Jacob Hinkle

Intrinsic Polynomials for Regression on Riemannian Manifolds

We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

A vector momenta formulation of diffeomorphisms for improved geodesic regression and atlas construction

This paper presents a novel approach for diffeomorphic image regression and atlas estimation that results in improved convergence and numerical stability. We use a vector momenta representation of a diffeomorphisms initial conditions instead of the standard scalar momentum that is typically used. The corresponding variational problem results in a closed-form update for template estimation in both the geodesic regression and atlas estimation problems. While we show that the theoretical optimal solution is equivalent to the scalar momenta case, the simplification of the optimization problem leads to more stable and efficient estimation in practice. We demonstrate the effectiveness of our method for atlas estimation and geodesic regression using synthetically generated shapes and 3D MRI brain scans.

Polynomial regression on riemannian manifolds

In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds. The theory enables parametric analysis in a wide range of applications, including rigid and non-rigid kinematics as well as shape change of organs due to growth and aging. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Bookstein and the analysis of the shape changes associated with aging of the corpus callosum from the OASIS Alzheimers study.

Investigating the Feasibility of Rapid MRI for Image-Guided Motion Management in Lung Cancer Radiotherapy

Cycle-to-cycle variations in respiratory motion can cause significant geometric and dosimetric errors in the administration of lung cancer radiation therapy. A common limitation of the current strategies for motion management is that they assume a constant, reproducible respiratory cycle. In this work, we investigate the feasibility of using rapid MRI for providing long-term imaging of the thorax in order to better capture cycle-to-cycle variations. Two nonsmall-cell lung cancer patients were imaged (free-breathing, no extrinsic contrast, and 1.5 T scanner). A balanced steady-state-free-precession (b-SSFP) sequence was used to acquire cine-2D and cine-3D (4D) images. In the case of Patient 1 (right midlobe lesion, ~40 mm diameter), tumor motion was well correlated with diaphragmatic motion. In the case of Patient 2, (left upper-lobe lesion, ~60 mm diameter), tumor motion was poorly correlated with diaphragmatic motion. Furthermore, the motion of the tumor centroid was poorly correlated with the motion of individual points on the tumor boundary, indicating significant rotation and/or deformation. These studies indicate that image quality and acquisition speed of cine-2D MRI were adequate for motion monitoring. However, significant improvements are required to achieve comparable speeds for truly 4D MRI. Despite several challenges, rapid MRI offers a feasible and attractive tool for noninvasive, long-term motion monitoring.

4D MAP Image Reconstruction Incorporating Organ Motion

Four-dimensional respiratory correlated computed tomography (4D RCCT) has been widely used for studying organ motion. Most current algorithms use binning techniques which introduce artifacts that can seriously hamper quantitative motion analysis. In this paper, we develop an algorithm for tracking organ motion which uses raw time-stamped data and simultaneously reconstructs images and estimates deformations in anatomy. This results in a reduction of artifacts and an increase in signal-to-noise ratio (SNR). In the case of CT, the increased SNR enables a reduction in dose to the patient during scanning. This framework also facilitates the incorporation of fundamental physical properties of organ motion, such as the conservation of local tissue volume. We show in this paper that this approach is accurate and robust against noise and irregular breathing for tracking organ motion. A detailed phantom study is presented, demonstrating accuracy and robustness of the algorithm. An example of applying this algorithm to real patient image data is also presented, demonstrating the utility of the algorithm in reducing artifacts.

4D CT image reconstruction with diffeomorphic motion model.

Four-dimensional (4D) respiratory correlated computed tomography (RCCT) has been widely used for studying organ motion. Most current RCCT imaging algorithms use binning techniques that are susceptible to artifacts and challenge the quantitative analysis of organ motion. In this paper, we develop an algorithm for analyzing organ motion which uses the raw, time-stamped imaging data to reconstruct images while simultaneously estimating deformation in the subjects anatomy. This results in reduction of artifacts and facilitates a reduction in dose to the patient during scanning while providing equivalent or better image quality as compared to RCCT. The framework also incorporates fundamental physical properties of organ motion, such as the conservation of local tissue volume. We demonstrate that this approach is accurate and robust against noise and irregular breathing patterns. We present results for a simulated cone beam CT phantom, as well as a detailed real porcine liver phantom study demonstrating accuracy and robustness of the algorithm. An example of applying this algorithm to real patient image data is also presented.

A hierarchical geodesic model for diffeomorphic longitudinal shape analysis

Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. In this paper, we propose a novel hierarchical geodesic model (HGM), which generalizes HLMs to the manifold setting. Our proposed model explains the longitudinal trends in shapes represented as elements of the group of diffeomorphisms. The individual level geodesics represent the trajectory of shape changes within individuals. The group level geodesic represents the average trajectory of shape changes for the population. We derive the solution of HGMs on diffeomorphisms to estimate individual level geodesics, the group geodesic, and the residual geodesics. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans.

Hierarchical Geodesic Models in Diffeomorphisms

Hierarchical linear models (HLMs) are a standard approach for analyzing data where individuals are measured repeatedly over time. However, such models are only applicable to longitudinal studies of Euclidean data. This paper develops the theory of hierarchical geodesic models (HGMs), which generalize HLMs to the manifold setting. Our proposed model quantifies longitudinal trends in shapes as a hierarchy of geodesics in the group of diffeomorphisms. First, individual-level geodesics represent the trajectory of shape changes within individuals. Second, a group-level geodesic represents the average trajectory of shape changes for the population. Our proposed HGM is applicable to longitudinal data from unbalanced designs, i.e., varying numbers of timepoints for individuals, which is typical in medical studies. We derive the solution of HGMs on diffeomorphisms to estimate individual-level geodesics, the group geodesic, and the residual diffeomorphisms. We also propose an efficient parallel algorithm that easily scales to solve HGMs on a large collection of 3D images of several individuals. Finally, we present an effective model selection procedure based on cross validation. We demonstrate the effectiveness of HGMs for longitudinal analysis of synthetically generated shapes and 3D MRI brain scans.

Quantifying variability in radiation dose due to respiratory-induced tumor motion

State of the art radiation treatment methods such as hypo-fractionated stereotactic body radiation therapy (SBRT) can successfully destroy tumor cells and avoid damaging healthy tissue by delivering high-level radiation dose that precisely conforms to the tumor shape. Though these methods work well for stationary tumors, SBRT dose delivery is particularly susceptible to organ motion, and few techniques capable of resolving and compensating for respiratory-induced organ motion have reached clinical practice. The current treatment pipeline cannot accurately predict nor account for respiratory-induced motion in the abdomen that may result in significant displacement of target lesions during the breathing cycle. Sensitivity of dose deposition to respiratory-induced organ motion represents a significant challenge and may account for observed discrepancies between predictive treatment plan indicators and clinical patient outcomes. Improved treatment-planning and delivery of SBRT requires an accurate prediction of dose deposition uncertainties resulting from respiratory motion. To accomplish this goal, we developed a framework that models both organ displacement in response to respiration and the underlying random variations in patient-specific breathing patterns. Our organ deformation model is a four-dimensional maximum a posteriori (MAP) estimation of tissue deformation as a function of chest wall amplitudes computed from clinically obtained respiratory-correlated computed tomography (RCCT) images. We characterize patient-specific respiration as the probability density function (PDF) of chest wall amplitudes and model patient breathing patterns as a random process. We then combine the patient-specific organ motion and stochastic breathing models to calculate the resulting variability in radiation dose accumulation. This process allows us to predict uncertainties in dose delivery in the presence of organ motion and identify tissues at risk of receiving insufficient or harmful levels of radiation.

Four-dimensional tissue deformation reconstruction (4D TDR) validation using a real tissue phantom

Calculation of four‐dimensional (4D) dose distributions requires the remapping of dose calculated on each available binned phase of the 4D CT onto a reference phase for summation. Deformable image registration (DIR) is usually used for this task, but unfortunately almost always considers only endpoints rather than the whole motion path. A new algorithm, 4D tissue deformation reconstruction (4D TDR), that uses either CT projection data or all available 4D CT images to reconstruct 4D motion data, was developed. The purpose of this work is to verify the accuracy of the fit of this new algorithm using a realistic tissue phantom. A previously described fresh tissue phantom with implanted electromagnetic tracking (EMT) fiducials was used for this experiment. The phantom was animated using a sinusoidal and a real patient‐breathing signal. Four‐dimensional computer tomography (4D CT) and EMT tracking were performed. Deformation reconstruction was conducted using the 4D TDR and a modified 4D TDR which takes real tissue hysteresis (4D TDRHysteresis) into account. Deformation estimation results were compared to the EMT and 4D CT coordinate measurements. To eliminate the possibility of the high contrast markers driving the 4D TDR, a comparison was made using the original 4D CT data and data in which the fiducials were electronically masked. For the sinusoidal animation, the average deviation of the 4D TDR compared to the manually determined coordinates from 4D CT data was 1.9 mm, albeit with as large as 4.5 mm deviation. The 4D TDR calculation traces matched 95% of the EMT trace within 2.8 mm. The motion hysteresis generated by real tissue is not properly projected other than at endpoints of motion. Sinusoidal animation resulted in 95% of EMT measured locations to be within less than 1.2 mm of the measured 4D CT motion path, enabling accurate motion characterization of the tissue hysteresis. The 4D TDRHysteresis calculation traces accounted well for the hysteresis and matched 95% of the EMT trace within 1.6 mm. An irregular (in amplitude and frequency) recorded patient trace applied to the same tissue resulted in 95% of the EMT trace points within less than 4.5 mm when compared to both the 4D CT and 4D TDRHysteresis motion paths. The average deviation of 4D TDRHysteresis compared to 4D CT datasets was 0.9 mm under regular sinusoidal and 1.0 mm under irregular patient trace animation. The EMT trace data fit to the 4D TDRHysteresis was within 1.6 mm for sinusoidal and 4.5 mm for patient trace animation. While various algorithms have been validated for end‐to‐end accuracy, one can only be fully confident in the performance of a predictive algorithm if one looks at data along the full motion path. The 4D TDR, calculating the whole motion path rather than only phase‐ or endpoints, allows us to fully characterize the accuracy of a predictive algorithm, minimizing assumptions. This algorithm went one step further by allowing for the inclusion of tissue hysteresis effects, a real‐world effect that is neglected when endpoint‐only validation is performed. Our results show that the 4D TDRHysteresis correctly models the deformation at the endpoints and any intermediate points along the motion path. PACS numbers: 87.55.km, 87.55.Qr, 87.57.nf, 87.85.Tu