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The main purpose of this project is to create a method for determining the absolute position of an accelerometer. Acceleration and angular speed were obtained from an accelerometer attached to a vehicle as it moves around. As the vehicle moves to collect information the orientation of the accelerometer changes, so a rotation matrix is applied to the data based on the angular change at each time. The angular change and distance are obtained by using the trapezoidal approximation of the integrals. This method was first validated by using simple sets of "true" data which are explicitly known sets of data to compare the results to. Then, an analysis of how different time steps and levels of noise affect the error of the results was performed to determine the optimal time step of 0.1 sec that was then used for the actual tests. The tests that were performed were: a stationary test for uses of calibration, a straight line test to verify a simple test, and a closed loop test to test the accuracy. The graphs for these tests give no indication of the actual paths, so the final results can only show that the data from the accelerometer is too noisy and inaccurate for this method to be used by this sensor. The future work would be to test different ways to get more accurate data and then use it to verify this methods. These ways could include using more sensors to interpolate the data, reducing noise by using a different sensor, or adding a filter. Then, if this method is considered accurate enough, it could be implemented into control systems.
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The prescription opioid Vicodin is the nation's most widely prescribed pain reliever. The majority of Vicodin abusers are first introduced via prescription, distinguishing it from other drugs in which the most common path to abuse begins with experimentation. I develop and analyze two mathematical models of Vicodin use and abuse, considering only those patients with an initial Vicodin prescription. Through adjoint sensitivity analysis, I show that focusing efforts on prevention rather than treatment has greater success at reducing the total population of abusers. I prove that solutions to each model exist, are unique, and are non-negative. I also derive conditions for which these solutions are asymptotically stable.
Verification and Validation (V&V) are necessary processes to ensure accuracy of computational methods. Simulations are essential for addressing impact cratering problems, because these problems often exceed experimental capabilities. I show that the Free Lagrange (FLAG) hydrocode, developed and maintained by Los Alamos National Laboratory, can be used for impact cratering simulations by verifying FLAG against two analytical models of aluminum-on-aluminum impacts at different impact velocities and validating FLAG against a glass-into-water laboratory impact experiment. My verification results show good agreement with the theoretical maximum pressures, and my mesh resolution study shows that FLAG converges at resolutions low enough to reduce the required computation time from about 28 hours to about 25 minutes.
Asteroid 16 Psyche is the largest M-type (metallic) asteroid in the Main Asteroid Belt. Radar albedo data indicate Psyche's surface is rich in metallic content, but estimates for Psyche's composition vary widely. Psyche has two large impact structures in its Southern hemisphere, with estimated diameters from 50 km to 70 km and estimated depths up to 6.4 km. I use the FLAG hydrocode to model the formation of the largest of these impact structures. My results indicate an oblique angle of impact rather than a vertical impact. These results also support previous claims that Psyche is metallic and porous.
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First, prostate cancer is modeled, where androgen is considered the limiting nutrient since most tumors depend on androgen for proliferation and survival. The model's accuracy for predicting the biomarker for patients on intermittent androgen deprivation therapy is tested by comparing the simulation results to clinical data as well as to an existing simpler model. The results suggest that a simpler model may be more beneficial for a predictive use, although further research is needed in this field prior to implementing mathematical models as a predictive method in a clinical setting.
Next, two chronic myeloid leukemia models are compared that consider Imatinib treatment, a drug that inhibits the constitutively active tyrosine kinase BCR-ABL. Both models describe the competition of leukemic and normal cells, however the first model also describes intracellular dynamics by considering BCR-ABL as the limiting nutrient. Using clinical data, the differences in estimated parameters between the models and the capacity for each model to predict drug resistance are analyzed.
Last, a simple model is presented that considers ovarian tumor growth and tumor induced angiogenesis, subject to on and off anti-angiogenesis treatment. In this environment, the cell quota represents the intracellular concentration of necessary nutrients provided through blood supply. Mathematical analysis of the model is presented and model simulation results are compared to pre-clinical data. This simple model is able to fit both on- and off-treatment data using the same biologically relevant parameters.