Principles

Ultrasonic waves are acoustic waves with frequencies above the upper audible limit of human hearing (>20kHz). In practice, the most common method to produce medical ultrasound is by mechanical vibrations of a specialised material when an external alternating electric field is applied across it, i.e., the piezoelectric effect. Acoustic waves are generated with devices known as transducers or probes, in which piezoelectric crystals are the core [18].

Placed between two electrodes that apply a voltage, these crystals are induced to expand and contract in a sinusoidal manner. By doing so, appropriate sound waves for diagnostic use are produced. When travelling through matter, a fraction of these waves is reflected or scattered upon finding an impedance mismatched in the boundary of two media while the remaining fraction is transmitted [18]. When acoustic backscatter waves strike the crystal, the piezoelectric effect is opposite, converting the induced vibrations in an electrical signal whose amplitude is proportional to the mechanical pressure [18]. The electric signal is detected by the electrodes attached to the crystal and converted to a digital signal to be further processed for imaging and characterising the medium [18], [19].

Mechanical waves are characterised by: 1) period (T), that is the time between two consecutive repetitions of an identical pattern; 2) frequency (f), the reciprocal ofT, defined as number of times a pattern repeats itself per second; and 3) wavelength (λ), which describes the spatial repetition of the pattern sequence [42], [43]. These concepts are related as:

λ=cT= c f

(2.1) wherecis the local speed of sound.

The relationship between speed of sound propagation (c), elastic modulus (K) and local density of a material (ρ) is described as:

c=

√K ρ

(2.2) This equation shows that the speed of sound is limited by the features of the material, indicating a time delay between the emitted and the detected acoustic waves, known as time of flight (TOF). These time delays provide practical information about physical properties of the tissues under study [42], [43].

In wave propagation, a more general term for the factor limiting the speed of sound is acoustic impedance (Z_A) and depends directly on the density of the material [42], [43]:

Z_A=ρc (2.3)

Regarding the interaction of ultrasound waves with matter, the most relevant concepts to consider are attenuation, reflection, refraction and scat-tering.

Indeed, attenuation is an essential physical phenomenon in medical ultrasound. Body tissue converts acoustic waves into motion as a result of friction, which translates into heat energy. This absorption reduces the intensity of sound waves that are reflected and, ultimately, detected by the transducer. Thus, the process of detection is more difficult in a deep region of interest as the attenuation is higher [42], [43]. The attenuation in the pressure signal,d P, is described as:

d P=αPd z (2.4)

Where P is the incident pressure; d z is the distance over which the absorption takes place; andαis the attenuation factor of the medium [42], [43]. Solving this equation forPin the case of a plane wave:

P(z) =P₀e^−αz (2.5)

Where P₀ is the pressure atz = 0. This equation is known as the Beer–Lambert–Bouguer law of attenuation and shows the exponential de-crease in amplitude of ultrasound as it passes through homogeneous tissue [42], [43].

Attenuation varies with tissue type (see table 2.1) and is stronger at higher frequencies, particularly at deeper locations, producing a low-pass filtering effect and limiting the frequency that can be used clinically in echocardio-graphy [42], [43].

Attenuation is often defined as the intensity ratio of a measured signal (P₁) to the source signal (P₀) expressed in logarithmic form:

d B=10l o g₁₀(P₁

P₀) (2.6)

Due to the high attenuation coefficient of air, usually a gel with a low attenuation coefficient is used as an interface between the transducer and

Table 2.1: Values of ultrasonic parameters for different biological tissues.

Adapted from [18].

Tissue Speed of Attenuation Acoustic impedance propagation [m/s] coefficient [cm⁻¹] [10⁴k g/m²s]

Blood 1540 0.0198 1.6

Bone 2240 3.01 3.8-7.4

Fat 1450 0.1 1.4

Muscle 1540 0.193 1.7

Lung 0.26

Plasma 0.0069 1.5

Water 1480

the skin during ultrasound measurements to avoid air gaps. Additionally, ultrasound is unable to image the lungs, since they contain air, or any struc-ture behind. Bone also has a high attenuation coefficient, making difficult to perform ultrasound measurements of tissues covered by them [42], [43].

Effectively, it is necessary to have a field of vision of the tissue without bones or lungs in the way. For instance, certain windows must be used to image the heart and avoid the lungs [42], [43].

After the transducer emitted acoustic waves, they propagate across the body tissue, finding acoustically different media. At these interfaces, the energy is partially reflected, and the remainder is transmitted into the second tissue, generating secondary waves [42], [43]. By definition, the angle of incidence (θ_i) is equal to the angle of reflection (θ_r) (see figure 2.2). In practice, the angle of incidence should be limited below 3^○to avoid reflected waves to miss the transducer [42], [43].

Figure 2.2: Graphical description of the Snell law. At the interface of two different media, the passing through of an incident (i) pressure wave generates a reflected (r) and a transmitted (t) pressure wave-front.

Reflection occurs at a smooth boundary of two media. How much of the incident wave is reflected can be calculated as the ratio of the reflected wave to the incident wave, i.e., the pressure reflection coefficient (r) [42], [43]:

r= P_r

P_i = Z_A2cosθ_i−Z_A1cosθ_t

Z_A2cosθ_i+Z_A1cosθ_t (2.7) Whereθ_i andθ_tare the incident and transmitted angles, respectively;

Z_A1andZ_A2are the acoustic impedances of both media; andP_iis the incident pressure andP_ris the reflected pressure. This equation shows that the pressure reflection coefficient is dependent on the acoustic impedance of both media [42], [43].

In turn, the ratio of the transmitted to reflected wave pressures, or the pressure transmission coefficient (t) is defined as:

t= P_t

P_i = 2Z_A2cosθ_i

Z_A2cosθ_i+Z_A1cosθ_t (2.8) Where transmitted pressure isP_t.

These equations establish that modest variations of acoustic impedance between two media generate noticeable alterations in the intensity of reflected and transmitted signals. In fact, biological tissues have comparable acoustic impedances, allowing most of the sound waves to be transmitted (see table 2.1).

Thus, ultrasound can penetrate the body to provide information on deep structures, but it can also “bounce” between interfaces, acting as a source of noise [42], [43].

Another source of noise associated with reflection is induced by the small size of the cells. Reflection at a microscopic scale (structures smaller than the wavelength of sound) undergoes scattering, which is defined as diffuse reflection, meaning the acoustic waves are sent in all directions [42], [43].

Scattering occurs at tissue inhomogeneities that induce the reflected waves to interfere with each other, producing a complex ultrasound pattern known as speckle. Because speckle is relatively constant over time, it is possible to track its motion to detect tissue deformation and strain. In sum, scattering is a form of attenuation and a complex source of noise in ultrasound imaging [42], [43].

Resolution is another relevant parameter of ultrasound imaging. There are two scanning directions, and thus two distinctive resolutions: axial and lateral [18]. Axial resolution is the capacity to distinguish two subsequent layers in the direction of the wave propagation and is a direct function of the emitted signal wavelength, whereas lateral resolution refers to the ability to

discern between two points perpendicular to the wave propagation and is determined by the ultrasound beam width and, hence, by the dimension of the transducer [42], [43].