In the name of GOD

1. In the name of GOD

2. ULTRASOUND Tavakolli.M.B,PhD Medical University of Isfahan Faculty of Medicine Department of Medical Physics and Medical Engineering

3. Chapter 1 • Introduction

4. History Research for ultrasonic imaging start from 1940 during world war two in parallel to radar and sonar First pilot ultrasound scan was purposed by Johan Reid and John wild (1950) and Holms and Howy in Clorado At the same time therapy use was also started in Illinois university From 1970 routine ultrasonic image was in progress During these days Doppler sono was started using pulse and CW Duplex systems were developed to measure both blood flow and imaging the vessels Today ultrasound in the second most widely used imaging method

5. Role of ultrasound in medicine • Not only it is a complement for other imaging methods, also it has several specific advantages. • Advantages: • 1-Use non-ionizing radiation • 2-Cheap compared with other modalities • 3-Real time • 4-Resolution is around 1 millimeter • 5-Data about blood flow • 6-Portable instrument • 7-Useable in most of medical filels • Disadvantages: • 1-Bone and air screen the image • 2-Window to study biological systems is limited • 3-Need proficient operator • 4-Same time image are not easily assessable

6. Chapter 2 Fundamental of Acoustic Propagation

7. Fundamental of Acoustic Propagation • Sound Wave: Is a type of mechanical energy that is transmitted through medium. • Propagation: Sound wave propagate through deformation of the elastic medium • Characterized by parameters such as: • Particle or medium velocity • Pressure • Particle displacement • Density • Temperature

8. Fundamental of Acoustic Propagation • Sound wave spectrum: Is divided into three region ofInfrasound(f<20Hz); Sound (f=20 to 20000Hz) and Ultrasound (f>20kHz) • To understand the propagation in the medium; the medium can be modeled as a three dimensional matrix of elements. In two dimensional we can show:

9. Sound velocity • C=fλ (λ is the wavelength in meter and f is the frequency in Hz)

10. Stress and Strain Relationship • Propagation of ultrasound depend on strain and stress of the material • Stress: It is the tensile force exerted on the incremental cube by other parts of the unit area • On a unit surface perpendicular to Z axis; the stress can be separated into three components of: • Kzx, Kyy, Kxy and Kzx, Kyx and Kxx denoted the stress acting on the Y- and X plane • The longitudinal stress in the z direction by the Z plane is:

11. The shear strain in the X direction by the Z-plane along the Z-axis is: • U, V and W denote the displacement in the X-, Y- and Z- direction respectively • Longitudinal strain of Z-plane in the Z-direction. • Shear strain of Z- plane in the Y-direction

12. Under the condition of small displacements, the stress-strain relationships are linear and: • ν and µ (Shear modulus) are Lame constants. • The lame constants are related to more conventional material constants such as Young,s modulus (E), Bulk modulus (B) and Possion,s ratio by the following: • Young,s modulus=stress/stain or Kzz/εzz • Poisson,s ratio=-strain in transverse direction/strain in longitudinal direction or –εyx/εzz • The bulk modulus=1?comperessibility. • Comperessibility (G): • P is the applied pressure • Pressure=-Kzz • In afluid in which approch to 0, B=>V and E=0

13. Acoustic wave equation • By applying Nerwton’s second law, equation of motion in Z direction of a cube of a medium is: • Compressional wave: • If shear stress is not resent Kzy and Kzx=0 and we have: • Substituting is equation a we have: • The solution for this equation is in the form of f(z±ct), where the negative sign indicates a wave travelling in +z direction. • The displacement W is in the same direction as the wave propagation. The type of wave is called compressional or longitudinal wave and is: • W- and W+ are displacement in positive and negative direction • ω is the angular frequency • K=ω/c is the wave number

14. Sound velocity is given by: • For fluid µ=0 and: • Shear waveFor a case in which kzz=kzy=0, a new type of wave in which the displacement W is perpendicular to the direction of propagation X is characterized by: • By sbstituting Kzx=µ() into preceding equation: • This equation describes a wave travelling in the X-direction with a displacement in the Z-direction. The solution is: • This type of wave is called shear or transverse wave

15. Characteristic Impedance The medium or particle velocity in the Z-direction uz is: It can be seen that the particle velocity is always 90 degree out of phase relative to the displacement The relation between stress and pressure is: so For longitudinal wave we can write: Replacing jωW by uc: Pressure like velocity is 90 degree out of phase relative to displacement. Pressure and velocity are in phase for positive traveling and 180 out of phase for negative traveling wave.

16. The specific acoustic impedance is defined as: Positive and negative are for positive and negative traveling wave

17. Intensity • Intensity is the energy carried by a wave per unit area normal to the direction of propagation. • Power is the intensity per unit time (P) • P=Force exerted by the pressure wavexmedium displacement=forcexmedium velocity • And • For sinusoidal wave the average intensity in a cycle is: • P0 and u0 are peak values of pressure and medium velocity. • Because Z=p/u=ρc then

18. Important parameters • Intensity within an ultrasound beam in general is not spatially uniform. Typical is: • Spatial average intensity ISA is the average intensity over the ultrasound beam • 2x is the beam width (spatial extent between two -3 or 6dB points) • Temporal average intensity is the average intensity over a pulse repitition period Tand is given by duty factor (pulse duration t/pulse duration T) and temporal peak intensity

19. Other parameters being used (AIUM and FDA) • Spatial average temporal average intensity (ISATA) • Spatial Peak temporal average intensity (ISPTA) • Spatial Peak pulse average intensity (ISPPA)

22. Cont. • Spatial averaging over the cross sectional area of the beam for each temporal intensity is also specified. • A cut off point of 0.25% times SP intensity has been established to limit the area over which the intensity is averaged. • Three combinations are possible: • I(SATP): Spatial Average Temporal Peak Intensity • I(SAPA): Spatial Average Pulse Average Intensity • I(SATA): Spatial Average Temporal Average Intensity • For non-focused transducers, SP is usually grater than the SA by a factor of 2 to 6 • The SP intensity is often converted to SA by multiplying the peak value by one third • In case of focussing transducer, the degree of focusing influencing this factor gratly • The ration of SP to Sa range from 5 to 50

24. Other Intensity descriptors • Instantaneous peak intensity (ip): Is the maximum intensity with respect to space and time and is the same as I(SPTP). • Imis refers to time averaged intensity over the largest half cycle in the pulse at the spatial peak. • For a perfect sine wave it is equal to I(SPPA) • Instantaneous intensity is determined from the measured acoustic pressure (p) using the equation: ip=p2/ρc • The peak negative pressure is also called the peak rarefactional pressure (sometime used)

25. Radiation force • For a plane wave of intensity and sound velocity of c, radiation force per unit area or radiation pressure is: • A is a constant depending on the acoustic properties of the target and its geometry • For a reflector with very high Z in water a=2 and if it is a perfect absorber a=1 • It can be understand from momentum transfer at boundary • For perfect reflector f should twice than perfect absorber • One way to measure radiation force is to use radiation force balance

26. Reflection and Refraction • As the ultrasound energy encounters and interface, partly reflected and partly diffracted. According to Snell’s law we have: • The pressure reflection and transmission are: • As p=zu, I=p02/2Z then: • For normal incidence

27. Attenuation • Pressure of a plane monoenergatic wave propagation in the z-direction decrease exponentially as: • P(z=0) is the pressure at z=0 • Attenuation coefficient is: • Att. Coeff. Is in nepers. In dB it is: • Typical attenuation coefficient for different materials are shown in table 1.

28. Absorption • Most of the absorbed energy changes to heat • Depends on : • Viscosity • Relaxation • Absorption depends on frequency of the wave • According to the definition of shear strain In z direction we have: • And the strain rate in fluid is: • ux is particle velocity in x direction and is the velocity gradient along z-axis • When a fluid with finite viscosity is subject to shear stress kxz, it shows a velocity gradient and the coefficient of viscosity with unit poises is defined as the ratio of shear stress to the resultant velocity gradient: or • It can be shown that for homogenous medium absorption coefficient is:

29. It can be seen that absorption coefficient depends on ω2 • In reality absorption coefficient in most biological tissue depends on ω and not ω2 • Absorption coefficient is dominated by relaxation • Relaxation is the process of return of a particle to its natural position when pushed by an ultrasonic wave • The required finite time for this process is called relaxation time • When relaxation time is short compared with wave period its effect is small • When it is comparable to wave period, the particle may not be able to return to its position completely before second wave arrive and hence more energy requires to reverse it • If the frequency is increased high enough that the molecules can not follow the wave motion, the relaxation effect become negligible. • Maximum absorption occurs when the relaxation motion of the particles is completely out of synchronization with wave motion

30. Mathematically, the relaxation process can be represented by :

31. Many components, giving rise to many relaxation frequencies, are in biological tissues. • Absorption coefficient in biological tissue can be expressed as: • A is constant and Bi and fi are relaxation constant and frequency. • Figure is a possible scenario for this equation in which many relaxation process may overlap resulting in a linear increase in diagnostic frequency range or a constant αλ.

32. Figure below shows absorption of ultrasound in various tissue. It is almost linear in most of the tissues in term of (αλ)

33. Scattering • As a wave is incident on an object part of the wave will be scattered • The scattering characteristic is depend on scattering cross section • Assuming incident plane wave pi(r)=e-jkr with k=ki (i is a unit vector in the incident direction) and r are vectors representing the wave number and the position • The scattered wave at rs due to a scatterer at r0 is given by: • Where o is a unit vector in the direction of observation, provided that the observation point is in the far field of the scatterer and

34. Term f(o,i) is called scattering amplitude function which describes the scattering properties of the object and depends on the direction of the incident and observation of the wave • The incident of the intensity in a medium with mechanical impedance Z is: • And scattered intensity is: • Substituting in the equation (a) we have: • Rearranging the above equation we have • σd(o,i)=IfI2 is called differential scattering cross section, which is the power scattered in the o direction with incident direction I in one solid angle per unit incident intensity • When o=-I, σd(i,-i) is called backscattering cross section

35. By integrating σd over 4п, σs (power scattered by the object unit incident intensity) is: • dΩ is the differential solid angle • Similarly absorption cross section σa can be defined as the total power absorbed by the object • Attenuation coefficient is 2α=σs +σa (2α is the intensity attenuation coefficient) • If n particle per unit volume present and n is small (n<1% of the volume); 2α=n(σs+σa) • If n is large multiple scattering occurs • To solve the equation for multiple scattering different method can be used (computer based and analytical based) • If the scatterer assumed to be sphare with radius much smaller than the wavelength (Morse and Ingard, 1968) then

36. For a dense scatterer object like tissue a parameter called backscattering coefficient in a unit centimeter-stradian-1 also called volumetric backscattering cross-section which is the power scattered by a unit volume of scatterers in one solid angle per unit incident intensityThe integrated backscattered (IB) is defined as the frequency average of the backscatter over the bandwidth of the signal and is: • To eliminate the dependence of backscatter signal from tissues characteristics, IB is usually expressed in dB • er(ω) is the reflected signal from the flat reflector at angular frequency of ω

37. Ultrasonic scattering depends on frequency, particle size and material, … • It is very attractive to obtain mechanical, physical and pathological information from scattered radiation

38. Nonlinearity parameter B/A • Spatial peak temporal peak intensity (Isptp) may reach the level of 100W/cm2, thus nonlinear acoustic phenomena must be considered • Reasons for study • 1-harmonic imaging • 2-tissue pattern characterization • 3-to understand the absorption by tissue • Non-linear behavior of a fluid can be assessed by factor B/A, the second order parameter • In an adiabatic process in which the entropy is constant or there is no flow energy, the relation between pressure and density can be expressed as a Tylor series expansion of pressure, p about the point of equilibrium density ρ0 and entropy is: • from these equation and the speed of sound equation: • we can show that:

39. The above equation can be converted to more easier using thermodynamic relationship • The first term represents change in sound speed per unit change in pressure at constant temperature and entropy • The second term represent the change in sound per unit change in temperature at constant pressure and entropy • Therefore B/A can be estimated from thermodynamic process • It can be estimated also by finite amplitude method in which the second pressure harmonic is measured

40. B/A value for a few tissues are: • Nonlinearity can cause distortion of the wave forms • The harmonic amplitude relative to the fundamental or first order harmonic is shown below

41. Doppler effect • It describe changes in frequency of the received wave when reflected by a moving target • Doppler shift is: • When c>>ν then • When direction of particle motion is not in the same direction of wave propagation: