Drug Modeling Analysis

Our project was specifically looking at modeling the absorption, circulation, and metabolism of morphine. Morphine is an addictive drug (opioid) used clinically as a powerful painkiller. Due to the chemical makeup of morphine, users develop a tolerance for this drug. This causes them to need more and more per each subsequent dosage. Because everyone's body has a different tolerance level for morphine, we were not able to nail down any of our coefficients in our model.

There are some averages published by the University of Lausanne in Switzerland on their pharmacokinetics research site (http://sepia.unil.ch/pharmacology/index.php?id=85). The oral bioavailability (absorption rate) is around 25%, so our k_1 value would be .25. Our k_2 value is the elimination rate, which is defined as either V_max / K_m or as the clearance / volume of distribution. Given that the clearance is 72L/h and the volume of distribution is 245L/70kg, we get a k_2 value of 20.571kg/hr. This volume of distribution is based off a 70kg man, so as we stated before, results vary greatly from person to person.

**Idea taken from Olivia Nguyen and Xiaoyu Shi's paper on modeling the absorption, circulation, and metabolism of tirapazamine.

Drug Modeling (Simplified)

Yellowstone Wolf Lab

Year	Time (years)	Wolf Population	Births	Deaths	Elk Killed	Available Data		Births/Population	Deaths/Population	Elk Killed/Population
1995	1	21	9	2	43	GYA	t(0)=14	0.4285714286	0.09523809524	2.047619048
1996	2	37	14	9	124	GYA		0.3783783784	0.2432432432	3.351351351
1997	3	80	49	16	234	GYA		0.6125	0.2	2.925
1998	4	83	36	10	197	GYA		0.4337349398	0.1204819277	2.373493976
1999	5	72	38	14	276	GYA		0.5277777778	0.1944444444	3.833333333
2000	6	119	71	14	281	Both		0.5966386555	0.1176470588	2.361344538
2001	7	132	43	6	311	Both		0.3257575758	0.04545454545	2.356060606
2002	8	148	65	4	291	Both		0.4391891892	0.02702702703	1.966216216
2003	9	174	59	16	313	GYA		0.3390804598	0.09195402299	1.798850575
2004	10	169	59	20	241	Both		0.349112426	0.1183431953	1.426035503
2005	11	118	22	25	244	Both		0.186440678	0.2118644068	2.06779661
2006	12	136	60	10	219	Both		0.4411764706	0.07352941176	1.610294118
2007	13	171	64	6	272	Both		0.3742690058	0.0350877193	1.590643275
2008	14	124	22	17	463	Both		0.1774193548	0.1370967742	3.733870968
2009	15	96	23	10	302	Both		0.2395833333	0.1041666667	3.145833333
2010	16	97	38	9	211	Both		0.3917525773	0.09278350515	2.175257732
2011	17	98	34	10	267	Both		0.3469387755	0.1020408163	2.724489796
2012	18	83	20	15	159	Both		0.2409638554	0.1807228916	1.915662651
2013	19	95	41	7	193	Both		0.4315789474	0.07368421053	2.031578947
2014	20	104	41	5	148	Both		0.3942307692	0.04807692308	1.423076923
							Average	0.3827547299	0.1156443443	2.342890475
							Variance	0.01381157161	0.003830577555	0.5263891894

This is the data we got from http://www.nps.gov/yell/learn/nature/wolfreports.htm which is the Yellowstone National Park website. They had yearly reports for wolf population, including births, deaths, and elk killed. (Population is total population, not change in population) We got the coefficient for W by averaging the number of deaths/population per year. This number signifies the natural decay of wolves. Since we had the change in population for that year, wolf population, and the natural death rate, we could solve for the coefficient for E (which is elk killed, not total elk population) for each year, then took the average to get a result of .07789.

dW/dt = -.116W + .07789E 

Since we don't have any information about the total elk population, we only have one equation for this predator-prey system of equations model. This equation is unsolvable because of the reason stated before. There is not enough historical data recorded on Yellowstone elk.


The following graph shows the number of wolf births and deaths per year in the Greater Yellowstone Area. In all but one of the years, births outnumbered deaths. However, due to emigration, wolf population in the area does not reflect continuous growth as we would expect looking at this graph.

We are allowed to use the mean as an estimator for our coefficient because our histogram of deaths per population is relatively close to normal. If the distribution was not close to normal, then the mean would not be an effective estimator for the wolf death rate per year.
 







This is a bar graph of the total number of wolves in the Greater Yellowstone Area. This population includes pups older than 6 months to adult wolves. These population numbers were taken in December of the year listed.

Analysis/Results

We set out with the goal of tracking the amount of height change in a pitcher's rise ball. We collected video evidence from three of Coe's softball pitchers at one of their indoor practices. We analyzed ten videos for each of the three pitchers using Tracker, a video analysis program. Tracker enabled us to plot the position of the ball in flight through each of the video's frames. We then created position time graphs for all thirty videos. However, we ran into difficulties trying to analyze our 3-dimensional calibration of 43 ft (mound to plate) in Tracker, which is a 2-dimensional program. Because our filming had to take place at an angle due to limited space constraints, we could not incorporate the z-axis into our calibration of the videos, so we were not able to track the change on the y-axis effectively. While we originally intended to compare the change on the y-axis between pitchers, we could only compare their x-axis motion.

We found that the pitches had cubic fits, so we analyzed each of the coefficients per pitcher. We calculated and plotted the averages for each pitcher. We had large standard deviations and variances for each pitcher, so a larger sample size would have been beneficial to obtain a more accurate sample mean. We found that Arran's rise balls traveled with a higher velocity earlier than both Kalyn's and Mariah's. Mariah's pitches sped up faster than Kalyn's as they got closer to the plate. While we were not able to analyze what we originally intended to, our data still led to relevant insight for the pitchers' rise balls.

Pitcher Comparison

Arran
Pitch #	A-value	B-value	C-value	D-value
1	356.000	-66.610	38.920	0.218
2	454.500	-154.900	45.200	-0.893
3	332.900	-76.980	38.980	-0.664
4	498.200	-177.600	44.810	-0.530
5	336.900	-92.800	42.760	-0.632
6	250.700	-29.370	36.400	0.634
7	407.100	-11.270	45.700	0.345
8	466.700	-138.700	47.790	0.481
9	366.300	-94.590	42.040	0.838
10	419.000	-114.200	42.530	0.794
Average	388.830	-95.702	42.513	0.059
Standard Deviation	74.589	52.850	3.552	0.668
Variance	5563.589	2793.100	12.616	0.446
Mariah
Pitch #	A-value	B-value	C-value	D-value
1	441.400	-196.300	46.660	-0.992
2	354.600	-44.450	31.700	0.266
3	184.000	-7.857	28.800	-0.300
4	284.100	-19.220	32.490	0.565
5	275.900	-2.838	30.530	0.964
6	285.900	-52.290	33.070	0.178
7	339.300	-112.300	42.150	0.116
8	282.800	-56.260	35.780	0.142
9	304.900	-23.470	33.120	1.163
10	489.500	-135.500	41.660	0.283
Average	324.240	-65.049	35.596	0.238
Standard Deviation	87.789	63.296	5.882	0.607
Variance	7706.929	4006.381	34.596	0.369
Kalyn
Pitch #	A-value	B-value	C-value	D-value
1	306.300	-88.300	36.340	-0.611
2	459.000	-152.300	45.840	0.219
3	217.700	-33.680	24.750	0.103
4	244.100	-27.460	34.680	0.327
5	227.300	-25.260	31.180	0.407
6	214.500	-19.580	29.830	0.651
7	240.800	-34.660	33.110	-0.146
8	213.000	-18.570	29.590	-0.394
9	360.500	-55.010	36.380	0.755
10	381.900	-74.910	33.770	0.913
Average	286.510	-52.973	33.547	0.222
Standard Deviation	86.561	42.196	5.587	0.496
Variance	7492.892	1780.489	31.215	0.246

Position-Time Graphs

Arran

Videos

Arran

Kalyn