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Phys Med Biol. Author manuscript; available in PMC 2010 March 30.

Published in final edited form as:

Published online 2006 September 22. doi: 10.1088/0031-9155/51/20/001

PMCID: PMC2847449

NIHMSID: NIHMS168161

Department of Biomedical Engineering, University of California, Davis, CA, 95616, USA

The publisher's final edited version of this article is available at Phys Med Biol

See other articles in PMC that cite the published article.

The existing models of the dynamics of ultrasound contrast agents (UCAs) have largely been focused on an UCA surrounded by an infinite liquid. Preliminary investigations of a microbubble’s oscillation in a rigid tube have been performed using linear perturbation, under the assumption that the tube diameter is significantly larger than UCA size. In the potential application of drug and gene delivery, it may be desirable to fragment the agent shell within small blood vessels and in some cases to rupture the vessel wall, releasing drugs and genes at the site. The effect of a compliant small blood vessel on the UCA’s oscillation and the microvessel’s acoustic response are unknown. The aim of this work is to propose a lumped-parameter model to study the interaction of a microbubble oscillation and compliable microvessels. Numerical results demonstrate that in the presence of UCAs, the transmural pressure through the blood vessel substantially increases and thus the vascular permeability is predicted to be enhanced. For a microbubble within an 8 to 40 micron vessel with a peak negative pressure of 0.1MPa and a center frequency of 1MHz, small changes in the microbubble oscillation frequency and maximum diameter are observed. When the ultrasound pressure increases, strong nonlinear oscillation occurs, with an increased circumferential stress on the vessel. For a compliable vessel with the range of diameters considered in this work, 0.2 MPa PNP at 1 MHz is predicted to be sufficient for microbubble fragmentation regardless the vessel diameter, however, for a rigid vessel 0.5 MPa PNP at 1 MHz may not be sufficient to fragment the bubbles. For a center frequency of 1MHz, a peak negative pressure of 0.5 MPa is predicted to be sufficient to exceed the stress threshold for vascular rupture in a small (diameter less than 15 μm) compliant vessel. As the vessel or surrounding tissue becomes more rigid, the UCA oscillation and vessel dilation decrease, however the circumferential stress is predicted to increase. Decreasing the vessel size or the center frequency increases the circumferential stress. For the two frequencies considered in this work, the circumferential stress does not scale as the inverse of the square root of the acoustic frequency *v _{a}* as in the Mechanical Index, but rather has a stronger frequency dependence, 1/

Ultrasound contrast agents (UCAs) are encapsulated microbubbles and were originally designed to image perfusion by improving the echo amplitude from the blood pool (Blomley et al. 2001; Correas et al. 2001). When driven by ultrasound pulses, the UCAs can generate unique echo signatures resulting from the bubbles’ nonlinear oscillations (Bloch et al. 2004). The applications of UCAs have recently expanded to include targeted imaging, thrombolysis, and drug and gene delivery (Dayton et al. 1999; Hynynen et al. 2001; Bloch et al. 2004; Korpanty et al. 2005; Chappell and Price 2006). In an ultrasound-assisted drug delivery system, the encapsulating shell prolongs the circulation time and the gas core increases the effect of ultrasound on the vehicle. Ultrasound-enhanced drug delivery may be useful in monitoring drug release and decreasing the systemic effects of toxic drugs. The encapsulated microbubbles for drug delivery usually have radii from 0.5 to 5*μm* with a shell-thickness between 10 and 250 nm and hence behave much like gas bubbles. There have been numerous theoretical investigations on the acoustic scattering and the nonlinear dynamics of UCAs in blood (Church 1995; Frinking et al. 1999; Allen et al. 2001; Allen et al. 2002; Hu et al. 2004; Stride and Saffari 2004; Qin et al. 2006).

For ultrasound-enhanced drug delivery, fragmentation of a vehicle also may be important for localizing delivery. Relative expansion cavitation thresholds ranging from 2.32 to 3.463 have been predicted (Apfel 1986). From experimental data, we have observed relative expansion fragmentation thresholds from 1.6 to 3 (Chomas et al. 2001a; Chomas et al. 2001b). While the pulse length required for fragmentation was observed to increase for a thick-shelled delivery vehicle (compared with a lipid-shelled bubble), the relative expansion fragmentation threshold was unchanged (May 2002).

Efforts in modeling the dynamics of UCAs have largely been focused on using various modified Rayleigh-Plesset bubble dynamics equations, of which the cornerstone assumption is that a single UCA is surrounded by an infinite fluid and remains spherical until it collapses. Comparing the theoretical predictions with experimental results demonstrates that the Rayleigh-Plesset equation and the various modified models work remarkably well for the dynamics of cavitation in an unbounded field or in large vessels (Prosperetti 1975; Morgan et al. 2000; Hynynen et al. 2001; Allen et al. 2002). However, this would appear to be a poor description of the conditions in small blood vessels and capillaries which constrain oscillation. Recently, increasing attention has been directed to the effect of small vessels on the microbubble’s oscillation (Yuan et al. 1999; Ory et al. 2000; Sassaroli and Hynynen 2004; Hu et al. 2005; Sassaroli and Hynynen 2005). In these models, the microvessel is usually simplified as a rigid tube and distinct characteristics observed for a microbubble’s oscillation in a tube as compared with oscillation in an unbounded field or in large vessels. Caskey et al (Caskey et al. 2006) have recently experimentally studied the oscillation of microbubbles in microvessel phantoms with diameters similar to those of capillaries. They found that the bubble’s expansion ratio in a rigid capillary phantom is substantially decreased as compared to the prediction of the Raleigh-Plesset model for microbubbles within an infinite liquid. As they noted, a major limitation of such experiments, as well as many other models, is that vessel phantoms are less compliant than true capillaries. An appropriate model including blood vessel compliance is necessary in theoretical and experimental studies (Hynynen et al 2001), particularly for microbubble oscillations in ultrasound-assisted thrombolysis, and drug and gene delivery.

Qin et al (Qin et al 2006) recently studied the asymmetric oscillations of a microbubble confined in compliable microvessels after experiencing a lithotripsy shock wave. In their model, the blood vessel compliance is characterized by a static nonlinear relationship between the intraluminal pressure and the expansion ratio of the vessel radius, which represents the variation of the vessel stiffness with the pressure of the interior liquid. The evolving configuration of the microbubble is pre-assumed to be ellipsoidal and the blood model is simplified as an incompressible inviscid fluid. They found that the oscillation of microbubbles in smaller vessels is more violent and induces substantial intraluminal pressure on the vessel wall.

Experimental studies of the dynamics of microbubbles both *in vitro* and *in vivo* indicate that microbubble oscillation can enhance vascular permeability and even locally damage vasculature (Skyba et al. 1998; May et al. 2002; Li et al. 2004; Hwang et al. 2005; Stieger et al. 2006). It is thus essential to examine the mechanism whereby vascular permeability is enhanced and vascular injuries are produced, so that the appropriate strategies are designed to improve local drug and gene delivery efficiency and to minimize any permanent damage to capillaries (Stieger et al. 2006). Theoretical predictions of bubble oscillation and the microvessel acoustic response can provide insight into this mechanism. In this work, we proposed a lumped-parameter model to study the interaction of UCAs with compliant microvessels. Because the blood viscosity greatly affects a microbubble’s oscillation in small blood vessels and capillaries (Khismatullin 2004), the blood viscosity has been considered in our model. Motivated by *in vitro* and *in vivo* optical observations of the evolving configuration of the microbubbles (Dayton et al. 2001; Chomas et al. 2001a; Patel et al. 2002), we proposed a numerical method to physically determine microbubble configuration by the interaction of the gas bubble and the surrounding liquid. In ultrasound-assisted drug delivery, the large deformation of the capillary wall can cause an additional perivascular reactive radial stress on the exterior vessel wall. The perivascular reactive radial stress of surrounding tissue and the inertial effect of the microvessel and surrounding tissue have also been included.

As shown in figure 1, an initially spherical microbubble lying at the center of a liquid-filled tubular vessel is insonified and vibrates in a compliable microvessel. Mass conservation and the balance of linear momentum for the viscous incompressible liquid within the vessel lead to following equations:

$$\begin{array}{l}\nabla \u2022\mathbf{v}=\mathbf{0}\\ \rho \left(\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}-\widehat{\mathbf{v}})\u2022\nabla \mathbf{v}\right)=-\nabla \u2022\mathbf{\sigma},\end{array}$$

(1)

where the stress tensor σ equals − *p*+*η*(**v**+**v*** ^{T}*). Unless otherwise mentioned, the notation used in this work is summarized in table 1. The gas inside the microbubble is assumed to obey a polytropic law, i.e.,

An illustration of the interaction of a compliable microvessel and an initially spherical microbubble at the center of a liquid-filled tubular microvessel after being insonified.

$${p}_{g}={p}_{ge}{\left(\frac{{V}_{e}}{V}\right)}^{\kappa}.$$

(2)

The boundary condition at the air-liquid interface is

$$-(\mathbf{n}\u2022\mathbf{\sigma}\u2022\mathbf{n})={p}_{g}-\sigma {\kappa}_{m}.$$

(3)

The unit normal vector **n** is defined to be positive from the gas to the liquid. Care is required to calculate the mean curvature *κ _{m}* because it requires computation of the second derivative of the surface. Considering that the ultrasound wavelength is much larger than the microvessel diameter and that the geometry is cylindrically symmetrical, this problem can be simplified. Thus, the mean curvature

$${\kappa}_{m}=\frac{1}{2}\left({\kappa}_{2}+\frac{{n}_{r}}{r}\right).$$

(4)

The following equivalent weak form of the governing equation is used to calculate the mean curvature *κ _{2}* by applying the divergence theorem (Weatherburn C.E. 1972) on the curve

$$\underset{l}{\int}{\kappa}_{2}\sigma \widehat{u}{\mathbf{n}}_{l}dl=-\underset{l}{\int}{\nabla}_{l}(\sigma \widehat{u})dl+\underset{\partial l}{\oint}\mathbf{m}(\sigma \widehat{u})d(\partial l).$$

(5)

The volume variance of the bubble is compensated by the vessel’s deformation and after subtracting the scattered wave from the microbubble, the liquid at the end of the vessel can be considered to be undisturbed, as considered in Sassaroli and Hynynen (2005). Since the acoustic impedances of the microvessel, blood pool and surrounding tissue are similar, the boundary conditions at the microvessel ends are approximated as:

$$p={p}_{i0}+{p}_{s}(t).$$

(6)

We found that a vessel length of 100μm is sufficiently long to obtain precise results within the typical computational interval of 5μs. If the computational time is long and the vessel short, the generated wave within the vessel could propagate to the vessel ends.

Based on previous measurements of the static response of a vessel, the microvessel compliance is described by a nonlinear relation between the transmural pressure of the vessel wall and the expansion ratio of vessel radius,

$${p}_{i}-{p}_{a}={\sigma}_{0}\frac{{e}^{\beta (\delta -1)}-1}{\gamma {\delta}^{2}-1/2},$$

(7)

where *σ _{0}* and

Dilation ratio of a microvessel as a function of transmural pressure for the experimental data of a exteriorized frog mesenteric capillary (Swayne G.T.G 1984) and the prediction of the model in equation (7) with *σ*_{0}= 4101 (2*k+*1) Pa, *β*=15.35( **...**

$${\rho}_{c}{h}_{c}{\ddot{r}}_{i}+{\rho}_{t}{h}_{t}{\ddot{r}}_{i}={p}_{i}-{p}_{a}-{\sigma}_{0}\frac{{e}^{\beta (\delta -1)}-1}{\gamma {\delta}^{2}-1/2}-Y\xi (1-{\nu}^{2}){E}_{\theta}({R}_{a},t),$$

(8)

where the cap double dot means the second derivative with respect to time *t*. The first term on the left-hand side of equation (8) is the inertial stress of the blood vessel; the second term on the left-hand side of equation (8) is the effective inertial stress of surrounding tissue. The term *p _{i}* −

$$\begin{array}{l}{\sigma}_{0}=4101(2k+1)Pa\\ \beta =15.35(k+1)\\ Y=8\times {10}^{5}(2k+1)Pa\end{array}$$

(9)

where *k* = 0 corresponds to the values previously measured for mesentery. The mapping of blood vessel particles during deformation can be described as

$$r=r(R,Z),\theta =\mathrm{\Theta},z=\lambda Z,$$

(10)

which leads to the incompressibility condition:

$$r=\sqrt{{r}_{i}^{2}+({R}^{2}-{R}_{i}^{2})/\lambda}.$$

(11)

With the use of equation (8) as one boundary equation, it is difficult to get numerically convergent solutions. Using equation (11), the following equivalent equation is obtained facilitating convergent solutions for the coupling boundary conditions between the microvessel and intravascular liquid:

$$({\rho}_{c}{h}_{c}+{\rho}_{t}{h}_{t})\upsilon \stackrel{.}{\zeta}=-({\rho}_{c}{h}_{c}+{\rho}_{t}{h}_{t}){\upsilon}^{\prime}{\zeta}^{2}+{p}_{i}-{p}_{a}+{p}_{\delta}+{p}_{t}$$

(12)

where the dot indicates the first derivative with respect to time, *t*. Other terms are defined as follows in (13).

$$\begin{array}{l}\delta ={\scriptstyle \frac{{r}_{m}}{{R}_{m}}}={\scriptstyle \frac{{r}_{i}+\sqrt{{r}_{i}^{2}+\left({R}_{a}^{2}-{R}_{i}^{2}\right)/\lambda}}{{R}_{a}+{R}_{i}}}\\ \zeta =\stackrel{.}{\delta}\\ \upsilon =\frac{{R}_{a}+{R}_{i}}{1+{r}_{i}/\sqrt{{r}_{i}^{2}+\left({R}_{a}^{2}-{R}_{i}^{2}\right)/\lambda}}\\ {\upsilon}^{\prime}=-\frac{({R}_{a}+{R}_{i})({R}_{a}^{2}-{R}_{i}^{2})}{\lambda {\left({r}_{i}+\sqrt{{r}_{i}^{2}+\left({R}_{a}^{2}-{R}_{i}^{2}\right)/\lambda}\right)}^{2}\sqrt{{r}_{i}^{2}+\left({R}_{a}^{2}-{R}_{i}^{2}\right)/\lambda}}\upsilon \\ {p}_{\delta}=-{\sigma}_{0}\frac{{e}^{\beta (\delta -1)}-1}{\gamma {\delta}^{2}-1/2}\\ {p}_{t}=-Y(1-{\nu}^{2})({h}_{t}/{R}_{a}){E}_{\theta}({R}_{m},t)\end{array}$$

(13)

During computation, *δ* in equation (12) is discretized as a function of coordinates as represented in the first expression in equation (13), and the nondimensional parameter *ζ* is treated as an independent variable. The normal velocity of the liquid and microvessel inner wall must be the same:

$$\begin{array}{l}{v}_{r}={\stackrel{.}{r}}_{i}=\zeta \upsilon \\ {v}_{z}=\zeta \upsilon \frac{{{n}_{z}}^{w}}{{{n}_{r}}^{w}}\end{array}$$

(14)

The unit normal vector (*n _{r}^{w}*,

When the value of *k* in equation (9) and the value of *h _{t}* in equation (12) are extremely large, a compliable vessel will reduce to a rigid wall, the coupling equation (12) will be diminished, and the boundary condition in equation (14) will be replaced by(

The effect of the vessel diameter on the radial oscillation and transmural pressure are first examined for compliant vessels within a low intensity ultrasound field, for which a comparison to the Rayleigh-Plesset equation can be made. With a low transmission pressure ultrasound field with a peak negative pressure (PNP) of 0.1 MPa and center frequency of 1 MHz, a microbubble’s oscillation in a 40*μm* diameter vessel is consistently in agreement with that predicted by the Rayleigh-Plesset equation (figure 3). However, with this low transmission pressure, the oscillation amplitude is decreased within an 8*μm* vessel as compared with the result within a 40*μm* vessel. Specifically, the maximum oscillation amplitude is decreased by 3.3% within the smaller vessel and the oscillation amplitude following the driving cycle (after 1 μs in figure 3) is reduced by 67.8%. Under these conditions, the microbubble’s oscillation frequency increases 16.1% from 2.73 MHz in a 40 *μm* vessel to 3.17 MHz in an 8*μm* vessel.

Comparison of expansion ratio of mean size (diameter) of a microbubble with *d*_{0}=3*μm* (*k*=0.0) confined within a microvessel of *D*_{0}=8*μm* and *D*_{0}=40*μm*, as a function of time *t* in an ultrasound field with PNP of 0.1MPa and center frequency **...**

For the 8*μm* diameter vessel without UCAs, the 0.1 MPa ultrasound pressure wave produces slight fluctuations in the transmural pressure with respect to the initial value of *20 mm Hg* (figure 4), however, the presence of UCAs can substantially increase the transmural pressure. For these parameters, the transmural pressure in the 8*μm* diameter vessel can be increased up to 7.6 times as compared with a 40*μm* vessel.

Comparison of the normalized transmural pressure at (*z=0*) as a function of time *t* in an ultrasound field with PNP of 0.1MPa and center frequency of 1MHz for a microbubble of *d*_{0}= 3*μm* (*k*=0.0) confined a microvessel of *D*_{0}=8*μm* and *D*_{0}=40*μm* **...**

Next, the asymmetric oscillation of a microbubble in a small vessel is demonstrated for a higher ultrasound pressure of 0.5 MPa. As the negative ultrasound pressure arrives in the region of interest, the microbubble and microvessel expand from initial vessel diameters of 8 and 20 microns (figures 5 and and6,6, respectively). The microvessel’s dilation reaches its maximum value at *t*= 0.32*μs* and *t*= 0.64*μs* for the 8 and 20*μm* diameter vessels, respectively while the microbubble reaches its maximum volume at 0.35 and 0.64*μs*, respectively. The asymmetrical oscillation of the microbubble occurs primarily during the process of contraction, (figure 5(d) and figure 6(d)). The peak circumferential stress travels along the vessel axis after peak expansion. The corresponding dilation frequency of the microvessel increases 100% from 0.39 MHz for the 20*μm* vessel to 0.78 MHz for the 8*μm* vessel. In comparison with figure 3, the microbubble’s oscillation frequency in figures 5 and and66 decreases 77.6% from 3.17 MHz to 0.71 MHz for the 8*μm* vessel and 85.7% from 2.73 MHz to 0.39 MHz for the 20*μm* vessel.

Intravascular pressure distribution and the deformation of the contrast agent of *d*_{0}= 3*μm* in a microvessel of *D*_{0}=8*μm* in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz at: (a) t=0.02*μs* ; (b) t=0.26*μs* **...**

Intravascular pressure distribution and the deformation of the contrast agent of *d*_{0}=3 *μm* in a microvessel of *D*_{0}=20 *μm* in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz at: (a) t=0.02*μs* ; (b) t=0.50*μs* **...**

The effect of ultrasound parameters on the oscillation of the microbubble and microvessel is next summarized for multiple vessel diameters and PNP values in figures 7, ,88 and and9.9. With a center frequency of 1 MHz and low transmission pressure 0.1MPa, the expansion ratio of the microbubble is equal to or slightly less than the values predicted by Rayleigh-Plesset equation, i.e.1.364 and 1.352, for the 40 and 8*μm* vessels, respectively. Similarly, for the 40*μm* vessel, the expansion ratio is similar to that predicted by Rayleigh-Plesset analysis in an unbounded medium for each pressure. For a transmitted PNP of 0.5MPa, the microbubble expansion is greatest near a ratio of *d*_{0}/*D*_{0} equal to 3/15, decreasing as the vessel diameter increases or decreases.

Maximum expansion ratio of the mean size (diameter) of the microbubble as a function of *d*_{0}/*D*_{0} for different pressure amplitudes and the expansion ratios predicted by the Rayleigh-Plesset equation at the corresponding pressures, shown at *d*_{0}/*D*_{0}= 3/40.

Maximum dilation ratio of the microvessel as a function of *d*_{0}/*D*_{0}for different pressure amplitudes.

Maximum induced circumferential stress in the vessel wall as a function of *d*_{0}/*D*_{0}for different pressure amplitudes.

The dilation ratio of the microvessel increases with an increase in *d*_{0}/*D*_{0} reaching a maximum near 1.6 for the PNP of 0.5 MPa (figure 8). The same trend holds for the induced circumferential stress in the vessel wall (figure 9). The increase of the circumferential stress with the increase of *d*_{0}/*D*_{0} in figure 9 indicates that for the same ultrasound parameters, small vessels experience a higher stress than large vessels. Figure 9 also shows that that the transmission pressure plays a vital role; the higher transmission pressure, the higher the circumferential stress. The vascular strength predicted by the literature is 0.80 MPa for microvessels such as those of interest in this work (Di Martino et al. 2006) and therefore 0.5 MPa 1 MHz ultrasound will induce a circumferential stress exceeding this value in vessels with a diameter between 8 and 15*μm*. For a PNP of 0.2MPa and center frequency of 1MHz, the circumferential stress is below the ultimate vascular strength for all values of *d*_{0}/*D*_{0} considered here.

Prior to evaluating the oscillation of a microbubble in vessels with varied compliance, the case of a very rigid vessel modeled by *k=100* is compared with that for a bubble near a rigid boundary (as described in the methods section) in figure 10. Both the predictions of the bubble’s expansion and induced pressure are in excellent agreement.

Comparison of the results in a vessel with *k*=100, *h*_{t}= 4*cm* and a rigid vessel for a microbubble of *d*_{0}=3*μm* confined within a microvessel of *D*_{0}= 8*μm* in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz: (a) the expansion **...**

The typical intravascular pressure distribution and the deformation of the contrast agent in vessels with a decreased compliance (*k*=0.5) show that the vessel’s dilation frequency increases 14.1% from 0.78 MHz for a compliant (*k*=0.0) vessel (figure 5) to 0.89 MHz for a decreased vessel compliance (*k*=0.5) (figure 11). The corresponding expansion ratio of the microbubble and dilation of the microvessel are decreased with an increase in the stiffness (figures 12 and and13,13, respectively). At the limit of a rigid vessel, the microbubble expansion ratios are 1.6 for an 8*μm* vessel, 2.2 for a 15*μm* vessel and 2.6 for 20*μm* vessel. The induced circumferential stress predicted by the model increases with the increase in the stiffness coefficient *k* (figure 14). In the compliant vessel *(k=0)* for transmission of 0.5MPa with a center frequency of 2MHz. the induced circumferential stress is 0.026 MPa, increasing to 0.125 for *k=0.7*, in each case far below the predicted ultimate vascular strength. Similar values are obtained for a center frequency of 1 MHz with a PNP of 0.25 MPa.

Intravascular pressure distribution and the deformation of the contrast agent of *d*_{0}=3*μm* in a microvessel of *D*_{0}=8*μm* in a less compliant vessel, *k*=0.5, in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz at: (a) t=0.02 **...**

Maximum expansion ratio of the mean size (diameter) of the microbubble of *d*_{0}=3*μm* as a function of nondimensional rigidity parameter *k* for in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz. The length of the rigid tube is 200 **...**

Maximum dilation ratio of the microvessel as a function of nondimensional parameter *k* for *d*_{0}=3*μm* in an ultrasound field with PNP of 0.5MPa and center frequency of 1MHz.

Maximum induced circumferential stress in the vessel wall as a function of nondimensional parameter *k* for a microbubble of *d*_{0}=3*μm* within a vessel of *D*_{0}=15*μm*.

In order to evaluate the dependence of the critical circumferential stress on the ultrasound transmission pressure and the center frequency, the stress was calculated and compared for three cases over a range of values of vascular compliance (figure 14). For example, we found that for two equivalent Mechanical Indices (MI, which is defined by
$(\mathit{PNP}/\mathit{MPa})/\sqrt{{\nu}_{a}/\mathit{MHz}})$) with 0≤ *k*≤ 0.7, the circumferential stress differs. For a PNP of 0.35MPa and center frequency of 1MHz, the circumferential stress in a 15 *μm* vessel (*k=0*) is 0.221MPa, however, for a PNP of 0.5MPa and center frequency of 2MHz the calculated stress in the vessel (*k=0*) is 0.026 MPa. Comparing two values with a constant ratio of
$(\mathit{PNP}/\mathit{MPa})/({\nu}_{a}/\mathit{MHz})$ (PNP of 0.5 MPa and center frequency of 2 MHz versus PNP of 0.25 and center frequency of 1 MHz), the calculated circumferential stress is similar (figure 14).

If the viscous damping is neglected, the microbubble’s natural frequency in an unbounded liquid is given by

$${f}_{0}=\frac{1}{2\pi}\sqrt{\frac{3\kappa ({p}_{0}+2\sigma /({d}_{0}/2))}{\rho {({d}_{0}/2)}^{2}}-\frac{2\sigma}{\rho {({d}_{0}/2)}^{3}}},$$

(15)

which leads to 2.86MHz for the 3 micron diameter agent considered here. The calculated oscillation frequency of the microbubble in a 40 *μm* vessel excited with a center frequency 1 MHz and PNP of 0.1 MPa is 2.73 MHz, according to figure 3. This implies that microbubble’s oscillation after insonation is harmonic with a low transmission pressure. For a higher PNP pressure of 0.5 MPa, the bubble’s oscillation frequency substantially decreases, as shown in figure 5 and and6.6. This nonlinear oscillation depends on the ultrasound pressure and frequency, and the boundaries in the vicinity of the bubbles (Flynn 1975; Oguz and Prosperetti 1998).

The microbubble’s oscillation amplitude decreases in small vessels, as shown in figure 3, consistent with the recent observation *in vitro* by Caskey et al (2006) that a microbubble’s expansion is reduced in small rigid vessels. A substantial decrease in oscillation amplitude at low transmission pressure could be caused by two effects. If the transmission pressure is low, the microvessel wall forms a strong constraint, represented by the small dilation of the vessel with the 0.1MPa 1MHz ultrasound in figure 8. The vessel wall therefore decreases the oscillation of the microbubble, which is shown by the decreased maximum oscillation amplitude in the small vessel (figure 3 and the 0.1MPa 1MHz curve in figure 7). Also, the liquid viscosity has a larger effect on the bubble’s oscillation (viscous damping) in the small vessel than in a large vessel. Thus the bubble’s oscillation amplitude in a small vessel decreases substantially after several cycles (figure 3). When the transmission pressure increases, the transmitted ultrasound wave forces the vessel wall to vibrate, as represented by the relatively large dilation of the vessel with the 0.2MPa 1MHz ultrasound in figure 8. Finally, when the transmission pressure increases further (0.5MPa 1MHz ultrasound) and the vessel’s dilation is correspondingly large, the nonlinear response of the blood vessel (as described in equation (7) and shown in figure 2) and the nonlinear response of the surrounding tissue (as described in the last relation in equation (13)) prevent the vessel from further dilating and thus decrease the bubble’s expansion ratio in the small vessel in figure 7.

Based on a relative expansion-fragmentation threshold of approximately 3 (Apfel 1986; Chomas et al. 2001a; Chomas et al. 2001b), for the compliant vessel represented by *k=0* and the range of diameters considered here, the relative expansion would result in fragmentation for UCAs insonified with a center frequency of 1 MHz and PNP of 0.2 MPa. For the lower pressure of 0.1 MPa, the smaller relative expansion is unlikely to produce fragmentation. As k increases, however, the relative expansion of the bubble decreases, and within a rigid vessel a PNP of 0.5 MPa may be insufficient to fragment the vehicle within the range of vessel diameters considered. This may have important consequences for drug delivery in tumors, for which the tissue stiffness and interstitial pressure may be very high.

The flux of water across the microvessel wall can be described by the classic Starling Equation as *J _{v}*=

(Stieger et al. 2006) demonstrates convective transport of fluorescent dextran through small gaps in the vascular wall in a manner consistent with this prediction, and with a pressure of 0.5 MPa for 1 MHz transmission. Extravasation of blood cells and UCAs into the interstitium has been demonstrated when tissue and UCA are exposed to MHz-frequency pulsed ultrasound (Price et al. 1998; Hwang et al. 2005; Stieger et al. 2006). In Stieger et al, the diameter of the vessels affected increases with increasing PNP, and this is also predicted by our analysis for vessels with a diameter of 8 to 20 microns and a transmission center frequency of 1 MHz.

The frequency dependence of the maximum circumferential stress does not scale with the MI in this study, instead decreasing the center frequency has a greater impact on the resulting stress than that predicted by MI. In other studies, the frequency dependence of vascular effects also was not predicted by the MI (Forsberg et al. 2006; Stieger et al. 2006). For the parameters employed here, the ratio $(\mathit{PNP}/\mathit{MPa})/({\nu}_{a}/\mathit{MHz})$ closely estimated the change in stress with frequency. Additional studies are required to determine whether this ratio is useful for the prediction of vascular damage over a range of parameters, tissues, or models.

For large vessels, the induced circumferential stress in the vessel wall is usually far below the vascular ultimate strength. The collapse of UCAs can generate a liquid jet and radiate shock waves which could damage endothelial cells (Philipp and Lauterborn 1998; Chomas et al. 2001b; Postema et al. 2004). The results presented here do not refute the long-standing hypothesis that jets and shock waves are likely mechanisms for enhancement of vascular permeability and endothelial cell damage in large vessels since the transmural pressure increase across the vessel wall in the presence of UCAs and the induced stress in the vessel are low.

Vascular compliance across a wide range of vessel sizes and tissue types has not been thoroughly characterized. The compliable vessels studied in this work (*γ =* 20.5, *h _{t}* = 0.1

In order to locally deliver shell material from a microbubble to the vessel wall in a predictable fashion, the oscillation of microbubbles in small compliant vessels must be characterized. Two fold or greater relative expansion is expected to be required to produce microbubble and shell rupture and free the trapped drug. Substantially greater microbubble oscillation may produce vascular rupture (ultimately enhancing delivery to the interstitium) as circumferential stress increases or rapid collapse occurs. In this work, we proposed a lumped-parameter model to study the interaction of microbubble oscillation and compliable microvessels. The blood viscosity has been included, and the microbubble configuration is physically determined by the interaction of the gas bubble and the surrounding liquid. The perivascular reactive radial stress of surrounding tissue and the inertial effect of the microvessel and surrounding tissue have also been included.

Numerical results demonstrate that in the presence of UCAs, the transmural pressure substantially increases. With a center frequency of 1 MHz and PNP of 0.1 MPa, comparing the oscillation of a microbubble within 8 and 40 micron vessels, small changes in the microbubble oscillation frequency and maximum diameter are observed. When the transmission pressure is large (0.5MPa PNP at 1MHz), the microbubble’s expansion changes with the ratio of microbubble and microvessel initial diameter and decreases substantially in a rigid vessel. For a compliable vessel, 0.2 MPa PNP at 1 MHz is predicted to be sufficient for microbubble fragmentation within the range of vessel diameters considered, however, for a rigid vessel 0.5 MPa PNP at 1 MHz may not be sufficient for the vessels considered here. Additional studies are required to compare the fragmentation threshold observed *in vivo* with that previously measured *in vitro*.

The circumferential stress and the likelihood of vessel rupture decrease with increasing vessel initial diameter. As the transmission pressure increases, the microvessel dilation and circumferential stress increase. Based on previously published values for microvessel strength, for a center frequency of 1MHz, 0.5MPa is predicted to be sufficient for rupture of a small compliable vessel (*D*_{0} ≤ 15*μm*). The frequency dependence of the circumferential stress does not scale according to the MI in this study. Instead, in this preliminary analysis, the stress is similar for two frequencies producing an equal ratio of
$(\mathit{PNP}/\mathit{MPa})/({\nu}_{a}/\mathit{MHz})$ over a range of vessel compliance.

The support of NIH CA 76062 and CA 103828 are gratefully appreciated.

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