Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.

Distinct functions of POT1 at telomeres.

The mammalian protein POT1 binds to telomeric single-stranded DNA (ssDNA), protecting chromosome ends from being detected as sites of DNA damage. POT1 is composed of an N-terminal ssDNA-binding domain and a C-terminal protein interaction domain. With regard to the latter, POT1 heterodimerizes with the protein TPP1 to foster binding to telomeric ssDNA in vitro and binds the telomeric double-stranded-DNA-binding protein TRF2. We sought to determine which of these functions-ssDNA, TPP1, or TRF2 binding-was required to protect chromosome ends from being detected as DNA damage. Using separation-of-function POT1 mutants deficient in one of these three activities, we found that binding to TRF2 is dispensable for protecting telomeres but fosters robust loading of POT1 onto telomeric chromatin. Furthermore, we found that the telomeric ssDNA-binding activity and binding to TPP1 are required in cis for POT1 to protect telomeres. Mechanistically, binding of POT1 to telomeric ssDNA and association with TPP1 inhibit the localization of RPA, which can function as a DNA damage sensor, to telomeres.

Hyperpolarized Xe MR imaging of alveolar gas uptake in humans.

BACKGROUND: One of the central physiological functions of the lungs is to transfer inhaled gases from the alveoli to pulmonary capillary blood. However, current measures of alveolar gas uptake provide only global information and thus lack the sensitivity and specificity needed to account for regional variations in gas exchange. METHODS AND PRINCIPAL FINDINGS: Here we exploit the solubility, high magnetic resonance (MR) signal intensity, and large chemical shift of hyperpolarized (HP) (129)Xe to probe the regional uptake of alveolar gases by directly imaging HP (129)Xe dissolved in the gas exchange tissues and pulmonary capillary blood of human subjects. The resulting single breath-hold, three-dimensional MR images are optimized using millisecond repetition times and high flip angle radio-frequency pulses, because the dissolved HP (129)Xe magnetization is rapidly replenished by diffusive exchange with alveolar (129)Xe. The dissolved HP (129)Xe MR images display significant, directional heterogeneity, with increased signal intensity observed from the gravity-dependent portions of the lungs. CONCLUSIONS: The features observed in dissolved-phase (129)Xe MR images are consistent with gravity-dependent lung deformation, which produces increased ventilation, reduced alveolar size (i.e., higher surface-to-volume ratios), higher tissue densities, and increased perfusion in the dependent portions of the lungs. Thus, these results suggest that dissolved HP (129)Xe imaging reports on pulmonary function at a fundamental level.

Reevaluation of mid-Pliocene North Atlantic sea surface temperatures

Multiproxy temperature estimation requires careful attention to biological, chemical, physical, temporal, and calibration differences of each proxy and paleothermometry method. We evaluated mid-Pliocene sea surface temperature (SST) estimates from multiple proxies at Deep Sea Drilling Project Holes 552A, 609B, 607, and 606, transecting the North Atlantic Drift. SST estimates derived from faunal assemblages, foraminifer Mg/Ca, and alkenone unsaturation indices showed strong agreement at Holes 552A, 607, and 606 once differences in calibration, depth, and seasonality were addressed. Abundant extinct species and/or an unrecognized productivity signal in the faunal assemblage at Hole 609B resulted in exaggerated faunal-based SST estimates but did not affect alkenone-derived or Mg/Ca-derived estimates. Multiproxy mid-Pliocene North Atlantic SST estimates corroborate previous studies documenting high-latitude mid-Pliocene warmth and refine previous faunal-based estimates affected by environmental factors other than temperature. Multiproxy investigations will aid SST estimation in high-latitude areas sensitive to climate change and currently underrepresented in SST reconstructions. Copyright 2008 by the American Geophysical Union.

Functional imaging of numerical processing in adults and 4-y-old children.

Adult humans, infants, pre-school children, and non-human animals appear to share a system of approximate numerical processing for non-symbolic stimuli such as arrays of dots or sequences of tones. Behavioral studies of adult humans implicate a link between these non-symbolic numerical abilities and symbolic numerical processing (e.g., similar distance effects in accuracy and reaction-time for arrays of dots and Arabic numerals). However, neuroimaging studies have remained inconclusive on the neural basis of this link. The intraparietal sulcus (IPS) is known to respond selectively to symbolic numerical stimuli such as Arabic numerals. Recent studies, however, have arrived at conflicting conclusions regarding the role of the IPS in processing non-symbolic, numerosity arrays in adulthood, and very little is known about the brain basis of numerical processing early in development. Addressing the question of whether there is an early-developing neural basis for abstract numerical processing is essential for understanding the cognitive origins of our uniquely human capacity for math and science. Using functional magnetic resonance imaging (fMRI) at 4-Tesla and an event-related fMRI adaptation paradigm, we found that adults showed a greater IPS response to visual arrays that deviated from standard stimuli in their number of elements, than to stimuli that deviated in local element shape. These results support previous claims that there is a neurophysiological link between non-symbolic and symbolic numerical processing in adulthood. In parallel, we tested 4-y-old children with the same fMRI adaptation paradigm as adults to determine whether the neural locus of non-symbolic numerical activity in adults shows continuity in function over development. We found that the IPS responded to numerical deviants similarly in 4-y-old children and adults. To our knowledge, this is the first evidence that the neural locus of adult numerical cognition takes form early in development, prior to sophisticated symbolic numerical experience. More broadly, this is also, to our knowledge, the first cognitive fMRI study to test healthy children as young as 4 y, providing new insights into the neurophysiology of human cognitive development.

An active learning approach for rapid characterization of endothelial cells in human tumors.

Currently, no available pathological or molecular measures of tumor angiogenesis predict response to antiangiogenic therapies used in clinical practice. Recognizing that tumor endothelial cells (EC) and EC activation and survival signaling are the direct targets of these therapies, we sought to develop an automated platform for quantifying activity of critical signaling pathways and other biological events in EC of patient tumors by histopathology. Computer image analysis of EC in highly heterogeneous human tumors by a statistical classifier trained using examples selected by human experts performed poorly due to subjectivity and selection bias. We hypothesized that the analysis can be optimized by a more active process to aid experts in identifying informative training examples. To test this hypothesis, we incorporated a novel active learning (AL) algorithm into FARSIGHT image analysis software that aids the expert by seeking out informative examples for the operator to label. The resulting FARSIGHT-AL system identified EC with specificity and sensitivity consistently greater than 0.9 and outperformed traditional supervised classification algorithms. The system modeled individual operator preferences and generated reproducible results. Using the results of EC classification, we also quantified proliferation (Ki67) and activity in important signal transduction pathways (MAP kinase, STAT3) in immunostained human clear cell renal cell carcinoma and other tumors. FARSIGHT-AL enables characterization of EC in conventionally preserved human tumors in a more automated process suitable for testing and validating in clinical trials. The results of our study support a unique opportunity for quantifying angiogenesis in a manner that can now be tested for its ability to identify novel predictive and response biomarkers.

Scaling limits of a model for selection at two scales

The dynamics of a population undergoing selection is a central topic in evolutionary biology. This question is particularly intriguing in the case where selective forces act in opposing directions at two population scales. For example, a fast-replicating virus strain outcompetes slower-replicating strains at the within-host scale. However, if the fast-replicating strain causes host morbidity and is less frequently transmitted, it can be outcompeted by slower-replicating strains at the between-host scale. Here we consider a stochastic ball-and-urn process which models this type of phenomenon. We prove the weak convergence of this process under two natural scalings. The first scaling leads to a deterministic nonlinear integro-partial differential equation on the interval $[0,1]$ with dependence on a single parameter, $\lambda$. We show that the fixed points of this differential equation are Beta distributions and that their stability depends on $\lambda$ and the behavior of the initial data around $1$. The second scaling leads to a measure-valued Fleming-Viot process, an infinite dimensional stochastic process that is frequently associated with a population genetics.

Perceived cannabis use norms and cannabis use among adolescents in the United States.

Due to changes in cannabis policies, concerns about cannabis use (CU) in adolescents have increased. The population of nonwhite groups is growing quickly in the United States. We examined perceived CU norms and their association with CU and CU disorder (CUD) for White, Black, Hispanic, Native-American, Asian-American, Native Hawaiian/Pacific Islander (NH/PI), and mixed-race adolescents. Data were from adolescents (12-17 years) in the 2004-2012 National Surveys on Drug Use and Health (N = 163,837). Substance use and CUD were assessed by computer-assisted, self-interviewing methods. Blacks, Hispanics, Native-Americans, and mixed-race adolescents had greater odds of past-year CU and CUD than Whites. Among past-year cannabis users (CUs), Hispanics and Native-Americans had greater odds of having a CUD than Whites. Asian-Americans had the highest prevalence of perceived parental or close friends' CU disapproval. Native-Americans and mixed-race adolescents had lower odds than Whites of perceiving CU disapproval from parents or close friends. In adjusted analyses, adolescent's disapproval of CU, as well as perceived disapproval by parents or close friends, were associated with a decreased odds of CU in each racial/ethnic group, except for NHs/PIs. Adolescent's disapproval of CU was associated with a decreased odds of CUD among CUs for Whites (personal, parental, and close friends' disapproval), Hispanics (personal, parental, and close friends' disapproval), and mixed-race adolescents (personal, close friends' disapproval). Racial/ethnic differences in adolescent CU prevalence were somewhat consistent with adolescents' reports of CU norm patterns. Longitudinal research on CU health effects should oversample nonwhite adolescents to assure an adequate sample for analysis and reporting.

Effects of chronic mild traumatic brain injury on white matter integrity in iraq and afghanistan war veterans

Mild traumatic brain injury (TBI) is a common source of morbidity from the wars in Iraq and Afghanistan. With no overt lesions on structural MRI, diagnosis of chronic mild TBI in military veterans relies on obtaining an accurate history and assessment of behavioral symptoms that are also associated with frequent comorbid disorders, particularly posttraumatic stress disorder (PTSD) and depression. Military veterans from Iraq and Afghanistan with mild TBI (n = 30) with comorbid PTSD and depression and non-TBI participants from primary (n = 42) and confirmatory (n = 28) control groups were assessed with high angular resolution diffusion imaging (HARDI). White matter-specific registration followed by whole-brain voxelwise analysis of crossing fibers provided separate partial volume fractions reflecting the integrity of primary fibers and secondary (crossing) fibers. Loss of white matter integrity in primary fibers (P < 0.05; corrected) was associated with chronic mild TBI in a widely distributed pattern of major fiber bundles and smaller peripheral tracts including the corpus callosum (genu, body, and splenium), forceps minor, forceps major, superior and posterior corona radiata, internal capsule, superior longitudinal fasciculus, and others. Distributed loss of white matter integrity correlated with duration of loss of consciousness and most notably with "feeling dazed or confused," but not diagnosis of PTSD or depressive symptoms. This widespread spatial extent of white matter damage has typically been reported in moderate to severe TBI. The diffuse loss of white matter integrity appears consistent with systemic mechanisms of damage shared by blast- and impact-related mild TBI that involves a cascade of inflammatory and neurochemical events. © 2012 Wiley Periodicals, Inc.

Using electronic health record data for substance use Screening, Brief Intervention, and Referral to Treatment among adults with type 2 diabetes: Design of a National Drug Abuse Treatment Clinical Trials Network study.

BACKGROUND: The Affordable Care Act encourages healthcare systems to integrate behavioral and medical healthcare, as well as to employ electronic health records (EHRs) for health information exchange and quality improvement. Pragmatic research paradigms that employ EHRs in research are needed to produce clinical evidence in real-world medical settings for informing learning healthcare systems. Adults with comorbid diabetes and substance use disorders (SUDs) tend to use costly inpatient treatments; however, there is a lack of empirical data on implementing behavioral healthcare to reduce health risk in adults with high-risk diabetes. Given the complexity of high-risk patients' medical problems and the cost of conducting randomized trials, a feasibility project is warranted to guide practical study designs. METHODS: We describe the study design, which explores the feasibility of implementing substance use Screening, Brief Intervention, and Referral to Treatment (SBIRT) among adults with high-risk type 2 diabetes mellitus (T2DM) within a home-based primary care setting. Our study includes the development of an integrated EHR datamart to identify eligible patients and collect diabetes healthcare data, and the use of a geographic health information system to understand the social context in patients' communities. Analysis will examine recruitment, proportion of patients receiving brief intervention and/or referrals, substance use, SUD treatment use, diabetes outcomes, and retention. DISCUSSION: By capitalizing on an existing T2DM project that uses home-based primary care, our study results will provide timely clinical information to inform the designs and implementation of future SBIRT studies among adults with multiple medical conditions.