Ultrasound Calculations

The table provides the data for the acoustic impedance and absorption coefficients for muscle and bone.

ultrasound 4

A parallel beam of ultrasound of intensity I_0  enters the muscle of thickness 3.0 cm as shown. The ultrasound is then reflected at the tissue boundary and returns to the surface of the muscle.

Calculate the intensity, in terms of I_0 , that is received when the ultrasound returns back to the surface of the muscle.


There are three important parts to solve this problem. The first part would be the attenuation of intensity inside the muscle. The second part would be the reflection of ultrasound at the muscle-bone boundary. The last part would be the intensity attenuation inside the muscle back to the surface.

I=I_0 e^{-21 times 0.03}

 

Types of Ultrasound Scans

We will investigate two types of ultrasound scans.

A-Scan

This is a simple type of scan. A pulse of ultrasound is sent into the tissue and the echoes are detected to determine the depth of the structure.

ultrasound 3

The thickness of the bone can be calculated using Δt , which is the time it takes for the ultrasound to travel twice the distance of the bone. Since the speed of ultrasound in bone is known, the thickness of the bone can be determined using the formula

t=frac{v times Delta t}{2}

where v is the speed of ultrasound in the medium.

A-scans are used for simple procedures such as measuring the thickness of bones.

B-Scan

This type of scan produces more detailed images than A-scan. A series of pulses are directed at the organ, and each reflected pulse is analysed to the depth and nature of the reflecting surface. The timing of the reflected pulse provides information to the depth of the surface while the intensity provides information to the type of reflecting surface.

ultrasound 4

In the above diagram, it can be seen that when the series of pulses are sent, a pattern of dots is obtained, which allows the computer to generate an image of the scanned organ.

Ultrasound Intensity Reflection and Attenuation

Intensity Reflection Coefficient

When a beam of ultrasound is directed into a body, a certain proportion of the initial intensity would be reflected at the boundary. The ratio of the reflected intensity to the initial intensity can be obtained using the formula

\begin{aligned}\frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}\end{aligned}

ultrasound

Air-Tissue Boundary

Using the formula from above, we can calculate that the intensity reflection coefficient of an air-tissue boundary is more than 99%.

\begin{aligned}\frac{I_r}{I_0}=\frac{(1.63 - 0.0004)^2}{(1.63 + 0.0004)^2}=0.999\end{aligned}

Note that the squaring of both the numerator and the denominator means that it does not matter which medium is assigned as Z_1 and Z_2 .

This is the reason why impedance matching need to be done when ultrasound is performed on a patient.

Intensity Attenuation

When ultrasound travels across a tissue, its intensity attenuates. This is because the wave loses energy to the tissue as it travels. The intensity of the wave at any particular depth of the tissue is determined by the formula

I=I_0 e^{-mu x}

where x is the depth of the point at which the intensity is determined, $latex μ $ is the absorption coefficient of the material and I_0 is the original intensity before the ultrasound wave enters the particular medium.

ultrasound 2

To find the intensity of the ultrasound that is received at the edge of medium 1, we can employ the following sequence of calculations:

  1. I_0 is the initial intensity.
  2. When the wave reaches the boundary of medium 1 and medium 2, it would have been attenuated by the amount I=I_0 e^{-\mu x} where x  is the depth of medium 1.
  3. This wave is then reflected at the boundary, where \begin{aligned} I_2 = \frac{(Z_2-Z_1)^2}{Z_2+Z_1)^2}\end{aligned}.
  4. Lastly, when the wave reaches the edge of medium 1, its final intensity I_3=I_2 e^{-\mu x}

Summary

  1. When you are calculating the reflection of ultrasound at a boundary, use \frac{I_r}{I_0}=\frac{(Z_2-Z_1)^2}{(Z_2+Z_1)^2}
  2. When you are calculating the intensity attenuation over a distance, use I=I_0 e^{-\mu x}

Problems

Using Ultrasound for Medical Imaging

In medical imaging, ultrasound is used mainly to detect the shape of a foetus. Ultrasound is also used for imaging the internal of a body. Since the wavelength of an ultrasound is in the order of about 1 cm, it cannot be used to detect structures that are small. Ultrasound is most effective with structures like the bone, or internal organs and foetus.

Principle of Using Ultrasound for Medical Imaging

The principle used in medical imaging is called echo sounding. Ultrasound is directed into the body. Since different materials have different acoustic impedance, when an ultrasound is incident on a boundary of two materials, there would be a reflected component of the ultrasound. This reflected component would then be received by the detector and used to compose the image of the reflected surfaces in the body.

Acoustic Impedance

Acoustic impedance is defined as the product of the density of medium and the speed of sound in the medium.

Mathematically,

 Z= \rho \times c

The unit of Z would be \text{kg m}^{-2}\text{ s}^{-1}

The meaning of acoustic impedance is how good the sound wave travels in that medium. Large value means the sound wave travels better. The acoustic impedance of some common materials are listed below:

Acoustic impedance of some common materials in the body
Acoustic impedance of some common materials in the body

There will be significant reflection of the ultrasound waves at boundaries of two materials that have widely different acoustic impedance. For example, significant reflection occurs between blood and done boundaries. Hence, we can use ultrasound to investigate the shape of the bone. This would not be a good way to investigate the structure of muscle because muscle and blood have similar acoustic impedance.

Use of Gel in Ultrasonic Scanning

When a pregnant woman performs an ultrasound scan, the doctor would always smear a layer of gel onto her abdomen before performing the scan. The boundary between air and the skin causes significant reflection. The acoustic impedance of air is 0.0004 but skin has an acoustic impedance  of about 1.6. Most of the incident ultrasonic waves would be reflected, leaving very little waves to penetrate the body to be reflected by the other internal structures of the body. Using gel removes the air medium such that the boundary would be between the gel and the skin, which have very similar acoustic impedance. Smearing of gel onto the skin is known as impedance matching.

Ultrasonic Production and Detection

Production

Sound is created when a surface vibrates, pushing the medium particles into a series of compressions and rarefactions. A human being would normally be able to hear sound with frequency between 20 Hz and 20000 Hz.

A piezo-electric  crystal is able to expand or contract according to the applied voltage. If an alternating current is applied across this crystal, it will expand and contract according to the a.c. frequency. When the a.c. frequency is beyond 20000 Hz, ultrasound would be produced.

Detection

When a piezo-electric crystal changes shape, electric voltages are produced. When the ultrasound is incident onto a piezo-electric crystal, the crystal would expand and contract according to the ultrasound frequency. The series of expansions and contractions produce very small alternating voltages, which can be amplified with an operational amplifier. The signal would then be converted to images.