Modeling of photoelastic imaging of mechanical stresses in transparent solids mimicking kidney stones

Abstract

Theoretical and numerical models were developed to calculate the polariscopic integrated light intensity that forms a projection of the dynamic stress within an axisymmetric elastic object. Although the model is general, this paper addressed its application to measurements of stresses in model kidney stones from a burst wave lithotripter for stone fragmentation. The stress was calculated using linear elastic equations, and the light propagation was modeled in the instantaneous case by integrating over the volume of the stone. The numerical model was written in finite differences. The resulting images agreed well with measured images. The measured images corresponded to the maximum shear stress distribution, although other stresses were also plotted. Comparison of the modeled and observed polariscope images enabled refinement of the photoelastic constant by minimizing the error between the calculated and measured fields. These results enable quantification of the stress within the polariscope images, determination of material properties, and the modes and mechanisms of stress production within a kidney stone. Such a model may help in interpreting elastic waves in structures, such as stones, toward improving lithotripsy procedures.

Figures

FIG. 1.

Orientation of the electromagnetic wave vectors relative to the coordinate axes.

FIG. 2.

(Color online) Geometry of the experiment. An acoustic source produces a focused ultrasound burst, which propagates along the axis of an elastic specimen (z axis). A photoelastic image is formed with the use of a circular polariscope consisting of a polarizer, two quarter-wave plates, and an analyzer. A collimated optical beam from a pulsed light-emitting diode (LED) is projected onto the specimen along the x axis.

FIG. 3.

(Color online) The cross-sectional geometry of an optical ray traversing the elastic specimen. The index of refraction of the liquid surrounding the cylinder has been matched to the index of refraction of the cylinder to minimize light refraction.

FIG. 4.

Pressure waveform at the focus measured by the fiber-optic hydrophone in water (upper) and obtained using finite-difference modeling (lower).

FIG. 5.

(Color online) Typical patterns for the finite-difference modeling of the maximum principal mechanical stress, σmax (upper without the specimen, center with the elastic specimen), and the polariscope light intensity normalized by its background value, I/I0 (lower). The acoustic source is positioned at the left-hand side, outside of the presented region. The upper image shows σmax in the liquid in the absence of the stone, and the center image shows the case when the specimen is present. The acoustic pressure in liquid is equal to the inverted σmax.

FIG. 6.

(Color online) Instantaneous distributions of various parameters of elastic perturbations in the specimen at consecutive time points. Time is counted from the moment the ultrasound pulse propagating to the right arrives at the proximal left face of the specimen. The value of the corresponding quantity on the color pictures changes from dark blue to dark red with the green color corresponding to the zero level. The color bar limits are ±0.0015 (Ωφ), ±0.0005 (D), and ±3.5 MPa (σmax and τmax). The normalized light intensity I/I0 changes from 0 (black) to 1 (white). The polariscope images best correlate with the maximum shear stress distribution.

FIG. 7.

Comparison of an experimental photoelastic image (upper) with the corresponding theoretical images calculated using different photoelastic constants: C0= 5.3 × 10−4 Pa−1 m−1 (center) and C0= 2.4 × 10−4 Pa−1 m−1 (lower). The ultrasound burst center frequency is 340 kHz, and the peak negative pressure at the focus in the absence of the stone is 4.5 MPa. The image represents mechanical stresses in the stone 14 μs after the burst enters the stone.

FIG. 8.

A sequence of photoelastic images representing wave propagation through the specimen at 3-μs snapshots (roughly one period of the 340 kHz tone burst) starting 5 μs after the ultrasound pulse reaches the specimen. The left column shows experimentally measured data, and the right column shows numerically calculated results obtained using the photoelastic constant, C0= 2.4 × 10−4 Pa−1 m−1.