K. Efthymiou, K. Tsiglifis & N. Pelekasis Laboratory of Fluid - PowerPoint PPT Presentation

Pulsations of a coated microbubble – Steady pulsations, transient break-up & effect of nearby surfaces K. Efthymiou, K. Tsiglifis & N. Pelekasis Laboratory of Fluid Mechanics and Turbomachinery Department of Mechanical Engineering University of Thessaly, Volos, GREECE Funding: “HERACLITUS” Program, Greek Ministry of Education 5th BIFD International Symposium, Haifa, Israel, 8-11 July 2013

Microbubbles (Contrast Agents)  Bubbles surrounded by an elastic membrane for PEG stability  Low density internal gas that is soluble in blood  Diameter from 1 to 10 μ m  Polymer, lipid or protein (e.g. albumin) monolayer shell of thickness from 1 to 30 nm Lipid monolayer Motivation p  Contrast perfusion imaging  check the • Ac MI  circulatory system by means of contrast f enhancers in the presence of ultrasound (Sboros et al. 2002, Frinking & de Jong, Postema et al., Ultrasound Med. Bio. 1998, 2004) Contrast enhanced perfusion imaging, via a sequence of low and high Mechanical Index (MI) ultrasound pulses

Sonoporation  reinforcement of drug delivery to • nearby cells that stretch open by oscillating contrast agents (Marmottant & Hilgenfeldt, Nature 2003) • Micro-bubbles act as vectors for drug or gene delivery to targeted cells (Klibanov et al., adv. Drug Delivery Rev., 1999, Ferrara et al. Annu. Rev. Biomed., 2007)  Need for specially designed contrast agents: • Controlled pulsation and break-up for imaging and perfusion measurements • Chemical shell treatment for controlled wall adhesion for targeted drug delivery  Need for models covering a wider range of CA behavior (nonlinear material behavior, shape deformation, buckling, interfacial mass transport etc., compression vs. expansion only behavior, nonlinear resonance frequency- thresholding)  Need to understand experimental observations and standardize measurements in order to characterize CA’s

Asymmetric oscillations of a microbubble near a wall (S. Zhao et al., Applied Physics, 2005) (H. J. Vos et al., Ultrasound in Med. & Biol., 2008)  Experiments have shown that the presence of a nearby wall affects the bubble’s oscillations. In particular its maximum expansion  Asymmetric oscillations, toroidal bubble shapes during jet inception have been observed  The bubble oscillates asymmetrically in the plane normal to the wall, while it oscillates symmetrically in the plane parallel to the wall (i.e. deformation has an orientation perpendicular to the wall)

Axisymmetric Pulsations            G S,2d = δ G S         P t P 1 sin  t  2 v    ,   st     f f f        R , 3 R / G o Characteristic space and time scales: n Eq Eq S.2d e s o Dimensionless parameters: Liquid ρ, P st r   s P G    f We  P St , S ,2 d   f θ ξ= 1 ξ= 0  Axis of symmetry  2 2  3 G / R G ,2 / R S d Eq S,2d Eq Gas σ Potential P G   k 3 G R G R Shell  incompressible B   B S ,2 d Eq S ,2 d Eq Re Re G s , μ s , k B , δ flow 2   s l G R 2 2 S ,2 d Eq S l  Axisymmetry  Ideal, irrotational flow of high Reynolds number  Incompressible surrounding fluid with a sinusoidal pressure change in the far field  Ideal gas in the microbubble undergoing adiabatic pulsations  Very thin viscoelastic shell whose behavior is characterized by the constitutive law (e.g. Hooke, Mooney-Rivlin or Skalak)  The shell exhibits bending modulus that determines bending stresses along with curvature variations  Shell parameters: area dilatation modulus χ=3 G s δ , dilatational viscosity μ s , degree of softness b for strain softening shells or area compressibility C for strain hardening ones and the bending modulus k B

 Shell viscosity dominates liquid viscosity, Re s <<Re l and we can drop viscous stresses on the liquid side  Therefore the tangential force balance is satisfied on the shell with the viscous and elastic stresses in the shell balancing each other  Force balance on the bubble’s interface :       n n 1 2 k          s     m r r : P I X n P n n F F , s L G   Re We We L              T qn F F n Fe T , n t s s El Vis  Surface gradient, Stress tensor : T : s   , : Elastic and viscous stress tensors El vis Transverse shear tensor due to bending moments qn :  Torque balance on the bubble’s interface :      Tensor of bending moments q m ( I nn ), m : s

Shell Constitutive Laws – Isotropic Tension  Linear behavior  Hooke’s law 2.0 Kelvin – Voigt law with viscous stresses: 1.5    1   A          H 2 2 s T G  1  K 1 K A 1.0   1 S 1 s 0.5 Κ: area dilatation modulus Stress 0.0 G s : shear modulus Strain softening, b=0 Strain softening, b=1 -0.5 ν s : surface Poisson ratio Strain hardening, C=1 Strain hardening, C=5 -1.0 ΔΑ/Α: relative area change Marmottant model Neo-Hookean -1.5 -2.0  Strain softening material (e.g. lipid -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 Isotropic Strain = ΔΑ/Α monolayer) 2D Mooney – Rivlin law:       4 2 G 1                  MR MR 2 2 T  1  1 , 0 1   1 6 Ψ=1 -b : degree of smoothness  Strain hardening material (e.g. red blood-cell membrane that consist of a lipid bilayer) Skalak law:               C: degree of area compressibility SK 2 2 2 T G  1 1  C 1 , 1 C ,  1 SK

Phase Diagrams of Polymeric and Lipid Shells Contrast agent BR14 subject to 2.4 MHz ultrasound -13 Nt m, b=0.5, χ=0.54Nt/m, μ s =1.54χ10 -8 Kg/s k b =1.5x10 5 4 3 R 0 in μm 2 P 4 P 3 Gas bubbles P 2 1 Static buckling BR14 Dynamic buckling after 6 periods Dynamic buckling after 4 periods 0 0 2 4 6 8 10 12 ε • Lower and upper solid lines -> threshold for Parametric excitation of shape modes buckling and transient break-up based on takes place at lower amplitudes than experiments (Bouakaz et al. 2005) those required for dynamic and static t • Lower dotted line, crosses and solid squares -> buckling buckling threshold obtained via static stability, Based on the amplitude threshold for finite element analysis and surface evolver • shape deformations k B can be estimated Upper dotted line -> static rupture criterion due to stretching at expansion (too high) • The behavior of polymeric shells, Open circles -> transient break-up based on the revised Marmottant model (requires unrealistically large area dilatation, conforms with large shell thickness) the concept of a viscoelastic solid • with stretching and bending Solid triangles -> Threshold of dynamic buckling based on linear stability and simulations stiffness

Parametric Stability – Resonance            R 3.6 m G , 80 MPa , 1 nm , 20 Pa s b , 0, 0.5, eq s s kg          998 , P 101325 Pa , 1.07, v 1.7 MHz K , 3.0 d 14 Nm 3 l st f BD m Saturation - Harmonic resonance Transient Break-up 1,4 1,4 1,2 1,2 1,0 1,0 0,8 0,8 0,6 R(t) 0,6 R(t) P 4 0,4 0,4 P 7 Stability , ε=3 Stability , ε=2 P 8 0,2 0,2 0,0 0,0 v 0 =1.1 MHz, v 4 =2 MHz A Strain Softening membrane -0,2 -0,2 M -0,4 P 1.4 1.4 L I 1.2 1.2 T 1 1 U D 0.8 0.8 P 0 E P 0 0.6 Simulation 0.6 P 2 P 4 P 6 0.4 ε=2 P 7 0.4 P 6 P 8 0.2 0.2 0 Simulation 0 -0.2 ε=3 -0.2 t -0.4 t 0 50 100 150 200 250 300 350 0 5 10 15 20 25 30 35 40 45