Convergence of numerical time-averaging and stationary measures via Poisson equations

Numerical approximation of the long time behavior of a stochastic di.erential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantages of this approach are its simplicity and universality. It works equally well for a range of explicit and implicit schemes, including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus, and we study only smooth test functions. However, we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit

We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31 (2008), pp. 334-368] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths and is close to a convection CFL condition for large mean free paths. Our analysis is based on very simple energy estimates. © 2010 Society for Industrial and Applied Mathematics.

Postoperative statin use and risk of biochemical recurrence following radical prostatectomy: results from the Shared Equal Access Regional Cancer Hospital (SEARCH) database.

OBJECTIVE: To investigate the effect of statin use after radical prostatectomy (RP) on biochemical recurrence (BCR) in patients with prostate cancer who never received statins before RP. PATIENTS AND METHODS: We conducted a retrospective analysis of 1146 RP patients within the Shared Equal Access Regional Cancer Hospital (SEARCH) database. Multivariable Cox proportional hazards analyses were used to examine differences in risk of BCR between post-RP statin users vs nonusers. To account for varying start dates and duration of statin use during follow-up, post-RP statin use was treated as a time-dependent variable. In a secondary analysis, models were stratified by race to examine the association of post-RP statin use with BCR among black and non-black men. RESULTS: After adjusting for clinical and pathological characteristics, post-RP statin use was significantly associated with 36% reduced risk of BCR (hazard ratio [HR] 0.64, 95% confidence interval [CI] 0.47-0.87; P = 0.004). Post-RP statin use remained associated with reduced risk of BCR after adjusting for preoperative serum cholesterol levels. In secondary analysis, after stratification by race, this protective association was significant in non-black (HR 0.49, 95% CI 0.32-0.75; P = 0.001) but not black men (HR 0.82, 95% CI 0.53-1.28; P = 0.384). CONCLUSION: In this retrospective cohort of men undergoing RP, post-RP statin use was significantly associated with reduced risk of BCR. Whether the association between post-RP statin use and BCR differs by race requires further study. Given these findings, coupled with other studies suggesting that statins may reduce risk of advanced prostate cancer, randomised controlled trials are warranted to formally test the hypothesis that statins slow prostate cancer progression.

Intracardiac acoustic radiation force impulse imaging: a novel imaging method for intraprocedural evaluation of radiofrequency ablation lesions.

BACKGROUND: Arrhythmia recurrence after cardiac radiofrequency ablation (RFA) for atrial fibrillation has been linked to conduction through discontinuous lesion lines. Intraprocedural visualization and corrective ablation of lesion line discontinuities could decrease postprocedure atrial fibrillation recurrence. Intracardiac acoustic radiation force impulse (ARFI) imaging is a new imaging technique that visualizes RFA lesions by mapping the relative elasticity contrast between compliant-unablated and stiff RFA-treated myocardium. OBJECTIVE: To determine whether intraprocedure ARFI images can identify RFA-treated myocardium in vivo. METHODS: In 8 canines, an electroanatomical mapping-guided intracardiac echo catheter was used to acquire 2-dimensional ARFI images along right atrial ablation lines before and after RFA. ARFI images were acquired during diastole with the myocardium positioned at the ARFI focus (1.5 cm) and parallel to the intracardiac echo transducer for maximal and uniform energy delivery to the tissue. Three reviewers categorized each ARFI image as depicting no lesion, noncontiguous lesion, or contiguous lesion. For comparison, 3 separate reviewers confirmed RFA lesion presence and contiguity on the basis of functional conduction block at the imaging plane location on electroanatomical activation maps. RESULTS: Ten percent of ARFI images were discarded because of motion artifacts. Reviewers of the ARFI images detected RFA-treated sites with high sensitivity (95.7%) and specificity (91.5%). Reviewer identification of contiguous lesions had 75.3% specificity and 47.1% sensitivity. CONCLUSIONS: Intracardiac ARFI imaging was successful in identifying endocardial RFA treatment when specific imaging conditions were maintained. Further advances in ARFI imaging technology would facilitate a wider range of imaging opportunities for clinical lesion evaluation.

Micro-Anatomical Quantitative Imaging Towards Enabling Automated Diagnosis of Thick Tissues at the Point of Care

Histopathology is the clinical standard for tissue diagnosis. However, histopathology has several limitations including that it requires tissue processing, which can take 30 minutes or more, and requires a highly trained pathologist to diagnose the tissue. Additionally, the diagnosis is qualitative, and the lack of quantitation leads to possible observer-specific diagnosis. Taken together, it is difficult to diagnose tissue at the point of care using histopathology.

Several clinical situations could benefit from more rapid and automated histological processing, which could reduce the time and the number of steps required between obtaining a fresh tissue specimen and rendering a diagnosis. For example, there is need for rapid detection of residual cancer on the surface of tumor resection specimens during excisional surgeries, which is known as intraoperative tumor margin assessment. Additionally, rapid assessment of biopsy specimens at the point-of-care could enable clinicians to confirm that a suspicious lesion is successfully sampled, thus preventing an unnecessary repeat biopsy procedure. Rapid and low cost histological processing could also be potentially useful in settings lacking the human resources and equipment necessary to perform standard histologic assessment. Lastly, automated interpretation of tissue samples could potentially reduce inter-observer error, particularly in the diagnosis of borderline lesions.

To address these needs, high quality microscopic images of the tissue must be obtained in rapid timeframes, in order for a pathologic assessment to be useful for guiding the intervention. Optical microscopy is a powerful technique to obtain high-resolution images of tissue morphology in real-time at the point of care, without the need for tissue processing. In particular, a number of groups have combined fluorescence microscopy with vital fluorescent stains to visualize micro-anatomical features of thick (i.e. unsectioned or unprocessed) tissue. However, robust methods for segmentation and quantitative analysis of heterogeneous images are essential to enable automated diagnosis. Thus, the goal of this work was to obtain high resolution imaging of tissue morphology through employing fluorescence microscopy and vital fluorescent stains and to develop a quantitative strategy to segment and quantify tissue features in heterogeneous images, such as nuclei and the surrounding stroma, which will enable automated diagnosis of thick tissues.

To achieve these goals, three specific aims were proposed. The first aim was to develop an image processing method that can differentiate nuclei from background tissue heterogeneity and enable automated diagnosis of thick tissue at the point of care. A computational technique called sparse component analysis (SCA) was adapted to isolate features of interest, such as nuclei, from the background. SCA has been used previously in the image processing community for image compression, enhancement, and restoration, but has never been applied to separate distinct tissue types in a heterogeneous image. In combination with a high resolution fluorescence microendoscope (HRME) and a contrast agent acriflavine, the utility of this technique was demonstrated through imaging preclinical sarcoma tumor margins. Acriflavine localizes to the nuclei of cells where it reversibly associates with RNA and DNA. Additionally, acriflavine shows some affinity for collagen and muscle. SCA was adapted to isolate acriflavine positive features or APFs (which correspond to RNA and DNA) from background tissue heterogeneity. The circle transform (CT) was applied to the SCA output to quantify the size and density of overlapping APFs. The sensitivity of the SCA+CT approach to variations in APF size, density and background heterogeneity was demonstrated through simulations. Specifically, SCA+CT achieved the lowest errors for higher contrast ratios and larger APF sizes. When applied to tissue images of excised sarcoma margins, SCA+CT correctly isolated APFs and showed consistently increased density in tumor and tumor + muscle images compared to images containing muscle. Next, variables were quantified from images of resected primary sarcomas and used to optimize a multivariate model. The sensitivity and specificity for differentiating positive from negative ex vivo resected tumor margins was 82% and 75%. The utility of this approach was further tested by imaging the in vivo tumor cavities from 34 mice after resection of a sarcoma with local recurrence as a bench mark. When applied prospectively to images from the tumor cavity, the sensitivity and specificity for differentiating local recurrence was 78% and 82%. The results indicate that SCA+CT can accurately delineate APFs in heterogeneous tissue, which is essential to enable automated and rapid surveillance of tissue pathology.

Two primary challenges were identified in the work in aim 1. First, while SCA can be used to isolate features, such as APFs, from heterogeneous images, its performance is limited by the contrast between APFs and the background. Second, while it is feasible to create mosaics by scanning a sarcoma tumor bed in a mouse, which is on the order of 3-7 mm in any one dimension, it is not feasible to evaluate an entire human surgical margin. Thus, improvements to the microscopic imaging system were made to (1) improve image contrast through rejecting out-of-focus background fluorescence and to (2) increase the field of view (FOV) while maintaining the sub-cellular resolution needed for delineation of nuclei. To address these challenges, a technique called structured illumination microscopy (SIM) was employed in which the entire FOV is illuminated with a defined spatial pattern rather than scanning a focal spot, such as in confocal microscopy.

Thus, the second aim was to improve image contrast and increase the FOV through employing wide-field, non-contact structured illumination microscopy and optimize the segmentation algorithm for new imaging modality. Both image contrast and FOV were increased through the development of a wide-field fluorescence SIM system. Clear improvement in image contrast was seen in structured illumination images compared to uniform illumination images. Additionally, the FOV is over 13X larger than the fluorescence microendoscope used in aim 1. Initial segmentation results of SIM images revealed that SCA is unable to segment large numbers of APFs in the tumor images. Because the FOV of the SIM system is over 13X larger than the FOV of the fluorescence microendoscope, dense collections of APFs commonly seen in tumor images could no longer be sparsely represented, and the fundamental sparsity assumption associated with SCA was no longer met. Thus, an algorithm called maximally stable extremal regions (MSER) was investigated as an alternative approach for APF segmentation in SIM images. MSER was able to accurately segment large numbers of APFs in SIM images of tumor tissue. In addition to optimizing MSER for SIM image segmentation, an optimal frequency of the illumination pattern used in SIM was carefully selected because the image signal to noise ratio (SNR) is dependent on the grid frequency. A grid frequency of 31.7 mm-1 led to the highest SNR and lowest percent error associated with MSER segmentation.

Once MSER was optimized for SIM image segmentation and the optimal grid frequency was selected, a quantitative model was developed to diagnose mouse sarcoma tumor margins that were imaged ex vivo with SIM. Tumor margins were stained with acridine orange (AO) in aim 2 because AO was found to stain the sarcoma tissue more brightly than acriflavine. Both acriflavine and AO are intravital dyes, which have been shown to stain nuclei, skeletal muscle, and collagenous stroma. A tissue-type classification model was developed to differentiate localized regions (75x75 µm) of tumor from skeletal muscle and adipose tissue based on the MSER segmentation output. Specifically, a logistic regression model was used to classify each localized region. The logistic regression model yielded an output in terms of probability (0-100%) that tumor was located within each 75x75 µm region. The model performance was tested using a receiver operator characteristic (ROC) curve analysis that revealed 77% sensitivity and 81% specificity. For margin classification, the whole margin image was divided into localized regions and this tissue-type classification model was applied. In a subset of 6 margins (3 negative, 3 positive), it was shown that with a tumor probability threshold of 50%, 8% of all regions from negative margins exceeded this threshold, while over 17% of all regions exceeded the threshold in the positive margins. Thus, 8% of regions in negative margins were considered false positives. These false positive regions are likely due to the high density of APFs present in normal tissues, which clearly demonstrates a challenge in implementing this automatic algorithm based on AO staining alone.

Thus, the third aim was to improve the specificity of the diagnostic model through leveraging other sources of contrast. Modifications were made to the SIM system to enable fluorescence imaging at a variety of wavelengths. Specifically, the SIM system was modified to enabling imaging of red fluorescent protein (RFP) expressing sarcomas, which were used to delineate the location of tumor cells within each image. Initial analysis of AO stained panels confirmed that there was room for improvement in tumor detection, particularly in regards to false positive regions that were negative for RFP. One approach for improving the specificity of the diagnostic model was to investigate using a fluorophore that was more specific to staining tumor. Specifically, tetracycline was selected because it appeared to specifically stain freshly excised tumor tissue in a matter of minutes, and was non-toxic and stable in solution. Results indicated that tetracycline staining has promise for increasing the specificity of tumor detection in SIM images of a preclinical sarcoma model and further investigation is warranted.

In conclusion, this work presents the development of a combination of tools that is capable of automated segmentation and quantification of micro-anatomical images of thick tissue. When compared to the fluorescence microendoscope, wide-field multispectral fluorescence SIM imaging provided improved image contrast, a larger FOV with comparable resolution, and the ability to image a variety of fluorophores. MSER was an appropriate and rapid approach to segment dense collections of APFs from wide-field SIM images. Variables that reflect the morphology of the tissue, such as the density, size, and shape of nuclei and nucleoli, can be used to automatically diagnose SIM images. The clinical utility of SIM imaging and MSER segmentation to detect microscopic residual disease has been demonstrated by imaging excised preclinical sarcoma margins. Ultimately, this work demonstrates that fluorescence imaging of tissue micro-anatomy combined with a specialized algorithm for delineation and quantification of features is a means for rapid, non-destructive and automated detection of microscopic disease, which could improve cancer management in a variety of clinical scenarios.

Genetic variants in the TEP1 gene are associated with prostate cancer risk and recurrence.

BACKGROUND: Telomere-related genes play an important role in carcinogenesis and progression of prostate cancer (PCa). It is not fully understood whether genetic variations in telomere-related genes are associated with development and progression in PCa patients. METHODS: Six potentially functional single-nucleotide polymorphisms (SNPs) of three key telomere-related genes were evaluated in 1015 PCa cases and 1052 cancer-free controls, to test their associations with risk of PCa. Among 426 PCa patients who underwent radical prostatectomy (RP), the prognostic significance of the studied SNPs on biochemical recurrence (BCR) was also assessed using the Kaplan-Meier analysis and Cox proportional hazards regression model. The relative telomere lengths (RTLs) were measured in peripheral blood leukocytes using real-time PCR in the RP patients. RESULTS: TEP1 rs1760904 AG/AA genotypes were significantly associated with a decreased risk of PCa (odds ratio (OR): 0.77, 95% confidence interval (CI): 0.64-0.93, P=0.005) compared with the GG genotype. By using median RTL as a cutoff level, RP patients with TEP1 rs1760904 AG/AA genotypes tended to have a longer RTL than those with the GG genotype (OR: 1.55, 95% CI: 1.04-2.30, P=0.031). A significant interaction between TEP1 rs1713418 and age in modifying PCa risk was observed (P=0.005). After adjustment for clinicopathologic risk factors, the presence of heterozygotes or rare homozygotes of TEP1 rs1760904 and TNKS2 rs1539042 were associated with BCR in the RP cohorts (hazard ratio: 0.53, 95% CI: 0.36-0.79, P=0.002 and hazard ratio: 1.67, 95% CI: 1.07-2.48, P=0.017, respectively). CONCLUSIONS: These data suggest that genetic variations in the TEP1 gene may be biomarkers for risk of PCa and BCR after RP.

Apoptotic variants as predictors of risk of oropharyngeal cancer recurrence after definitive radiotherapy.

Single nucleotide polymorphisms (SNPs) in the promoter region of FAS and FASLG may alter their transcriptional activity. Thus, we determined the associations between four FAS and FASLG promoter variants (FAS1377G>A, rs2234767; 670A>G, rs1800682; FASLG844T>C, rs763110 and 124A>G, rs5030772) and the risk of recurrence of squamous cell carcinoma of the oropharynx (SCCOP). We evaluated the associations between FAS and FASLG genetic variants and the risk of recurrence in a cohort of 1,008 patients. The log-rank test and multivariate Cox models were used to evaluate the associations. Compared with patients with common homozygous genotypes of FAS670 and FASLG844 polymorphisms, patients with variant genotypes had lower disease-free survival rates (log-rank p < 0.0001 and p < 0.0001, respectively) and an approximately threefold higher risk of SCCOP recurrence (HR, 3.2;95% CI, 2.2-4.6; and HR, 3.1; 95% CI, 2.2-4.4, respectively) after multivariate adjustment. Furthermore, among patients with HPV16-positive tumors, those with variant genotypes of these two polymorphisms had lower disease-free survival rates (log-rank, p < 0.0001 and p < 0.0001, respectively) and a higher recurrence risk than did patients with common homozygous genotypes (HR, 12.9; 95% CI, 3.8-43.6; and HR, 8.1; 95% CI, 3.6-18.6, respectively), whereas no significant associations were found for FAS1377 and FASLG124 polymorphisms. Our findings suggest that FAS670 and FASLG844 polymorphisms modulate the risk of recurrence of SCCOP, particularly in patients with HPV16-positive tumors. Larger studies are needed to validate these results.

Enhancing the Hodgkin-Huxley Equations: Simulations Based on the First Publication in the Biophysical Journal.

The experiments in the Cole and Moore article in the first issue of the Biophysical Journal provided the first independent experimental confirmation of the Hodgkin-Huxley (HH) equations. A log-log plot of the K current versus time showed that raising the HH variable n to the sixth power provided the best fit to the data. Subsequent simulations using n(6) and setting the resting potential at the in vivo value simplifies the HH equations by eliminating the leakage term. Our article also reported that the K current in response to a depolarizing step to ENa was delayed if the step was preceded by a hyperpolarization. While the interpretation of this phenomenon in the article was flawed, subsequent simulations show that the effect completely arises from the original HH equations.