Suslov Sergey A, Ooi Andrew, Manasseh Richard
Mathematics, H38, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Jun;85(6 Pt 2):066309. doi: 10.1103/PhysRevE.85.066309. Epub 2012 Jun 12.
The nonlinear dynamic behavior of microscopic bubbles near a rigid wall is investigated. Oscillations are driven by the ultrasonic pressure field that arises in various biomedical applications such as ultrasound imaging or targeted drug delivery. It is known that, when bubbles approach a blood-vessel wall, their linear dynamic response is modified. This modification may be very useful for real-time detection of bubbles that have found targets; in future therapeutic technologies, it may be useful for controlled release of medical agents encapsulating microbubbles. In this paper, the nonlinear response of microbubbles near a wall is studied. The Keller-Miksis-Parlitz equation is adopted, but modified to account for the presence of a rigid wall. This base model describes the time evolution of the bubble surface, which is assumed to remain spherical, and accounts for the effect of acoustic radiation losses owing to liquid compressibility in the momentum conservation. Two situations are considered: the base case of an isolated bubble in an unbounded medium, and a bubble near a rigid wall. In the latter case, the wall influence is modeled by including a symmetrically oscillating image bubble. The bubble dynamics is traced using a numerical solution of the model equation. Subsequently, Floquet theory is used to accurately detect the bifurcation point where bubble oscillations stop following the driving ultrasound frequency and undergo period-changing bifurcations. Of particular interest is the detection of the subcritical period-tripling and -quadrupling transition. The parametric bifurcation maps are obtained as functions of nondimensional parameters representing the bubble radius, the frequency and pressure amplitude of the driving ultrasound field, and the distance from the wall. It is shown that the presence of the wall generally stabilises the bubble dynamics, so that much larger values of the pressure amplitude are needed to generate nonlinear responses. Thus, a clinical protocol in which selected nonlinear harmonics are examined under varying insonation amplitudes may be useful in detecting microbubble proximity to walls.
研究了刚性壁附近微观气泡的非线性动力学行为。振荡由超声压力场驱动，该压力场出现在各种生物医学应用中，如超声成像或靶向药物递送。众所周知，当气泡靠近血管壁时，其线性动力学响应会发生改变。这种改变对于实时检测已找到目标的气泡可能非常有用；在未来的治疗技术中，对于封装微气泡的药物的控释可能也很有用。本文研究了壁附近微气泡的非线性响应。采用了凯勒 - 米克斯 - 帕利茨方程，但对其进行了修改以考虑刚性壁的存在。这个基础模型描述了气泡表面的时间演化，假设气泡表面保持球形，并在动量守恒中考虑了由于液体可压缩性导致的声辐射损失的影响。考虑了两种情况：无界介质中孤立气泡的基础情况，以及刚性壁附近的气泡。在后一种情况下，通过包含一个对称振荡的镜像气泡来模拟壁的影响。使用模型方程的数值解来追踪气泡动力学。随后，利用弗洛凯理论准确检测气泡振荡跟随驱动超声频率停止并经历周期变化分岔的分岔点。特别令人感兴趣的是检测亚临界周期三倍化和四倍化转变。作为表示气泡半径、驱动超声场的频率和压力振幅以及与壁的距离的无量纲参数的函数，获得了参数分岔图。结果表明，壁的存在通常会使气泡动力学稳定，因此需要大得多的压力振幅值才能产生非线性响应。因此，一种在不同声压幅度下检查选定非线性谐波的临床方案可能有助于检测微气泡与壁的接近程度。
