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.

Evaluating intra-articular drug delivery for the treatment of osteoarthritis in a rat model.

Osteoarthritis (OA) is a degenerative joint disease that can result in joint pain, loss of joint function, and deleterious effects on activity levels and lifestyle habits. Current therapies for OA are largely aimed at symptomatic relief and may have limited effects on the underlying cascade of joint degradation. Local drug delivery strategies may provide for the development of more successful OA treatment outcomes that have potential to reduce local joint inflammation, reduce joint destruction, offer pain relief, and restore patient activity levels and joint function. As increasing interest turns toward intra-articular drug delivery routes, parallel interest has emerged in evaluating drug biodistribution, safety, and efficacy in preclinical models. Rodent models provide major advantages for the development of drug delivery strategies, chiefly because of lower cost, successful replication of human OA-like characteristics, rapid disease development, and small joint volumes that enable use of lower total drug amounts during protocol development. These models, however, also offer the potential to investigate the therapeutic effects of local drug therapy on animal behavior, including pain sensitivity thresholds and locomotion characteristics. Herein, we describe a translational paradigm for the evaluation of an intra-articular drug delivery strategy in a rat OA model. This model, a rat interleukin-1beta overexpression model, offers the ability to evaluate anti-interleukin-1 therapeutics for drug biodistribution, activity, and safety as well as the therapeutic relief of disease symptoms. Once the action against interleukin-1 is confirmed in vivo, the newly developed anti-inflammatory drug can be evaluated for evidence of disease-modifying effects in more complex preclinical models.

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.

In vivo luminescence imaging of NF-κB activity and serum cytokine levels predict pain sensitivities in a rodent model of osteoarthritis.

OBJECTIVE: To investigate the relationship between NF-κB activity, cytokine levels, and pain sensitivities in a rodent model of osteoarthritis (OA). METHODS: OA was induced in transgenic NF-κB-luciferase reporter mice via intraarticular injection of monosodium iodoacetate (MIA). Using luminescence imaging we evaluated the temporal kinetics of NF-κB activity and its relationship to the development of pain sensitivities and serum cytokine levels in this model. RESULTS: MIA induced a transient increase in joint-related NF-κB activity at early time points (day 3 after injection) and an associated biphasic pain response (mechanical allodynia). NF-κB activity, serum interleukin-6 (IL-6), IL-1β, and IL-10 levels accounted for ∼75% of the variability in pain-related mechanical sensitivities in this model. Specifically, NF-κB activity was strongly correlated with mechanical allodynia and serum IL-6 levels in the inflammatory pain phase of this model (day 3), while serum IL-1β was strongly correlated with pain sensitivities in the chronic pain phase of the model (day 28). CONCLUSION: Our findings suggest that NF-κB activity, IL-6, and IL-1β may play distinct roles in pain sensitivity development in this model of arthritis and may distinguish the acute pain phase from the chronic pain phase. This study establishes luminescence imaging of NF-κB activity as a novel imaging biomarker of pain sensitivities in this model of OA.

Energy recovery in individuals with knee osteoarthritis.

OBJECTIVE: Pathological gaits have been shown to limit transfer between potential (PE) and kinetic (KE) energy during walking, which can increase locomotor costs. The purpose of this study was to examine whether energy exchange would be limited in people with knee osteoarthritis (OA). METHODS: Ground reaction forces during walking were collected from 93 subjects with symptomatic knee OA (self-selected and fast speeds) and 13 healthy controls (self-selected speed) and used to calculate their center of mass (COM) movements, PE and KE relationships, and energy recovery during a stride. Correlations and linear regressions examined the impact of energy fluctuation phase and amplitude, walking velocity, body mass, self-reported pain, and radiographic severity on recovery. Paired t-tests were run to compare energy recovery between cohorts. RESULTS: Symptomatic knee OA subjects displayed lower energetic recovery during self-selected walking speeds than healthy controls (P = 0.0018). PE and KE phase relationships explained the majority (66%) of variance in recovery. Recovery had a complex relationship with velocity and its change across speeds was significantly influenced by the self-selected walking speed of each subject. Neither radiographic OA scores nor subject self-reported measures demonstrated any relationship with energy recovery. CONCLUSIONS: Knee OA reduces effective exchange of PE and KE, potentially increasing the muscular work required to control movements of the COM. Gait retraining may return subjects to more normal patterns of energy exchange and allow them to reduce fatigue.

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.

Biomarkers of PTA

© Springer Science+Business Media New York 2015.Prognostic biomarkers may indicate the likelihood of disease development and speed of progression or may serve as predictive indicators of responsiveness to treatment. Joint injuries, particularly severe injuries, may result in post-traumatic osteoarthritis (PTOA), and pre- and post-injury prognostic biomarkers are needed to enhance primary and secondary prevention approaches for PTOA. Several macromolecules from joint structures found in serum, urine, and synovial fluid are promising biochemical markers for monitoring joint metabolism and health before and after joint injury. The use of metabolic profiling (analysis of small molecules) as a predictive tool for osteoarthritis (OA) has increased in the past decade. Although there is some question as to whether PTOA and idiopathic OA are comparable conditions, there is some evidence to suggest that components of their pathogenesis are similar. Potentially, biomarkers important to the high-risk PTOA profile translate to idiopathic OA. Further work is needed to confirm the utility of macromolecules and metabolites as biomarkers for PTOA, particularly focusing on those strongly correlated to clinical efficacy measures important to the patient (e.g., symptoms, physical function, and quality of life) and the causal pathway of PTOA.

Colchicine effectiveness in symptom and inflammation modification in knee osteoarthritis (COLKOA): study protocol for a randomized controlled trial.

BACKGROUND: Despite the high prevalence and global impact of knee osteoarthritis (KOA), current treatments are palliative. No disease modifying anti-osteoarthritic drug (DMOAD) has been approved. We recently demonstrated significant involvement of uric acid and activation of the innate immune response in osteoarthritis (OA) pathology and progression, suggesting that traditional gout therapy may be beneficial for OA. We therefore assess colchicine, an existing commercially available agent for gout, for a new therapeutic application in KOA. METHODS/DESIGN: COLKOA is a double-blind, placebo-controlled, randomized trial comparing a 16-week treatment with standard daily dose oral colchicine to placebo for KOA. A total of 120 participants with symptomatic KOA will be recruited from a single center in Singapore. The primary end point is 30% improvement in total Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) score at week 16. Secondary end points include improvement in pain, physical function, and quality of life and change in serum, urine and synovial fluid biomarkers of cartilage metabolism and inflammation. A magnetic resonance imaging (MRI) substudy will be conducted in 20 participants to evaluate change in synovitis. Logistic regression will be used to compare changes between groups in an intention-to-treat analysis. DISCUSSION: The COLKOA trial is designed to evaluate whether commercially available colchicine is effective for improving signs and symptoms of KOA, and reducing synovial fluid, serum and urine inflammatory and biochemical joint degradation biomarkers. These biomarkers should provide insights into the underlying mechanism of therapeutic response. This trial will potentially provide data to support a new treatment option for KOA. TRIAL REGISTRATION: The trial has been registered at clinicaltrials.gov as NCT02176460 . Date of registration: 26 June 2014.

Elucidating the Molecular Composition of Cartilage by Proteomics.

Articular cartilage consists of chondrocytes and two major components, a collagen-rich framework and highly abundant proteoglycans. Most prior studies defining the zonal distribution of cartilage have extracted proteins with guanidine-HCl. However, an unextracted collagen-rich residual is left after extraction. In addition, the high abundance of anionic polysaccharide molecules extracted from cartilage adversely affects the chromatographic separation. In this study, we established a method for removing chondrocytes from cartilage sections with minimal extracellular matrix protein loss. The addition of surfactant to guanidine-HCl extraction buffer improved protein solubility. Ultrafiltration removed interference from polysaccharides and salts. Almost four-times more collagen peptides were extracted by the in situ trypsin digestion method. However, as expected, proteoglycans were more abundant within the guanidine-HCl extraction. These different methods were used to extract cartilage sections from different cartilage layers (superficial, intermediate, and deep), joint types (knee and hip), and disease states (healthy and osteoarthritic), and the extractions were evaluated by quantitative and qualitative proteomic analyses. The results of this study led to the identifications of the potential biomarkers of osteoarthritis (OA), OA progression, and the joint specific biomarkers.

ENIGMA and the individual: Predicting factors that affect the brain in 35 countries worldwide.

In this review, we discuss recent work by the ENIGMA Consortium (http://enigma.ini.usc.edu) - a global alliance of over 500 scientists spread across 200 institutions in 35 countries collectively analyzing brain imaging, clinical, and genetic data. Initially formed to detect genetic influences on brain measures, ENIGMA has grown to over 30 working groups studying 12 major brain diseases by pooling and comparing brain data. In some of the largest neuroimaging studies to date - of schizophrenia and major depression - ENIGMA has found replicable disease effects on the brain that are consistent worldwide, as well as factors that modulate disease effects. In partnership with other consortia including ADNI, CHARGE, IMAGEN and others(1), ENIGMA's genomic screens - now numbering over 30,000 MRI scans - have revealed at least 8 genetic loci that affect brain volumes. Downstream of gene findings, ENIGMA has revealed how these individual variants - and genetic variants in general - may affect both the brain and risk for a range of diseases. The ENIGMA consortium is discovering factors that consistently affect brain structure and function that will serve as future predictors linking individual brain scans and genomic data. It is generating vast pools of normative data on brain measures - from tens of thousands of people - that may help detect deviations from normal development or aging in specific groups of subjects. We discuss challenges and opportunities in applying these predictors to individual subjects and new cohorts, as well as lessons we have learned in ENIGMA's efforts so far.