Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System

It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, $\hbar \to 0$, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator, $C(t)$, for the classical and quantum kicked rotor -- a textbook driven chaotic system -- and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength $K$, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as $K \to 0$, while the OTOC's growth rate may decrease much slower showing higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time $t_E$: transitioning from a time-independent value of $t^{-1} \ln{C(t)}$ at $t < t_E$ to its monotonous decrease with time at $t>t_E$. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.Comment: 5+2 pages, 5+2 figures; accepted for publication in Phys. Rev. Lett., Vol. 118 (2017

MR Imaging Texture Analysis in the Abdomen and Pelvis

Texture analysis (TA) is a form of radiomics and refers to quantitative measurements of the histogram, distribution and/or relationship of pixel intensities or gray scales within a region of interest on an image. TA can be applied to MRI of the abdomen and pelvis, with the main strength being quantitative analysis of pixel intensities and heterogeneity rather than subjective/qualitative analysis. There are multiple limitations of MR texture analysis (MRTA) including a dependency on image acquisition and reconstruction parameters, non-standardized approaches without or with image filtration, diverse software methods and applications, and statistical challenges relating numerous texture analysis results to clinical outcomes in retrospective pilot studies with small sample sizes. Despite these limitations, there is a growing body of literature supporting MRTA. In this review, the application of MRTA to the abdomen and pelvis will be discussed, including tissue or tumor characterization and response evaluation or prediction of outcomes in various tumors

Retrospective CT/MRI Texture Analysis of Rapidly Progressive Hepatocellular Carcinoma

Rapidly progressive hepatocellular carcinoma (RPHCC) is a subset of hepatocellular carcinoma that demonstrates accelerated growth, and the radiographic features of RPHCC versus non-RPHCC have not been determined. The purpose of this retrospective study was to use baseline radiologic features and texture analysis for the accurate detection of RPHCC and subsequent improvement of clinical outcomes. We conducted a qualitative visual analysis and texture analysis, which selectively extracted and enhanced imaging features of different sizes and intensity variation including mean gray-level intensity (mean), standard deviation (SD), entropy, mean of the positive pixels (MPP), skewness, and kurtosis at each spatial scaling factor (SSF) value of RPHCC and non-RPHCC tumors in a computed tomography (CT) cohort of n = 11 RPHCC and n = 11 non-RPHCC and a magnetic resonance imaging (MRI) cohort of n = 13 RPHCC and n = 10 non-RPHCC. There was a statistically significant difference across visual CT irregular margins p = 0.030 and CT texture features in SSF between RPHCC and non-RPHCC for SSF-6, coarse-texture scale, mean p = 0.023, SD p = 0.053, MPP p = 0.023. A composite score of mean SSF-6 binarized + SD SSF-6 binarized + MPP SSF-6 binarized + irregular margins was significantly different between RPHCC and non-RPHCC (p = 0.001). A composite score ≥3 identified RPHCC with a sensitivity of 81.8% and specificity of 81.8% (AUC = 0.884, p = 0.002). CT coarse-texture-scale features in combination with visually detected irregular margins were able to statistically differentiate between RPHCC and non-RPHCC. By developing an image-based, non-invasive diagnostic criterion, we created a composite score that can identify RPHCC patients at their early stages when they are still eligible for transplantation, improving the clinical course of patient care

CT texture analysis: a potential tool for prediction of survival in patients with metastatic clear cell carcinoma treated with sunitinib

BACKGROUND: To assess CT texture based quantitative imaging biomarkers in the prediction of progression free survival (PFS) and overall survival (OS) in patients with clear cell renal cell carcinoma undergoing treatment with Sunitinib. METHODS: In this retrospective study, measurable lesions of 40 patients were selected based on RECIST criteria on standard contrast enhanced CT before and 2 months after treatment with Sunitinib. CT Texture analysis was performed using TexRAD research software (TexRAD Ltd, Cambridge, UK). Using a Cox regression model, correlation of texture parameters with measured time to progression and overall survival were assessed. Evaluation of combined International Metastatic Renal-Cell Carcinoma Database Consortium Model (IMDC) score with texture parameters was also performed. RESULTS: Size normalized standard deviation (nSD) alone at baseline and follow-up after treatment was a predictor of OS (Hazard ratio (HR) = 0.01 and 0.02; 95% confidence intervals (CI): 0.00 – 0.29 and 0.00 – 0.39; p = 0.01 and 0.01). Entropy following treatment and entropy change before and after treatment were both significant predictors of OS (HR = 2.68 and 87.77; 95% CI = 1.14 – 6.29 and 1.26 – 6115.69; p = 0.02 and p = 0.04). nSD was also a predictor of PFS at baseline and follow-up (HR = 0.01 and 0.01: 95% CI: 0.00 – 0.31 and 0.001 – 0.22; p = 0.01 and p = 0.003). When nSD at baseline or at follow-up was combined with IMDC, it improved the association with OS and PFS compared to IMDC alone. CONCLUSION: Size normalized standard deviation from CT at baseline and follow-up scans is correlated with OS and PFS in clear cell renal cell carcinoma treated with Sunitinib

Metastases or benign adrenal lesions in patients with histopathological verification of lung cancer: Can CT texture analysis distinguish?

INTRODUCTION: Distant metastases are found in the many of patients with lung cancer at time of diagnosis. Several diagnostic tools are available to distinguish between metastatic spread and benign lesions in the adrenal gland. However, all require additional diagnostic steps after the initial CT. The purpose of this study was to evaluate if texture analysis of CT-abnormal adrenal glands on the initial CT correctly differentiates between malignant and benign lesions in patients with confirmed lung cancer. MATERIALS AND METHODS: In this retrospective study 160 patients with endoscopic ultrasound-guided biopsy from the left adrenal gland and a contrast-enhanced CT in portal venous phase were assessed with texture analysis. A region of interest encircling the entire adrenal gland was used and from this dataset the slice with the largest cross section of the lesion was analyzed individually. RESULTS: Several texture parameters showed statistically significantly difference between metastatic and benign lesions but with considerable between-groups overlaps in confidence intervals. Sensitivity and specificity were assessed using ROC-curves, and in univariate binary logistic regression the area under the curve ranged from 36 % (Kurtosis 0.5) to 69 % (Entropy 2.5) compared to 73 % in the best fitting model using multivariate binary logistic regression. CONCLUSION: In lung cancer patients with abnormal adrenal gland at imaging, adrenal gland texture analyses appear not to have any role in discriminating benign from malignant lesions

The clinical utility of prostate cancer heterogeneity using texture analysis of multiparametric MRI

Purpose To determine if multiparametric MRI (mpMRI) derived filtration-histogram based texture analysis (TA) can differentiate between different Gleason scores (GS) and the D’Amico risk in prostate cancer. Methods We retrospectively studied patients whose pre-operative 1.5T mpMRI had shown a visible tumour and who subsequently underwent radical prostatectomy (RP). Guided by tumour location from the histopathology report, we drew a region of interest around the dominant visible lesion on a single axial slice on the T2, Apparent Diffusion Coefficient (ADC) map and early arterial phase post-contrast T1 image. We then performed TA with a filtration-histogram software (TexRAD -Feedback Medical Ltd, Cambridge, UK). We correlated GS and D’Amico risk with texture using the Spearman’s rank correlation test. Results We had 26 RP patients with an MR-visible tumour. Mean of positive pixels (MPP) on ADC showed a significant negative correlation with GS at coarse texture scales. MPP showed a significant negative correlation with GS without filtration and with medium filtration. MRI contrast texture without filtration showed a significant, negative correlation with D’Amico score. MR T2 texture showed a significant, negative correlation with the D’Amico risk, particularly at textures without filtration, medium texture scales and coarse texture scales. Conclusion ADC map mpMRI TA correlated negatively with GS, and T2 and post-contrast images with the D’Amico risk score. These associations may allow for better assessment of disease prognosis and a non-invasive method of follow-up for patients on surveillance. Further, identifying clinically significant prostate cancer is essential to reduce harm from over-diagnosis and over-treatment

CT texture analysis can help differentiate between malignant and benign lymph nodes in the mediastinum in patients suspected for lung cancer

BACKGROUND: In patients with non-small-cell lung carcinoma NSCLC the lymph node staging in the mediastinum is important due to impact on management and prognosis. Computed tomography texture analysis (CTTA) is a postprocessing technique that can evaluate the heterogeneity of marked regions in images. PURPOSE: To evaluate if CTTA can differentiate between malignant and benign lymph nodes in a cohort of patients with suspected lung cancer. MATERIAL AND METHODS: With tissue sampling as reference standard, 46 lymph nodes from 29 patients were analyzed using CTTA. For each lymph node, CTTA was performed using a research software "TexRAD" by drawing a region of interest (ROI) on all available axial contrast-enhanced computed tomography (CT) slices covering the entire volume of the lymph node. Lymph node CTTA comprised image filtration-histogram analysis undertakes two stages: the first step comprised an application of a Laplacian of Gaussian filter to highlight fine to coarse textures within the ROI, followed by a quantification of textures via histogram analysis using mean gray-level intensity from the entire volume of the lymph nodes. RESULTS: CTTA demonstrated a statistically significant difference between the malignant and the benign lymph nodes (P = 0.001), and by binary logistic regression we obtained a sensitivity of 53% and specificity of 97% in the test population. The area under the receiver operating curve was 83.4% and reproducibility was excellent. CONCLUSION: CTTA may be helpful in differentiating between malignant and benign lymph nodes in the mediastinum in patients suspected for lung cancer, with a low intra-observer variance

Can intracranial time-of-flight-MR angiography predict extracranial carotid artery stenosis?

Objectives: Extracranial stenosis of the internal carotid artery (ICA) is an important cause of ischemic stroke and transient ischemic attack (TIA). It can be diagnosed using contrast-enhanced CT or MR angiography (MRA) as well as Doppler ultrasound. In this study, we assessed the diagnostic value of intracranial time-of-flight (TOF) MRA to predict extracranial ICA stenosis (ICAS). Methods: We retrospectively analyzed consecutive patients with acute ischemic stroke or TIA and middle- (50-69%) or high-grade (70-99%) unilateral extracranial ICAS according to NASCET criteria assessed by ultrasound between January 2016 and August 2018. The control group consisted of patients without extracranial ICAS. Intraluminal signal intensities (SI) of the intracranial ICA on the side of the extracranial stenosis were compared to the contralesional side on TOF-MRA source images. SI ratios (SIR) of contralesional:lesional side were compared between groups. Results: In total, 151 patients were included in the main analysis. Contralesional:lesional SIR in the intracranial C4-segment was significantly higher in patients with ipsilateral extracranial ICA stenosis (n = 51, median 74 years, 57% male) compared to the control group (n = 100, median 68 years, 48% male). Mean SIR was 1.463 vs. 1.035 (p < 0.001) for right-sided stenosis and 1.362 vs. 1.000 (p < 0.001) for left-sided stenosis. Receiver-operating characteristic curve demonstrated a cut-off value of 1.086 for right-sided [sensitivity/specificity 75%/81%; area under the curve (AUC) 0.81] and 1.104 for left-sided stenosis (sensitivity/specificity 70%/84%; AUC 0.80) in C4 as a good predictor for high-grade extracranial ICAS. Conclusions: SIR on TOF-MRA can be a marker of extracranial ICAS