathit{a}}$ | $\mathit{dx}$ ${}^{\mathit{b}}$ (mm) | ${\mathit{t}}_{1}$ ${}^{\mathit{c}}$ (s) | ${\mathit{t}}_{2}$ ${}^{\mathit{d}}$ (s) | N${}^{\mathit{a}}$ | $\mathit{dx}$ ${}^{\mathit{b}}$ (mm) | ${\mathit{t}}_{1}$ ${}^{\mathit{c}}$ (s) | ${\mathit{t}}_{2}$ ${}^{\mathit{d}}$ (s) | |

7 | 32${}^{3}$ | 1 | 20.21 | 149.18 | 64${}^{3}$ | 1 | 168.29 | 132.14 |

7.5 | 64${}^{3}$ | 0.5 | 168.88 | 149.83 | 128${}^{3}$ | 0.5 | 2533.10 | 135.21 |

7.25 | 128${}^{3}$ | 0.25 | 2535.67 | 147.26 | 256${}^{3}$ | 0.25 | >18,000 | 134.63 |

7.1 | 320${}^{3}$ | 0.1 | — ^{e} | 146.46 | 640${}^{3}$ | 0.1 | — ^{e} | 133.16 |

^{e}—: Data is not available.

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**MDPI and ACS Style**

Mohammadi, L.; Behnam, H.; Tavakkoli, J.; Avanaki, M.R.N.
Skull’s Photoacoustic Attenuation and Dispersion Modeling with Deterministic Ray-Tracing: Towards Real-Time Aberration Correction. *Sensors* **2019**, *19*, 345.
https://doi.org/10.3390/s19020345

**AMA Style**

Mohammadi L, Behnam H, Tavakkoli J, Avanaki MRN.
Skull’s Photoacoustic Attenuation and Dispersion Modeling with Deterministic Ray-Tracing: Towards Real-Time Aberration Correction. *Sensors*. 2019; 19(2):345.
https://doi.org/10.3390/s19020345

**Chicago/Turabian Style**

Mohammadi, Leila, Hamid Behnam, Jahan Tavakkoli, and Mohammad R. N. Avanaki.
2019. "Skull’s Photoacoustic Attenuation and Dispersion Modeling with Deterministic Ray-Tracing: Towards Real-Time Aberration Correction" *Sensors* 19, no. 2: 345.
https://doi.org/10.3390/s19020345