Resident Physics Lectures Attenuation Math

Attenuation Reduction in amplitude & intensity as sound travels through medium Causes absorption sound energy converted to heat dominant influence in soft tissue reflection scattering

Absorption Units dB indicates signal gain decibels (dB) dB indicates signal gain “+” indicates signal gets larger “-” indicates signal gets smaller ultrasound absorption is always negative dB sound always loses intensity negative sometimes implied dB indicates fraction of intensity lost

x = log10(y) means 10 to what power = y ? or 10x = y Logarithm Review x = log10(y) means 10 to what power = y ? or 10x = y

Logarithms Review log 1 = 0 log 10 = 1 log 100 = 2 log 10n = n log (1/10) = 10-1 = -1 log (1/100) = 10-2 = -2 log (1/1000) = 10-3 = -3

Gain & Decibels dB =10 X log10 [power out / power in] Tissue (attenuation) Power Out decibel definition dB =10 X log10 [power out / power in] Power Ratio = Power Out / Power In dB =10 X log10 [power ratio]

Gain & Decibels Power Ratio < 1 Power Ratio = Power Out / Power In Tissue (attenuation) Power Out Power Ratio = Power Out / Power In                  Power Ratio > 1 Amplifier Power Out > Power In Log [Power ratio] >0 Power Ratio < 1 Absorber / Attenuator Power Out < Power In Log [Power ratio] <0 dB =10 X log10 [power ratio]

Power Ratio Decibel calculation Power Ratio = Power Out / Power In dB =10 X log10 [power ratio] Decibel calculation Power ratio dB 1 0 10 10 100 20 1/100 -20 10 n n X 10 2 3 logarithms log 1 = 0 log 10 = 1 log 100 = 2 log 10n = n log (1/10) = 10-1 = -1 log (1/100) = 10-2 = -2 log (1/1000) = 10-3 = -3

dB Attenuation dB / 10 indicates # of powers of ten attenuation Every increase of 10 dB indicates another factor of 10 attenuation

dB: Try Again 10 dB = 1 power of 10 = 10 20 dB = 2 powers of 10 = 100 10 dB: 1 factor of 10 or 10 you morons 60 dB: 6 factors of 10 or 1,000,000, nyuk, nyuk, nyuk 20 dB: 2 factors of 10 or 10 X 10 or 100 10 dB = 1 power of 10 = 10 20 dB = 2 powers of 10 = 100 60 dB = 6 powers of 10 = 1,000,000

Logarithm Law Log(A x B) = Log(A) + Log(B) Log(20) = Log(10) + Log(2)

Logarithm Law - 16 dB means signal is reduced by a factor of 40 16 dB = 10 dB + 3 dB + 3 dB X10 X2 X2 = X40 - 16 dB means signal is reduced by a factor of 40

Attenuation & Frequency Attenuation affected by medium frequency As frequency increases, so does attenuation bass sound carries farther than treble high frequency = poorer penetration

Attenuation In Soft Tissue Rule of Thumb 0.5 dB / cm attenuation for each MHz frequency “cm” refers to distance of sound travel other texts may say 1 dB / cm depth / MHz 1 cm depth equivalent to 2 cm sound travel

Rule of Thumb 0.5 dB/cm/MHz To calculate attenuation (dB) simply multiply rule of thumb by round trip distance & by frequency 5 MHz sound; 10 cm sound travel attenuation = 0.5 dB/cm/MHz X 10 cm X 5 MHz = 25 dB 3.5 MHz sound; 4 cm sound travel attenuation = 0.5 dB/cm/MHz X 4 cm X 3.5 MHz = 7 dB

Attenuation Coefficient Attenuation Coefficient = 0.5 * Freq. (dB/cm) (dB/cm/MHz) * (MHz) indicates fraction of beam intensity lost per unit distance of sound traval

Attenuation Coefficient Attenuation Coefficient = 0.5 * Freq. (dB/cm) (dB/cm/MHz) * (MHz) Frequency (MHz) Attenuation Coefficient (dB/cm) 1 0.5 2 1.0 5 2.5 10

Attenuation Coefficient Comments Attenuation Coefficient = 0.5 * Freq. (dB/cm) (dB/cm/MHz) (MHz) Longer path increased attenuation Higher frequency  increased attenuation coefficient Higher attenuation coefficient  more attenuation

dB vs. Intensity Ratio dB Intensity Fraction atten. Ratio atten. dB attenuation =10 X log10 [intensity ratio] Fraction attenuated = 1 - intensity ratio dB Intensity Fraction atten. Ratio atten. 1 .79 .21 2 .63 .37 3 .50 .50 4 .40 .60 5 .32 .68 10 .1 .90 20 .01 .99 30 .001 .999

Soft Tissue Attenuation Calculation Attenuation = Attenuation Coefficient X Path Length Freq. Atten Coef. Atten(dB). % Int. Red. Atten(dB) % Int. Red. dB / cm 1 cm 1 cm 10 cm 10 cm 2.0 1.0 1 21 10 90 3.5 1.8 1.8 34 18 98 5.0 2.5 2.5 44 25 99.7 7.5 3.8 3.8 58 38 99.98 10.0 5.0 5.0 68 50 99.999

Attenuation Why dB? Rule of thumb doesn’t always work dB’s can be added together Rule of thumb doesn’t always work Attenuation higher in lung & bone than in soft tissue Attenuation in lung and bone not proportional to frequency Class during lecture on attenuation

Attenuation Coefficients 0.5 dB/cm/MHz is soft tissue average assumed by scanner Tissue Attenuation Coefficient (dB/cm/MHz) Fat 0.6 Brain 0.6 Liver 0.5 Kidney 0.9 Muscle 1.0 Heart 1.1

HID = 3 dB / Attenuation Coefficient Half Intensity Depth Decreases with increasing frequency HID = 3 dB / Attenuation Coefficient HID = 3 dB / Freq (MHz) * 2 Frequency Atten Coef. HID (MHz) dB/cm cm ------------------------------------------------------ 1 0.5 6.0 2 1.0 3.0 5 2.5 1.2 10 5.0 0.6

Attenuation half intensity depth (HID) depth where intensity = 50% of original corresponds to 3dB attenuation 180 150 HID 100 66 39

Practical Implications of Attenuation limits maximum imaging depth higher frequencies result in increased attenuation decreased imaging depth improved axial resolution