Modulation of the interstitial fluid pressure by high intensity focused ultrasound as a way to alter local fluid and solute movement: insights from a mathematical model.

High intensity focused ultrasound (HIFU) operated in thermal mode has been reported to reduce interstitial fluid pressure and improve the penetration of large macromolecules and nanoparticles in tumor and normal tissue. Little is understood about how the interstitial fluid pressure and velocity as well as the interstitial macromolecule transport are affected by HIFU exposure. A mathematical model is presented here which sheds light on the initial biophysical changes brought about HIFU. Our continuum model treats tissue as an effective poro-elastic material that reacts to elevated temperatures with a rapid drop in interstitial elastic modulus. Using parameters from the literature, the model is extrapolated to derive information on the effect in tumors, and to predict its impact on the convective and diffusive transport of macromolecular drugs. The model is first solved using an analytical approximation with step-wise changes at each boundary, and then solved numerically starting from a Gaussian beam approximation of the ultrasound treatment. Our results indicate that HIFU causes a rapid drop in interstitial fluid pressure that may be exploited to facilitate convection of macromolecules from vasculature to the exposed region. However, following a short recovery period in which the interstitial fluid pressure is normalized, transport returns to normal and the advantages disappear over time. The results indicate that this effect is strongest for the delivery of large molecules and nanoparticles that are in the circulation at the time of treatment. The model may be easily applied to more complex situations involving effects on vascular permeability and diffusion.

Age estimation by pulp/tooth ratio in lateral and central incisors by peri-apical X-ray.

Since 2004, several papers on the analysis of the apposition of secondary dentine have been published. The aim of this paper was to study a sample of peri-apical X-ray images of upper and lower incisors, both lateral and medial, to examine the application of pulp/tooth area ratio as an indicator of age. A sample of 116 individuals, 62 men and 54 women, aged between 18 and 74 years, was studied. Data were fitted with age as a linear function of the pulp/tooth ratio of incisors. The total variance explained by the regression equation ranged from 51.3% of age, when lower lateral incisors were used as explanatory variable, to 81.6% when upper lateral incisors were used. The accuracy of the corresponding regression model yielded ME = 8.44 and 5.34 years, respectively. These results show that, although incisors are less reliable than canines or lower premolars, they can be used to estimate age-at-death when the latter are absent.

Modeling focused ultrasound exposure for the optimal control of thermal dose distribution.

Preclinical studies indicate that focused ultrasound at exposure conditions close to the threshold for thermal damage can increase drug delivery at the focal region. Although these results are promising, the optimal control of temperature still remains a challenge. To address this issue, computer-simulated ultrasound treatments have been performed. When the treatments are delivered without taking into account the cooling effect exerted by the blood flow, the resulting thermal dose is highly variable with regions of thermal damage, regions of underdosage close to the vessels, and areas in between these two extremes. When the power deposition is adjusted so that the peak thermal dose remains close to the threshold for thermal damage, the thermal dose is more uniformly distributed but under-dosage is still visible around the thermally significant vessels. The results of these simulations suggest that, for focused ultrasound, as for other delivery methods, the only way to control temperature is to adjust the average energy deposition to compensate for the presence of thermally significant vessels in the target area. By doing this, we have shown that it is possible to reduce the temperature heterogeneity observed in focused ultrasound thermal treatments.

Linear behavior of a preformed microbubble containing light absorbing nanoparticles: insight from a mathematical model.

Microbubbles are used as ultrasonic contrast agents in medical imaging because of their highly efficient scattering properties. Gold nanoparticles absorb specific wavelengths of optical radiation very effectively with the subsequent generation of thermo-acoustic waves in the surrounding medium. A theoretical and numerical analysis of the possibility of inducing radial oscillations in a pre-existing spherical microbubble, through the laser excitation of gold nanoparticles contained within, is presented. A description of such a system can be obtained in terms of a confined two-phase model, with the nanoparticles suspended in a confined region of gas, surrounded by a liquid. The Rayleigh-Plesset equation is assumed to be valid at the boundary between the gas and the liquid. The confined two-phase model is solved in linear approximation. The system is diagonalized and the general solution is obtained. This solution is in the form of exponentially decaying oscillatory functions for the temperature and pressure inside the bubble, and radial oscillations of the bubble boundary. It was found that, for the right size of bubbles, the oscillatory behavior takes place in the low megahertz range, which is ideal for medical applications. This study suggests the possibility of new applications of microbubbles in photoacoustic imaging.

Age estimation by pulp/tooth area ratio in canines: study of a Portuguese sample to test Cameriere's method.

Age estimation in adults is an important problem in both anthropological and forensic fields, and apposition of secondary dentine is often used as an indicator of age. In recent papers, Cameriere et al. studied the pulp/tooth area ratio of canines for this purpose. The present study examines the application of the pulp/tooth area ratio by peri-apical X-ray images as an age indicator in a Portuguese identified sample. The statistical model was then compared with results from an Italian identified sample, to establish whether a common regression model for both samples could be developed. The Portuguese sample consisted of 126 canines of male and 132 of female from subjects 20 to 84 years old, from the osteological collection of the Museum of Anthropology at Coimbra University. The Italian sample consisted of 114 canines of male and 86 of female from subjects 20 to 79 years old, analyzed in Cameriere et al. (2007), and came from the Frassetto osteological collection of Sassari (Sardinia), now housed in the Museum of Anthropology, Department of Experimental and Evolutionistic Biology, University of Bologna. Statistical analysis was performed in order to obtain multiple regression formulas for dental age calculation, with chronological age as dependent variable, and gender and pulp/tooth area ratio on upper (RA(u)) and lower canines (RA(l)) as independent variables. ANCOVA analysis showed that gender was not significant but that variables RA(u) and RA(l) were. The regression model for the Portuguese sample yielded the following equations: Age=101.3-556.68 RA(u) (upper canines) and Age=92.37-492.05 RA(l) (lower canines). Both models explained about 97% of total variance, and mean prediction errors were ME=2.37 years and 2.55 years, respectively. Comparisons between the equation referring to the Portuguese sample and the equivalent linear equations proposed by Cameriere et al. for the Italian sample did not reveal significant differences between the linear models, suggesting that a common regression model could be applied for both samples. The common regression model, describing age as a linear function of RA(u) and RA(l), yielded the following linear regression formulas: Age=100.598-544.433 RA(u); Age=91.362-480.901 RA(l), and explained 86% and 93% of total variance, respectively. Mean prediction errors were ME=2.68 years and 2.73 years, respectively.

Numerical investigation of heating of a gold nanoparticle and the surrounding microenvironment by nanosecond laser pulses for nanomedicine applications.

We have modeled, by finite element analysis, the process of heating of a spherical gold nanoparticle by nanosecond laser pulses and of heat transfer between the particle and the surrounding medium, with no mass transfer. In our analysis, we have included thermal conductivity changes, vapor formation, and changes of the dielectric properties as a function of temperature. We have shown that such changes significantly affect the temperature reached by the particle and surrounding microenvironment and therefore the thermal and dielectric properties of the medium need to be known for a correct determination of the temperature elevation. We have shown that for sufficiently low intensity and long pulses, it is possible to establish a quasi-steady temperature profile in the medium with no vapor formation. As the intensity is increased, a phase-change with vapor formation takes place around the gold nanoparticle. As phase-transition starts, an additional increase in the intensity does not significantly increase the temperature of the gold nanoparticle and surrounding environment. The temperature starts to rise again above a given intensity threshold which is particle and environment dependent. The aim of this study is to provide useful insights for the development of molecular targeting of gold nanoparticles for applications such as remote drug release of therapeutics and photothermal cancer therapy.

Resonance frequency of microbubbles in small blood vessels: a numerical study.

Microbubbles are currently used as ultrasound contrast agents. Their potential therapeutic applications are also under investigation. This work is designed to provide some insight into the mechanisms of energy absorption and deposition by a preformed gas bubble in the microvasculature to optimize its efficacy. In the linear regime, the most favourable condition for the transfer of energy from an ultrasonic field to a gas bubble occurs when the centre frequency of the ultrasonic field equals the resonance frequency of the bubble. The resonance frequency of gas microbubbles has been investigated up to now mainly in unbounded liquids; however when bubbles are confined in small regions, their resonance frequency is strongly affected by the surrounding boundaries. A parametric study on how the resonance frequency of microbubbles in blood vessels is affected by the bubble radius, vessel radius and the bubble position in the vessel is presented. The resonance frequency decreases below its free value with decreasing vessel radius for vessels smaller than 200-300 microm depending on the bubble size. This model suggests the possibility of using ultrasound in a range of frequencies that are, in general, lower than the ones used now for therapeutic and diagnostic applications of ultrasound (a few MHz). When microbubbles oscillate at their resonance frequency they absorb and therefore emit more energy. This energy may allow specific blood vessels to be targeted for both diagnostic and therapeutic applications of ultrasound.

Forced linear oscillations of microbubbles in blood capillaries.

A theoretical investigation of the forced linear oscillations of a gas microbubble in a blood capillary, whose radius is comparable in size to the bubble radius is presented. The natural frequency of oscillation, the thermal and viscous damping coefficients, the amplitude resonance, the energy resonance, as well as the average energy absorbed by the system, bubble plus vessel, have been computed for different kinds of gas microbubbles, containing air, octafluropropane, and perflurobutane as a function of the bubble radius and applied frequency. It has been found that the bubble behavior is isothermal at low frequencies and for small bubbles and between isothermal and adiabatic for larger bubbles and higher frequencies, with the viscous damping dominating over the thermal damping. Furthermore, the width of the energy resonance is strongly dependent on the bubble size and the natural frequency of oscillation is affected by the presence of the vessel wall and position of the bubble in the vessel. Therefore, the presence of the blood vessel affects the way in which the bubble absorbs energy from the ultrasonic field. The motivation of this study lies in the possibility of using gas microbubbles as an aid to therapeutic focused ultrasound treatments.