streamingiopFig1_SourcePatternsFig2a_SphericalStreamingFig2b_SphericalDistanceFig3a_CylindricalStreamingFig3b_CylindricalDistanceFig4a_FocusedStreamingFig4b_StreamingTimesFocusedComparisonsFig5a_FocusedStreamingFullRangeFig5b_StreamingTimesFocusedSourceCloseTheory for Acoustic Streaming in Soft P orousMatterRaghu RaghavanTher ata xis, LLC , John s Hop k ins East ern Bu ilding,1101 East 33rd Street, Baltimore, MD 21218, USA.Nov em ber 11, 2014AbstractAcoustic streaming in bulk fluids has a vast literature. People arenow investigating acoustic enhancement of delivery of therapeutics, suchas drug molecules or other particulates introduce d directly into brainparenchyma, for exam ple. This paper exam ines acoustic streaming insoftporousmediasuchasbiologicaltissuetoinvestigateifitisofefficacyin enhancing fluid transport. The principal results of this paper are:(i) new streaming e quations for porous media, which show interestinglysignificant differences from those that describe streaming in pure fluids;(ii) the Green functions obtained for the se e quations in isotropic, homoge-neous media; and (iii) approximate evaluation of the st reaming velocitiesand resulting particle trajectories for simple forms of acoustic sources. Itis seen that m oderate power (∼ 10 watts) of the sou nd wave in tissueresults in s treaming velocities that can be larger than mo derately highflow rates (∼ 5 L min) used in c urrent convection—enhanced deliveryof drugs in b rain. The many gaps in this treatment of the theory forstreaming in porous media that remain to be filled are also discussed.1IntroductionThis paper deals with acoustic streaming in porous media, particularly softmaterials such as brain tissue, adapting the standard derivation of streaming inpure fluids to the case of porous media. Ultimately, our purpose is to pro videuseful guidance in certain drug delivery applications, where the convection offluids which are directly injected into thetissue—andtheadvectionofther-apeutic molecules in suspension therein — is assisted with the application ofsonication. (We will provide references in the more detailed discussions below.)We emphasize that this distinct from several other applications of ultrasound inbiomedicine such as therapeutic ultrasound which is aimed at destroying tissue,diagnostic ultrasound for imaging, or ultrasound for opening the blood—brainbarrier for drug delivery. In the applications we envisage, the purpose is simply1to assess whether ultrasound will assist streaming. The motivation for thisstems from looking at the current practice in intraparenchymal infusions. Thisis done by inserting one or sev eral catheters and pumping the therapeutic solu-tion through. The tissue is a highly resistive medium for fluid flow, and thereare many pathways that may lead the flow to undesired areas. Since there isonly place of control (the port of the catheter for single-port catheters, or theports in a multiport catheter), the fluid is at the mercy of the medium to guideits path once it leaves the catheter. If streaming were effectiv e, the sonicationwo uld be able to focus and direct the acoustic beams as we desire, and guidethe fluid and particles to reach the target, and avoid other areas. We call thishoped for application Acoustic Shepherding. We also mention that we expectsuch applications in other areas as well, perhaps in environmental or geophys-ical applications. However, the purpose of this paper is more modest. Wedevelop the basic theory for acoustic streaming in porous media and comparepreliminarily with some experiments. The limitations of the previous work onthis, and of our o wn, will be discussed.Our central simplifying assumption, further elucidated below, will be con-sistent with that of the imaging community; namely that there is a longitudinalw ave mode that propagates as if tissue were a homogeneous continuum. Follow-ing this Introduction, there are three principal sections in this paper. The goalof the first is to propose the equations for streaming, and that of the second isto write the Green function solutions for these equations (in isotropic, homoge-neous, infinite media). We discuss in detail the simplifying assumptions made,and this necessitates some review of known material from the correspondingtreatments of acoustic propagation and streaming in pure fluids. The thirdsection provides some calculations for specific acoustic sources, and comparesour solutions with the results for enhancement of particle transport in brain thathave been reported. There is a short conclusions section, highlighting some ofthe man y lacunae left in our treatment. Appendices discuss relevant materialthat would impede the discussion if placed in the main text.All quantities are measured in cgs units, unless otherwise mentioned, andthroughout we shall assume we are dealing with a harmonic component of anacoustic signal, i.e., the time dependence of the signal is assumed of the form−, where the angular frequency is real. Also,weusethenotation:= or =:to mean that the quan tity facing the colon is defined to be the quantity facingthe equals sign. The usual unsymmetric convention is adopted for the Fouriertransform, e.g.,˜ (k):=Z3 exp (−k · x) (x) (1)so that 1 (2)3occursintheinverse,andsimilarlyforthetime—frequencytransforms. Also we use the usual adjectives Eulerian and the ahistorical La-grangian as synonyms for the more descriptive but less used spatial and material,respectively. Our treatment will b e in the Eulerian pictur e.22 Acoustic streaming in fluids and porous mediaIn this section, we shall arrive at some equations for acoustic streaming in aporous medium which contains fluid—filled interstices. These interstices are re-ferred to as connected pore space in the geophysical literature and as interstitialspace for applications to biological tissue. We assume they fill up a volume frac-tion of the total space. We are primarily concerned with live brain tissue inour applications, ( & 02) and with ultrasonic frequencies between 1 −10 MHz.The interstitial fluidinbrainflows in channels whose widths have been esti-mated of the order of 50 nm [1], though we caution that this is by no meanscertain. The skin depth is of the order ofp,where ≈ 0007 cm2Á sec isthe kinematic viscosity of the fluid, i.e., water at body temperature. It thusexceeds the channel width even for frequencies as high as 10
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