Unit 2 Physical Principles Flashcards

Fundamental Relationships and Metric Units

• Fundamental Mathematical Relationships (Reference: Edelman, pages 2-4):

  • Unrelated: Two items that are not associated with each other.

  • Directly Related or Proportional: Two items are associated such that when one item increases, the other increases.

  • Related or Proportional: Two items are associated or affiliated, but the relationship is not specified as direct or inverse.

  • Inversely Related or Proportional: Two items are associated such that when one item increases, the other decreases.

  • Reciprocal Relationship: A specific type of inverse relationship where the two items multiplied together equal 1. (e.g., $x \times \frac{1}{x} = 1$).

• Metric System and Unit Conversions (Reference: Edelman, pages 5-10):

  • Giga (G): $10^9$ (e.g., $1,000,000,000$)

  • Mega (M): $10^6$ (e.g., $1,000,000$)

  • Kilo (k): $10^3$ (e.g., $1,000$)

  • Hecto (h): $10^2$ (e.g., $100$)

  • Deca (da): $10^1$ (e.g., $10$)

  • Base Units: Meters, Seconds, Hertz, Liters, Pascals, Rayls, Watts. Scientific value is $10^0$ (e.g., $1$).

  • Deci (d): $10^{-1}$ (e.g., $0.1$)

  • Centi (c): $10^{-2}$ (e.g., $0.01$)

  • Milli (m): $10^{-3}$ (e.g., $0.001$)

  • Micro (\mu): $10^{-6}$ (e.g., $0.000001$)

  • Nano (n): $10^{-9}$ (e.g., $0.000000001$)

Characteristics of Sound Waves

• General Definition: Sound waves are traveling variations of energy carrying quantities from one place to another.

• Medium Requirement: Sound requires a medium to travel through; it cannot travel in a vacuum.

• Wave Type: Sound waves are longitudinal and mechanical pressure waves.

• Particle Motion: Results in back-and-forth particle motion that is parallel to the direction of wave travel.

• Note on Transverse Waves: These travel perpendicular to the direction of wave propagation.

• Regions of Compression and Rarefaction:

  • Compressions:

    • Areas of high pressure and high density.

    • Medium molecules are squeezed together.

    • Represent the positive amplitude of the wave.

    • Related to media stiffness.

  • Rarefactions:

    • Areas of low pressure and low density.

    • Medium molecules are stretched apart.

    • Represent the negative amplitude of the wave.

    • Related to media elasticity.

Acoustic Variables and Interference

• Acoustic Variables: Variation in these variables cause energy transfer in sound waves. Changes in these variables can lead to media damage called bioeffects.

  • Pressure: Concentration of force in an area ($\text{Force / Area}$). Measured in Pascals ($Pa$).

  • Density: Concentration of mass (matter) in a volume. Measured in $g/cm^3$ or $kg/cm^3$.

  • Particle Motion or Distance: Measured in $cm$ or $mm$.

  • Temperature: Measured in Celsius ($^{\circ}C$) or Fahrenheit ($^{\circ}F$).

• Wave Interference:

  • In-Phase: Peaks and troughs of the wave occur at the same time and/or same location.

  • Out-of-Phase: Peaks and troughs occur at different times and/or locations.

  • Constructive Interference: Created by in-phase waves that result in a single wave with a larger amplitude than either of the individual waves.

  • Destructive Interference: Created by out-of-phase waves that result in a single wave with a smaller amplitude than either individual wave.

  • Complete Destructive Interference: Occurs if equal quantity positive and negative amplitude sound waves interact, canceling each other out.

  • Note: Different sound wave frequencies cause both constructive and destructive interference.

Parameters of Continuous Wave (CW) Ultrasound

• Period (T):

  • Definition: Time it takes for one cycle to occur (start to start).

  • Units: Second ($s$), millisecond ($ms$), microsecond ($\mu s$).

  • Formula: $T(\mu s) = \frac{1}{f(MHz)}$

  • Adjustability: Not adjustable by the sonographer.

  • Practice Question: What is the period for a $5\,MHz$ transducer? (Answer: $0.2\,\mu s$).

• Frequency (f):

  • Definition: Number of cycles per unit of time (second). A cycle is one complete variation of an acoustic variable.

  • Units: Hertz ($Hz$), $kHz$, $MHz$.

  • Formula 1: $f(MHz) = \text{Propagation speed (mm/}\mu\text{s)} \times \text{Wavelength (mm)}$

  • Formula 2: $f \times T = 1$ (Reciprocals).

  • Adjustability: Not adjustable by the sonographer.

  • Practice Question: In soft tissue, what is the frequency of a wavelength measuring $0.1\,mm$?

• Frequency Ranges:

  • Human hearing: $20\,Hz$ to $20,000\,Hz$.

  • Infrasound: Less than $20\,Hz$.

  • Ultrasound: Higher than $20,000\,Hz$.

  • Diagnostic Ultrasound: $2\,MHz$ to $15\,MHz$.

  • TTE (Transthoracic Echocardiogram): $2\,MHz$ to $4\,MHz$.

  • TEE (Transesophageal Echocardiogram): $5\,MHz$.

• Amplitude:

  • Definition: Maximum variation occurring in an acoustic variable (positive or negative). It indicates the relative strength/intensity of the wave.

  • Measurement: Difference between average value and the maximum/minimum value.

  • Compression Amplitude: Normal to maximum.

  • Rarefaction Amplitude: Normal to minimum.

  • Units: Pascals ($Pa$), $kg/m$, $cm$, Celsius, and Decibels ($dB$).

  • Source: Initially determined by the sound source (probe/system), but decreases as it propagates.

  • Adjustability: Adjustable by sonographer via output power controls.

• Power:

  • Definition: Rate of energy transfer or rate of work performed.

  • Units: Watts ($W$) or milliwatts ($mW$).

  • Proportionality: Power is proportional to $\text{Amplitude}^2$ and proportional to Intensity.

  • Bioeffects: Higher power is related to increased risk of bioeffects.

  • Adjustability: Adjustable by the sonographer.

  • Power Values:

    • M-Mode: $1 - 5\,mW$

    • Gray-scale imaging: $1 - 20\,mW$

    • Color Doppler: $2 - 20\,mW$

    • Pulsed Doppler: $2 - 20\,mW$

• Intensity:

  • Definition: Concentration of energy in a sound beam; the rate at which energy passes through a unit area.

  • Units: $mW/cm^2$ and $W/cm^2$.

  • Formula: $\text{Intensity (W/cm}^2\text{)} = \frac{\text{Power (W)}}{\text{Area (cm}^2\text{)}}$

  • Relationships: Intensity and Area are inversely related. If beam area doubles (power constant), intensity is reduced to $50\%$. If beam area halves, intensity doubles.

  • Proportionality: Intensity is proportional to $\text{Amplitude}^2$.

  • Source/Propagation: Initially source-determined; decreases with propagation due to attenuation.

  • Adjustability: Adjustable via output power and electronic focusing. Intensity is higher at shallow focal zones than deep ones.

  • Intensity Values (Estimated ranges):

    • Gray-scale: $1 - 200\,mW/cm^2$

    • M-mode: $70 - 130\,mW/cm^2$

    • Color Doppler: $10 - 230\,mW/cm^2$

    • Pulsed Doppler: $20 - 290\,mW/cm^2$

• Wavelength (\lambda):

  • Definition: Length of one cycle (start to start).

  • Unit: Millimeter ($mm$).

  • Determination: The only parameter determined by both the source and the medium.

  • Formula: $\lambda(mm) = \frac{c(mm/\mu s)}{f(MHz)}$

  • Adjustability: Not adjustable by the sonographer.

• Propagation Speed (c):

  • Definition: The speed at which a wave moves through a medium.

  • Units: $m/s$ or $mm/\mu s$.

  • Formula: $c(m/s) = f(Hz) \times \lambda(m)$.

  • Adjustability: Not adjustable; changes ONLY if the medium changes.

  • Speed Determination Factors:

    • Stiffness (Bulk Modulus): Ability to resist compression and maintain shape. Stiffness and speed are directly related.

    • Density: Weight of a medium. Density and speed are inversely related.

    • Elasticity (Young's Modulus): Inversely related to stiffness; elastic media distort under pressure.

  • Tissue Propagation Speeds:

    • Air / Lung: $330\,m/s$ / $550\,m/s$

    • Fat / Water: $1450\,m/s$ / $1480\,m/s$

    • Soft Tissue Average: $1540\,m/s$ ($1.54\,mm/\mu s$)

    • Liver and Blood: $1560\,m/s$

    • Muscle / Tendon: $1600\,m/s$ / $1700\,m/s$

    • Bone: $3500\,m/s$

    • Metals: $2000 - 7000\,m/s$

    • Speed Trend: Slower in gases < liquids < fastest in solids.

Pulsed Wave (PW) Parameters

• Spatial Pulse Length (SPL):

  • Definition: Length of space that one pulse takes up (beginning to end).

  • Unit: Millimeter ($mm$).

  • Formula: $\text{SPL (mm) = Number of cycles} \times \lambda(mm)$.

  • Relationship: Inversely proportional to frequency. Short SPL yields better image quality.

  • Depth Effect: Remains constant at all depths.

  • Determination: Source and Medium.

  • Practice Question: What is the SPL of a three-cycle pulse with a wavelength of $0.41\,mm$?

• Pulse Duration (PD):

  • Definition: The time required for one pulse to occur (start to end); the "transmit," "talking," or "on" time.

  • Unit: Microsecond ($\mu s$).

  • Formula 1: $\text{PD (}\mu\text{s) = Number of cycles} \times T(\mu s)$.

  • Formula 2: $\text{PD (}\mu\text{s) = } \frac{\text{Number of cycles}}{f(MHz)}$.

  • Pulse cycles: Sonographic pulses are usually $2$ or $4$ cycles; Doppler pulses are typically $5$ to $30$ cycles.

  • Note: Shorter pulses create more accurate images.

• Pulse Repetition Period (PRP):

  • Definition: Time from the start of one pulse to the start of the next (transmit time + receive time).

  • Unit: Millisecond ($ms$).

  • Formula: $\text{PRP (ms) = } \frac{1}{\text{PRF (kHz)}}$.

  • Adjustability: Adjustable when the sonographer adjusts depth of view. PRP is directly proportional to depth.

  • Depth Increase: PRP increases; listening time increases, but pulse duration remains constant.

• Pulse Repetition Frequency (PRF):

  • Definition: Number of pulses occurring in one second.

  • Unit: $Hz/sec$ or $kHz/sec$.

  • Relationship: PRP and PRF are reciprocals. PRF is inversely proportional to depth.

  • Depth Increase: PRF decreases.

  • Adjustability: Adjustable via depth.

  • Note: Unrelated to sound frequency.

• Duty Factor (DF):

  • Definition: Percentage or fraction of time the pulsed ultrasound is on.

  • Unit: Unitless (expressed as a decimal or percentage).

  • Formula: $\text{Duty Factor = } \frac{\text{Pulse Duration (}\mu\text{s)}}{\text{Pulse Repetition Period (}\mu\text{s)}} \times 100$

  • Adjustability: Adjusted by depth. DF is inversely related to depth; directly proportional to PRF.

  • Imaging Defaults: PW sonography is typically $0.2\%$; CW sonography is $1.0$ ($100\%$).

  • Shallow vs. Deep Imaging Summary:

    • Shallow: Higher DF, Higher PRF, Lower PRP.

    • Deep: Lower DF, Lower PRF, Higher PRP.

Intensity Measurements and Mathematical Concepts

• Spatial Intensity (Distribution over space):

  • Spatial Peak (SP): Maximum intensity at the beam center.

  • Spatial Average (SA): Mean intensity across the entire beam.

  • Note: Larger beam area yields lower intensity.

• Temporal Intensity (Distribution over time):

  • Temporal Peak (TP): Maximum intensity when pulse is on.

  • Temporal Average (TA): Average intensity over the entire PRP.

  • Pulse Average (PA): Average intensity during the pulse duration.

• Measurement Methods (Units: $W/cm^2$):

  • SPTP: Spatial peak, temporal peak (highest value).

  • SATA: Spatial average, temporal average (lowest value).

  • SPTA: Spatial peak, temporal average (Standard for thermal bioeffects).

  • Beam Uniformity Ratio: $\frac{SP}{SA}$.

• Logarithms:

  • Definition: The exponent to which a base must be raised to produce a number.

  • Log 100 = $2$; Log 1000 = $3$.

  • Increasing log by $1$ increases the number $10$-fold; by $2$ increases $100$-fold.

• Decibels (dB):

  • Used for attenuation (intensities before/after) and amplification (image brightness).

  • Positive Decibels (Intensity increasing):

    • $+3\,dB$ = Final intensity is $2 \times$ original.

    • $+10\,dB$ = Final intensity is $10 \times$ original. (Hint: Add a zero for every $10\,dB$).

  • Negative Decibels (Intensity decreasing):

    • $-3\,dB$ = Intensity reduced to half ($1/2$).

    • $-6\,dB$ = Intensity reduced to quarter ($1/4$).

    • $-10\,dB$ = Intensity reduced to tenth ($1/10$).

Attenuation, Reflection, and Refraction

• Attenuation:

  • Definition: Weakening of sound intensity as it propagates.

  • Determinants: Path length (distance) and Frequency. Both are directly related to attenuation.

  • Relationships: No relation to propagation speed.

  • Medium Attenuation Levels:

    • Water: Extremely low.

    • Fat, Blood, Urine: Low.

    • Soft Tissue: Intermediate.

    • Muscle: Higher.

    • Bone, Lung: Even higher.

    • Air: Extremely high.

• Components of Attenuation:

  • Reflection:

    • Specular: Occurs at large smooth boundaries (e.g., diaphragm). One direction; angle of incidence equals angle of reflection. Returns to probe only at $90^{\circ}$.

    • Diffuse (Backscatter): Occurs at large rough boundaries. Random/disorganized angles. Most reflects return to probe but are weaker.

  • Scattering: Redirection in many directions. Interface smaller than beam. High frequency scatters more.

    • Rayleigh Scattering: Occurs when reflectors are much smaller than wavelength (e.g., Red Blood Cell). Omnidirectional distribution. Proportional to $f^4$.

  • Absorption: Primary cause of attenuation ($80\%$). Conversion of sound to heat. Directly related to frequency and depth.

• Calculations:

  • Attenuation Coefficient: Attenuation per cm of travel. In soft tissue: $0.5\,dB/cm/MHz$.

  • Formula: $\text{Atten Coef. (dB/cm) = } \frac{f(MHz)}{2}$.

  • Total Attenuation: $\text{Total Atten (dB) = Path length (cm)} \times \text{Atten Coef. (dB/cm)}$.

  • Half Value Layer (HVL) Thickness: Distance sound travels to reduce intensity to half ($-3\,dB$).

  • Formula: $\text{HVL = } \frac{3}{\text{Atten Coef.}}$ or $\frac{6}{f}$.

• Impedance (Z):

  • Definition: Acoustic resistance to sound travel.

  • Unit: Rayls ($Z$).

  • Formula: $Z(rayls) = \text{Density (kg/m}^3\text{)} \times \text{Propagation speed (m/s)}$.

  • Normal Incidence: Striking boundary at exactly $90^{\circ}$. Reflection correlates to difference in Z.

  • Oblique Incidence: Striking at angles other than $90^{\circ}$. Reflected sound does not return to probe.

  • IRC% (Intensity Reflection Coefficient): $\text{IRC\%} = \left[ \frac{Z_2 - Z_1}{Z_2 + Z_1} \right]^2 \times 100$ (Percentage reflected).

  • ITC% (Intensity Transmission Coefficient): Percentage transmitted.

  • Conservation of Energy: $\text{IRC + ITC = 100\%}$.

  • Boundary Reflections:

    • Soft tissue to air: $99\%$

    • Soft tissue to bone: $50\%$

    • Soft tissue to soft tissue: < 1\%

• Refraction:

  • Definition: Bending of the sound beam during transmission.

  • Requirements: Oblique incidence AND different propagation speeds.

  • Snell's Law: $\frac{\sin(\text{transmission angle})}{\sin(\text{incident angle})} = \frac{c_2}{c_1}$.

  • If $c_2 = c_1$: No refraction.

  • If c_2 > c_1: Transmission angle > Incident angle.

  • If c_2 < c_1: Transmission angle < Incident angle.

Range Equation and Resolution

• Range Equation:

  • Go-return time (Time-of-flight): Time for pulse to go to reflector and back.

  • Relationship: Depth is directly related to time-of-flight.

  • Formula: $\text{depth (mm) = } \frac{\text{propagation speed (mm/}\mu\text{s)} \times \text{go-return time (}\mu\text{s)}}{2}$.

• 13-Microsecond Rule:

  • In soft tissue, every $13\,\mu s$ of go-return time indicates the reflector is $1\,cm$ deep ($2\,cm$ total travel).

  • $13\,\mu s = 1\,cm$ deep; $26\,\mu s = 2\,cm$ deep.

• Resolution Types:

  • Axial Resolution (LARRD): Longitudinal, Axial, Range, Radial, Depth.

    • Definition: Minimum separation required parallel to the beam path.

    • Formula: $\text{Axial Res (mm) = } \frac{SPL}{2}$.

    • Soft Tissue: \frac{0.77 \times \text{# cycles in pulse}}{f(MHz)}.

    • Improved by: Higher frequency, fewer cycles in pulse (less ringing).

  • Lateral Resolution (LATA): Lateral, Angular, Transverse, Azimuthal.

    • Definition: Minimum separation perpendicular to beam side-to-side.

    • Determination: Beam width. Best at the focus (focal zone).

    • Formula: $\text{Lateral Res (mm) = beam diameter (mm)}$.

  • Resolution Comparison:

    • Axial is generally superior to lateral because pulses are shorter than they are wide.

    • Axial remains same with depth; Lateral changes with depth.

Real Time Imaging and Temporal Resolution

• Frame Rate (FR):

  • Number of image frames stored per second ($15 - 60\,Hz$).

  • Determining factors: Depth and pulses per image (# scan lines, sector size, density).

  • Fundamental limit: Propagation speed of sound in medium.

• Temporal Resolution:

  • Ability to accurately locate moving structures in time.

  • Improved by high Frame Rate.

• Factors Affecting Number of Pulses per Frame:

  1. Number of Focal Points: Single focus (higher FR/Temp Res) vs. Multi-focus (lower FR/Temp Res, better Lateral Res).

  2. Sector Size (FOV): Narrow sector (fewer pulses, higher FR) vs. Wide sector (more pulses, lower FR).

  3. Line Density: Low line density (higher FR, poor spatial resolution) vs. High line density (lower FR, better spatial resolution).

• Depth and FR: Inverse relationship. Shallower depth = higher FR.

• Elevation Resolution (Slice Thickness Resolution):

  • Measured perpendicular to the imaging plane.

  • Best with $1.5$ dimensional arrays or annular arrays. Poor in standard array transducers.

Questions & Discussion

• Practice Problem: What is the speed of sound if the depth is $10\,cm$ and the round trip time is $2\,seconds$? ($\text{Speed = Total Distance / Time}$. $10\,cm$ depth = $20\,cm$ total distance. $20\,cm / 2\,s = 10\,cm/s$.)

• Sid Edelman Quote on -9 dB: "You do not ask why. Just know it!"

• Resolution Trade-off Summary: As frequency increases, axial resolution improves (The Good), but penetration decreases (The Bad). Sonographers must balance resolution with penetration (The Ugly).