Mechanical haemolysis in shock wave lithotripsy (SWL): I. Analysis of cell deformation due to SWL flow-fields

Abstract

This work analyses the interaction of red blood cells (RBCs) with shock-induced and bubble-induced flows in shock wave lithotripsy (SWL), and calculates, in vitro, the lytic effects of these two flows. A well known experimentally observed fact about RBC membranes is that the lipid bilayer disrupts when subjected to an areal strain (ΔA/A)c of 3%, and a corresponding, critical, isotropic tension, Tc, of 10 mN m-1 (1 mN m-1 = 1 dyne cm-1). RBCs suspended in a fluid medium tend to deform in accordance with the deformation of the surrounding fluid medium. The fluid flow-field is lytically effective if the membrane deformation exceeds the above threshold value. From kinematic analysis, motion of an elementary fluid particle can always be decomposed into a uniform translation, an extensional flow (e.g. vec u∞(x,y,z) = (k(t)x,-k(t)y,0)) along three mutually perpendicular axes, and a rigid rotation of these axes. However, only an extensional flow causes deformation of a fluid particle, and consequently deforms the RBC membrane. In SWL, a fluid flow-field, induced by a non-uniform shock wave, as well as radial expansion/implosion of a bubble, has been hypothesized to cause lysis of cells. Both the above flow-fields constitute an unsteady, extensional flow, which exerts inertial as well as viscous forces on the RBC membrane. The transient inertial force (expressed as a tension, or force/length), is given by Tiner~ρrc3k/τ, where τ is a timescale of the transient flow and rc is a characteristic cell size. When the membrane is deformed due to inertial effects, membrane strain is given by ΔA/A~kτ. The transient viscous force is given by Tvisc~ρ(ν/τ)1/2rc2k, where ρ and ν are the fluid density and kinematic viscosity. For the non-uniform shock, the extensional flow exerts an inertial force, Tinerapprox64 mN m-1, for a duration of 3 ns, sufficient to induce pores in the RBC membrane. For a radial flow-field, induced by bubble expansion/implosion, the inertial forces are of a magnitude 100 mN m-1, which last for a duration of 1 µs, sufficient to cause rupture. Bubble-induced radial flow is predicted to be lytically more effective than shock-induced flow in typical in vitro experimental conditions.

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