This page contains a list of my papers and conference presentations. You can view the abstract of a paper by clicking its title. The manuscripts and full texts of several papers are available in PDF format for downloading (see the corresponding links following the abstract). Note that the full texts can be downloaded only if your library is subscribed to electronic journals of American Institute of Physics (OJPS portal) and Elsevier Science (ScienceDirect portal). It should be remarked that the co-authors of papers No. 1, 2 and 4 - 7 are listed alphabetically (Cyrillic alphabet), as is customary in Russia.
I.Sh. Akhatov, N.K. Vakhitova, G.Ya. Galeeva, R.I. Nigmatulin, and D.B. Khismatullin, "Weak oscillations of a gas bubble in a spherical volume of compressible liquid," J. Appl. Maths Mechs 61(6), 921 - 930 (1997):
The following spherically symmetric problem is considered: a single gas bubble at the centre of a spherical flask filled with a compressible liquid is oscillating in response to forced radial excitation of the flask walls. In the long-wave approximation at low Mach numbers, one obtains a system of differential-difference equations generalizing the Rayleigh-Lamb-Plesseth equation. This system takes into account the compressibility of the liquid and is suitable for describing both free and forced oscillations of the bubble. It includes an ordinary differential equation analogous to the Herring-Flinn-Gilmore equation describing the evolution of the bubble radius, and a delay equation relating the pressure at the flask walls to the variation of the bubble radius. The solutions of this system of differential-difference equations are analysed in the linear approximation and numerical analysis is used to study various modes of weak but non-linear oscillations of the bubble, for different laws governing the variation of the pressure or velocity of the liquid at the flask wall. These solutions are compared with numerical solutions of the complete system of partial differential equations for the radial motion of the compressible liquid around the bubble.
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I.Sh. Akhatov and D.B. Khismatullin, "Long-wave - short-wave interaction in bubbly liquids," J. Appl. Maths Mechs 63(6), 917 - 926 (1999):
The interaction of long and short waves in a rarefied monodisperse mixture of a weakly compressible liquid containing bubbles of gas is considered. It is shown that the equations describing the dynamics of the perturbations in the bubbly liquid admit of the existence of long-wave-short-wave Benney-Zakharov resonance. A special modification of the multiple-scale method is employed to derive the interaction equations. In the non-resonant case, the interaction equations reduce to the non-linear Schr”dinger equation in the form of the short-wave envelope while, in the resonance case, they reduce to the well-known system of Zakharov equations. The characteristics of long-wave-short-wave interaction in a bubbly liquid lie in the fact that, at certain values of the frequency of the short wave, the interaction coefficients vanish ("interaction degeneracy"). A class of new interaction models is constructed in the case of "degeneracy". Degenerate resonance interaction in a bubbly liquid is investigated numerically using these models.
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I.Sh. Akhatov and D.B. Khismatullin, "Effect of dissipation on the interaction between long and short waves in bubbly liquids," Fluid Dynamics 35(4), 573 - 583 (2000):
The interaction of long and short waves in a rarefied monodisperse mixture of a weakly compressible viscous liquid and gas bubbles is considered. For taking the dissipation effects into account an effective-viscosity scheme is used. Four cases of dissipation are distinguished, namely, strong, medium, weak, and very weak dissipation. In the cases of moderate, weak, and very weak dissipation the equations of resonance and non-resonance wave interaction are derived using the multiscale method. The effect of "degeneration" of the interaction is detected in certain of the models constructed. In the case of "degeneration" a class of new models of the dissipative resonance interaction is constructed and investigated numerically.
I.Sh. Akhatov and D.B. Khismatullin, "Two-dimensional mechanismsof interaction between ultrasound and sound in bubbly liquids: interaction equations," Acoust. Phys. 47(1), 10 - 15 (2001):
The interaction of long (sound) and short (ultrasound) waves propagating in a rarefied monodisperse mixture of a weakly compressible liquid with gas bubbles is considered. Using the multiscale method, the Davey-Stewartson system of equations is derived as a model of two-dimensional interaction. It is shown that, for some values of parameters, this system is reduced to an integrable form (the Davey-Stewartson I equations) and has localized solutions in the form of dromions (exponentially decaying waves of the short-wave envelope). One of the most important properties of dromions is their ability to move according to the law that governs the variations of the boundary conditions set at infinity for the long wave. It is suggested that these solutions be used for controlling the effects of ultrasound on bubbly liquids.
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I.Sh. Akhatov and D.B. Khismatullin, "Mechanisms of interaction between ultrasound and sound in liquids with bubbles: singular focusing," Acoust. Phys. 47(2), 140 - 144 (2001):
A two-dimensional interaction between long-wave (sound) and short-wave (ultrasound) pressure perturbations in a rarefied monodisperse mixture of a weakly compressible liquid with gas bubbles is considered. The conditions at which this interaction leads to a singular focusing (an explosive instability) of ultrasound are determined. A numerical study of the defocusing and the singular focusing in a bubbly liquid is carried out. The effect of the long-wave-short-wave resonance on the development of two-dimensional disturbances is studied.
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I.S. Akhatov and D.B. Khismatullin, "Mechanisms for the interaction of ultrasound with sound in liquids with bubbles. Singular focusing," Russian Ultrasonics 31(1), 15 - 25 (2001):
The paper considers the two-dimensional interaction between long wavelength (sound) and short wavelength (ultrasound) pressure perturbations in a rarified monodispersive mixture of a weakly compressible liquid with gas bubbles. The conditions are derived for which this interaction leads to singular focusing (explosive instability) of the ultrasound. A numerical study is made of the defocusing and singular focusing in a bubbly liquid. The effect of the long-wave- short-wave resonance on the development of two-dimensional perturbations is investigated.
Note: This paper is another translation of my second paper in Akusticheskii Zhurnal (original translation is paper No. 6). It is translated and published by "Multi-Science" (Great Britain). The information about "Russian Ultrasonics" can be found here.
D.B. Khismatullin and A. Nadim, "Shape oscillations of a viscoelastic drop," Phys. Rev. E 63, 061508 (2001) (10 pages):
Small-amplitude axisymmetric shape deformations of a viscoelastic liquid drop in microgravity are theoretically analyzed. Using the Jeffreys constitutive equation for linear viscoelasticity, the characteristic equation for the frequency and decay factor of the shape oscillations is derived. Asymptotic analysis of this equation is performed in the low- and high-viscosity limits and for the cases of small, moderate, and large elasticities. Elastic effects are shown to give rise to a type of shape oscillation that does not depend on the surface tension. The existence of such oscillations is confirmed by numerical solution of the characteristic equation in various regimes. A method for determining the viscoelastic properties of highly viscous liquids based upon experimental measurements of the frequency and damping rate of such shape oscillations is suggested.
[ Manuscript Full text access ]
D.B. Khismatullin and I.Sh. Akhatov, "Sound-ultrasound interaction in bubbly fluids: Theory and possible applications," Phys. Fluids 13(12), 3582 - 3598 (2001):
The interaction between sound and ultrasound waves in a weakly compressible viscous liquid with gas bubbles is considered. Using the method of multiple scales one- and two-dimensional nonlinear interaction equations are derived. The degeneracy of the interaction is found in bubbly fluids. This phenomenon lies in the fact that the interaction coefficients vanish at a certain frequency of ultrasound. We demonstrate that the integrable Davey-Stewartson I (DSI) system of equation can describe the two-dimensional sound-ultrasound evolution. The DSI equations are remarkable by their solutions referred to as dromions. In bubbly fluids the dromion represents the localized focused ultrasound wave which can alter the direction of its motion under changes in the boundary conditions for the sound wave. The condition of singular focusing of ultrasound in bubbly fluids is obtained. By numerical analysis of the interaction models, we reveal such processes as intensification of ultrasound by sound, nonlinear instability of a sound profile, and prove the validity of the singular focusing condition. Finally, possible applications of the results are outlined.
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D.B. Khismatullin and A. Nadim, "Radial oscillations of encapsulated microbubbles in viscoelastic liquids," Phys. Fluids 14(10), 3534 - 3557 (2002):
The small-amplitude radial oscillations of a gas microbubble encapsulated by a viscoelastic solid shell and surrounded by a slightly compressible viscoelastic liquid are studied theoretically. The Kelvin-Voigt and 4-constant Oldroyd models are used to describe the viscoelastic properties of the shell and liquid, respectively. The equation for radial oscillation is derived using the method of matched asymptotic expansions. Based on this equation, we present the expressions for damping coefficients and scattering cross-sections at the fundamental frequency and at twice that frequency. The numerical maximization of the amplitude-frequency response function shows that the resonance frequency for the encapsulated microbubble highly depends on viscous damping and therefore significantly differs from the undamped natural frequency. The effects of the shell and liquid parameters on the resonance frequency and scattering cross-sections are analyzed.
[ Manuscript Full text access ]
D.B. Khismatullin, Y.Y. Renardy, and V. Cristini, "Inertia-induced breakup of highly viscous drops subjected to simple shear," Phys. Fluids 15(5), 1351 - 1354 (2003):
We investigate the inertia-driven breakup of viscous drops suspended in a less viscous liquid under simple shear. For Stokes flow, it is known that there is a critical value of the viscosity ratio, beyond which breakup does not occur. We find that for viscosity ratios larger than this, inertia can be used as a mechanism of breakup. Inertia increases the angle of tilt of the drops and effectively leads to emulsification for a wider range of viscosity ratios than in Stokes flow.
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D.B. Khismatullin, "Resonance frequency of microbubbles: Effect of viscosity," J. Acoust. Soc. Am. 116, 1463 - 1473 (2004):
The transmitted frequency at which a gas bubble of millimeter or submillimeter size oscillates resonantly in a low-viscosity liquid is approximately equal to the undamped natural frequency (referred to as the Minnaert frequency if surface tension effects are disregarded). Based on a theoretical analysis of bubble oscillation, this paper shows that such an approximation cannot be validated for microbubbles used in contrast-enhanced ultrasound imaging. The contrast-agent microbubbles represent either encapsulated bubbles of size less than 10 mm or free (nonencapsulated) bubbles of submicron size. The resonance frequency of the microbubbles deviates significantly from the undamped natural frequency over the whole range of microbubble sizes due to the increased viscous damping coefficient. The difference between these two frequencies is shown to have a tremendous impact on the resonant backscatter by the microbubbles. In particular, the first and second harmonics of the backscattered signal from the microbubbles are characterized by their own resonance frequencies, equal to neither the microbubble resonance frequency nor the undamped natural frequency.
[ Manuscript ] [ Full text access ]
D.B. Khismatullin and G.A. Truskey, "3D numerical simulation of receptor-mediated leukocyte adhesion to surfaces: Effects of cell deformability and viscoelasticity," Phys. Fluids (Accepted in April 2004).
Computational fluid dynamics is used to investigate the effects of cell deformability and viscoelasticity on receptor-mediated leukocyte adhesion to endothelium or a ligand coated surface in a parallel plate flow chamber. In the three-dimensional numerical code, a leukocyte is modeled as a compound viscoelastic drop (a nucleus covered by a thick layer of cytoplasm). The nucleus, cytoplasm, and extracellular fluid are considered as Newtonian or viscoelastic liquids of high viscosity. The receptor-ligand interaction is incorporated into the code by using the spring-peeling kinetic model under the assumption that leukocyte receptors are located on the tips of cylindrical microvilli distributed over the leukocyte membrane. The code is based on the Volume-of-Fluid method and the Giesekus constitutive equation is implemented in the code to capture viscoelasticity of the cytoplasm and nucleus. Numerical simulations demonstrate the formation and breakup of membrane tethers observed in vitro and suggest that the elasticity of the cytoplasm is responsible for a tear-drop shape of rolling leukocytes in vivo. We show that the leukocyte membrane can be extended and disrupted under high shear if the receptor-ligand bonds live in a stressed state for a sufficiently long time. If the shear rate is low, the leukocyte rolls along the surface. The rolling velocity of the viscoelastic cell is smaller than that of the Newtonian cell. This is due to the increased deformability of the viscoelastic cell and, as a result, the decreased torque acting on this cell.
[ Manuscript ]
D.B. Khismatullin and G.A. Truskey, "A 3D numerical study of the effect of channel height on leukocyte deformation and adhesion in parallel-plate flow chambers," Microvasc. Res. 68, 188 - 202 (2004).
The effect of channel height on leukocyte adhesion to a lower plate in a parallel-plate flow chamber is studied by direct numerical simulations in three dimensions. The numerical model takes into account deformability and viscoelasticity of the leukocyte, membrane ruffles (microvilli), and the presence of mechanically different regions inside the cell (nucleus and cytoplasm). Leukocyte adhesion is assumed to be mediated by interactions of adhesion molecules on the tips of microvilli with their counterparts on the lower plate. Results of this study indicate that an adherent leukocyte experiences much less drag than a rigid sphere due to its deformation and transient stress growth. While overall leukocyte deformation is modest at shear stresses encountered in the microcirculation, deformation in the contact region is significant. At fixed wall shear stress, the contact area of the cell membrane with the substrate increases with increasing the ratio of cell diameter to channel height, leading to greater adhesion. This suggests that in vitro flow chamber studies typically underestimate leukocyte adhesion that occurs in the microcirculation.
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I.Sh. Akhatov, R.I. Nigmatulin, D.B. Khismatullin, and K.R. Khusnutdinova, "Resonant interaction of long and short pressure waves in bubbly liquids" in Proceedings of the Third International Conference on Multiphase Flow (ICMF'98), Lyon, France (1998):
The interaction of long and short pressure waves in a weakly compressible ideal liquid with gas bubbles is considered. It is shown that the equations for dynamics of a bubbly liquid allows the long wave / short wave resonance, when the group velocity of a short wave equals the phase velocity of a long wave. By the method of multiple scales the equations of wave interaction are derived. In the nonresonant case these equations reduce to the well-known nonlinear Schr\"odinger equation for the envelope of a short wave. In resonant case it leads to the Zakharov's system of equations. The specific character of the interaction in the bubbly liquid is revealed. It consists in vanishing the interaction coefficients at a certain frequency of a short wave (the degeneration of the interaction). In the case of the degeneration of the resonant interaction new systems of the interaction equations are deduced. For the equations of the degenerate resonant interaction the numerical investigation of the nonlinear structure formation processes is presented.
D.B. Khismatullin and I.Sh. Akhatov, "New interaction models of long and short pressure waves in bubbly liquids" in Proceedings of the X session of Russian Acoustical Society, Vol. 1, 73 - 78, GEOS, Moscow, Russia (2000):
The interaction of long-wave (sonic) and short-wave (ultrasonic) perturbations in a weakly compressible viscous liquid with gas bubbles is considered. By the method of multiple scales the equations of one- and two-dimensional wave interaction are derived. It is revealed that interaction coefficients of these equations vanish for some values of a short-wave frequency ("degeneration" of interaction). In the degenerate case a class of new systems of one-dimensional interaction is constructed. The non-resonant evolution in two spatial dimensions of long and short waves is described by the equations reduced, without regard to dissipation, to the Davey-Stewartson system. It is shown that the system is transformed to the integrable form (Davey-Stewartson I equations) and has the dromion solutions for particular choices of the parameters of a bubbly liquid. It is offered to make use of the dromion capacity to travel according to the evolution of boundary conditions for the long wave at infinity for controlling ultrasonic action processes in a bubbly liquid. The conditions by which a two-dimensional sound-ultrasound interaction in this mixture leads to singular focusing of disturbances (blow-up instability) are derived. The existence of singular focusing in bubbly liquids is confirmed by numerical experiments. The influence of a long-wave / short-wave resonance on two-dimensional wave interactions is analyzed. http://www.akin.ru/Docs/Rao/Ses10/PH15.PDF
D.B. Khismatullin and I.Sh. Akhatov, "Interaction of sound and ultrasound waves in bubbly systems" in Dynamics of multiphase systems (M. Ilgamov, I. Akhatov and S. Urmancheev, Eds.), 187 - 196, Gilem Publisher & Pol Publisher, Ufa (2000):
The interaction of long-wave (sonic) and short-wave (ultrasonic) perturbations in a weakly compressible viscous liquid with gas bubbles is considered. It is shown that the equation describing dynamics of weak perturbations in a bubbly liquid allows the Benney-Zakharov long-wave/short-wave resonance. By the method of multiple scales the equations of one- and two-dimensional wave interaction are derived. Non-resonant, resonant and degenerate cases are discussed. The degenerate case is the specific of long-wave/short-wave interaction in a bubbly liquid. The 'degeneration' phenomenon consists in vanishing of the interaction coefficients at a certain frequency of the short wave. The conditions by which a two-dimensional sound--ultrasound interaction in bubbly mixtures leads to singular focusing of disturbances (blowup instability) are derived. The numerical investigations of degenerate resonant interaction between sound and ultrasound and of singular focusing of ultrasound in these media are done. The influence of dissipation on one- and two-dimensional wave interactions is analysed. The role of a long-wave/short-wave resonance in the propagation of two-dimensional disturbances is studied.
D.B. Khismatullin and I.Sh. Akhatov, "Two-dimensional long-wave/short-wave interaction in bubbly liquids" in Proceedings of the Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation (A. Bermúdez, D. Gómez, C. Hazard, P. Joly, and J. E. Roberts, Eds.), SIAM (2000):
The interaction of long and short pressure waves in a weakly compressible liquid with gas bubbles is considered. By the method of multiple scales the Davey-Stewartson system of equations is derived. It is shown that this system is transformed to the integrable Davey-Stewartson I equations for particular choices of the parameters of a bubbly liquid and thus has localized dromion solutions (exponentially decaying envelopes of short waves). One of the most important features of the dromion is its capacity to travel on the tracks described by the boundaries of a long wave profile. Therefore, such solutions may be of interest for controlling ultrasonic action processes in bubbly liquids. The conditions by which a two-dimensional interaction between long and short waves in bubbly liquids leads to singular focusing of disturbances (blow-up instability) are found. The numerical investigation of this phenomenon is done. The influence of a long-wave/short-wave resonance on the growth of the blow-up instability is analyzed.
D.B. Khismatullin and I.Sh. Akhatov, "Sound-ultrasound interaction in bubbly liquids" in Proceedings of the Fourth International Conference on Multiphase Flow (ICMF'2001), New Orleans, USA (2001):
Note: This paper was among the best 30 publications of ICMF'2001 and was recommended for publication in the special issue of the International Journal of Multiphase Flow. However, the Editorial Board of IJMF decided not to proceed with its publication because a more comprehensive paper on the same subject was published in Physics of Fluids (paper No. 9) prior to the make-up of the special issue.The interaction between sound and ultrasound waves in a weakly compressible viscous liquid with gas bubbles is considered. Using the method of multiple scales one- and two-dimensional nonlinear interaction equations are derived. The degeneracy of the interaction is found in bubbly liquids. This phenomenon lies in the fact that the interaction coefficients vanish at a certain frequency of ultrasound. We demonstrate that the integrable Davey--Stewartson I (DSI) system of equation can describe the two-dimensional sound-ultrasound evolution. The DSI equations are remarkable by their solutions referred to as dromions. In bubbly liquids the dromion represents localized focused ultrasound which can alter the direction of its motion under changes in boundary conditions for sound. The condition of singular focusing of ultrasound in bubbly liquids is obtained. By numerical analysis of the interaction models we prove the validity of the singular focusing condition. Finally, possible applications of the results are outlined.
D.B. Khismatullin and A. Nadim, "Shape oscillations of a viscoelastic drop: acoustic rheometry", open-forum presentation, Fourth International Conference on Multiphase Flow (ICMF'2001), New Orleans, USA (May 2001):
Small-amplitude axisymmetric shape deformations of a viscoelastic liquid drop in microgravity are theoretically analyzed. Using the Jeffreys constitutive equation for linear viscoelasticity, the characteristic equation for the frequency and decay factor of the shape oscillations is derived. Asymptotic analysis of this equation is performed in the low- and high-viscosity limits and for the cases of small, moderate, and large elasticities. Elastic effects are shown to give rise to a new type of shape oscillation which does not depend on the surface tension. The existence of such oscillations is confirmed by numerical solution of the characteristic equation in various regimes. A method for determining the viscoelastic properties of highly viscous liquids based upon experimental measurements of the frequency and damping rate of such shape oscillations is suggested. [Work supported by NSF-NATO.]
D.B. Khismatullin, "Encapsulated microbubbles as drug/gene delivery agents (new project)," open-forum presentation, Fourth International Conference on Multiphase Flow (ICMF'2001), New Orleans, USA (May 2001):
Encapsulated gas microbubbles are well known as contrast agents for medical ultrasound imaging. However, they are able to do more than just image blood pool and tissue perfusion. Such bubbles, with an average size less than that of a red blood cell, are capable to penetrate even into the smallest capillaries and release drugs and genes, incorporated either inside them or on their surface, under the action of ultrasound. Moreover, the microbubbles will transport a specific drug to a specific site within the body (for instance, an anticancer drug to a specific tumor) if their surface contains ligands. The ligands (biotin or antibody) will bind to the receptors (avidin or antigen) situated at the blood vessel walls of the target site and force the microbubble to attach to the walls. Commercial development of these ideas is in its initial phase, but methods for preparing such microbubbles have already been patented. From a practical point of view, the targeted drug/gene delivery agents should:
- be able to reach the target site and flow through it;
- remain stable long enough to circulate and accumulate at the binding site;
- not detach from the target in the flowing blood (their binding to the target should be firm);
- rupture under exposion to ultrasound to achieve targeted drug release;
- not lead to deleterious bioeffects.
The goals of this project are to model the dynamics of drug- or gene-containing microbubbles in blood flow and, based on the conditions above, to identify the mechanical properties of the microbubble and its coating by which drug/gene delivery will be effective and safe. For this, we plan to study the mechanisms of bubble instability in blood flow, investigate the effects of hydrodynamic and acoustic radiation forces on bubble dynamics during the flow of the microbubble through the target site, consider radial oscillations of a single microbubble and a group of microbubbles (a bubble cloud) near blood vessel walls, and analyze the bioeffects that can arise due to the interaction between ultrasound and the oscillating microbubble.
It should be noted that a rigorous theoretical description for the behavior of the microbubbles covered with a biocompatible surface-active layer in blood flow is not available, despite the fact that these microbubbles have been the subject of intense experimental research and commercial development for use as contrast agents for medical ultrasound diagnostics. Existing theoretical models are based upon various forms of the Rayleigh-Plesset (RP) equation for spherical bubble oscillations, and attempt to take into account, often on the basis of unjustified conjectures, the elasticity and viscosity of the surfactant layer which is treated as a viscoelastic solid shell. It is known that the standard RP equation holds only if the liquid surrounding a gas bubble is Newtonian, incompressible and of infinite extent. These assumptions may be reasonable in certain inorganic aqueous media but not for living matter and, in particular, human tissue and blood. Nonetheless, the RP-based models are claimed to be "validated by extensive experimental results" by fitting the models, i.e. the a priori unknown values of shell elasticity and viscosity, to experimental measurements. In our view, such models cannot be accurately used to interpret in vivo measurements and we intend to provide a more accurate description that can actually be used with confidence for this purpose.
D.B. Khismatullin, "Encapsulated microbubbles in blood flow: new method for drug/gene delivery": 73rd Annual Meeting of the Society of Rheology, October 21 - 25, 2001 — Bethesda, Maryland:
Encapsulated gas microbubbles are well known as contrast agents for medical ultrasound imaging. Such bubbles, with an average size less than that of a red blood cell, are capable to penetrate even into the smallest capillaries and release drugs and genes, incorporated either inside them or on their surface, under the action of ultrasound. Moreover, the microbubbles will transport a specific drug to a specific site within the body if their surface contains ligands. Commercial development of these ideas is in its initial phase, but methods for preparing such microbubbles have already been patented. From a practical point of view, the targeted drug/gene delivery agents should: 1) be able to reach the target site and flow through it; 2) remain stable long enough to circulate and accumulate at the binding site; 3) not detach from the target in the flowing blood (their binding to the target should be firm); 4) rupture under exposion to ultrasound to achieve targeted drug release; 5) not lead to deleterious bioeffects.
The goals of this project are to model the dynamics of drug- or gene-containing microbubbles in blood flow and, based on the conditions above, to identify the mechanical properties of the microbubble and its coating by which drug/gene delivery will be effective and safe. For this, we plan to study the mechanisms of bubble instability in blood flow, investigate the effects of hydrodynamic and acoustic radiation forces on bubble dynamics during the flow of the microbubble through the target site, consider radial oscillations of a single microbubble and a group of microbubbles (a bubble cloud) near blood vessel walls, and analyze the bioeffects that can arise due to the interaction between ultrasound and the oscillating microbubble.
D.B. Khismatullin and A. Nadim, "Elasticity-driven shape oscillations of a non-Newtonian drop": 73rd Annual Meeting of the Society of Rheology, October 21 - 25, 2001 — Bethesda, Maryland:
Small-amplitude axisymmetric shape deformations of a non-Newtonian liquid drop in microgravity are theoretically analyzed. Using the Jeffreys constitutive equation for linear viscoelasticity, the characteristic equation for the frequency and decay factor of the shape oscillations is derived. Asymptotic analysis of this equation is performed in the low- and high-viscosity limits and for the cases of small, moderate, and large elasticities. Elastic effects are shown to give rise to a new type of shape oscillation which does not depend on the surface tension. The existence of such oscillations is confirmed by numerical solution of the characteristic equation in various regimes. A method for determining the viscoelastic properties of highly viscous liquids based upon experimental measurements of the frequency and damping rate of such shape oscillations is suggested.
D.B. Khismatullin and A. Nadim, "Radial oscillations of an encapsulated microbubble in a viscoelastic liquid": 2001 Division of Fluid Dynamics Meeting, November 18 - 20, 2001 — San Diego, California:
The small-amplitude radial oscillations of a microbubble encapsulated by a viscoelastic solid shell and surrounded by a slightly compressible viscoelastic liquid are studied theoretically. The Kelvin-Voigt and corotational Jeffreys models are used to describe the viscoelastic properties of the shell and liquid, respectively. The equation for radial oscillation is derived using the method of matched asymptotic expansions. Based on this equation, the expressions for damping coefficients and scattering cross-sections at the fundamental frequency and at twice that frequency are derived. The numerical maximization of the amplitude-frequency response function shows that the resonance frequency for the encapsulated microbubble highly depends on viscous damping and therefore significantly differs from the undamped natural frequency. The effects of the shell and liquid parameters on the resonance frequency and scattering cross-sections are analyzed and the implications for the use of such microbubbles as ultrasound contrast agents are highlighted.
D.B. Khismatullin and A. Nadim, "Radial dynamics of encapsulated microbubbles in biological fluids": 2002 Annual Meeting of AIChE, November 3 - 8, 2002 — Indianapolis, Indiana:
Despite the fact microbubbles covered with a biocompatible surface-active layer have been the subject of intense experimental research and commercial development for use as contrast agents for medical ultrasound diagnostics, a rigorous theoretical description for the pulsations of such encapsulated bubbles in blood flow is not available. In this talk, we present and analyze a mathematical model for radial oscillations of the encapsulated microbubble in blood. The Kelvin-Voigt and 4-constant Oldroyd models are used to describe the viscoelastic properties of the shell and the medium, respectively. Analytical expressions are obtained for frequencies, damping rates, and first- and second-harmonic scattering cross-sections of the microbubbles. We also review other biological applications of microbubbles, including drug/gene delivery with the help of microbubbles. The encapsulated bubbles with an average size less than that of a red blood cell are capable of penetrating even the smallest capillaries and releasing drugs and genes, which have been incorporated either within them or on their surfaces, under the action of ultrasound. These microbubbles can transport a specific drug to a specific site within the body (e.g., an anticancer drug to a specific tumor) if their surfaces contain ligands (biotin or antibody) that can bind to receptors (avidin or antigen) situated at the blood vessel walls of the target site.
D.B. Khismatullin, Y.Y. Renardy, and V. Cristini, "Inertia-induced breakup of highly viscous drops in shear flow": 2002 Annual Meeting of AIChE, November 3 - 8, 2002 — Indianapolis, Indiana:
For Stokes flow, a drop undergoing simple shear attains a stationary shape when the drop to matrix viscosity ratio is larger than 3.1. We show that beyond this viscosity ratio, inertia can be used as a mechanism for breakup. Inertia lifts a drop to a higher angle with respect to the horizontal. The drop then experiences higher shear, and breaks even at high drop to matrix viscosity ratios. Conditions for criticality are obtained for viscosity ratio larger than 1. With the addition of inertia, the limiting viscosity ratio moves to higher values. Results at O(1) Reynolds number will be shown. Direct numerical simulations are done for the three-dimensional problem with a volume-of-fluid continuous-surface-force formulation. This research is supported by NSF, ACS-PRF and NCSA.
D.B. Khismatullin and G.A. Truskey, "3D numerical simulation of receptor-mediated leukocyte adhesion to planar surfaces: Effects of cell deformability and viscoelasticity": 4th Colloquium on Cellular and Molecular Biomechanics, September 12 - 13, 2003 — University of Virginia, Charlottesville, Virginia:
Computational fluid dynamics is used to investigate the effects of cell deformability and viscoelasticity on the receptor-mediated leukocyte adhesion to the lower wall in a parallel wall flow chamber. In the developed three-dimensional numerical code, a leukocyte is modeled as a compound viscoelastic drop (a nucleus covered by a thick layer of cytoplasm). The nucleus, cytoplasm, and extracellular fluid are considered as Newtonian or viscoelastic liquids of high viscosity. The receptor-ligand interaction is incorporated into the code by using the spring-peeling kinetic model (Dembo et al 1988) under the assumption that leukocyte receptors are located on the tips of cylindrical microvilli distributed over the leukocyte membrane. Numerical simulations demonstrate the formation and breakup of membrane tethers [observed in vitro by Schmidtke and Diamond (2000)] and suggest that the elasticity of the cytoplasm is responsible for a tear-drop-like shape of rolling leukocytes. The authors thank R. M. Hochmuth, K. F. Ley, R. D. Kamm, and G. W. Schmid-Sch”nbein for stimulating and helpful discussions. The work was supported by the National Center for Supercomputing Applications and National Institutes of Health Grant HL-57446.
D.B. Khismatullin and G.A. Truskey, "3D numerical simulation of receptor-mediated leukocyte adhesion to surfaces: Effects of cell deformability and viscoelasticity": 1st US National Symposium on Frontiers in Biomechanics, September 30 - October 1, 2003 — Nashville, Tennessee:
Computational fluid dynamics is used to investigate the effects of cell deformability and viscoelasticity on leukocyte adhesion to the lower wall in a parallel wall flow chamber. In the developed three-dimensional numerical code, a leukocyte is modeled as a compound viscoelastic drop (a nucleus covered by a thick layer of cytoplasm). The nucleus, cytoplasm, and extracellular fluid are considered as Newtonian or viscoelastic liquids of high viscosity. The receptor-ligand interaction is incorporated into the code by using the spring-peeling kinetic model (Dembo et al 1988) under the assumption that leukocyte receptors are located on the tips of cylindrical microvilli distributed over the leukocyte membrane. Numerical simulations demonstrate the formation and breakup of membrane tethers [observed in vitro by Schmidtke and Diamond (2000)] and suggest that the elasticity of the cytoplasm is responsible for a tear-drop-like shape of rolling leukocytes. Acknowledgements: The author thanks R. M. Hochmuth, K. F. Ley, R. D. Kamm, and G. W. Schmid-Sch”nbein for stimulating and helpful discussions. The work was supported by the National Center for Supercomputing Applications and National Institutes of Health Grant HL-57446.