Theory for Acoustic Streaming in Soft Porous Matter Raghu Raghavan Therataxis LLC Johns Hopkins Eastern Building 1101 East 33rd Street Baltimore MD 21218 USA November 11 2014 Abstract Acoustic streaming in bulk fluids has a vast literature People are now investigating acoustic enhancement of delivery of therapeutics such as drug molecules or other particulates introduced directly into brain parenchyma for example This paper examines acoustic streaming in soft porous media such as biological tissue to investigate if it is of e cacy in enhancing fluid transport The principal results of this paper are i new streaming equations for porous media which show interestingly significant di erences from those that describe streaming in pure fluids ii the Green functions obtained for these equations in isotropic homogeneous media and iii approximate evaluation of the streaming velocities and resulting particle trajectories for simple forms of acoustic sources It is seen that moderate power 10 watts of the sound wave in tissue results in streaming velocities that can be larger than moderately high flow rates 5 L min used in current convection enhanced delivery of drugs in brain The many gaps in this treatment of the theory for streaming in porous media that remain to be filled are also discussed 1 Introduction This paper deals with acoustic streaming in porous media particularly soft materials such as brain tissue adapting the standard derivation of streaming in pure fluids to the case of porous media Ultimately our purpose is to provide useful guidance in certain drug delivery applications where the convection of fluids which are directly injected into the tissue and the advection of therapeutic molecules in suspension therein is assisted with the application of sonication We will provide references in the more detailed discussions below We emphasize that this distinct from several other applications of ultrasound in biomedicine such as therapeutic ultrasound which is aimed at destroying tissue diagnostic ultrasound for imaging or ultrasound for opening the blood brain barrier for drug delivery In the applications we envisage the purpose is simply 1 to assess whether ultrasound will assist streaming The motivation for this stems from looking at the current practice in intraparenchymal infusions This is done by inserting one or several catheters and pumping the therapeutic solution through The tissue is a highly resistive medium for fluid flow and there are many pathways that may lead the flow to undesired areas Since there is only place of control the port of the catheter for single port catheters or the ports in a multiport catheter the fluid is at the mercy of the medium to guide its path once it leaves the catheter If streaming were e ective the sonication would be able to focus and direct the acoustic beams as we desire and guide the fluid and particles to reach the target and avoid other areas We call this hoped for application Acoustic Shepherding We also mention that we expect such applications in other areas as well perhaps in environmental or geophysical applications However the purpose of this paper is more modest We develop the basic theory for acoustic streaming in porous media and compare preliminarily with some experiments The limitations of the previous work on this and of our own will be discussed Our central simplifying assumption further elucidated below will be consistent with that of the imaging community namely that there is a longitudinal wave mode that propagates as if tissue were a homogeneous continuum Following this Introduction there are three principal sections in this paper The goal of the first is to propose the equations for streaming and that of the second is to write the Green function solutions for these equations in isotropic homogeneous infinite media We discuss in detail the simplifying assumptions made and this necessitates some review of known material from the corresponding treatments of acoustic propagation and streaming in pure fluids The third section provides some calculations for specific acoustic sources and compares our solutions with the results for enhancement of particle transport in brain that have been reported There is a short conclusions section highlighting some of the many lacunae left in our treatment Appendices discuss relevant material that would impede the discussion if placed in the main text All quantities are measured in cgs units unless otherwise mentioned and throughout we shall assume we are dealing with a harmonic component of an acoustic signal i e the time dependence of the signal is assumed of the form where the angular frequency is real Also we use the notation or to mean that the quantity facing the colon is defined to be the quantity facing the equals sign The usual unsymmetric convention is adopted for the Fourier transform e g Z k 3 exp k x x 1 3 so that 1 2 occurs in the inverse and similarly for the time frequency transforms Also we use the usual adjectives Eulerian and the ahistorical Lagrangian as synonyms for the more descriptive but less used spatial and material respectively Our treatment will be in the Eulerian picture 2 2 Acoustic streaming in fluids and porous media In this section we shall arrive at some equations for acoustic streaming in a porous medium which contains fluid filled interstices These interstices are referred to as connected pore space in the geophysical literature and as interstitial space for applications to biological tissue We assume they fill up a volume fraction of the total space We are primarily concerned with live brain tissue in our applications 0 2 and with ultrasonic frequencies between 1 10 MHz The interstitial fluid in brain flows in channels whose widths have been estimated of the order of 50 nm 1 though we p caution that this is by no means certain The skin depth is of the order of where 0 007 cm2 sec is the kinematic viscosity of the fluid i e water at body temperature It thus exceeds the channel width even for frequencies as high as 10 MHz The viscous fluid is therefore always in an inner boundary layer in terms of boundary layer theory terminology There are at least two important di erences between tissue and the usual porous medium description of rocks in geophysics One is that the porous solid frame itself consisting largely of cells with a weak connecting network has about 80 of occluded fluid within it The matter being treated is very soft The second distinction is
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