Displacement Probability Research Articles

MO‐E‐BRA‐04: Application of Robust Optimization in Prostate Cancer

The knowledge about patient geometry at the exact time of radiation delivery is rarely complete. The result of radiotherapy should not depend sensitively on inevitable uncertainties. This quality of robustness of a treatment can be enforced during dose optimization by a variety of means, which can be classified by the frequency with which patient information is acquired, the timespan between acquisition and delivery, and the nature of the image information. For prostate radiotherapy, random target and normal tissue motion poses the greatest challenge as it requires high‐quality volume imaging and the information content may decay quickly.Even with today's on‐board imaging systems, a fair amount of uncertainty about the patient geometry remains, which is best described by probability distributions (PD) of pointwise displacements. Here, various off‐line and quasi on‐line image‐guided protocols differ mostly in how these PDs are constructed and how frequently it is updated. The PDs can be used to compute the expected values of dose or dose effect at each point of the patient model. This model may either be defined in the treatment room (and dose) coordinate system (TCS) or may be associated with the patient reference geometry and deform along with the anatomy. While the former is the traditional model for dose planning, the latter shifts the focus to the accumulation of dose in the tissue, hence the term tissue‐eye‐view (TEV).The most basic probabilistic patient model in TCS is the coverage probability model, where each volume element in a rigid reference patient geometry is weighted with the cumulative probability that some volume of interest can be found there. This information quantifies the relevance of a point in the CTV‐to‐PTV margin. Despite its apparent simplicity, it is possible to alleviate the common PTV‐overlaps‐organ paradox to an extent that allows iso‐toxic dose escalation by about 10 per cent. Moving to a probabilistic patient model in TEV abandons the PTV concept altogether, at the price of more image information and the need for deformable registration. The potential for iso‐toxic dose escalation lies at more than 20 per cent.Both optimization concepts rely on an a‐priori estimate of the pointwise displacement probabilities. A bias or time trend in these PDs would be potentially fatal. Hence, it is essential to update the PD during the course of treatment to minimize the “uncertainty in the estimates of uncertainties”. In consequence, robust off‐line adaptive protocols require some extent of monitoring while on‐line protocols require basically off‐line probabilistic models predicated (in a Bayesian sense) on the geometry of the day. Apart from the insufficiency in the input image data, another risk arises from the high specificity with which individual source of error influence the optimized dose distribution: a large margin could compensate for many uncertainties, while margin‐less optimization schemes need to quantify all of them. This limits the theoretical benefit of the most sophisticated models (daily on‐line imaging Bayesian TEV) significantly. The specific cost benefit ratio of various protocols remains to be evaluated In practice, in larger populations of patients.

A novel tensor distribution model for the diffusion-weighted MR signal

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

Hybrid diffusion imaging

Diffusion measurements in the human central nervous system are complex to characterize and a broad spectrum of methods have been proposed. In this study, a comprehensive diffusion encoding and analysis approach, hybrid diffusion imaging (HYDI), is described. The HYDI encoding scheme is composed of multiple concentric “shells” of constant diffusion weighting, which may be used to characterize the signal behavior with low, moderate and high diffusion weighting. HYDI facilitates the application of multiple data analyses strategies including diffusion tensor imaging (DTI), multi-exponential diffusion measurements, diffusion spectrum imaging (DSI) and q-ball imaging (QBI). These different analysis strategies may provide complementary information. DTI measures (mean diffusivity and fractional anisotropy) may be estimated from either data in the inner shells or the entire HYDI data. Fast and slow diffusivities were estimated using a nonlinear least squares bi-exponential fit on geometric means of the HYDI shells. DSI measurements from the entire HYDI data yield empirical model-independent diffusion information and are well-suited for characterizing tissue regions with complex diffusion behavior. DSI measurements were characterized using the zero displacement probability and the mean-squared displacement. The outermost HYDI shell was analyzed using QBI analysis to estimate the orientation distribution function (ODF), which is useful for characterizing the directions of multiple fiber groups within a voxel. In this study, an HYDI encoding scheme with 102 diffusion-weighted measurements was obtained over most of the human cerebrum in under 30 min.

Bundling, Entry Deterrence, and Specialist Innovators*

We examine a mechanism by which bundling may inefficiently deter entry into the market. The model considers an incumbent monopolist in two complementary components that faces a series of entry attempts by rivals. It is shown that the incumbent can practice bundling to buttress its monopoly position by keeping specialist innovators out of the market. Bundling prevents specialist rivals from coordinating in the dynamic entry process, reducing the probability of an eventual displacement of the incumbent. The specialization decisions of rivals are also distorted. Bundling may lead to lower customer and total economic welfare.

Q‐space analysis of lung morphometry in vivo with hyperpolarized3He spectroscopy

To examine the utility of a (3)He spectroscopic q-space technique for detecting changes in lung morphometry in vivo. A diffusion-weighted spectroscopy sequence was used to collect global diffusion data from healthy adults (N = 11), healthy children (N = 5), and chronic obstructive pulmonary disease (COPD) patients (N = 2) using 40 cc of hyperpolarized (3)He gas within a two second breathhold. Displacement probability profiles (DPP) were obtained by Fourier transformation of diffusion data with respect to q. A bi-Gaussian model was used to decompose the DPPs into narrow and broad components, characterized by root-mean-square displacements X(rms1) and X(rms2), respectively. In healthy adults, the narrow component (X(rms,1)) of the DPP had a mean displacement of 188 +/- 10 microm, slightly less than the reported average size of the alveoli. The broad component (X(rms,2)) had a mean value of 474 +/- 44 microm, comparable to the diameter of the respiratory bronchioles in the acinus. In children, both X(rms1) (167 +/- 4 microm) and X(rms2) (382 +/- 22 microm) compared to healthy adults (P < 0.01). In COPD patients, the mean displacements were elevated (X(rms1): 265 +/- 71 microm; X(rms2): 530 +/- 109 microm) compared to healthy adults. Excellent correlation was found between rms displacements and age (age vs. X(rms,1): r = 0.78, P < 0.001; age vs. X(rms,2): r = 0.90, P < 0.001). The q-space parameters agreed remarkably well with published alveolar morphometry data. The results suggest that the technique may be sensitive to disease, as evident from the elevated mean displacements in COPD patients compared to healthy volunteers. Detailed lung microstructural information can be obtained using a very low volume of inhaled (3)He.

TU‐C‐330A‐06: Evaluation of Internal Lung Motion Based On Extended Time Ultra‐Fast MRI Scan

Purpose: In order to develop radiation treatment planning based on a displacement probability function for lung tumors we developed a dynamic MRI protocol to image the internal human respiratory lung motion during a several minute period that simulates the duration of a radiation treatment, evaluated the long time displacement probability distribution function (PDF) of pulmonary vessels as surrogate tumors, and assessed its reproducibility with repeat imaging. Methods and Materials: TrueFISP (fast imaging with steady‐state precession) sequence was adapted to acquire real time MR images of human lungs during 5 minute scans in both sagittal and coronal planes. A total of 26 pulmonary vessels from different regions (upper, middle and lower) in 3 healthy subjects were examined. Motion profile and displacement PDF of each tracked pulmonary vessel were evaluated. Experiments were repeated after 2‐3 weeks to test the reproducibility. Results: Motion profiles and displacement PDF of the same subject showed similarity, but great variation between different subjects. Displacement PDFs varied tremendously but tended to stabilize during the 5‐minute scan, and were reproduced reasonably well to various degrees in the repeated experiments after a subject specific stabilizing time (270s, 120s, 200s for Subject 1, 2, 3 respectively). Conclusions: Experiments for the first time using ultra‐fast real time MRI in extended time scans produce stable and reproducible displacement PDF of internal pulmonary structures, weakly depending upon different individual breathing patterns. This methodology is being investigated in a clinical trial at our institution to determine in a larger scale whether the reproducibility of motion is statistically significant and whether patients with lung tumor exhibit similar predictable breathing characteristics.

Response probability density functions of strongly non-linear systems by the path integration method

The paper discusses challenges in numerical analysis and numerical/analytical results for strongly non-linear systems—systems with “signum”-type non-linearities. Such non-linearities are implemented for instantaneous variations of the systems’ parameters, to reduce their mean energy response when subjected to random excitations. Numerical results for displacement and velocity response probability density functions (PDFs), energy response PDFs and various order moments are obtained by the path integration technique. Attention is also given to evaluation of mean upcrossing rate, related to the system's half period, via Rice's formula informally applied to discontinuous response PDFs.

Mapping the intracellular fraction of water by varying the gradient pulse length in q-space diffusion MRI

Finite gradient pulse lengths are traditionally considered a nuisance in q-space diffusion NMR and MRI, since the simple Fourier relation between the acquired signal and the displacement probability is invalidated. Increasing the value of the pulse length leads to an apparently smaller value of the estimated compartment size. We propose that q-space data at different gradient pulse lengths, but with the same effective diffusion time, can be used to identify and quantify components with free or restricted diffusion from multiexponential echo decay curves obtained on cellular systems. The method is demonstrated with experiments on excised human brain white matter and a series of model systems with well-defined free, restricted, and combined free and restricted diffusion behavior. Time-resolved diffusion MRI experiments are used to map the spatial distribution of the intracellular fraction in a yeast cell suspension during sedimentation, and observe the disappearance of this fraction after a heat treatment.

Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT)

This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffusion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this technique's ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system.

Plasma diffusion across a magnetic field observed by collective light scattering: Experimental evidence for Lévy stable distributions

On a toroidal magnetized plasma discharge, ToriX, a collective light-scattering device has been set to investigate plasma turbulence and transport. The scattering device output signal is proportional to the space Fourier transform of the plasma density. The two-dimensional static form factor, at the scale of the scattering wave number k, is measured in absolute units as a function of k. It shows that fluctuations are large and mainly situated at large scales. By adding a small vertical component to the main toroidal magnetic field, a uniform plasma has been achieved and the form factor is significantly reduced. The time autocorrelation function of the scattered signal as a function of k is shown to be the Fourier transform of the displacement probability distribution. It is investigated at different times as a function of k and found to be consistent with a Lévy walk model with a characteristic exponent α closed to 1. The displacement probability distribution is thus close to a Lorentz function instead of a Gaussian.

Self-Diffusion Anisotropy of Small Penetrant Molecules in Deformed Elastomers

The self-diffusion anisotropy of n-hexane, n-heptane, n-octane, and n-decane in uniaxially compressed natural rubber samples with different cross-link densities was detected by pulsed-gradient stimulated spin−echo NMR. The effective diffusion coefficients and the displacement probabilities of alkane penetrant molecules were measured along and perpendicular to the direction of the compression force. For all solvents, the diffusion anisotropy increases with increasing compression. The microscopic theory of diffusion on the basis of the free volume is adapted to predict the dependence of the diffusion anisotropy on the deformation ratio. The theoretical dependence is in good agreement with the measurements on toluene and n-hexane swollen in cross-linked natural rubber samples. The anisotropy of diffusion of small penetrant molecules in elastomers could be used for NMR investigation of the deformation of polymer networks by the changes in the shape of the free volume.

Diffusional kurtosis imaging: The quantification of non‐gaussian water diffusion by means of magnetic resonance imaging

A magnetic resonance imaging method is presented for quantifying the degree to which water diffusion in biologic tissues is non-Gaussian. Since tissue structure is responsible for the deviation of water diffusion from the Gaussian behavior typically observed in homogeneous solutions, this method provides a specific measure of tissue structure, such as cellular compartments and membranes. The method is an extension of conventional diffusion-weighted imaging that requires the use of somewhat higher b values and a modified image postprocessing procedure. In addition to the diffusion coefficient, the method provides an estimate for the excess kurtosis of the diffusion displacement probability distribution, which is a dimensionless metric of the departure from a Gaussian form. From the study of six healthy adult subjects, the excess diffusional kurtosis is found to be significantly higher in white matter than in gray matter, reflecting the structural differences between these two types of cerebral tissues. Diffusional kurtosis imaging is related to q-space imaging methods, but is less demanding in terms of imaging time, hardware requirements, and postprocessing effort. It may be useful for assessing tissue structure abnormalities associated with a variety of neuropathologies.

Prostate gland motion assessed with cine-magnetic resonance imaging (cine-MRI)

To quantify prostate motion during a radiation therapy treatment using cine-magnetic resonance imaging (cine-MRI) for time frames comparable to that expected in an image-guided radiation therapy treatment session (20-30 min). Six patients undergoing radiation therapy for prostate cancer were imaged on 3 days, over the course of therapy (Weeks 1, 3, and 5). Four hundred images were acquired during the 1-h MRI session in 3 sagittal planes through the prostate at 6-s intervals. Eleven anatomic points of interest (POIs) have been used to characterize prostate/bony pelvis/abdominal wall displacement. Motion traces and standard deviation for each of the 11 POIs have been determined. The probability of displacement over time has also been calculated. Patients were divided into 2 groups according to rectal filling status: full vs. empty rectum. The displacement of POIs (standard deviation) ranged from 0.98 to 1.72 mm for the full-rectum group and from 0.68 to 1.04 mm for the empty-rectum group. The low standard deviations in position (2 mm or less) would suggest that these excursions have a low frequency of occurrence. The most sensitive prostate POI to rectal wall motion was the mid-posterior with a standard deviation of 1.72 mm in the full-rectum group vs. 0.79 mm in the empty-rectum group (p = 0.0001). This POI has a 10% probability of moving more than 3 mm in a time frame of approximately 1 min if the rectum is full vs. approximately 20 min if the rectum is empty. Motion of the prostate and seminal vesicles during a time frame similar to a standard treatment session is reduced compared to that reported in interfraction studies. The most significant predictor for intrafraction prostate motion is the status of rectal filling. A prostate displacement of <3 mm (90%) can be expected for the 20 min after the moment of initial imaging for patients with an empty rectum. This is not the case for patients presenting with full rectum. The determination of appropriate intrafraction margins in radiation therapy to accommodate the time-dependent uncertainty in positional targeting is a topic of ongoing investigations for the on-line image guidance model.

Worker displacement during the transition: Experience from Slovenia1

AbstractThe Slovenian transition created labor displacements that were bigger than those experienced in North America in the 1980s. In Slovenia, probability of both layoffs and quits fell with worker tenure, firm profitability and expected severance costs. Individuals facing a higher probability of displacement accepted slower wage growth than otherwise comparable workers. The incentives to avoid displacement were strong – workers that actually were displaced faced a slow process of transiting out of unemployment with only one‐third finding re‐employment. Correcting for selection, real wage losses for displaced workers are comparable to those reported for displaced workers in North America.

Worker Displacement during the Transition: Experience from Slovenia

The transition to market in Slovenia created labor displacements that were on par or greater than that experienced in North America in the 1980s. A simple theoretical model suggests that factors which raise the probability of layoff should also increase the probability of a quit, predictions that are borne out in data. Probability of both layoffs and quits fell with worker tenure, firm profitability and expected severance costs. Individuals facing a higher probability of displacement accepted slower wage growth than otherwise comparable workers. The incentives to avoid displacement were strong - workers that actually were displaced faced a slow process of transiting out of unemployment with only one-third finding reemployment. Correcting for selection, real wage losses for displaced workers are comparable to those reported for displaced workers in North America.

Investigation of phase trapping by secondary imbibition within Fontainebleau sandstone

Pulsed magnetic field gradient stimulated echo (PGSTE) NMR is used to investigate the simultaneous flow of two phases (an aqueous phase and an hydrocarbon phase) within a strongly water-wet sample of Fontainebleau sandstone. The Fontainebleau sandstone is prepared in increasing steady-state water saturations by a secondary imbibition process. The increase in the water saturation causes an increasing fraction of the oil phase (non-wetting phase) to become trapped within the sample. The stimulated echo dependence on the gradient pulse area, q, is used to derive the displacement probability, P( X), for a fixed observation time. These displacement probabilities clearly show the progressive trapping of the oil phase with increasing steady-state water saturations. Quantitative measurements of the fraction of the oil phase trapped were made from the echo attenuation function E Δ ( q), both as a function of water saturation and observation time.

Ion beam induced coherent displacement in Al on Au system

Molecular dynamics simulations of ion bombardment of heavy metals revealed a few years ago a surprising new damage formation mechanism, the coherent displacement of atoms towards surfaces [Nature 398 (1999) 49]. In principle this mechanism could be used to produce large ordered adatom islands on surfaces with low-fluence irradiation. In practice, however, under the irradiation conditions studied so far most of the surface damage was still produced by chaotic mechanisms, leading to disordered adatom formations. In this paper we examine whether a suitable kind of irradiation of an Al/Au multilayer system would produce surface damage mostly by coherent displacement. We achieve an increase of the coherent displacement probability from the previous 10–30%, giving encouragement for further attempts at optimizing the irradiation conditions.

Definition of displacement probability and diffusion time in q-space magnetic resonance measurements that use finite-duration diffusion-encoding gradients

In q-space diffusion NMR, the probability P( r ,t d ) of a molecule having a displacement r in a diffusion time t d is obtained under the assumption that the diffusion-encoding gradient g has an infinitesimal duration. However, this assumption may not always hold, particularly in human MRI where the diffusion-encoding gradient duration δ is typically of the same order of magnitude as the time offset Δ between encoding gradients. In this case, finite- δ effects complicate the interpretation of displacement probabilities measured in q-space MRI, and the form by which the signal intensity relates to them. By considering the displacement-specific dephasing, 〈 r | e iφ 〉 , of a set of spins accumulating a constant displacement vector r in the total time Δ+ δ during which diffusion is encoded, the probability recovered by a finite- δ q-space experiment can be interpreted. It is shown theoretically that a data analysis using a modified q-space index q ̃ =γδη g , with γ the gyromagnetic ratio and η= (Δ−δ/3)/(Δ+δ) , recovers the correct displacement probability distribution if diffusion is multi-Gaussian free diffusion. With this analysis, we show that the displacement distribution P( r ,t exp ) is measured at the experimental diffusion-encoding time t exp= Δ+ δ, and not at the reduced diffusion time t r= Δ− δ/3 as is generally assumed in the NMR and MRI literature. It is also shown that, by defining a probability P ( y ,Δ) that a time t< δ exists such that a displacement y occurs from time t to t+ Δ, it is possible to describe the physical significance of the result obtained when we use the q-space formalism valid for infinitesimal δ when δ is not infinitesimal. These deductions were confirmed by simulations for homogeneous Gaussian diffusion and for heterogeneous diffusion in permeable microscopic Gaussian domains that are homogeneous on the μm scale. The results also hold for diffusion inside restricted spherical reflecting domains, but only if the radius of the domain is larger than a critical size. The simulations of the displacement-specific dephasing obtain that if δ> δ c then η≠ (Δ−δ/3)/(Δ+δ) which implies that we can no longer obtain the correct displacement probability from the displacement distribution. In the case that | g |=18 mT/m and Δ− δ=5 ms, the parameter δ c in ms is given by “ δ c=0.49 a 2+0.24” where a is the sphere’s radius expressed in μm. Simulation of q-space restricted diffusion MRI experiments indicate that if η= (Δ−δ/3)/(Δ+δ) , the recovered displacement probability is always better than the Gaussian approximation, and the measured diffusion coefficient matches the diffusion coefficient at time t exp= Δ+ δ better than it matches the diffusion coefficient at time t r= Δ− δ/3. These results indicate that q-space MRI measurements of displacement probability distributions are theoretically possible in biological tissues using finite-duration diffusion-encoding gradients provided certain compartment size and diffusion encoding gradient duration constraints are met.

Self-Diffusion Anisotropy of Small Penetrants in Compressed Elastomers

The anisotropy of self-diffusion of toluene in uniaxially compressed natural rubber samples with different cross-link densities was detected using the pulsed-gradient stimulated spin-echo NMR method. The effective diffusion coefficients and the displacement probabilities of toluene were measured along and perpendicular to the direction of the compression force. The diffusion anisotropy increases with increasing compression and cross-link density. The anisotropy of diffusion can be used for NMR investigations of the deformation of the polymer networks by the changes in the shape of the free volume.

Diffraction patterns of stacked layer crystals

The relation between diffraction and planar faulting is studied. The layer displacement probability correlation function is shown to be related to the Fourier coefficients of the decomposed diffraction pattern. Peak displacement can be considered as a consequence of the departure of the faulted structure from the original periodicity, while peak broadening is associated with the loss of correlation. Several definitions of distance are introduced to compare stacking sequence and measure their degree of randomness. A run-length encoding procedure is considered, well suited for the identification of mixed polytypes and as a measure of disorder. The problem of identification of faulting complexes is discussed in terms of the pair correlation function of the binary sequence representing the layer stacks.