In elastography, quantitative elastograms are desirable as they are system and operator independent. Such quantification also facilitates more accurate diagnosis, longitudinal studies and studies performed across multiple sites. In optical elastography (compression, surface-wave or shear-wave), quantitative elastograms are typically obtained by assuming some form of homogeneity. This simplifies data processing at the expense of smearing sharp transitions in elastic properties, and/or introducing artifacts in these regions.
Recently, we proposed an inverse problem-based approach to compression OCE that does not assume homogeneity, and overcomes the drawbacks described above. In this approach, the difference between the measured and predicted displacement field is minimized by seeking the optimal distribution of elastic parameters. The predicted displacements and recovered elastic parameters together satisfy the constraint of the equations of equilibrium. This approach, which has been applied in two spatial dimensions assuming plane strain, has yielded accurate material property distributions.
Here, we describe the extension of the inverse problem approach to three dimensions. In addition to the advantage of visualizing elastic properties in three dimensions, this extension eliminates the plane strain assumption and is therefore closer to the true physical state. It does, however, incur greater computational costs. We address this challenge through a modified adjoint problem, spatially adaptive grid resolution, and three-dimensional decomposition techniques. Through these techniques the inverse problem is solved on a typical desktop machine within a wall clock time of ~ 20 hours. We present the details of the method and quantitative elasticity images of phantoms and tissue samples.
It is now well recognized that a host of imaging modalities (a list that includes Ultrasound, MRI, Optical Coherence Tomography, and optical microscopy) can be used to “watch” tissue as it deforms in response to an internal or external excitation. The result is a detailed map of the deformation field in the interior of the tissue. This deformation field can be used in conjunction with a material mechanical response to determine the spatial distribution of material properties of the tissue by solving an inverse problem. Images of material properties thus obtained can be used to quantify the health of the tissue. Recently, they have been used to detect, diagnose and monitor cancerous lesions, detect vulnerable
plaque in arteries, diagnose liver cirrhosis, and possibly detect the onset of Alzheimer’s disease. In this talk I will describe
the mathematical and computational aspects of solving this class of inverse problems, and their applications in biology
In particular, I will discuss the well-posedness of these problems and quantify the amount of displacement data necessary to obtain a unique property distribution. I will describe an efficient algorithm for solving the resulting inverse problem. I will also describe some recent developments based on Bayesian inference in estimating the variance in the estimates of material properties. I will conclude with the applications of these techniques in diagnosing breast cancer and in characterizing the mechanical properties of cells with sub-cellular resolution.
Quantitative elasticity imaging, which retrieves elastic modulus maps from tissue, is preferred to qualitative strain imaging for acquiring system- and operator-independent images and longitudinal and multi-site diagnoses.
Quantitative elasticity imaging has already been demonstrated in optical elastography by relating surface-acoustic and shear wave speed to Young’s modulus via a simple algebraic relationship. Such approaches assume largely homogeneous samples and neglect the effect of boundary conditions.
We present a general approach to quantitative elasticity imaging based upon the solution of the inverse elasticity problem using an iterative technique and apply it to compression optical coherence elastography. The inverse problem is one of finding the distribution of Young’s modulus within a sample, that in response to an applied load, and a given displacement and traction boundary conditions, can produce a displacement field matching one measured in experiment. Key to our solution of the inverse elasticity problem is the use of the adjoint equations that allow the very efficient evaluation of the gradient of the objective function to be minimized with respect to the unknown values of Young’s modulus within the sample. Although we present the approach for the case of linear elastic, isotropic, incompressible solids, this method can be employed for arbitrarily complex mechanical models.
We present the details of the method and quantitative elastograms of phantoms and tissues. We demonstrate that by using the inverse approach, we can decouple the artefacts produced by mechanical tissue heterogeneity from the true distribution of Young’s modulus, which are often evident in techniques that employ first-order algebraic relationships.
Near-field interference lithography is a promising variant of multiple patterning in semiconductor device fabrication
that can potentially extend lithographic resolution beyond the current materials-based restrictions on the
Rayleigh resolution of projection systems. With H<sub>2</sub>O as the immersion medium, non-evanescent propagation
and optical design margins limit achievable pitch to approximately 0.53λ/<i>n</i>H<sub>2</sub>O = 0.37λ. Non-evanescent images
are constrained only by the comparatively large resist indices (typically1.7) to a pitch resolution of 0.5/<i>n</i>resist
(typically 0.29). Near-field patterning can potentially exploit evanescent waves and thus achieve higher spatial
resolutions. Customized near-field images can be achieved through the modulation of an incoming wavefront
by what is essentially an in-situ hologram that has been formed in an upper layer during an initial patterned
exposure. Contrast Enhancement Layer (CEL) techniques and Talbot near-field interferometry can be considered
special cases of this approach.
Since the technique relies on near-field interference effects to produce the required pattern on the resist, the
shape of the grating and the design of the film stack play a significant role on the outcome. As a result, it is
necessary to resort to full diffraction computations to properly simulate and optimize this process.
The next logical advance for this technology is to systematically design the hologram and the incident wavefront
which is generated from a reduction mask. This task is naturally posed as an optimization problem, where
the goal is to find the set of geometric and incident wavefront parameters that yields the closest fit to a desired
pattern in the resist. As the pattern becomes more complex, the number of design parameters grows, and the
computational problem becomes intractable (particularly in three-dimensions) without the use of advanced numerical
techniques. To treat this problem effectively, specialized numerical methods have been developed. First,
gradient-based optimization techniques are used to accelerate convergence to an optimal design. To compute
derivatives of the parameters, an adjoint-based method was developed. Using the adjoint technique, only two
electromagnetic problems need to be solved per iteration to evaluate the cost function and all the components
of the gradient vector, independent of the number of parameters in the design.