Physics Formulas Bible Notes

Chapter 1: Doppler Shift

• Doppler displays gather information about motion, specifically blood flow in vessels.
• Ultrasound pulse goes into tissues at an initial frequency ($F_i$).
• Returning echoes have a reflected frequency ($F_r$) that is higher or lower than the initial frequency, depending on blood flow direction.

Doppler Shift Formulas

• Doppler Shift ($F_D$):
  • $F<em>D = F</em>r - F_i$
  • Units: MHz (megahertz)
  • Where:
    • $F_r$ = Reflected or received frequency (MHz)
    • $F_i$ = Initial frequency (MHz)
• Doppler Shift Detailed Formula:
  • $F<em>D = \frac{2(f</em>o)(V)COS(\Theta)}{C}$
  • Where:
    • $F_D$ = Doppler shift (MHz)
    • $f_o$ = Operating frequency (same as initial frequency) (MHz)
    • $V$ = Velocity of red blood cells (m/s)
    • $\Theta$ = Doppler angle (unit less)
    • $COS(\Theta)$ = Cosine of the Doppler angle (unit less)
    • $C$ = Propagation speed of sound in soft tissue (1540 m/s)

Chapter 2: Frequency, Period, Wavelength, Propagation Speed, Harmonic Frequencies

Frequency (F or f)

• Definition: Number of times something happens in a given period of time.
  • Example: Voting for the U.S. president every four years.
• In Ultrasound: Number of cycles (compression and rarefaction) per second.
  • Compression: Area of high particle density and high pressure.
  • Rarefaction: Area of low particle density and low pressure.

Frequency Formulas

1. F = \frac{\text{#cycles}}{\text{time to complete}}
2. $F = \frac{1}{T}$
   • F = Frequency (Hz, kHz, MHz)
   • T = Period (sec, ms, µsec)
3. $F = \frac{C}{\lambda}$
   • F = Frequency (Hz, kHz, MHz)
   • C = Propagation speed of sound in soft tissue (1540 m/s or 1.54 mm/µsec)
   • $\lambda$ = Wavelength (mm)

Period (T)

• Definition: Time it takes to complete one cycle (one compression and one rarefaction).
• Units of sound pressure: Pascal (Pa) or Megapascal (MPa)

Period Formula

4. $T = \frac{1}{F}$
   • T = Period (sec, ms, µsec)
   • F = Frequency (Hz, kHz, MHz)

Wavelength (λ)

• Definition: Distance measurement for one complete cycle.

Wavelength Formula

5. $\lambda = \frac{C}{F}$
   • $\lambda$ = Wavelength (mm)
   • C = Propagation speed of sound in soft tissue (1.54 mm/µsec) - Important: Use this value to keep units consistent.
   • F = Frequency (MHz) - Important: Use MHz to convert to µsec.

Propagation Speed (C)

• Definition: Speed of a wave as it moves through a medium (e.g., ultrasound through body tissues).
• Constant value in a particular medium.

Propagation Speed Formulas

6. $C = f(\lambda)$
7. $C = \frac{B}{\rho}$
   • C = Propagation speed of sound in soft tissue (1540 m/s or 1.54 mm/µsec or 0.154 cm/µsec)
   • f = Frequency (MHz)
   • $\lambda$ = Wavelength (mm)
   • B = Stiffness, hardness, or bulk modulus (N/m²)
   • $\rho$ = Density (mass/volume) (g/ml)

Harmonic Frequencies ($H_f$)

• Calculated by multiplying the fundamental frequency by all odd and even numbers.

Harmonic Frequencies Formula

8. $H_f = \text{Even and odd numbers} \times \text{fundamental Frequency}$
   • $H_f$ = Harmonic frequency (MHz)
   • Even and odd numbers = 1, 2, 3, 4, 5, 6, 7…
   • Fundamental frequency = transducer's frequency (MHz)

Pulse Repetition Period (PRP)

• Definition: Time from the beginning of one pulse to the beginning of the next pulse (includes pulse duration and listening time).

Pulse Repetition Period Formulas

9. $PRP = \frac{1}{PRF}$
10. $PRP = PD + Lt$
    • PRP = Pulse repetition period (sec, ms, µsec)
    • PRF = Pulse repetition frequency (Hz, kHz, MHz)
    • PD = Pulse duration (sec, ms, µsec)
    • Lt = Listening time (sec, ms, µsec)

Pulse Repetition Frequency (PRF)

• Definition: Number of pulses that occur in one second.

Pulse Repetition Frequency Formulas

11. PRF = \frac{\text{# of pulses}}{\text{time}}
12. $PRF = \frac{1}{PRP}$
    • PRF = Pulse repetition frequency (Hz, kHz, MHz)
    • PRP = Pulse repetition period (sec, ms, µsec)

Pulse Duration (PD)

• Definition: Time it takes to complete one pulse (from beginning to end).

Pulse Duration Formulas

13. $PD = n \times T$
14. $PD = PRP - Lt$
15. $PD = \frac{n}{F}$
    • PD = Pulse duration (sec, ms, µsec)
    • n = Number of cycles per pulse (unit less)
    • T = Period (sec, ms, µsec)
    • PRP = Pulse repetition period (sec, ms, µsec)
    • F = Frequency (Hz, kHz, MHz)

Duty Factor (DF)

• Definition: Percentage of time the machine is transmitting ultrasound.

Duty Factor Formula

16. $DF = \frac{PD}{PRP} \times 100$
    • DF = Duty factor (%)
    • PD = Pulse duration (sec, ms, µsec)
    • PRP = Pulse repetition period (sec, ms, µsec)

• Imaging pulse wave ultrasound: DF < 0.01 or < 1% (machine spends most time listening).
• Non-imaging continuous wave ultrasound: DF = 100% (sound beam is always on).

Period (T) - New Formula

*New Formula:

17. $T = \frac{PD}{n}$

Listening Time (Lt)

• Definition: Time when the machine is off and not transmitting pulses (dead time or reception time).
• Time when the machine is listening for returning echoes and interpreting their brightness and location.

Listening Time Formula

18. $Lt = PRP - PD$

Wavelength (λ) - New Formula

*New Formula:

19. $\lambda = \frac{SPL}{n}$
    • $\lambda$ = Wavelength (mm)
    • SPL = Spatial Pulse Length (mm)
    • n = Number of cycles per pulse (unit less)

Spatial Pulse Length (SPL)

• Definition: Length or distance measurement for one pulse (from beginning to end).

Spatial Pulse Length Formula

20. $SPL = n \times \lambda$
    • SPL = Spatial Pulse Length (mm)
    • n = Number of cycles per pulse (unit less)
    • $\lambda$ = Wavelength (mm)

Bandwidth (BW)

• Definition: Range of useful frequencies created in a short pulse.
• Short pulses are created using backing or damping material to limit crystal ringing.
• Shorter pulse = wider bandwidth.
• Longer pulse = narrower bandwidth.

Bandwidth Formula

21. $BW = \frac{f_o}{Q}$
    • BW = Bandwidth (MHz)
    • $f_o$ = Operating frequency (MHz)
    • Q = Quality factor (unit less)

• Bandwidth is inversely related to Quality factor.

Fractional Bandwidth (FB)

• Describes how large the bandwidth is compared to the operating frequency.

Fractional Bandwidth Formula

22. $FB = \frac{BW}{f_o}$
    • FB = Fractional bandwidth (unit less)
    • BW = Bandwidth (MHz)
    • $f_o$ = Operating frequency (MHz)

• Fractional bandwidth is directly related to bandwidth.

Quality Factor (Q)

• Relates a transducer's operating frequency to its bandwidth.
• Short pulse (pulse wave) = wider bandwidth and low quality factor.
• Long pulse (continuous wave) = narrow bandwidth and high quality factor.

Quality Factor Formula

23. $Q = \frac{f_o}{BW}$
    • Q = Quality factor (unit less)
    • $f_o$ = Operating frequency (MHz)
    • BW = Bandwidth (MHz)

• Quality factor is inversely related to BW.
• Quality factor for a short pulse is the same as the number of cycles in that pulse (n).

Attenuation

• Definition: Decrease of amplitude, power, or intensity of a sound wave as it propagates through media.
• Amplitude, power, and intensity are measurements of sound energy.
• Sound loses strength as it travels deeper into the body.

Attenuation Formulas

24. $A_t = \alpha \times Pl$
25. $A_t = \frac{F}{2} \times Pl$
26. $\alpha = \frac{F}{2}$
    • $A_t$ = Attenuation or total attenuation (dB) – negative dB indicates a loss.
    • $\alpha$ = Attenuation coefficient (dB/cm) - price sound pays to propagates through tissues.
    • Pl = Path length or depth (cm)
    • F = Frequency of the transducer (MHz)

Amplitude

• Definition: Maximum variation that occurs in an acoustic variable (sound wave). The mean variation is always zero (baseline).

Amplitude formula

27. $Amp = \text{max. variation - mean variation}$

• The units of amplitude are in pressures (Pa and MPa) or decibels (dB).

Intensity

• Definition: the energy or power per unit area.
• Intensity varies across the sound beam; the highest intensity is located at the focus.

Intensity Formulas

28. $I = \frac{P}{a}$
29. $I \alpha A^2$
30. $I<em>{spta} = I</em>{sppa} \times DF$

• I = intensity (units W/cm² or mW/cm²)
• a = area (units cm²)
• P = power (units W or Mw)
• A = amplitude (units MPa, Pa, or dB)
• $I_{spta}$ = intensity with considerations of spatial peak and temporal average (units W/cm² or mW/cm²)
• $I_{sppa}$ = intensity with considerations to spatial peak and pulse average (units W/cm² or mW/cm²)
• DF = duty factor (unit less). Value of 0.01 or 1%.

Attenuation (-dB)

• Formulas used in calculation of the final intensity ($I<em>f$) from the initial intensity ($I</em>i$).

1. 0dB: $I<em>f = 1 \times I</em>i$
2. -3dB: $I<em>f = \frac{I</em>i}{2}$
3. -6dB: $I<em>f = \frac{I</em>i}{4}$
4. -9dB: $I<em>f = \frac{I</em>i}{8}$
5. -10dB: $I<em>f = \frac{I</em>i}{10}$
6. -20dB: $I<em>f = \frac{I</em>i}{100}$
7. -30dB: $I<em>f = \frac{I</em>i}{1000}$

Acoustic Impedance (Z)

• Definition: Measurement of a medium's resistance to sound propagation.

Acoustic Impedance Formula

38. $Z = \rho \times C$

• Z = impedance (units rayls or MRayls)
• $\rho$ = density of the medium (units kg/liters)
• C = prop speed of sound in soft tissue (units m/s)
• The average impedance of soft tissue is 1,630,000 rayls or 1.63MRayls

Intensity Reflection Coefficient (IRC)

• The percentage of the U/S intensity that is bounced back when the sound beam encounters a medium interface.

IRC Formulas

39. $IRC = [\frac{(Z<em>2 - Z</em>1)}{(Z<em>2 + Z</em>1)}]^2 \times 100$
40. $IRC = \frac{I<em>r}{I</em>i}$
41. $I<em>r = IRC \times I</em>i$

• $Z_2$ = impedance of medium 2 (units Rayls or MRayls)
• $Z_1$ = impedance of medium 1 (units Rayls or MRayls)
• $I_r$ = reflected or received intensity (units mW/cm² or W/cm²)
• $I_i$ = initial or incident intensity (units mW/cm² or W/cm²)
• If we are talking about percentages: IRC + ITC = 100%
• If we are talking about decimals: IRC + ITC = 1

Intensity Transmission Coefficient (ITC)

• The percentage of U/S intensity that is allowed to pass through when the beam reaches an interface between 2 different media.

ITC Formulas:

42. $ITC = 1 - IRC$ or $ITC = 100\% - IRC(\%)$
43. $ITC = \frac{I<em>t}{I</em>i}$
44. $I<em>t = ITC \times I</em>i$

• $Z_2$ = impedance of medium 2 (units Rayls or MRayls)

• $Z_1$ = impedance of medium 1 (units Rayls or MRayls)

• $I_t$ = transmitted intensity (units mW/cm² or W/cm²)

• $I_i$ = initial or incident intensity (units mW/cm² or W/cm²)

• When we consider normal incidence we must first look at the angel of incidence, which must be 90º. Second we must consider acoustic impedance of the two different media. Third, only two things can happen reflection and transmission.

• No reflection will occur if the two different media have the same impedance.

• Small reflection will occur if the differences in the two impedances are slight.

• Large reflection will occur if the differences in the two impedances are considerable.

Snell's Law

• The physics of refraction is governed by Snell's law.

Snell's Law Formula:

45. $SL = \frac{Sin \text{ (transmission angle)}}{Sine; \text{ (incident angle)}} = \frac{C<em>2 \text{ (prop. Speed of medium 2)}}{C</em>1 \text{ (prop. Speed of medium 1)}}$

• If we have soft tissue representing the media, what is an estimate of the ultrasound frequency? (Hint, look at attenuation coefficient).

Soft Tissue Frequency Formula:

46. $F = A_c(2)$

• The units of rayls represent what property, and how is it determined? Impedance.

Rayleigh Scattering

• A special form of scattering that occurs when the structure of interest is much smaller than the beam's wavelength. The sound wave is equally scattered in all directions.
• A perfect example of this is red blood cells, when the sound beam hits the red blood cell, the returning scatter is uniform and in all direction.
• Rayleigh Scattering increases dramatically with increasing frequency.

Rayleigh Scattering Formula:

47. $RS = F^4$

• When Frequency is doubled the Rayleigh scatter is 16 times greater (2x2x2x2=16).

Range Equation

• In order for us to get an anatomic image on the screen, the TX emits a sound pulse. This sound pulse then travels in the body to a boundary of an organ, or in other words a reflector. After the sound pulse encounters the reflector, parts of the original pulse return to the TX.
• There are three ways in which we calculate depth:

1) 13 microsecond (us) Rule - For every 13us of go-return time, the reflector or object being scanned is 1 cm deeper in soft tissue

• 13us = 1cm = 2cm

• 26us = 2cm = 4cm

• 39us = 3cm = 6cm

• 52us = cm = cm

• Total distance traveled is calculated by calculating the depth of a reflector and multiplying it by 2.

Depth Formula 1:

48. $D = \frac{t}{13\frac{\mu s}{cm}}$

• D = depth (units cm)
• $13\frac{\mu s}{cm}$ = constant value.
• t = time of flight or the go return time (units us)

Time of Flight Formula 1:

49. $t = D \times 13\frac{\mu s}{cm}$

2) Range equation - is the formula used by the ultrasound machine in order to calculate the depth to a reflector and the time of flight, which is the time measurement of the sound pulse to go into the tissues of the body and return as echoes.

Depth Formula 2:

50. $D = \frac{Ct}{2}$

Time of Flight Formula 2:

51. $t = \frac{2D}{C}$

• D= depth (units mm or cm)
• C = Propagation speed in soft tissue 1.54 mm/us or 0.154 cm/us
• t = time (us)

3) Simplified range equation: These formulas are the simplified versions of the range equations. We can simplify the range equation because the prop. Speed in soft tissue is a fixed rate that is 1.54mm/us or 0.154cm/us and the number 2 in the denominator is a constant number in the equation. Therefore if we carry out this division part of the equation the simplified formulas are as follows:

Depth Formula 3:

52. $D = 0.77\frac{mm}{\mu s}(t)$
53. $D = 0.077\frac{cm}{\mu s}(t)$

Time of Flight Formula 3:

54. $t = \frac{D}{0.77\frac{mm}{\mu s}}$
55. $t = \frac{D}{0.077\frac{cm}{\mu s}}$

Chapter 3: Operating Frequency, Near Zone Length, Diameter of the Beam, Axial Resolution, Lateral Resolution

Operating Frequency of the Transducer

• PZT - The piezoelectric elements convert electric voltage into ultrasound pulses and converts returning echoes back into voltages where the ultrasound interprets them into images on our monitors.

Operating Frequency Formula:

56. $f<em>o = \frac{C</em>t}{2 \times Th}$

• $f_o$ = operating freq (units MHz)
• 2 = constant value unit less
• Th=thickness of the crystal (units mm)
• $C_t$ = prop speed of sound in the crystal which ranges between 4 - 6mm/us

Near Zone Length (NZL) / Focal Length

• The NZL of a beam is determined by the size and operating frequency of the element. The NZL increases as frequency or element size increases. Element size is also known as the aperture denoted (ap) or diameter of element in the TX.

Simplified NZL Formula:

57. $NZL = \frac{(D_t)^2 \times f}{6 \frac{mm}{\mu s}}$

• units of NZL = mm

Diameter of the Beam (DB)

• Memorize these values in order to calculate problems, DB stands for diameter of beam, units:mm

DB Formulas:

58. DB at 1 NZL = ½ $D<em>t$ or 0.5($D</em>t$)
59. DB at 2 NZL = $D_t$
60. DB at 3 NZL = 1½ $D<em>t$ or 1.5($D</em>t$)
61. DB at 4 NZL = 2 $D_t$

Axial Resolution

• This type of resolution measures the ability of a system to display two structures that are very close together. That is when the structures are parallel to the sound beam's main axis.
• The one with the shorter pulse is able to image more accurately the closely spaced reflectors.

Axial Resolution Formulas:

62. $Ax. Res. = \frac{SPL}{2}$
63. $Ax. Res = \frac{n \times \lambda}{2}$
64. $Ax. Res.= \frac{0.77 \frac{\mu s}{mm}}{F} \times n$

• Ax. Res = axial resolution (units mm)
• SPL = Spatial pulse length (units mm)
• n = number of cycles per pulse (unit less)
• $\lambda$ = wavelength (units mm)
• F = frequency of the tx (units MHz)

Lateral Resolution

• This type of resolution measures the ability of a system to display two structures that are very close together. That is when the structures are perpendicular to the sound beam's main axis.

Lateral Resolution Formula:

65. Lateral Resolution (mm) = Beam Diameter (mm)

• DB at 1 NZL = ½ $D<em>t$ or 0.5($D</em>t$)
• DB at 2 NZL = $D_t$
• DB at 3 NZL = 1½ $D<em>t$ or 1.5($D</em>t$)
• DB at 4 NZL = 2 $D_t$

Chapter 4: Time and Pulse Repetition Frequency

• To avoid echo misplacement, all the echoes from the previous pulse must be received before the next pulse is emitted. When we are scanning deeper structures, echoes take longer to return. The system will automatically lower the PRF for gray scale imaging, and this will also lower the number of images that are generated each second called frame rate (FR). Imaging depth or in other words penetration (pen) in the units of cm, multiplied by PRF (kHz) must not exceed 77 in order to avoid echo misplacement.

Minimum time between pulses formula:

66. $t = \frac{pen}{0.77\frac{mm}{\mu s}}$

• t = minimum allowable time between pulses (units: μs)
• pen = penetration or depth (units: mm)
• $0.77\frac{mm}{\mu s}$ = constant value.

Maximum allowable PRF formula:

67. $PRF = \frac{770 \frac{mm}{ms}}{pen}$

• PFR = maximum allowable PRF to avoid echo misplacement (units: kHz)
• pen = penetration or depth (units: mm)
• $770 \frac{mm}{ms}$ = constant value.
• Amplification is sonographer adjustable, by means of the 2D-gain, overall gain, or receiver gain (all different names for the same knob).
• The units of amplification are in dB, because attenuation units are in dB, and we correct for attenuation with the use of amplification, therefore units are the same.

Amplification (+ dB)

• Formulas used in calculation of the Final intensity ($I<em>f$) from the initial intensity ($I</em>i$).
• Amplitude ratio multiply by input voltage in order to calculate output voltage

1. 0dB: $I<em>f = 1 \times I</em>i$ = 1.0
2. +3dB: $I<em>f = 2 \times I</em>i$ = 1.4
3. +6dB: $I<em>f = 4 \times I</em>i$ = 2.0
4. +9dB: $I<em>f = 8 \times I</em>i$ = 2.8
5. +10dB: $I<em>f = 10 \times I</em>i$ = 3.2
6. +20dB: $I<em>f = 100 \times I</em>i$ = 10
7. +30dB: $I<em>f = 1000 \times I</em>i$ = 32
8. +60dB: $I<em>f = 1,000,000 \times I</em>i$ = 1000
9. +100dB: $I<em>f = 10,000,000,000 \times I</em>i$ = 100,000

• The amplifiers typically have ranges from 60 dB to 100 dB. For a 60 dB gain amplifier the output power in 1million times the power output, and the output voltage is 1000 times the input. This means that if we have a 10 μV (micro-volt = 1/1,000,000 of one volt) voltage input, the output voltage is 10 mV (mili-volt = 1/1000 of one volt) because 10μV is the same as 0.000010 V and if we multiply this by 1000 it would equal 0.01V or 10 mV.
• For a 100 dB gain amplifier the output power in 10 billion times the power output, and the output voltage is 100,000 times the input voltage This means that if we have a 10 μV (micro-volt = 1/1,000,000 of one volt) voltage input, the output voltage is 1V because 10μV is the same as 0.000010 V and if we multiply this by 100,000 it would equal 1V.

Shades of Gray

Shades of Gray Formula

77. $SG = 2^n$

• SG = shades of gray (unit less)
• 2 = constant value representing (0,1) unit less
• n = number of memory panels used, unit less

Frame Rate

Time for One Frame Formula

78. $T_{frame} = \frac{1}{Frame Rate}$

Frame Rate Formula

79. $Frame Rate = \frac{1}{T_{frame}}$

80. $T_{frame} = Odd Field + Even Field$

Chapter 5: Poiseuille's Equation, Reynold's Number, Continuity Rule, Bernoulli Effect

Poiseuille's Equation for volumetric flow rate

Poiseuille's Equation Formula:

81. $Q = \frac{\Delta P \times \pi \times d^4}{128 \times L \times \eta}$

• Q = Volume flow rate (ml/s)
• $\Delta P$ = Pressure difference ($dyne/cm^2$) most commonly (mmHg)
• $\pi$ = Pie a constant 3.14
• d = Diameter to the fourth power (cm)
• 128 = Constant
• L = Length of vessel (cm)
• $\eta$ = Viscosity, The viscosity of normal blood is 0.035 poise
• This equation is useful when we are concerned with steady flow in a long straight tube with no stenosis. This equation serves as an approximation of the blood flow conditions in our circulatory system.

Pressure

• The driving force behind flow. A pressure gradient is needed in order to for blood to flow. The greater the pressure difference, the greater the volume flow rate. We measure pressure in units of ($dyne/cm^2$) or millimeters of mercury (mmHg).

Pressure Formula

82. $\Delta p = P<em>1 - P</em>2$

• $\Delta p$ = change in pressure or pressure gradient (units: mmHg)
• $P_1$ = proximal pressure (units: mmHg)
• $P_2$ = Distal pressure (units: mmHg)

Reynold's Number

• This number predicts whether flow in vessels is laminar or turbulent.

Reynold's Number Formula:

83. $R<em>n = \frac{V</em>a \times d \times \rho}{\eta}$

• $R_n$ = Reynold's Number
• $V_a$ = Average flow speed (units: centimeters per second (cm/s))
• d = Tube diameter (Units: centimeters (cm))
• $\rho$ = Density (Units: grams/milliliters 1.05 (g/mL))
• $\eta$ = Viscosity (Units: 0.035 Poise or kilograms per meter-second (kg/m-s), one poise is 1 g/cm-s or 0.1 kg/m-s.)
• Reynold's Number for laminar flow is less than 1500.
• Reynold's Number for turbulent flow is greater than 2000.
• Reynold's Number for disturbed flow is (1500-2000)

Continuity Rule

• When we are concerned with a vessel in our circulatory system.
• In the case of a narrowing or stenosis in a vessel, the average flow speed in the area of the stenosis must be greater than the proximal and distal sections relative to the stenosis.
• This increase of the average flow speed is in place so that the volumetric flow rate is constant throughout the vessel.

Average flow speed Formula:

84. $V \alpha = \frac{\Delta P \times d^2}{32 \times L \times \eta}$

• Va = Average flow speed (cm/s)
• $\Delta P$ = Pressure difference ($dyne/cm^2$) most commonly (mmHg)
• d² = Diameter to the second power (cm)
• 32 = Constant unit less
• L = Length (cm)
• $\eta$ = Viscosity, The viscosity of normal blood is 0.035 poise

Bernoulli Effect: Vascular

• The Bernoulli Effect describes the affect at the stenosis in respect to pressure. At the stenosis the pressure must be less than the pressure proximal and distal to the stenosis. If we think about it this is necessary to allow the blood to accelerate into the stenosis and decelerate out of it and this is done because energy balance must be maintained.
• From a modified form of the Bernoulli's equation we get the magnitude of the decrease in pressure that results from the increasing flow speed at the stenosis.

Change in pressure Formulas:

85. $\Delta P = \frac{1}{2} \times \rho \times (v)$

• or

85. $\Delta P = \frac{\rho \times v^2}{2}$

• $\Delta P$ = pressure difference or change in pressure (units = mmHg)
• $\rho$ = density, The density of blood is 1.05g/mL
• v = flow speed (units = m/s)

Bernoulli Effect: Cardiac

• Form yet another modified form of the Bernoulli's equation we can calculate the pressure drop from a stenotic heart valve, which can reduce cardiac output. We use the following form of the equation to carry out this calculation.

Change in pressure Formula Cardiac:

87. $\Delta P = 4(v)^2$

Doppler Effect

• This effect describes the change in receiver frequency of a sound as a result of relative motion. The frequency changes as a result of relative motion between the sound source and the receiver. Another way of saying this is the frequency of sound changes when the sound source and the receiver move closer together or further apart.
• Doppler shift (MHz) = reflected frequency - transmitted frequency

Doppler shift Formula

88. $F<em>D = F</em>r - F_i$

• $F_D$ = Doppler shift (MHz)
• $F_r$=Received frequency, or reflected frequency (MHz)
• $F_i$ = Initial frequency or transmitted frequency (MHz)
• Increased reflected frequency = movement towards
• Decreased reflected frequency = movement away

The Doppler Equation

Doppler shift Formula

89. $F<em>D = \frac{2 \times f</em>o \times V \times Cos \Theta}{C}$

• $F_D$ = Doppler shift (MHz)
• $f_o$ = Operating frequency, or transducer frequency (MHz