Medical Imaging Injecting Mathematics into the Problem of Bubbly Blood. Sarah McBurnie Prof Jon Chapman OCIAM, University of Oxford

Transcription

1 Medical Imaging Injecting Mathematics into the Problem of Bubbly Blood Sarah McBurnie Prof Jon Chapman OCIAM, University of Oxford

2 Diagnostic Ultrasound 03/02/2009 2

3 Ultrasound Contrast Agents Encapsulated gas bubbles Injected into the body to enhance ultrasound imaging of blood flow Characteristic nonlinear echo A micrograph of Optison with red blood cells 03/02/2009 3

4 And it really does make a difference M Main and P Grayburn, American Heart Journal. 137(1): , January /02/2009 4

5 Bubble, bubble, toil and trouble UCAs improve the image obtained but Imaging artefacts can occur Can we improve the model describing the nonlinear passage of ultrasound through the bubbly blood, to get a better image from UCA-aided scanning? Courtesy of Dr H. Becher & A. Ehlgen 03/02/2009 5

6 What is the acoustic response of a bubbly liquid? Bubble interactions? Bubble centres drift? Asymmetric oscillations? Resonance? Bubble size and separation Incoming wave s frequency and speed 03/02/2009 6

7 Parameter Sizes Microbubble size: ~ 3 µm Bubble separation: ~ 0.1 mm Wavelength of 3 MHz diagnostic ultrasound in blood: ~ 0.5 mm Image length: ~ 3 cm 10 cm for cardiac imaging 03/02/2009 7

8 Seek effective equations Want to know where regions of bubbly blood are, not the location of every individual bubble an introduction to mathematical homogenization 03/02/2009 8

9 Close up of an attractive pattern 03/02/2009 9

10 zooming out 03/02/

11 and again 03/02/

12 I. Homogenization via M.A.E.s We are seeking a continuum equation for the effective medium on the outer region scale outer region intermediate region O(1) O( ν 1/ 3 ) ν 1 β = O(ν 2 ) 1 inner region O(ν ) 03/02/

13 One Bubble Model Our building block is the response of one bubble: Compressible fluid of density ρ f, constant kinematic viscosity µ f, p p ( x, t) speed of sound c. p B (t) R(t) f as r Encapsulative shell modelled as a membrane (no bending stiffness and isotropic) Bubble assumed spherically symmetric initially 0 03/02/

14 System to Solve Conservation of mass: Conservation of momentum: subject to b.c. at infinity kinematic condition at bubble s surface dynamic condition at bubble s surface where S is the surface tension and the bubble pressure is given by 03/02/

15 Near Field Solution In the near field we have: One bubble, which is effectively in incompressible fluid (since length scale << wavelength), And sees only a uniform incoming pressure (again since length scale << wavelength) This is a well known problem, which leads to: The (generalised) Rayleigh Plesset Equation 03/02/

16 Far Field Solution By expanding (p=p 0 + ν p 1 + L) and matching we can show that, in the far field, the pressure perturbation p 1 satisfies: Equivalently: So, in the multibubble problem, each bubble acts as a point source to all the other bubbles 03/02/

17 Multibubble Model For the bubbly liquid on the outer scale we have: Homogenizing, by matching between the 3 different regions, we obtain: where bubble number density 03/02/

18 Comments Linearised solution recovers Foldy s scattering result Nonlinear case returns Van Wijngaarden s phenomenological equations Could (and will) analyse the acoustic response near resonance Thus far only looked at small volume fraction cases For large volume fraction only have two scales (as bubble size and separation are the same) need a different technique to handle this 03/02/

19 II. Homogenization via multiple scales Toy problem:, are assumed 1-periodic in their second argument Define a microscale variable Seek as an expansion: in which the are 1-periodic in 03/02/

20 Result and satisfies: 03/02/

21 An example 10 a(x/ε) x where y 0.2 exact solution for ε =0.5 exact solution for ε =0.1 exact solution for ε =0.05 homogenized solution x 03/02/

22 Extending this idea for our purposes 1D 3D a and f discontinuous system of equations (connecting bubble pressure, liquid pressure, bubble radii) aperiodic media 03/02/

23 An example 10 a(φ(x)/ε) x where y 0.2 exact solution for ε =0.1 exact solution for ε =0.05 exact solution for ε =0.01 exact solution for ε =0.005 homogenized solution x 03/02/

24 State of play Small volume fraction Linear Multiple Scales or M.A.E.s Nonlinear M.A.E.s Large volume fraction Multiple Scales 03/02/

25 Any Questions? 03/02/