31545 Medical Imaging systems

Transcription

1 31545 Medical Imaging systems Lecture 2: Ultrasound physics Jørgen Arendt Jensen Department of Electrical Engineering (DTU Elektro) Biomedical Engineering Group Technical University of Denmark September 1, Outline of today 1. Discussion assignment on B-mode imaging 2. Derivation of wave equation and role of speed of sound 3. Wave types and properties 4. Intensities and safety limits 5. Scattering of ultrasound 6. Questions for exercise 1 2

2 Design an ultrasound B-mode system Assume that a system can penetrate 3 wavelengths It should penetrate down to 1 cm in a liver 1. What is the largest pulse repetition frequency possible? 2. What is the highest possible transducer center frequency? 3. What is the axial resolution? 4. What is the lateral resolution for an F-number of 2? 3 Wave types x= x= λ/2 x= λ x= 3λ/2 One wavelength Direction of propagation Longitudinal wave Disturbance propagates. No transport of mass. Only longitudinal waves are possible in soft tissue. waves attenuate very quickly. Shear or transverse 4

3 Derivation of the wave equation in one dimension 5 Forces on volume element p - Acoustic pressure due to wave u - Particle velocity For this element: Newton s second law: F = ma 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 6

4 Preservation of mass Change in volume due to change in particle velocity: Assume linear relation to pressure κ - Compressibility of material in [m 2 /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 7 Combining the two equations: p x = ρ u t u x = κ p t Apply / t on second eqn.: Apply / x on first eqn.: Combining gives: leading to u x t = 2 u x t = p κ 2 t 2 2 u p t = ρ x2 x 2 u t x = 1 2 p ρ x 2 κ 2 p t = 1 2 p 2 ρ x 2 2 p x p 2 ρκ 2 t = 2 8

5 The linear wave equation: Introduce the speed of sound c: 1 c = ρκ Finally gives the linear wave equation: 2 p x p c 2 t 2 = 9 Solutions to the wave equation Assume harmonic solution: Insert into equation: p(t, x) = p sin(ω (t x c )) 2 p 1 2 p x 2 c 2 t = ( 2 ω ) 2 1 ( p + p c c 2ω2 ) sin(ω (t x 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 1

6 Link between speed of sound and delay Normalized pressure Normalized pressure g(t x/c) for x = g(t x/c) for x = 1.54 mm x x/c Time [s] x 1 6 Propagation of 2 cycle 5 MHz pulse in a medium with a speed of sound of 154 m/s. 11 Table of speed of sound and densities Speed of Characteristic Medium Density sound acoustic impedance kg/m 3 m/s kg/[m 2 s] Air Blood Bone Brain Fat Kidney Lung Liver Muscle Spleen Distilled water Approximate densities, sound speeds, and characteristic acoustic impedances of human tissues (data from the compilation by Goss et al. (1978) and (198)). The speed of sound has been calculated from the density and characteristic acoustic impedance data. 12

7 Reflection and convolution Received signal for three events: y(t) = r 1 (t 1 )g(t 2t 1 ) + r 2 (t 2 )g(t 2t 2 ) + r 3 (t 3 )g(t 2t 3 ) +.. = N r i (t i )g(t 2t i ) i=1 Convolution model: y(t) = + r(θ)g(t θ)dθ = r(t) g(t) 13 Types of waves 14

8 Types of waves: Plane wave Plane 5 MHz wave propagating in the z direction, t=1 µs 1.5 Pressure x direction [mm] z direction [mm] p(t, r) = p sin(2πf (t z c )) Propagates and oscillates in z-direction. Constant phase in x and y direction. Used for intensity measures and description of reflection. 15 Types of waves: Spherical wave Propagates radially from a center point. p(t, r) = p r sin(2πf (t r c )) Used for describing diffraction, focusing, and ultrasound fields. Both are mathematical abstractions used in describing ultrasound. 16

9 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) = 1 p ρ x dt = +1 ρ p 1 ω cos(ω (t x c c ))dt = 1 ρc p sin(ω (t x p(t) )) = c ρc = p(t)/z z = ρc - Characteristic acoustic impedance of medium Ohm s law of acoustics: p(t) = zu(t) 17 Table of speed of sound and densities Speed of Characteristic Medium Density sound acoustic impedance kg/m 3 m/s kg/[m 2 s] Air Blood Bone Brain Fat Kidney Lung Liver Muscle Spleen Distilled water Approximate densities, sound speeds, and characteristic acoustic impedances of human tissues (data from the compilation by Goss et al. (1978) and (198)). The speed of sound has been calculated from the density and characteristic acoustic impedance data. 18

10 Example Typical values: c = 15 m/s, ρ = 1 kg/m 3 z = kg/[m 2 s] = 1.5 MRayl p = 1 kpa, ω = 2π rad/s gives u = 1 13 =.6 m/s Displacement: u(t)dt = u ω = 2.1 nm Actual particle displacements are very, very small. 19 Safety and intensity of ultrasound 2

11 Intensities Defined as power per unit area [W/m 2 ] For a plane wave: For I = 1 T T p(t) u(t)dt p(t, x) = p sin(ω (t x c )) u(t, x) = p(t, x)/z = p /z sin(ω (t x c )) this gives I = p2 2z in [W/m 2 ] or more convenient in ultrasound [mw/cm 2 ] 21 Intensity measures Instantaneous intensity: I i (t, r) = p(t, r)2 z Spatial Peak Temporal Average intensity: I spta (t, r) = 1 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 22

12 Example of intensities 1 I sptp.8 Instantaneous intensity [W/m ] I sppa I sppa I spta Time [s] x 1 5 Different measures of intensity for a typical ultrasound pulse. The pulse repetition time is set artificially low at 17.5 µs. 23 FDA safety limits I spta I sppa I m mw/cm 2 W/cm 2 W/cm 2 Use In Situ Water In Situ Water In Situ Water Cardiac Peripheral vessel Ophthalmic Fetal imaging (a) 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) 24

13 Example Plane wave with an intensity of 73 mw/cm 2 (cardiac, water) Pulse emitted: f = 5 MHz, M = 2 periods, T prf = 2µs Pressure emitted: I spta = 1 p = T prf M/f I spta2zt prf M/f I i (t, r)dt = M/f T prf P 2 2z (Rock concert at 12 db (pain level) is 5 Pa) = Pa = 33.5 atm The normal atmospheric pressure is 1 kpa. 25 Example continued The particle velocity is U = p Z = = 2.18 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 = 2.18/(2π ) = 69 nm. 26

14 Typical intensities for fetal scanning in water 2 Focus at 6 mm, elevation focus at 4 mm (Fetal) Intensity: Ispta [mw/cm 2 ] Axial distance [mm] 2.5 Peak pressure [MPa] 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. 27 Typical intensities for fetal scanning in tissue 5 Focus at 6 mm, elevation focus at 4 mm (Fetal) Intensity: Ispta [mw/cm 2 ] Axial distance [mm] Peak pressure [MPa] Axial distance [mm] 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. 28

15 Scattering and imaging 29 Fetus Head Amniotic fluid Amniotic fluid Boundary of placenta Mouth Foot Spine Ultrasound image of 13th week fetus. The markers at the border of the image indicate one centimeter. 3

16 Speckle pattern in ultrasound image 4 x 4 cm image of a human liver for a 28-year old male. Speckle pattern in liver. The dark areas are blood vessels. 31 Backscattering Power of backscattered signal: P s = I i σ sc I i - Incident intensity [W/m 2 ] σ sc - Backscattering cross-section [m 2 ] Received power of backscattered signal at transducer: r - Transducer radius R - Distance to scattering region P r = I i σ sc r 2 4R 2 32

17 Backscattering coefficient Backscattering coefficient [1/(cm sr)] Human liver... Bovine liver. Bovine pancreas Bovine kidney x Bovine heart o Bovine spleen Bovine blood, H = 8 %... Bovine blood, H = 23 %. Bovine blood, H = 34 % Bovine blood, H = 44 % * Human blood, H = 26 % Frequency [MHz] 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 Description of received signal 34

18 Theoretical model of received signal [ ρ( r1 ) p r ( r 5, t) = v pe (t) 2 c( r ] 1) h pe ( r t 1, r 5, t)d 3 r 1 V ρ c = v pe (t) h pe ( r t 1, t) f m ( r r 1 ) = PSF pe ( r 1, t) f m ( r r 1 ) Electrical impulse response: v pe (t) Transducer spatial response: h pe ( r 1, r 5, t) = h t ( r 1, r 5, t) h r ( r t 5, r 1, t) Scattering term: f m ( r 1 ) = ρ( r 1) 2 c( r 1) ρ c Point spread function PSF pe ( r 1, t) r 1 Position of transducer r 5 Position of scatterers c Speed of sound ρ Density of medium h( r 5, r 1, t) Spatial impulse response ρ, c Perturbations in density and speed of sound 35 Point spread function x1-5 Envelope of time response, Pulse-echo field Time [s] x [mm] Basic model: p r ( r 5, t) = PSF pe ( r 1, t) r f m ( r 1 ) The topic of exercise 2 to investigate the influence of the point spread function 36

19 Linear Acoustic System 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 ultrasound system - Exercise 1 Display Scan converter Memory Detector Time-gain compensation Image area Transmit/ receive Control Sweeping beam Beam steering Signal processing in a simple ultrasound system with a single element ultrasound transducer. 38

20 Signal processing 1. Acquire RF data (load from file) 2. Perform Hilbert transform to find envelope: env(n) = y(n) 2 + H{y(n)} 2 3. Compress the dynamic range to 5 db: log env(n) = 2 log 1 (env(n)) 4. Scale to the color map range (128 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 39 Polar to rectangular mapping and interpolation 1 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 4

21 % % 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 1.1, 11/9-21, JAJ: Help text corrected 41 Geometry angle roc x start_angle start_depth z Relation between the different variables 42

22 Polar to rectangular mapping and interpolation 2 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 32 bits values (uint32) % % Output: img_data - The final image as 32 bits values % % Calling: img_data = make_interpolation (envelope_data); 43 Resulting image Axial position [mm] Lateral position [mm] Ultrasound image of portal veins in the liver. 44

23 Useful Matlab commands Loading of files: load ([ in_vivo_data/882e_b_mode_invivo_frame_no_,num2str(j)]) Making a movie: for j=1:66 image(randn(2)) colormap(gray(256)) axis image F(j)=getframe; end % Play the movie 5 times at 22 fr/s movie(f,5, 22) 45 Discussion for next time Use the ultrasound system from the first discussion assignment 1. What is the largest pressure tolerated for cardiac imaging? 2. What displacement will this give rise to in tissue? Prepare your Matlab code for exercise 1. 46