Ultrasound Physics

Parameters of  Sound (Continuous Wave)

• Frequency:

  • Number of certain events that occur in particular time duration.

  • In diagnostic ultrasound the frequency of a wave is described as the number of cycles of an acoustic variable that occur in one second. 

  • Frequency = How many cycles / second.

  • Cycle = One complete variation of a sound wave (acoustic variable)

Frequency

• Unit = Hertz (Hz)

  • 1 cycle/sec. = 1 Hz

  • Kilo (1000) cycles/sec. = 1 KHz

  • Million (1,000,000) cycles/sec. = 1 MHz

• Range of diagnostic ultrasound = 2 - 13 MHz 

• Determined by the sound source = transducer

• Formula = 1 cycle / 1 second → f = 1/time

  • Anything in the denominator position will have inverse relationship with the main variable

  • Anything in the numerator position will have a direct relationship/proportional with the main variable

• Inverse relationship with time; high frequency = less time 

Frequency (f)

• How often something happens

• CW: Number of cycles per second; a continuous wave

• Cycle: One complete variation in pressure or acoustic variable

Period: aka Time

• Definition: The time required to complete a single cycle.

• Period can also be described as the time from the start of a cycle to the start of the next cycle.

• Unit = Seconds, microsecond.

• Formula: Time = 1/f

• Determined by the sound source = transducer

• Average period in ultrasound is = 0.1 to 0.5 μsec

Period (T)

• Time it takes for one cycle to occur

• Units: second, microsecond (μs)

Wavelength

• Length or distance of a single cycle

• The distance between 2 consecutive points in the same phase of a wave cycle

• It is beginning and end points of one cycle

• It is represented by Lambda (λ)

• Unit = meters, mm

• Formula → λ (mm) = propagation speed (mm/μs) / frequency (MHz)

  • λ= c/f

Wavelength (λ)

• Length of space one cycle takes up

• Unit: millimeter (m)

• Wavelength has an inverse relationship with frequency

  • Higher the frequency the shorter the wavelength - good for imaging superficial structures

  • Lower the frequency the longer the wavelength - good for imaging deeper structures

• It influences longitudinal resolution (image quality)

• Determined by the sound source and medium

• Typical values in soft tissue = 0.1 to 0.8 mm 

Propagation Speed

• Definition: the speed at which a wave moves through a medium

• Speed of sound is set by the medium

• Represented by C

• Unit = meters / sec. ; mm. / μsec.

• Formula → C = f (λ)

• Determined by only the medium

• Depends up density and stiffness (properties of a medium)

Density (Mess)

• Mass per unit volume

• The concentration of matter

• Density is inversely proportional to the velocity of ultrasound in a medium

  • High density = less propagation speed

  • Low density = more propagation speed

  • Traffic is high → speed is slow

Stiffness /Hardness

• The resistance of a medium or material to compression

• Stiffness is determined by the bonds that hold the material together.

• Stiffness is directly proportional to the propagation speed

  • High stiffness = high propagation speed

  • Low stiffness = less propagation speed

• Stiffness dominates over density when determining propagation speed.

• Density is related to weight.

• Stiffness is related to ‘squishability’.

• Compressibility and elasticity are opposites of stiffness. 

Amplitude (Visual Display → Strength of Signal)

• Definition: the difference between the average value and the maximum value of an acoustic variable.

• The maximum height or variable of the sound wave

• Unit = varies (depends upon the variable) ; ex: cm/s, mm, dB (decibel)

• Can be changed by changing the TGC

• Determined by = sound source

• Maximum variation that occurs in an acoustic variable

• Units: dependent on acoustic variable

Power

• The rate that energy is transferred in an area.

• Power decreases as sound propagates through the body.

• Unit = Watts

• Determined by the sound source or manufacturer

• Can be changed by the sonographer.

Intensity

• Definition: concentration of energy in a sound beam that passes through a unit area

• Intensity is greatest at the focal zone

• Area is smaller, the intensity is higher

• Intensity decreases as sound propagates through the body (attenuation)

• Intensity is proportional to the power and amplitude square

• Formula → I = power (mw) / beam area (cm²) ; Intensity = Amplitude²

• Unit = watts/cm², mw/cm²

• Determined by = sound source (initially)

• Changed by the sonographer if needed.

• The rate at which energy passes through a unit area

• Units: milliwatts per centimeter squared (mW/cm²) and W/cm²

 Pulsed Ultrasound

• Definition: collection of pulses that travel together that are sent to the body through Transducer or probe, which are separated by gaps of no ultrasound i.e., listening time

• Anatomy of Pulsed U/S → Sending Pulse → 1% → Receiving Echo → 99%

• It consists of pulses separated by gaps in time for listening to the echoes (listening time).

• In a continuous wave, there is no break in cycles, but in pulsed waves there are breaks in cycles.

  • Transmission = On ; Reception = Off

• n = number of cycles/pulses (typically 2 to 3 pulses are used for imaging)

  • Less # of cycles in a pulse is good for image resolution

Parameters of Pulse Ultrasound

• PRF → Pulse Repetition Frequency

• PRP → Pulse Repetition Period

• PD → Pulse Duration

• DF → Duty Factor

• LT → Listening Time

Pulse Repetition Frequency: PRF

• Definition: the number of pulses occurring in one second.

• Expressed in = KHZ

• Formula → PRF = 1/PRP

• Inversely related to PRP

• The number of pulses occurring in one second

• Unit: kilohertz (kHz)

• If pulse repetition frequency decreases, the pulse repetition period increases

• PRF is AKA Sampling Rate or Scale

Pulse Repetition Period: (PRP)

• Definition: It is the time from beginning of one pulse to the beginning of the next pulse including listening time.

• Expressed in units = milliseconds (thousandth of a second) ms

• The time from the beginning of one pulse to the beginning of the next

• Unit: millisecond (ms)

• Formula: PRP (ms) = 1/PRF (kHz)

Pulse Duration:

• Definition: The time that pulse is actually on/occurs

  • Time taken by a single pulse

• Formula: PD (μs) = n X T (μs). (The number of cycles in a pulse x Period (the time for one cycle))

• Expressed in microseconds (μs)

• Unit: microsecond

• Sonographic pulses are 2 - 3 cycles long

• Shorter pulses → improved quality of sonographic images

• Pulse duration decreases, if the number of cycles in a pulse decreased or frequency increased

Spatial Pulse Length

• Definition: The length of space that a pulse takes up

• It is the distance from the start of the pulse to the end of the pulse

• Units: mm

• Determines the resolution of the image

• Short SPL → Improved axial resolution

• More cycles → Longer SPL

• Formula → SPL (mm) = n (# of cycles) X λ (mm)

Duty Factor: (DF)

• Definition: It is the fraction or % of time that pulsed Ultrasound is on

• Longer pulses increase the duty factor

• Increase in PRF → Increased DF → Decreased PRP

• Increase in PD → Increase in DF

• Unit less

• Fraction of time that pulsed ultrasound is on

• Typically DF for sonography = 0.1% to 1%

• Typically DF for Doppler ultrasound = 0.5% to 5%

• Formula → DF = PD/PRP X 100

• Ranges → 1 or <1

• Determined by → sound source (transducer) and can be changed by changing the depth

• For Doppler ultrasound the range = 0.5% to 5%

Listening Time (Reception of Echoes/Reflections)

• Definition: Listening time when echoes are returning to the transducer 

• In real world → PD = < 1%; LT = >99%

• Formula: Listening Time = PRP - PD

• Unit = microseconds (μs)

Physics Lecture 3

Attenuation (Loss of Intensity)

• Definition: The decrease in intensity, power, and amplitude of a sound wave as it travels.

  • The further the sound travels, the more the attenuation occurs.

• Units: dB decibel → decibel is a ratio of beginning intensity compared to final intensity.

• Components of attenuation:

  • Absorption: Conversion of sound energy into heat energy. 

  • Reflection: A certain amount is reflected back to the transducer as returning echoes.

  • Scattering: The sound wave is redirected to different directions depending upon the density of tissue or scanning angle of the sonographer.

• Relationship with frequency:

  • Attenuation of sound in soft tissue depends on the wave’s

    • Frequency 

    • Distance the wave travels

  • Greater the frequency, greater the attenuation

  • Increase the path length, greater the attenuation

Attenuation Coefficient: (Ac)

• Definition: The amount of attenuation that occurs with each cm the sound wave travels

• Unit of measurement: dB/cm

• Equation: 

  • Ac = ½ f

  • Total attenuation (dB) = path length (cm) X attenuation coefficient (dB/cm)

• 0.5 dB of attenuation per centimeter for each megahertz of frequency

• Relationship”

  • With higher frequency → attenuation increases

  • With higher frequency → attenuation coefficient increases

  • With increased path length → attenuation coefficient does not change

  • With increased path length → total attenuation increases

  • With increased Ac → total attenuation increases

Intensity (Related to POWER → expressed via amplitude)

• Amplitude and intensity are the indicators of the strength of the sound

• Amplitude → is the maximum variation that occurs in an acoustic variable

• Intensity → is the concentration of the power in a sound beam

  • Unfocused areas = low intensity

    • 1. Near zone

    • 2. Far zone

  • Focused beam is at the focal point and focal zone because of smaller area = increased intensity

  • It is the rate at which energy passes through a unit area

  • Intensity = Power/area

  • Unit = Watts/cm²

  • Power -- the rate at which energy is transferred

  • Energy -- capability of doing work

  • Increase in area = Decrease in intensity

  • Intensity is proportional to (Amplitude)²

  • Double the amplitude = intensity is increased 4 times

  • Half the amplitude = intensity is quartered

  • Intensity is not constant with in pulses

Levels of Intensity

• Temporal (Time) Intensity: A pulsed ultrasound does not have the same intensity at different times

• Average Intensity: Includes the highest, lower, lowest, and no intensity

• Five Key Words:

  • Peak = the maximum value

  • Average = the mean value

  • Spatial = referring to distance or space

  • Temporal = referring to all time (transmit & receive)

  • Pulse = referring only to the pulse existence (transmit only)

• Spatial Prak (SP) is the greatest intensity found across the beam, which usually is at the center

• Spatial Average (SA) is the average for all values found across the beam, including the larger values found near the center or smaller values near the periphery. 

• Temporal Peak (TP) is the greatest intensity found in the pulse as it passes by.

• Temporal Average (TA) is the lowest value because it includes the dead time between pulses where there is zero intensity.

• Pulse Average (PA) is averaged over the pulse duration. It is the average for all the values found in a pulse, including the larger values in the beginning and the lowest values in the end of the pulse. 

• Six resulting intensity descriptions:

  • These intensities describe the same wave in different ways:

    • Highest to lowest

      • SPTP → Spatial peak, Temporal peak

      • SATP → Spatial Average, Temporal peak

      • SPPA → Spatial peak, Pulse average

      • SAPA → Spatial average, Pulse average

      • SPTA → Spatial peak, Temporal average

      • SATA → Spatial average, Temporal average

• In continuous wave = Beam is always on

• So, TP = TA

• SPTP = SPTA

• SATP = SATA

Logarithms

• A technique of rating numbers

• The logarithm of any number represents the number of time that “10” has to be multiplied together to create the original number

• Always work in base 10

• Ex: 1000 = How many multiples of ‘10’ does it takes to get 1000

  • 10 x 10 x 10

  • Log = 3

  • So log of 1000 = 3

Decibels: A logarithmic scale

• A relative scale -- Ratio of the final intensity to the initial strength

• 2 intensities are needed to calculate dB

• Positive decibels

• 3 dB = 50%

• 3 dB = 2 times

• 6 dB = 2 x 2 = 4 times

• 9 dB = 2 x 2 x 2 = 8 times

• 3 dB decrease corresponds to a 50% reduction in intensity

• 6 dB decrease corresponds to a 75% reduction in intensity

• 10 dB decrease corresponds to a 90% reduction in intensity

Intensity Ratio

• Fraction of the original intensity that remains at the end of the path

• If the intensity at the beginning is known, the intensity at the end may be found by multiplying the beginning intensity by the intensity ratio

Decibels: A logarithmic scale

• Negative decibels

• -3dB = 50% loss

• -6dB = 75% loss

• -9dB = 87% loss

• -10dB = 90% loss

Note: This rule only applies to Powers and Intensities

Physics Lecture 4

Phase Relationships

• Positive Interference/ (IN Phase):

  • When two wave overlap each other in the same phase, they combine to form one stronger wave with greater amplitude

  • AKA → constructive interference or reinforcing interference or in phase

  • This wave helps in reflection and transmission of sound

• Negative Interference/ (OUT of phase):

  • When two waves overlap each other in different phases, they combine to form one weaker wave with smaller amplitude

  • AKA → Destructive interference or weakening interference or out of phase

  • This wave results in scattering and absorption of sound

Harmonics

• Harmonics create an image from reflection which is twice the frequency of the transmitted sound. (ex: 2 MHZ → harmonics = 4, 6, 8 MHz or 5 MHz → harmonics = 10, 15, 20 MHz)

• 2nd harmonics is always twice the fundamental frequency and it can be even and odd multiples of the fundamental frequency

• Transmission of sound at a certain frequency but reception at higher frequency

• Creation of harmonics reflection depends on non-linear behavior of the wave or non-linear behavior of the contrast agents as it travels through the body (positive interference)

• Harmonic frequency echoes improve the quality of sonographic images

• Even and odd multiples of the fundamental frequency

• Harmonic frequencies are generated as sound travels through tissue

Type of Harmonics

• Tissue Harmonics

  • Tissue harmonics are created during transmission

  • Dependence of propagation speed on pressure causes strong sound waves to change shape

  • High pressure are tends to travel faster than the low pressure area, which changes Sine wave to Sawtooth (sinusoidal shape to non sinusoidal shape)

  • Sound distorts (sine → sawtooth) since the compressions and rarefactions travel at different speeds, this creates harmonic energy

  • As scanning depth increases, harmonic signal increases

  • Harmonic signals are not presents at the surface of the transducer, because it is created during transmission

    • Sine Wave/Sinusoidal Shape:

      • One constant frequency

      • Called the fundamental frequency

    • Saw Tooth / Non - Sinusoidal Shape:

      • Contains additional frequencies which are multiples of the main fundamental frequency

      • As wave becomes saw tooth, harmonics become stronger

      • High frequency = stronger the harmonics (more forward tilt in the compression region)

Harmonics

• Higher pressure portions of the wave travel faster than the lower pressure portions

• The wave changes shape as it travels

• This change in the sinusoidal shape introduces harmonics

Contrast Harmonics

• Contrast harmonics are created by distortion of the contrast agents

• Contrasting Agents:

  • These agents create strong reflections that actually “light up” blood vessels or chambers when injected into the circulation

  • They increase echogenicity by generating harmonic reflections and that improve lesion detection and also improve Doppler signals

• Contrasting agents must be: safe (non-toxic), easily administered, small enough to pass through capillaries, stable enough to complete the study or procedure

• As the sound wave encounters the micros bubbles of the contrast agent, the bubble becomes distorted and this non-linear behavior of the contrast agent creates harmonic energy

• Use of agents improves: lesion detection, lesion characterization, and doppler detection

• Sound wave contains variations in pressure during compression and rarefaction, as the sound wave encounters the bubble, the pressure causes the bubble to contract and expand which generates harmonic energy

• The sound wave has to have the strength of pressure to change the shape of the bubble

• The mechanical index has to be > 0.1 for creating harmonic energy

• High intensity beam can create a higher MI which causes more harmonics and strong reflection which improves the vision and quality of the image with contrast agents

Echoes

• “Reflected & scattered sound waves produce the echoes.” (reflected = seen; scattered = absorbed)

• Reflection:

  • Specular Reflection (Mirror Zone):

    • Specular reflectors are large and smooth surfaces like mirror

    • Reflections from specular reflector come from smooth tissue or organ surfaces, like thyroid, liver, spleen, uterus, prostate, or diaphragm

    • It occurs when sound wavelength is much smaller than irregularities in the boundary or surface of the structure, making it organized

    • Is it seen best at 90 degrees angle (angle dependant)

• Backscatter (Diffuse) Reflection (return to the transducer)

  • A reflection that returns to the Transducer directly, that is random and disorganized or diffused reflection

  • Occurs when boundary or surface has irregularities of the same size as the wavelength of sound

Scattering

• Non-Specular Scatter (Rough Surfaces):

  • Scatter flectors are small and rough surfaces (like kidneys)

  • Reflections from scatter reflector are weaker than specular reflector

  • If the boundary between two media has irregularities with size similar to or a bit smaller than the pulse wavelength then the wave may be chaotically redirected in all directions

• Rayleigh Scattering (Scattered in all directions):

  • This scattering happens in all directions, ex: RBCs

  • If the reflector is much smaller than the wavelength, the sound is uniformly diverted in all directions making it systematic scattering

  • Usually occurs in high frequency transducers (frequency dependent)

Physics Lecture 5

Intensities

• Definitions:

  • Incident Intensity: Sound that travels from the transducer to the medium

  • Reflected Intensity: Sound that leaves transducer and returns back in same direction after hitting the medium

  • Transmitted Intensity: Sound that leaves the transducer, hit the medium, and continue to move through the 1st medium and go to the 2nd medium or get absorbed

  • Equation: Incident intensity = reflected intensity + transmitted intensity

  • Units: W/cm² (applies to all intensities)

Impedance

• Definition: It determines how much of an incident sound wave is reflected back through the first medium and how much is transmitted into the 2nd medium

• Units = Rayls (Z)

• Impedance = density x propagation speed

• Impedance mismatch = reflection

• Impedance math = transmission

Intensity Reflection Coefficient (IRC)

• Intensity Reflection Coefficient (IRC): The % of the ultrasound pulse is bounced back (reflected) when the sound beam passes from one medium to another

• Dividing the reflected (echo) intensity by the incident intensity that is reflected will tell how much of the pulse is reflected

• IRC = (Z2 - Z1)²/(Z2 +Z1)² x 100

Intensity Transmission Coefficient (ITC)

• Intensity Transmission Coefficient: The % of ultrasound intensity that is allowed to pass through when the beam reaches an interface between two media

• Dividing the transmitted intensity by the incident intensity yields the fraction of the incident intensity that is transmitted into the 2nd medium (indicates how much of the pulse is transmitted)

• ITC = 1 - IRC

• Increased IRC = Decreased ITC

• Units = None (%); both IRC and ITC are unitless

  • Note: Coefficients, percentage, ratios, and factors are usually unitless

CheckPoint

• Q1: At the boundary between 2 media:

  • If IRC and ITC are added, we must obtain 100% intensity

  • If reflected and transmitted intensities are add, we must obtain the incident intensity

• Q2: A sound wave with an intensity of 30 W/cm² strikes a boundary and is totally reflected:

  • What is the intensity reflection coefficient?

    • 100%

  • What is the intensity transmission coefficient?

    • 0%

  • What is the reflected intensity?

    • 30 W/cm²

Bandwidth

• The range of frequencies contained in a pulse

• The shorter the pulse, the broader the bandwidth

• Formula: BW = Operating frequency / Quality factor

• Unit = MHz

• Definition: The range of frequencies contained in a pulsed ultrasound below and above the main frequency

• The bandwidth is the range of frequencies between the highest and the lowest frequency emitted from the transducer

• An ultrasound pulse used for imaging contains frequencies in a wide range, a “wide bandwidth” because of the backing material

• The process of damping increases the range of frequencies present in a pulse

• Shorter the pulse, the greater the range of frequencies present

• Shorter pulses have broader bandwidth and fractional bandwidths

• Fractional bandwidth is the bandwidth divided by the operation frequency

• Unit: unitless

• Range of frequencies must have 50% of amplitudes of main frequency

Fractional Bandwidth

• Described how the bandwidth is compared with the operating frequency

• Unitless

• Formula: Fractional Bandwidth = Bandwidth / Operating frequency

Quality Factor

• A unitless # representing the degree of damping

• Q factor represents # of cycles in a pulse

• Shorter pulses = Low Q factor

• Low Q factor = Broader bandwidth

• When Q factor is low:

  • Damping is substantial

  • Pulse length and duration are short

  • Bandwidth is wide

  • Image is good

Normal Incidence

• Definition: The incident sound may be reflected back into the 1st medium or transmitted into the 2nd medium, most often - both occur

• It occurs at 90 deg

• Sound wave reflects back at the boundary

• Transmission occurs into the body

• It is also known as:

  • P → Perpendicular 

  • O → Orthogonal

  • R → Right angle

  • N → Ninety degree

  • N → Normal incidence

• Reflection and transmission is determined by the impedance differences in the two media

• Different acoustic impedances = Reflection

• Impedance match = Transmission

• Angle of incident is always equal to angle of reflection 

Incidence and Reflection Angles

• Direction of travel with respect to the boundary

• Incidence angle always equals the reflection angle

Oblique Incidence

• Oblique incidence: Denotes a direction of travel of the incident sound that is NOT perpendicular to the boundary between 2 mediums

• Sound wave comes at acute or obtuse angle

• There is an incident angle other than 90 deg

• Sound bends and take a new path

• Some of it is reflected at the surface

• Angle of incident is always equal to angle of reflection

• Also known as = LATA

  • L → Lateral

  • A → Azimuthal

  • T → Transverse 

  • A → Angular

• In short Oblique incidence:

  • Reflection (not main concern)

  • Transmission

  • Refraction

• Angle of incidence = Angle of reflection

Refraction

• Refraction: Occurs when two conditions are met: 

  • Has an oblique angle

  • Different propagation speeds

• Can not occur with normal incidence or identical propagation speeds

• Governed by Snell’s law

• Snell’s Law: Equation → Sine θ (transmission angle) / Sine θ (incident angle) = Propagation speed 2 / Propagation speed 1

• C! = C2 = no refraction

• C2 > C1 = > transmission angle

• C2 < C1 = < transmission angle

Range Equation

• Range Equation → (Go -Return Time) or (Round trip of sound wave) or (Time of Flight)

• Definition: Calculates the time it takes for a pulse to travel into the body, interacts with organs; tissues and returns to the transducer

• This information of location determines the place of the dot on the screen or display monitor

• The strength of the returning echo determines the brightness of the dot

• Formula:

  • 13 μs rule

  • Depth of reflector = T/13

• Units: cm

• Every 13 μs the reflector is 1 cm deeper in the body

• Depth of reflector = ½ C x T

• Depth of reflector = 0.77 x T

• Depth to the boundary = Go - Return time x C / 2

• Units: mm

  • Note: Range equation shows how far the reflector is placed in the body

Transmission Angle

• Dependent on the incidence angle and the media propagation speeds

• If propagation speed through the second medium is greater than medium one, the transmission angle is greater than the incidence angle and vice versa