Counting Cards: Combinatorics, Group Theory, and Probability in War

My project involved taking a look at the card game War, searching for patterns which develop in the game, and then looking for Mathematical explanations for those patterns. The project began when my advisor, Professor Marc Brodie, was playing War with his children and began to notice patterns and ask questions about them. The questions I set out to answer were: What is the probability of playing a game of war in which a loop develops? If we know the size of the deck we are using, can we determine what loop lengths are possible? How are cards cycling between players within a loop? What patterns of winning occur within a loop? What effect does changing the number of suits in the deck or the number of players have on the cycling of cards within a loop and the loop length? We found at least partial answers to all of these questions and more using Mathematica programs of simulated games along with basic theory from Combinatorics, Group Theory, and Probability

Influence of threshold selection and image sequence in in-vivo segmentation of enlarged perivascular spaces

BACKGROUND: Growing interest surrounds perivascular spaces (PVS) as a clinical biomarker of brain dysfunction given their association with cerebrovascular risk factors and disease. Neuroimaging techniques allowing quick and reliable quantification are being developed, but, in practice, they require optimisation as their limits of validity are usually unspecified.NEW METHOD: We evaluate modifications and alternatives to a state-of-the-art (SOTA) PVS segmentation method that uses a vesselness filter to enhance PVS discrimination, followed by thresholding of its response, applied to brain magnetic resonance images (MRI) from patients with sporadic small vessel disease acquired at 3 T.RESULTS: The method is robust against inter-observer differences in threshold selection, but separate thresholds for each region of interest (i.e., basal ganglia, centrum semiovale, and midbrain) are required. Noise needs to be assessed prior to selecting these thresholds, as effect of noise and imaging artefacts can be mitigated with a careful optimisation of these thresholds. PVS segmentation from T1-weighted images alone, misses small PVS, therefore, underestimates PVS count, may overestimate individual PVS volume especially in the basal ganglia, and is susceptible to the inclusion of calcified vessels and mineral deposits. Visual analyses indicated the incomplete and fragmented detection of long and thin PVS as the primary cause of errors, with the Frangi filter coping better than the Jerman filter.COMPARISON WITH EXISTING METHODS: Limits of validity to a SOTA PVS segmentation method applied to 3 T MRI with confounding pathology are given.CONCLUSIONS: Evidence presented reinforces the STRIVE-2 recommendation of using T2-weighted images for PVS assessment wherever possible. The Frangi filter is recommended for PVS segmentation from MRI, offering robust output against variations in threshold selection and pathology presentation.</p

Relationship Between Venules and Perivascular Spaces in Sporadic Small Vessel Diseases

Background and Purpose— Perivascular spaces (PVS) around venules may help drain interstitial fluid from the brain. We examined relationships between suspected venules and PVS visible on brain magnetic resonance imaging. Methods— We developed a visual venular quantification method to examine the spatial relationship between venules and PVS. We recruited patients with lacunar stroke or minor nondisabling ischemic stroke and performed brain magnetic resonance imaging and retinal imaging. We quantified venules on gradient echo or susceptibility-weighted imaging and PVS on T2-weighted magnetic resonance imaging in the centrum semiovale and then determined overlap between venules and PVS. We assessed associations between venular count and patient demographic characteristics, vascular risk factors, small vessel disease features, retinal vessels, and venous sinus pulsatility. Results— Among 67 patients (69% men, 69.0±9.8 years), only 4.6% (range, 0%–18%) of venules overlapped with PVS. Total venular count increased with total centrum semiovale PVS count in 55 patients after accounting for venule-PVS overlap (β=0.468 [95% CI, 0.187–0.750]) and transverse sinus pulsatility (β=0.547 [95% CI, 0.309–0.786]) and adjusting for age, sex, and systolic blood pressure. Conclusions— Despite increases in both visible PVS and suspected venules, we found minimal spatial overlap between them in patients with sporadic small vessel disease, suggesting that most magnetic resonance imaging-visible centrum semiovale PVS are periarteriolar rather than perivenular

Magnetic Resonance Imaging Tissue Signatures Associated With White Matter Changes Due to Sporadic Cerebral Small Vessel Disease Indicate That White Matter Hyperintensities Can Regress

Background White matter hyperintensities (WMHs) might regress and progress contemporaneously, but we know little about underlying mechanisms. We examined WMH change and underlying quantitative magnetic resonance imaging tissue measures over 1 year in patients with minor ischemic stroke with sporadic cerebral small vessel disease. Methods and Results We defined areas of stable normal‐appearing white matter, stable WMHs, progressing and regressing WMHs based on baseline and 1‐year brain magnetic resonance imaging. In these areas we assessed tissue characteristics with quantitative T1, fractional anisotropy (FA), mean diffusivity (MD), and neurite orientation dispersion and density imaging (baseline only). We compared tissue signatures cross‐sectionally between areas, and longitudinally within each area. WMH change masks were available for N=197. Participants' mean age was 65.61 years (SD, 11.10), 59% had a lacunar infarct, and 68% were men. FA and MD were available for N=195, quantitative T1 for N=182, and neurite orientation dispersion and density imaging for N=174. Cross‐sectionally, all 4 tissue classes differed for FA, MD, T1, and Neurite Density Index. Longitudinally, in regressing WMHs, FA increased with little change in MD and T1 (difference estimate, 0.011 [95% CI, 0.006–0.017]; −0.002 [95% CI, −0.008 to 0.003] and −0.003 [95% CI, −0.009 to 0.004]); in progressing and stable WMHs, FA decreased (−0.022 [95% CI, −0.027 to −0.017] and −0.009 [95% CI, −0.011 to −0.006]), whereas MD and T1 increased (progressing WMHs, 0.057 [95% CI, 0.050–0.063], 0.058 [95% CI, 0.050 –0.066]; stable WMHs, 0.054 [95% CI, 0.045–0.063], 0.049 [95% CI, 0.039–0.058]); and in stable normal‐appearing white matter, MD increased (0.004 [95% CI, 0.003–0.005]), whereas FA and T1 slightly decreased and increased (−0.002 [95% CI, −0.004 to −0.000] and 0.005 [95% CI, 0.001–0.009]). Conclusions Quantitative magnetic resonance imaging shows that WMHs that regress have less abnormal microstructure at baseline than stable WMHs and follow trajectories indicating tissue improvement compared with stable and progressing WMHs

“Getting pregnant during COVID-19 was a big risk because getting help from the clinic was not easy”: COVID-19 experiences of women and healthcare providers in Harare, Zimbabwe

The COVID-19 pandemic and associated measures may have disrupted delivery of maternal and neonatal health services and reversed the progress made towards dual elimination of mother-to-child transmission of HIV and syphilis in Zimbabwe. This qualitative study explores the impact of the pandemic on the provision and uptake of prevention of mother-to-child transmission (PMTCT) services from the perspectives of women and maternal healthcare providers. Longitudinal in-depth interviews were conducted with 20 pregnant and breastfeeding women aged 20–39 years living with HIV and 20 healthcare workers in two maternity polyclinics in low-income suburbs of Harare, Zimbabwe. Semi-structured interviews were held after the second and third waves of COVID-19 in March and November 2021, respectively. Data were analysed using a modified grounded theory approach. While eight antenatal care contacts are recommended by Zimbabwe’s Ministry of Health and Child Care, women reported only being able to access two contacts. Although HIV testing, antiretroviral therapy (ART) refills and syphilis screening services were accessible at first contact, other services such as HIV-viral load monitoring and enhanced adherence counselling were not available for those on ART. Closure of clinics and shortened operating hours during the second COVID-19 wave resulted in more antenatal bookings occurring later during pregnancy and more home deliveries. Six of the 20 (33%) interviewed women reported giving birth at home, assisted by untrained traditional midwives as clinics were closed. Babies delivered at home missed ART prophylaxis and HIV testing at birth despite being HIV-exposed. Although women faced multiple challenges, they continued to attempt to access services after delivery. These findings underline the importance of investing in robust health systems that can respond to emergency situations to ensure continuity of essential HIV prevention, treatment, and care services

Impact of the COVID-19 pandemic on the provision and uptake of services for the prevention of mother-to-child transmission of HIV in Zimbabwe

Zimbabwe is targeting elimination of mother-to-child transmission of HIV by December 2025, however the COVID-19 pandemic challenged health service delivery globally. Monthly aggregated data were extracted from DHIS-2 for all facilities delivering antenatal care (ANC). ZIMSTAT and Spectrum demographic estimates were used for population-level denominators. Programme indicators are among those in HIV care and population indicators reflect the total population. The mean estimated proportion of pregnant women booking for ANC per month did not change (91% pre-pandemic vs 91% during pandemic, p = 0.95), despite dropping to 47% in April 2020. At a programme-level, the estimated proportion of women who received at least one HIV test fell in April 2020 (3.6% relative reduction vs March (95% CI 2.2–5.1), p<0.001) with gradual recovery towards pre-pandemic levels. The estimated proportion of women who were retested among those initially negative in pregnancy fell markedly in April 2020 (39% reduction (32–45%), p<0.001) and the subsequent increase was much slower, only reaching 39% by September 2021 compared to average 53% pre-pandemic. The mean estimated proportion of pregnant women with HIV on ART was unchanged at programme-level (98% vs 98%, p = 0.26), but decreased at population-level (86% vs 80%, p = 0.049). Antiretroviral prophylaxis coverage decreased among HIV-exposed infants, at programme- (94% vs 87%, p = 0.001) and population-levels (76% vs 68%, p<0.001). There was no significant change in HIV-exposed infants receiving EID (programme: 107% vs 103%, p = 0.52; population: 87% vs 79%, p = 0.081). The estimated proportion of infants with HIV diagnosed fell from 27% to 18%, (p<0.001), while the estimated proportion on ART was stable at a programme (88% vs 90%, p = 0.82) but not population (22% vs 16%, p = 0.004) level. Despite a drop at the start of the pandemic most programme indicators rapidly recovered. At a population-level indicators were slower to return, suggesting less women with HIV identified in care

Effects of the COVID-19 pandemic on the outcomes of HIV-exposed neonates: a Zimbabwean tertiary hospital experience

Introduction: The COVID-19 pandemic has globally impacted health service access, delivery and resources. There are limited data regarding the impact on the prevention of mother to child transmission (PMTCT) service delivery in low-resource settings. Neotree (www.neotree.org) combines data collection, clinical decision support and education to improve care for neonates. Here we evaluate impacts of COVID-19 on care for HIV-exposed neonates. Methods: Data on HIV-exposed neonates admitted to the neonatal unit (NNU) at Sally Mugabe Central Hospital, Zimbabwe, between 01/06/2019 and 31/12/2021 were analysed, with pandemic start defined as 21/03/2020 and periods of industrial action (doctors (September 2019-January 2020) and nurses (June 2020-September 2020)) included, resulting in modelling during six time periods: pre-doctors’ strike (baseline); doctors’ strike; post-doctors’ strike and pre-COVID; COVID and pre-nurses’ strike; nurses’ strike; post nurses’ strike. Interrupted time series models were used to explore changes in indicators over time. Results: Of 8,333 neonates admitted to the NNU, 904 (11%) were HIV-exposed. Mothers of 706/765 (92%) HIV-exposed neonates reported receipt of antiretroviral therapy (ART) during pregnancy. Compared to the baseline period when average admissions were 78 per week (95% confidence interval (CI) 70–87), significantly fewer neonates were admitted during all subsequent periods until after the nurses’ strike, with the lowest average number during the nurses’ strike (28, 95% CI 23–34, p 0.22) in numbers of admissions or mortality by HIV exposure status. Fewer HIV-exposed neonates received a PCR test during the pandemic (23%) compared to the pre-pandemic periods (40%) (RR 0.59, 95% CI 0.41–0.84, p < 0.001). The proportion of HIV-exposed neonates who received antiretroviral prophylaxis during admission was high throughout, averaging between 84% and 95% in each time-period. Conclusion: While antiretroviral prophylaxis for HIV-exposed neonates remained high throughout, concerning data on low admissions and increased mortality, similar in HIV-exposed and unexposed neonates, and reduced HIV testing, suggest some aspects of care may have been compromised due to indirect effects of the pandemic

Antioxidant properties of MitoTEMPOL and its hydroxylamine

Piperidine nitroxides such as TEMPOL have been widely used as antioxidants in vitro and in vivo. MitoTEMPOL is a mitochondria-targeted derivative of TEMPOL designed to protect mitochondria from the oxidative damage that they accumulate, but once there is rapidly reduced to its hydroxylamine, MitoTEMPOL-H. As little is known about the antioxidant efficacy of hydroxylamines, this study has assessed the antioxidant activity of both MitoTEMPOL and MitoTEMPOL-H. The hydroxylamine was more effective at preventing lipid-peroxidation than MitoTEMPOL and decreased oxidative damage to mitochondrial DNA caused by menadione. In contrast to MitoTEMPOL, MitoTEMPOL-H has no superoxide dismutase activity and its antioxidant actions are likely to be mediated by hydrogen atom donation. Therefore, even though MitoTEMPOL is rapidly reduced to MitoTEMPOL-H in cells, it remains an effective antioxidant. Furthermore, as TEMPOL is also reduced to a hydroxylamine in vivo, many of its antioxidant effects may also be mediated by its hydroxylamine

Preventing foot ulceration in diabetes:systematic review and meta-analyses of RCT data

Aims/hypothesis: Foot ulceration is a serious complication for people with diabetes that results in high levels of morbidity for individuals and significant costs for health and social care systems. Nineteen systematic reviews of preventative interventions have been published, but none provides a reliable numerical summary of treatment effects. The aim of this study was to systematically review the evidence from RCTs and, where possible, conduct meta-analyses to make the best possible use of the currently available data. Methods: We conducted a systematic review and meta-analysis of RCTs of preventative interventions for foot ulceration. OVID MEDLINE and EMBASE were searched to February 2019 and the Cochrane Central Register of Controlled Trials to October 2018. RCTs of interventions to prevent foot ulcers in people with diabetes who were free from foot ulceration at trial entry were included. Two independent reviewers read the full-text articles and extracted data. The quality of trial reporting was assessed using the Cochrane Risk of Bias tool. The primary outcome of foot ulceration was summarised using pooled relative risks in meta-analyses. Results: Twenty-two RCTs of eight interventions were eligible for analysis. One trial of digital silicone devices (RR 0.07 [95% CI 0.01, 0.55]) and meta-analyses of dermal infrared thermometry (RR 0.41 [95% CI 0.19, 0.86]), complex interventions (RR 0.59 [95% CI 0.38, 0.90], and custom-made footwear and offloading insoles (RR 0.53 [95% CI 0.33, 0.85]) showed beneficial effects for these interventions. Conclusions/interpretation: Four interventions were identified as being effective in preventing foot ulcers in people with diabetes, but uncertainty remains about what works and who is most likely to benefit