Outline of today 3545 Medical Imaging systems. Discussion assignment on B-mode imaging Lecture : Ultrasound physics. Derivation of wave equation and role of speed of sound Jørgen Arendt Jensen Department of Electrical Engineering (DTU Elektro) Biomedical Engineering Group Technical University of Denmark September, 7 3. Wave types and properties 4. Intensities and safety limits 5. Scattering of ultrasound 6. Questions for exercise Design an ultrasound B-mode system Wave types Assume that a system can penetrate 3 wavelengths x= x= λ/ x= λ x= 3λ/ It should penetrate down to cm in a liver. What is the largest pulse repetition frequency possible? One wavelength. What is the highest possible transducer center frequency? Direction of propagation Longitudinal wave 3. What is the axial resolution? Disturbance propagates. No transport of mass. 4. What is the lateral resolution for an F-number of? Only longitudinal waves are possible in soft tissue. waves attenuate very quickly. Shear or transverse 3 4
Forces on volume element p - Acoustic pressure due to wave u - Particle velocity Derivation of the wave equation in one dimension Newton s second law: F = ma For this element: A p = Aρ x u t p - Difference in pressure from x to x + x. Negative pressure gives larger volume and positive pressure gives smaller volume Find the limit for x gives the first equation: p x = ρ u t 5 6 Preservation of mass Change in volume due to change in particle velocity: Assume linear relation to pressure κ - Compressibility of material in [m /N] vol = A u t vol = κ(a x) p If the pressure on the volume is positive, then the volume gets smaller and the volume change is negative. Combined: After limit gives the second equation: A u t = κ(a x) p u x = κ p t Combining the two equations: Apply / t on second eqn.: Apply / x on first eqn.: Combining gives: leading to p x = ρ u t u x t = u x t = p κ t u p t = ρ x x u u x = κ p t t x = p ρ x κ p t = p ρ x p x p ρκ t = 7 8
The linear wave equation: Introduce the speed of sound c: c = Finally gives the linear wave equation: ρκ p x p c t = Solutions to the wave equation Assume harmonic solution: Insert into equation: p(t, x) = p sin(ω (t x c )) p p x c t = ( ω ) ( p + p c c ) sin(ω (t x ω c )) = which shows this is a solution. Note that c is speed of sound, which links spatial position and time. General solutions: p(t, x) = g(t ± x c ) x c is time delay 9 Link between speed of sound and delay Normalized pressure Normalized pressure g(t x/c) for x =..4.6.8..4.6.8 g(t x/c) for x =.54 mm x 6 x/c..4.6.8..4.6.8 Time [s] x 6 Propagation of cycle 5 MHz pulse in a medium with a speed of sound of 54 m/s. Table of speed of sound and densities Speed of Characteristic Medium Density sound acoustic impedance kg/m 3 m/s kg/[m s] Air. 333.4 3 Blood.6 3 566.66 6 Bone.38.8 3 7 535 3.75 7.38 6 Brain.3 3 55 6.55.66 6 Fat.9 3 446.33 6 Kidney.4 3 567.6 6 Lung.4 3 65.6 6 Liver.6 3 566.66 6 Muscle.7 3 54 66.65.74 6 Spleen.6 3 566.66 6 Distilled water. 3 48.48 6 Approximate densities, sound speeds, and characteristic acoustic impedances of human tissues (data from the compilation by Goss et al. (978) and (98)). The speed of sound has been calculated from the density and characteristic acoustic impedance data.
Reflection and convolution Received signal for three events: y(t) = r (t )g(t t ) + r (t )g(t t ) + r 3 (t 3 )g(t t 3 ) +.. = N r i (t i )g(t t i ) i= Types of waves Convolution model: + y(t) = r(θ)g(t θ)dθ = r(t) g(t) 3 4 Types of waves: Plane wave Types of waves: Spherical wave Plane 5 MHz wave propagating in the z direction, t= µs.5 Pressure.5 x direction [mm].5 z direction [mm].5 p(t, r) = p sin(πf (t z c )) Propagates and oscillates in z-direction. Constant phase in x and y direction. Used for intensity measures and description of reflection. Propagates radially from a center point. p(t, r) = p r sin(πf(t r c )) Used for describing diffraction, focusing, and ultrasound fields. Both are mathematical abstractions used in describing ultrasound. 5 6
Velocity of particle Relation between pressure and particle velocity: p x = ρ u t Finding the relation between pressure and particle velocity for harmonic plane wave p(t, x) = p sin(ω (t x c )): u(t) = p ρ x dt = + ρ p ω cos(ω (t x c c ))dt = ρc p sin(ω (t x p(t) )) = c ρc = p(t)/z z = ρc - Characteristic acoustic impedance of medium Table of speed of sound and densities Speed of Characteristic Medium Density sound acoustic impedance kg/m 3 m/s kg/[m s] Air. 333.4 3 Blood.6 3 566.66 6 Bone.38.8 3 7 535 3.75 7.38 6 Brain.3 3 55 6.55.66 6 Fat.9 3 446.33 6 Kidney.4 3 567.6 6 Lung.4 3 65.6 6 Liver.6 3 566.66 6 Muscle.7 3 54 66.65.74 6 Spleen.6 3 566.66 6 Distilled water. 3 48.48 6 Ohm s law of acoustics: p(t) = zu(t) Approximate densities, sound speeds, and characteristic acoustic impedances of human tissues (data from the compilation by Goss et al. (978) and (98)). The speed of sound has been calculated from the density and characteristic acoustic impedance data. 7 8 Example Typical values: c = 5 m/s, ρ = kg/m 3 z =.5 6 kg/[m s] =.5 MRayl p = kpa, ω = π 5 6 rad/s gives u = 3 =.6 m/s.5 6 Safety and intensity of ultrasound Displacement: u(t)dt = u ω =. nm Actual particle displacements are very, very small. 9
Intensities Defined as power per unit area [W/m ] For a plane wave: For this gives I = T p(t) u(t)dt T p(t, x) = p sin(ω (t x c )) u(t, x) = p(t, x)/z = p /z sin(ω (t x c )) I = p z in [W/m ] or more convenient in ultrasound [mw/cm ] Intensity measures Instantaneous intensity: I i (t, r) = p(t, r) z Spatial Peak Temporal Average intensity: I spta (t, r) = T T I i(t, r)dt Spatial Peak Temporal Peak intensity: I sptp (t, r) = max{i i (t, r)} I sppa - Spatial Peak Pulsed Average intensity: Mean over pulse duration Example of intensities I sptp FDA safety limits.8 Instantaneous intensity [W/m ].6.4. I sppa I sppa I spta I sppa I m mw/cm W/cm W/cm Use In Situ Water In Situ Water In Situ Water Cardiac 43 73 65 4 6 6 Peripheral vessel 7 5 65 4 6 6 Ophthalmic 7 68 8 5 Fetal imaging (a) 46 7 65 4 6 6 I spta.5.5.5 3 3.5 Time [s] x 5 Highest known acoustic field emissions for commercial scanners as stated by the United States FDA (The use marked (a) also includes intensities for abdominal, intra-operative, pediatric, and small organ (breast, thyroid, testes, neonatal cephalic, and adult cephalic) scanning) Different measures of intensity for a typical ultrasound pulse. The pulse repetition time is set artificially low at 7.5 µs. 3 4
Example Plane wave with an intensity of 73 mw/cm (cardiac, water) Pulse emitted: f = 5 MHz, M = periods, T prf = µs Pressure emitted: I spta = M/f I i (t, r)dt = M/f P T prf T prf z p = I sptazt prf = 3.35 6 Pa = 33.5 atm M/f (Rock concert at db (pain level) is 5 Pa) Example continued The particle velocity is U = p Z = 3.35 6.48 6 =.8 m/s. The particle velocity is the derivative of the particle displacement, so z(t) = U sin(ω t kz)dt = U cos(ω t kz). ω The displacement is z = U /ω =.8/(π 5 6 ) = 69 nm. The normal atmospheric pressure is kpa. 5 6 Typical intensities for fetal scanning in water Typical intensities for fetal scanning in tissue Focus at 6 mm, elevation focus at 4 mm (Fetal) 5 Focus at 6 mm, elevation focus at 4 mm (Fetal) Intensity: Ispta [mw/cm ] 5 5 Intensity: Ispta [mw/cm ] 4 3 3 4 5 6 7 8 9 Axial distance [mm] 3 4 5 6 7 8 9 Axial distance [mm].5.4 Peak pressure [MPa].5.5 3 4 5 6 7 8 9 Axial distance [mm] Peak pressure [MPa]..8.6.4. 3 4 5 6 7 8 9 Axial distance [mm] Simulated intensity profile in water for fetal scanning using a using a 65 elements 5 MHz linear array focused at 6 mm with an elevation focus at 4 mm. 7 Simulated intensity profile in tissue for fetal scanning using a using a 65 elements 5 MHz linear array focused at 6 mm with an elevation focus at 4 mm. 8
Fetus Head Amniotic fluid Scattering and imaging Amniotic fluid Mouth Boundary of placenta Foot Spine Ultrasound image of 3th week fetus. The markers at the border of the image indicate one centimeter. 9 3 Speckle pattern in ultrasound image Backscattering Power of backscattered signal: P s = I i σ sc I i - Incident intensity [W/m ] σ sc - Backscattering cross-section [m ] Received power of backscattered signal at transducer: P r = I i σ sc r 4R 4 x 4 cm image of a human liver for a 8-year old male. Speckle pattern in liver. The dark areas are blood vessels. r - Transducer radius R - Distance to scattering region 3 3
Backscattering coefficient Description of received signal 3 Backscattering coefficient [/(cm sr)] 4 5 6 Human liver... Bovine liver. Bovine pancreas Bovine kidney x Bovine heart o Bovine spleen Bovine blood, H = 8... Bovine blood, H = 3. Bovine blood, H = 34 Bovine blood, H = 44 7 Frequency [MHz] * Human blood, H = 6 Measured backscattering coefficients for human liver tissue, bovine myocardium, bovine kidney cortex, human spleen, and bovine and human blood as a function of frequency Backscattering coefficient - power received for a volume of scatterers. 33 34 Theoretical model of received signal Point spread function x -5 Envelope of time response, Pulse-echo field [ ρ( r ) p r ( r 5, t) = v pe (t) c( r ] ) h pe ( r t, r 5, t)d 3 r V ρ c = v pe (t) h pe ( r t, t) f m ( r r ) = PSF pe ( r, t) f m ( r r ) Electrical impulse response: v pe(t) Transducer spatial response: h pe( r, r 5, t) = h t( r, r 5, t) h r( r t 5, r, t) Scattering term: f m( r ) = ρ( r) ρ Point spread function PSF pe( r, t) c( r) c r Position of transducer r 5 Position of scatterers c Speed of sound ρ Density of medium h( r 5, r, t) Spatial impulse response ρ, c Perturbations in density and speed of sound Basic model: Time [s] 9.35 9.3 9.5 9. 9.5 9. -6-4 - 4 6 x [mm] p r ( r 5, t) = PSF pe ( r, t) r f m ( r ) The topic of exercise to investigate the influence of the point spread function 35 36
Linear Acoustic System Signal processing in ultrasound system - Exercise Display Scan converter Memory Detector Time-gain compensation Image area Transmit/ receive Control Sweeping beam Beam steering Impulse response at a point in space (spatial impulse response). We will return to this in the next lecture on ultrasound fields, focusing and imaging. 37 Signal processing in a simple ultrasound system with a single element ultrasound transducer. 38 Signal processing. Acquire RF data (load from file). Perform Hilbert transform to find envelope: env(n) = y(n) + H{y(n)} 3. Compress the dynamic range to 5 db: log env(n) = log (env(n)) 4. Scale to the color map range (8 gray levels) 5. Make polar to rectangular mapping and interpolation 6. Display the image 7. Load a number of clinical images and make a movie Polar to rectangular mapping and interpolation Two step procedure to set-up tables and then perform the interpolation. Table setup: Function for calculating the different weight tables for the interpolation. The tables are calculated according to the parameters in this function and are stored in the internal C-code. Input parameters: start_depth - Depth for start of image in meters image_size - Size of image in meters start_of_data - Depth for start of data in meters delta_r - Sampling interval for data in meters N_samples - Number of samples in one envelope line theta_start - Angle for first line in image delta_theta - Angle between individual lines N_lines - Number of acquired lines 39 4
Geometry scaling - Scaling factor from envelope to image Nz - Size of image in pixels Nx - Size of image in pixels Output: Nothing, everything is stored in the C program Calling: make_tables (start_depth, image_size,... start_of_data, delta_r, N_samples,... theta_start, delta_theta, N_lines,... scaling, Nz, Nx); Version., /9-, JAJ: Help text corrected angle z roc start_angle start_depth x Relation between the different variables 4 4 Resulting image Polar to rectangular mapping and interpolation Perform the actual interpolation: Function for making the interpolation of an ultrasound image. The routine make_tables must have been called previously. Input parameters: envelope_data - The envelope detected and log-compressed data as an integer array as 3 bits values (uint3) Output: img_data - The final image as 3 bits values Calling: img_data = make_interpolation (envelope_data); Axial position [mm] 5 5 6 4 4 6 Lateral position [mm] Ultrasound image of portal veins in the liver. 43 44
Useful Matlab commands Loading of files: load ([ in_vivo_data/88e_b_mode_invivo_frame_no_,numstr(j)]) Making a movie: for j=:66 image(randn()) colormap(gray(56)) axis image F(j)=getframe; end Play the movie 5 times at fr/s Discussion for next time Use the ultrasound system from the first discussion assignment. What is the largest pressure tolerated for cardiac imaging?. What displacement will this give rise to in tissue? Prepare your Matlab code for exercise. movie(f,5, ) 45 46