Finite-element Modeling of Elastic Surface Modes and Scattering from Spherical Objects

A finite-element model of wave propagation using COMSOL Multiphysics (COMSOL Inc., Burlington, MA) was developed to solve the problem of high frequency ultrasound scattering from spheres. This model is used to predict ultrasound backscatter from cells for ultrasound tissue characterization. In this work, the backscatter from an elastic sphere was used to validate the computational model against analytical solutions (Faran theory). Agreements between analytical and finite element solutions were found in the scattered far-field over a range of frequencies of interest (10 70 MHz). Oscillations of the elastic sphere at various resonance frequencies (peaks in the power spectrum) were also investigated. The resonance frequencies predicted by the analytical solutions corresponded to surface modes. A systematic relationship between the resonance frequency and its corresponding surface mode was found. The oscillations of the elastic sphere were visualized at these resonances. An ultrasound scattering model by a single cell is also presented. The model treats the cell as an elastic sphere (nucleus) surrounded by a fluid shell (cytoplasm). Comparison of the theoretical backscatter predicted by the model and experimental measurements for Acute Myeloid Leukemia (AML) cell is also shown. Finally, the implications of these results on the prediction of ultrasound backscatter from cells, and on ultrasound tissue characterization techniques are discussed.


Introduction
It has been shown that high frequency ultrasound (20MHz -60MHz) can be used to detect structural and physical changes in cell ensembles during cell death (Sherar et al. 1987), including apoptosis (Czarnota et al. 1997;Czarnota et al. 1999;Kolios et al. 2004). Ultrasonic backscatter from cell ensembles treated with the chemotherapeutic cisplatin, which induces apoptosis, increased the ultrasound backscatter by 9-13dB resulting in much brighter images. However, the precise physical mechanism that causes this increase is not known.
A theoretical model of acoustic wave scattering is required to better understand scattering from cells, which would then allow us to determine the proportion of cells undergoing apoptosis in a given ensemble. The effect of the morphological changes during apoptosis on ultrasound backscatter cannot be modeled using analytical methods. Most scattering models used for lower frequency ultrasonic tissue characterization assume a random distribution of scatterers (Lizzi et al. 1983;Insana et al. 1990;Oelze et al. 2002). These models have been used to diagnose various tissue pathologic states (Lizzi et al. 1988;Lizzi et al. 1997;Ursea et al. 1998;Feleppa et al. 2000). However, they do not take into account the elastic nature of cells, which may be significant at higher frequencies.
There is evidence of an elastic nature to cells with a variation in elasticity within the cell (Caille et al. 2002;Zinin et al. 2005). For example, the cell nucleus has been shown to be stiffer (Tseng et al. 2004) and more viscous (Guilak et al. 2000) than the cytoplasm.
Moreover, there is an increasing interest in the modeling of ultrasound contrast agents' oscillations, collapse, and scattering in ultrasound fields for a better understanding of contrast agent imaging (Stride et al. 2003). Current models used cannot easily take into account features such as non-radial bubble oscillations due to inhomogeneity and the interactions of bubbles with their surroundings.
Analytical solutions to the problem of wave scattering from simple, isotropic, and homogeneous three dimensional structures such as spheres have been studied extensively in the past (Anderson 1950;Faran 1951;Hickling 1962). However, these formulations are not flexible enough to be extended to wave scattering from complex scatterers such as cells which exhibit irregular shape and heterogeneous composition. Baddour et al. (2005) made the first attempt to build such a model for OCI-AML-5 (Acute Myeloid Leukemia) (Wang et al. 1991) cells based on Faran's formulation (Faran 1951) by assuming that scattering occurs from the nucleus of a cell. They assumed that the nucleus was homogenous and spherical. The analytical model failed to accurately predict the experimentally measured backscatter from cells. Therefore, a more complete cell model which incorporates the mechanical properties of various cell components is needed.
In our previous work (Falou et al. 2005), a finite-element model of acoustic wave propagation was developed to solve the problem of acoustic scattering from spheres. The model was shown to accurately predict near-field scattering.
In this work, a comparison of finite-element and analytical solutions in the far-field is given. Moreover, a study of surface modes exhibited by an elastic microsphere subject to various frequencies is presented. Finally, a scattering model of a single cell which treats the cell as an elastic nucleus surrounded by a fluid cytoplasm is shown and compared with experimental measurements.

Far-field Backscatter Calculation
Given the pressure and its normal derivative on a spherical surface surrounding the scatterer, the Helmholtz Integral equation can be used to obtain the far-field backscattered pressure. The Helmholtz Integral equation (Junger et al. 1993) is given by: where, R and R 0 are the position vectors locating field and source points, respectively.
is the outward pointing unit vector normal to the radiating spherical surface, is the density of the medium, and is the normal component of acceleration at the surface S 0 . g is the free-space Green's function defined as:  Table 1 gives the physical properties of the polystyrene microsphere.

Surface Modes of a Polystyrene Microsphere
The BSTF versus frequency for a 43 micron polystyrene microsphere was plotted to determine the microsphere resonance frequencies corresponding to peaks in the plot. Oscillations representing surface modes of the microsphere at various resonance frequencies were then visualized by plotting particle's displacement at the sphere's surface using COMSOL Multiphysics 3.3a (COMSOL Inc., Burlington, MA).

Proposed Cell Model
This section presents an ultrasound scattering model by a single AML cell. The model treats the cell as an elastic sphere (nucleus) surrounded by a fluid shell (cytoplasm). Table 2 lists the physical properties of the cell used in the simulations.
Comparison of the theoretical backscatter predicted by the model and experimental measurements (Baddour et al. 2007) for Acute Myeloid Leukemia (AML) cell is also shown.    Figure 3 depicts the deformations of the polystyrene microsphere immersed in PBS at 5 resonance frequencies which corresponded to surface modes often represented by an integer n, where n = 0 corresponds to the radial oscillation, n = 2 to an ellipsoid, etc. It can be seen that there is a systematic relationship between the resonance frequency and its corresponding surface mode. The higher the resonance frequency, the more complex the oscillation pattern, and hence, the higher the mode number. Knowledge of such deformations is important since it may help in understanding the process of bubble rupture in the study of ultrasound contrast agents. Figure 4 compares the theoretical BSTF versus frequency predicted by the finite-element model with experimental measurements for AML cells. It is evident that the proposed model does not accurately predict the backscatter from AML cells. This may due to several reasons: firstly, assumptions were made regarding the physical properties of AML cells as precise measurements are not available in the literature. For instance, Poisson's ratio (the ratio of strain in the lateral direction to strain in the axial direction) of AML cells assumed to be the same as that of another cell type, chondrocytes, which may not be the case. Our work has shown that a small variation in the Poisson's ratio may substantially affect the backscattered field. Secondly, this model doesn't take into account the viscosity of the AML cell and nucleus which may affect the backscatter through the process of damping. Lastly, the proposed model may be oversimplified since does not take into account the presence of the cell membranes and large organelles in the cytoplasm such as, mitochondria, which may also affect the backscatter.

Conclusions and Future work
Agreement between analytical and finiteelement solutions was found in the scattered farfield over a range of frequencies of interest (10 -70 MHz). Oscillations of an elastic sphere at various resonance frequencies were investigated. A systematic relationship between the resonance frequency and its corresponding surface mode was found. An ultrasound scattering model by a single cell was also presented. The model treats the cell as an elastic sphere (nucleus) surrounded by a fluid shell (cytoplasm). Comparison of the theoretical backscatter predicted by the model and experimental measurements for Acute Myeloid Leukemia (AML) cell did not show a good agreement. Reasons for such discrepancies were discussed.
Future work includes the modeling of backscatter and oscillations from ultrasound contrast agents and the refinement of the finiteelement cell model.