Mathematical Economics: A Reader

This paper is modeled as a hypothetical first lecture in a graduate Microeconomics or Mathematical Economics Course. We start with a detailed scrutiny of the notion of a utility function to motivate and describe the common patterns across Mathematical concepts and results that are used by economists. In the process we arrive at a classification of mathematical terms which is used to state mathematical results in economics. The usefulness of the classification scheme is illustrated with the help of a discussion of fixed-point theorems and Arrow's impossibility theorem. Several appendices provide a step-wise description of some mathematical concepts often used by economists and a few useful results in microeconomics.Mathematics, Set theory, Utility function, Arrow's impossibility theorem

Determining the Impact of Increased Physical Activity on Improving Sleep Quality in Young Adults

Determining the Impact of Increased Physical Activity on Improving Sleep Quality in Young Adults Disturbed sleep, defined as any alteration to normal sleep patterns, has been linked to poor cardiovascular health and an increase in cardiovascular disease (CVD) risk. These negative sleep patterns are highly prevalent with 35% to 41% of individuals in the United States reported some form of disturbed sleep. Although high amounts of physical activity (PA) are often associated with high sleep quality, little is known about PA’s effectiveness to improve different aspects of sleep (e.g. duration vs quality) and the mechanisms to which it can improve sleep quality. Purpose: The study sought to determine the ability of increased PA to improve sleep efficiency in healthy young adults. Methods: Nineteen young adults (25±4 yrs) were recruited for this study. Subjects wore an accelerometer (Actigraph GT3x-BT) for a total of three weeks to record daily physical activity (step count; low, moderate, and vigorous physical activity) and sleep variables (efficiency, wake after sleep onset, number of nightly awakenings, time per awakening, and total sleep time). Subjects maintained normal physical activity levels for the first week (BL), then increased their step count by an average of 5,000 steps/day across the next two weeks (W1 and W2). Heart rate variability (HRV) and venous blood draws were collected weekly to assess sympathetic activity and inflammation, respectively. Results: The physical activity intervention resulted in significant increases (p \u3c 0.001) in step-count for both W1 (13163 ± 3184) and W2 (12168 ± 3619) when compared to BL (8648 ± 2615 steps/day). No significant differences from BL were observed when examining sleep efficiency (BL: 83.8 ± 6.4; W1: 85.5 ± 4.0; W2: 84.2 ± 6.1 %), sympathetic-vagal balance, and inflammatory marker concentrations in W1 and W2. A significant correlation was revealed when assessing the change in sleep efficiency from BL to W1 (r = 0.81, p \u3c 0.001) and BL to W2 (r = 0.52, p = 0.02) when compared to initial sleep efficiency values. Conclusion: This study revealed that although young healthy individuals appear to lack improvements in sleep efficiency with an increase in physical activity, those who reported the lowest sleep quality had the greatest improvements in sleep efficiency following an increase in physical activity. Therefore, the findings of the study suggest that although increasing physical activity can improve sleep quality, a potential “ceiling effect” may occur, as when sleep quality is adequate, augmenting physical activity no longer has a substantial effect.https://scholarscompass.vcu.edu/gradposters/1058/thumbnail.jp

Analysis of carotid lumen surface morphology using three-dimensional ultrasound imaging

Carotid plaque surface irregularity and ulcerations play an important role in the risk of ischemic stroke. Ulcerated or fissured plaque, characterized by irregular surface morphology, exposes thrombogenic materials to the bloodstream, possibly leading to life- or brain-threatening thrombosis and embolization. Therefore, the quantification of plaque surface irregularity is important to identify high-risk plaques that would likely lead to vascular events. Although a number of studies have characterized plaque surface irregularity using subjective classification schemes with two or more categories, only a few have quantified surface irregularity using an objective and continuous quantity, such as Gaussian or mean curvature. In this work, our goal was to use both Gaussian and mean curvatures for identifying ulcers from 3D carotid ultrasound (US) images of human subjects. Before performing experiments using patient data, we verified the numerical accuracy of the surface curvature computation method using discrete spheres and tori with different sampling intervals. We also showed that three ulcers of the vascular phantom with 2 mm, 3 mm and 4 mm diameters were associated with high Gaussian and mean curvatures, and thus, were easily detected. Finally, we demonstrated the application of the proposed method for detecting ulcers on luminal surfaces, which were segmented from the 3D US images acquired for two human subjects

Quantification of carotid vessel wall and plaque thickness change using 3D ultrasound images

Quantitative measurements of carotid plaque burden progression or regression are important in monitoring patients and in evaluation of new treatment options. 3D ultrasound (US) has been used to monitor the progression or regression of carotid artery plaques. This paper reports on the development and application of a method used to analyze changes in carotid plaque morphology from 3D US. The technique used is evaluated using manual segmentations of the arterial wall and lumen from 3D US images acquired in two imaging sessions. To reduce the effect of segmentation variability, segmentation was performed five times each for the wall and lumen. The mean wall and lumen surfaces, computed from this set of five segmentations, were matched on a point-by-point basis, and the distance between each pair of corresponding points served as an estimate of the combined thickness of the plaque, intima, and media (vessel-wall-plus-plaque thickness or VWT). The VWT maps associated with the first and the second US images were compared and the differences of VWT were obtained at each vertex. The 3D VWT and VWT-Change maps may provide important information for evaluating the location of plaque progression in relation to the localized disturbances of flow pattern, such as oscillatory shear, and regression in response to medical treatments

The Effects of a High Fat Meal on Blood Flow Regulation during Arm Exercise

A diet high in saturated fats results in endothelial dysfunction and can lead to atherosclerosis, a precursor to cardiovascular disease. Exercise training is a potent stimulus though to mitigate the negative effects of a high saturated fat diet; however, it is unclear how high-saturated fat meal (HSFM) consumption impacts blood flow regulation during a single exercise session. PURPOSE: This study sought to examine the impact of a single HSFM on peripheral vascular function during an acute upper limb exercise bout. METHODS: Ten young healthy individuals completed two sessions of progressive handgrip exercise. Subjects either consumed a HSFM (0.84 g of fat/kg of body weight) 4 hours prior or remained fasted before the exercise bout. Progressive rhythmic handgrip exercise (6kg, 12kg, 18kg) was performed for 3 minutes per stage at rate of 1 Hz. The brachial artery (BA) diameter and blood velocity was obtained using Doppler Ultrasound (GE Logiq e) and BA blood flow was calculated with these values. RESULTS: BA blood flow and flow mediated dilation (normalized for shear rate) during the handgrip exercise significant increased from baseline in all workloads, but no differences were revealed in response to the HSFM consumption. CONCLUSION: Progressive handgrip exercise augmented BA blood flow and flow mediated dilation in both testing days; however, there was no significant differences following the HSFM consumption. This suggests that upper limb blood flow regulation during exercise is unaltered by a high fat meal in young healthy individuals.https://scholarscompass.vcu.edu/gradposters/1060/thumbnail.jp

Quantification of Cerebral Ventricle Volume Change of Preterm Neonates Using 3D Ultrasound Images

Intraventricular hemorrhage (IVH) is a major cause of brain injury in preterm neonates. Quantitative measurement of ventricular dilation or shrinkage is important for monitoring patients and in evaluation of treatment options. 3D ultrasound (US) has been used to monitor the ventricle volume as a biomarker for ventricular dilation. However, volumetric quantification does not provide information as to where dilation occurs. The location where dilation occurs may be related to specific neurological problems later in life. For example, posterior horn enlargement, with thinning of the corpus callosum and parietal white matter fibres, could be linked to poor visuo-spatial abilities seen in hydrocephalic children. In this work, we report on the development and application of a method used to analyze local surface change of the ventricles of preterm neonates with IVH from 3D US images. The technique is evaluated using manual segmentations from 3D US images acquired in two imaging sessions. The surfaces from baseline and follow-up were registered and then matched on a point-by-point basis. The distance between each pair of corresponding points served as an estimate of local surface change of the brain ventricle at each vertex. The measurements of local surface change were then superimposed on the ventricle surface to produce the 3D local surface change map that provide information on the spatio-temporal dilation pattern of brain ventricles following IVH. This tool can be used to monitor responses to different treatment options, and may provide important information for elucidating the deficiencies a patient will have later in life

Nonrigid registration of three-dimensional ultrasound and magnetic resonance images of the carotid arteries

Atherosclerosis at the carotid bifurcation can result in cerebral emboli, which in turn can block the blood supply to the brain causing ischemic strokes. Noninvasive imaging tools that better characterize arterial wall, and atherosclerotic plaque structure and composition may help to determine the factors which lead to the development of unstable lesions, and identify patients at risk of plaque disruption and stroke. Carotid magnetic resonance (MR) imaging allows for the characterization of carotid vessel wall and plaque composition, the characterization of normal and pathological arterial wall, the quantification of plaque size, and the detection of plaque integrity. On the other hand, various ultrasound (US) measurements have also been used to quantify atherosclerosis, carotid stenosis, intima-media thickness, total plaque volume, total plaque area, and vessel wall volume. Combining the complementary information provided by 3D MR and US carotid images may lead to a better understanding of the underlying compositional and textural factors that define plaque and wall vulnerability, which may lead to better and more effective stroke prevention strategies and patient management. Combining these images requires nonrigid registration to correct the nonlinear misalignments caused by relative twisting and bending in the neck due to different head positions during the two image acquisition sessions. The high degree of freedom and large number of parameters associated with existing nonrigid image registration methods causes several problems including unnatural plaque morphology alteration, high computational complexity, and low reliability. Thus, a twisting and bending model was used with only six parameters to model the normal movement of the neck for nonrigid registration. The registration technique was evaluated using 3D US and MR carotid images at two field strengths, 1.5 and 3.0 T, of the same subject acquired on the same day. The mean registration error between the segmented carotid artery wall boundaries in the target US image and the registered MR images was calculated using a distance-based error metric after applying a twisting and bending model based nonrigid registration algorithm. An average registration error of 1.4 +/- 0.3 mm was obtained for 1.5 T MR and 1.5 +/- 0.4 mm for 3.0 T MR, when registered with 3D US images using the nonrigid registration technique presented in this paper. Visual inspection of segmented vessel surfaces also showed a substantial improvement of alignment with this nonrigid registration technique compared to rigid registration