Acoustic impedance and boundary reflection

A sound wave can travel differently in different tissues. When it reaches a boundary between them, the difference between their acoustic impedances determines how much of the wave is reflected and how much is transmitted.

The acoustic impedance $$Z$$ of a medium depends on its density $$\rho$$ and the speed of sound $$c$$ in it:

Acoustic impedance is measured in $$\mathrm{kg,m^{-2},s^{-1}}$$, a unit also called the rayl.

Reflection at a tissue boundary

When a sound wave reaches a boundary between tissues with different impedances, part of its intensity is reflected back into the first tissue and part is transmitted into the second. For normal incidence, where the wave meets the boundary perpendicularly, the reflected fraction is:

Here, $$I_i$$ and $$I_r$$ are the incident and reflected intensities, while $$Z_1$$ and $$Z_2$$ are the impedances of the first and second tissues.

The greater the impedance mismatch between the tissues, the greater the fraction of incident intensity reflected. If their impedances are equal, $$Z_2-Z_1=0$$, so no intensity is reflected at the boundary.

If absorption at the boundary is neglected, the incident intensity is divided between reflection and transmission. The transmitted fraction is therefore:

Here, $$I_t$$ is the transmitted intensity. A larger reflected fraction therefore leaves a smaller fraction transmitted.

Ultrasound attenuation

As an ultrasound wave travels through tissue, its intensity decreases: less sound energy passes through each unit area per second. This decrease is called attenuation.

Tissue absorbs some of the wave’s energy, transferring it to the tissue’s internal energy. Some sound is also scattered, meaning it is redirected away from the original beam. Both processes reduce the intensity travelling onwards in the original direction.

Attenuation and echo depth

As the ultrasound travels deeper, attenuation leaves less intensity available to produce echoes from deeper structures. An echo is then weakened further as it travels back through tissue to the transducer. The total distance therefore includes both the outward and return journeys.

Attenuation generally increases with distance travelled and with ultrasound frequency. Higher-frequency ultrasound is therefore more strongly attenuated, limiting the depth from which echoes strong enough to be useful can be obtained.

Piezoelectric ultrasound transducers

An ultrasound transducer contains a material that can convert electrical signals into vibrations and vibrations back into electrical signals. This allows the same transducer to transmit an ultrasound pulse and receive its returning echoes.

The transducer uses a piezoelectric material. When a potential difference is applied across it, the material changes shape. The reverse also happens: when the material is mechanically stressed, such as by compression or stretching, it produces a potential difference.

Transmitting and receiving ultrasound

To transmit, a short electrical pulse is applied across the piezoelectric element. The element changes shape and vibrates, producing an ultrasound pulse.

To receive, a returning ultrasound echo compresses and stretches the element. This mechanical stress produces an electrical signal, which can be processed to obtain information about the echo. The transducer therefore acts as both a transmitter and a receiver, using the two reciprocal effects of the piezoelectric material.

Ultrasound A-scans

An A-scan shows the echoes received after an ultrasound pulse is transmitted along a single line. Its horizontal axis shows the time elapsed since transmission, and its vertical axis shows the amplitude of the received echo.

As the pulse travels, it can produce echoes at boundaries between structures. Each returning echo appears as a peak on the A-scan. The position of a peak shows when the echo arrived; its height indicates the strength of the received echo.

Finding depth from an A-scan

If the speed of sound is known and taken as constant, the time taken by an echo can be used to find the depth of the boundary. The pulse travels to the boundary and back, so the total distance travelled is twice the depth:

Here, $$d$$ is the boundary’s depth, $$c$$ is the sound speed and $$t$$ is the time between transmitting the pulse and receiving its echo. A later echo therefore corresponds to a greater depth along the beam.

An A-scan gives depth information along one line, not a two-dimensional image. In the eye, peaks from different structures can be used to find their depths; the difference between those depths gives the distance between the structures along the beam.

Ultrasound B-scans

An ultrasound B-scan builds a two-dimensional image of a slice through the body. It uses echoes produced when ultrasound pulses reach boundaries between different tissues. Each echo provides information about the position and appearance of a boundary along the beam’s path.

From echoes to a B-scan image

A pulse travels into the body and an echo returns from a reflecting boundary. The time between sending the pulse and receiving the echo indicates the boundary’s depth: a later echo has travelled farther to the boundary and back, so it comes from deeper within the body.

The strength of the echo determines the brightness shown at that position in the image. A stronger echo is displayed more brightly; a weaker echo is displayed less brightly.

A single beam path gives information along one line. By combining echoes collected from different beam positions or directions, the scanner builds a cross-sectional image showing the arrangement of internal structures.

B-scans are used, for example, to image a fetus during pregnancy and to examine internal organs.

Ultrasound imaging compared with other methods

Ultrasound imaging sends high-frequency sound waves into the body and uses their returning echoes to form an image. The sound waves are non-ionising: unlike the ionising radiation used in X-ray and CT imaging, they do not have enough energy to ionise atoms. MRI also does not use ionising radiation, but it uses a different method to produce images.

Ultrasound is useful for imaging soft tissues. It can show images in real time, so movement can be observed as it happens. Ultrasound equipment is also relatively portable compared with some other imaging equipment.

Ultrasound resolution and penetration

Image resolution is the ability to distinguish two nearby structures as separate. Higher-frequency ultrasound has a shorter wavelength and can therefore provide better resolution. However, higher-frequency waves are generally attenuated more strongly: they lose more of their energy as they travel through tissue. Less energy returns as an echo from deep structures, so penetration is reduced.

The frequency chosen is therefore a compromise. A higher frequency can reveal more detail near the surface, while a lower frequency can reach greater depths but generally provides less detail.

Limits on ultrasound images

Air and bone can obstruct ultrasound waves, making structures behind them difficult to image. Image quality also depends on the depth and type of structure being examined: echoes from deeper structures are weaker because the waves travel farther through tissue.

For these reasons, ultrasound may be less suitable or provide less detail than X-ray, CT or MRI for some structures. The best method depends on what needs to be examined.

Ultrasound exposure

Although ultrasound avoids ionising radiation, it can deposit energy in tissue. This can cause heating and potentially other biological effects. Ultrasound exposure is therefore controlled, including by limiting the strength of the waves and the duration of exposure.