Team:Slovenia/Model
Modeling of ultrasound propagation
Ultrasound are sound waves with frequencies higher than upper audible limit of human hearing, i.e. higher than 20 kHz.
In the frequency range between 0.8 and 5 Mhz, the sound wave length is between 2 and 0.3 mm. Ultrasound is mainly used for diagnostic
purposes. Unlike light ultrasound can easily penetrate the tissue and is at those intensities completely harmless. It is therefore used also in
prenatal diagnostics where non-invasivness is of vital importance. Short wavelengths additionally enable us to create small regions of high intensity focused ultrasound waves at the focal point,
which can be used for therapeutic purposes
Based on our improvement of the ultrasound responsiveness of the cells it would be feasible to introduce a genetic device into the target tissue and then selectively activate only specific cells by using focused ultrasound, causing a sufficient intensity of the acoustic pressure. This would for example enable the stimulation of neurons in the selected brain section to achieve non-invasive deep brain stimulation or to stimulate cells to produce different hormones.
For this purpose we designed a model of the ultrasound propagation. The model can guide us in the process of the ultrasound device calibration with the potential to use multiple ultrasound probes.
Modeling
Sound modeling with wave equation
The code for the ultrasound propagation model is written in C++ programming language with OpenGL library. The model was parallelized using OpenMP to take advantage of multi-core processor architecture and to achieve better performance. Source code is available on GitHub.
Sound is a vibration that propagates as a mechanical wave of pressure and displacement through the medium. The speed of its propagation also depends on the properties of the medium.
This means that the speed of sound varies greatly between different types of tissues
| Name | Speed (m/s) |
|---|---|
| Skin | 1730 |
| White matter | 1570 |
| Grey matter | 1570 |
| Skull | 4080 |
| Cerebrospinal Fluid | 1500 |
| Air | 343 |
In order to achieve a realistic simulation of the sound propagation through brain an MR image represented in 3 was used
The green color represents skin, blue color represents bone, cyan color represents the cerebrospinal fluid, dark grey color represents grey brain matter and light grey color represents white brain matter. Since the speed of sound does not vary much between dark and white grey matter it is sufficient to model only brain matter in general.
Although focusing is not good, one can notice the elongation of a focused area. This is the case even in simulation with higher frequencies. This suggests that efficient focusing in the tissue requires at least two probes.
The probes in our model are capable of emitting sound waves at frequencies ranging from 0.5 Mhz to 5 Mhz at acoustic pressure from 10 kPa to 100 kPa. The focused probes that we used have focal length of 7 cm. Besides focused probes we also modelled unfocused probes. Both types of probes have an aperture of 4.4 cm. In general propagation of sound from the focused probe can be approximated with an analytic function of a Gaussian beam (6), but this representation fails when we want to model wave interferences from multiple probes. For such cases wave equation model needs to be established.
Acoustic wave equation is a second-order linear partial differential equation that governs the propagation of acoustic waves through the medium. The general form of the equation is:
\begin{equation}
\frac{\partial^2 p}{\partial t^2} = c^2(\nabla^2 p)
\end{equation}
where $c$ is speed of sound in the medium, $p$ is acoustic pressure (the deviation from ambient pressure) and $\nabla^2$ is Laplacian operator (sum of second order partial derivatives with respect to each independent variable). In two dimensional Cartesian coordinate system we get the following equation:
\begin{equation}
\frac{\partial^2 p}{\partial t^2} = c^2(\frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2})
\end{equation}
While wave equation cannot be solved analytically, one can approximate it with finite difference method. Let's presume we want to approximate the solution on a squared area. First, we need to discretize the area to obtain the grid of evenly distributed points $(x_{i}, y_{j})$, $i,j = 1,...,n$. The Laplacian operator can now be approximated as:
\begin{equation}
\nabla^2 p_{i, j} \approx \frac{1}{h^2}(p_{i + 1, j} + p_{i-1, j} + p_{i, j + 1} + p_{i, j - 1} - 4p_{i,j})
\end{equation}
where $h$ is Euclidean distance between two adjacent points. Since the frequencies between 0.8 and 5 Mhz correspond to the wavelength between 2 and 0.3 mm, $h$ does not need to be smaller than 0.3 mm. We now obtain a $n*n$ system of the second order ordinary differential equations:
\begin{equation}
\ddot{p}_{i, j} = \frac{c^2}{h^2}(p_{i + 1, j} + p_{i-1, j} + p_{i, j + 1} + p_{i, j - 1} - 4p_{i,j})
\end{equation}
These equation can be simplified to the first order differential equations with the introduction of a new variable that represents the speed: $(v_{i, j} = \dot{p}_{i, j})$. The new system now has the following form:
\begin{equation}
\begin{aligned}
\dot{p}_{i, j} = v_{i, j} \\
\dot{v}_{i, j} = \frac{c^2}{h^2}(p_{i + 1, j} + p_{i-1, j} + p_{i, j + 1} + p_{i, j - 1} - 4p_{i,j})
\end{aligned}
\end{equation}
The dispersion of the waves is already included in the wave equation. However, the absorption of sound waves in tissue is not included. The absorption can be approximated with the introduction of the attenuation coefficient $k$. We fitted attenuation coefficient to experimental data
Finally we can solve the given system with iterative numeric methods such as Euler's method or Runge-Kutta method. The most widely known method of the Runge-Kutta family is RK4, which was also used in our model. While RK4 introduces some computational overhead it also significantly increases the numerical stability of the system in comparison to the Euler's method. The most basic steps used for the calculation of ultrasound propagation are displayed in (5).
We model a propagation of the ultrasound waves on a 500 x 500 grid, where distances between points are exactly 0.3 cm. This translates to a stimulated area of 15 by 15 cm. The model computes propagation of waves in 2 dimensions. While the computational complexity of calculating the solutions of wave equation in 3 dimensions is asymptotically much higher than in 2 dimensional space, it is unnecessary to demonstrate the ability to focus the ultrasound by modulation of the intensity, frequency and geometry of the ultrasound probes. However, in 3 dimensional space the intensity in the focal point should increase, since addition of sound waves is coming from all directions not just from left and right. Three or more probes could be combined to define the focal point in 3D. In our model it is also possible to observe the main sound characteristics such as the attenuation, reflection and occlusion. Probes are modelled as the set of finite elements. The pressure of every finite element on the probe is set according to sine function with the selected amplitude and frequency, thus every finite element acts like independent source of the acoustic pressure.
Results
Simulations computed with our model (7) clearly show that it is possible to target small regions of brain tissue. We can also observe the one dimensional cut of acoustic pressure, where focusing is even more evident (8). The probe on the right is slightly displaced due to the asymmetry of the tissue. This clearly shows that models need to be applied to calculate the optimal position of ultrasound probes. Such software will include automatic tissue segmentation, modeling of propagation of ultrasound waves and will assist experts in probe positioning. Our model is a step in that direction, since it can be used as a tool for positioning the probes. In the model that we generated focused and unfocused probes can be added or removed and positioned. Additionally, probe's acoustic pressure and frequency can be set.
One can observe the small focused area in the middle of the brain (Fornix). We can also detect green specks in the bone region, which are caused by the reflection of the sound waves. Since the speed of sound in the bone is much higher than the speed in soft tissue, the acoustic wavelengths in the bone are increased.
The waves that travel from the right to the left, are slightly deformed on the right side due to the wave interferences from two separate probes.
Conclusion and outlook
We generated a model for the propagation of the ultrasound waves through tissue. In our case a transversal cut of the middle head area was taken. Two probes were adjusted in such a manner that the focusing on small area was obtained. With this model we can confirm the potential to use the ultrasound as a platform for therapeutic applications, but for more advanced use the software tools needs to be further developed. Further development will include automatic tissue segmentation from patient medical images, computing propagation of ultrasound waves in 3 dimensions aimed to guide the experts with the probe positioning. The low intensity ultrasound image could be used to receive the feedback of the sonogram of the tissue for each probe that could be used to drive ultrasound stimulation for cell activation in the desired area. It is likely that the specific shape of the stimulated tissue could be selected rather than just a single focal point.
References
