609_Final_Presentation
Raghu Ramnath
5- May-2018
DATA 609 Mathematical Modeling Techniques
Agenda
- Introduction
- Problem Description
- Assumptions
- Model Construction
- Methodology
- Least Squares Criterion
- Chebyshev Criterion
- Conclusion
Introduction
In this project, I will investigate the relationship of the body weights of mammals and their pulse rates.
Problem Description
(a) Construct a model relating blood flow through the heart to body weight.
(b) The following data relate weights of some mammals to their heart rate measured in beats per minute. Construct a model that relates heart rate to body weight. Discuss the assumptions of your model. Use the data to check your model.
mammals body_weight pulse_rate
1 Vespergo pipistrellas 4 660
2 Mouse 25 670
3 Rat 200 420
4 Guinea pig 300 300
5 Rabbit 2000 205
6 Little dog 5000 120
7 Big dog 30000 85
8 Sheep 50000 70
9 Man 70000 72
10 Horse 450000 38
11 Ox 5e+05 40
12 Elephant 3e+06 48
Assumptions
It is understood that smaller the mammals have much higher resting pulse rates than large ones. For examples, a mouse, with a body mass of 25 grams, may have a heart rate of 670 beats/minute, as compared to 48 beats/minute for a Elephant.
Warm-blooded animals at rest use large quantities of energy to maintain body temperature, due to heat loss through the body surface. Let us make the following assumptions about energy balance over a fixed time interval (1 minute):
A. Energy lost is proportional to the surface area of the body.
B. Energy gained is proportional to the amount of blood flowing through the lungs, the source of oxygen for the body.
C. Energy lost = energy gained for a body in temperature equilibrium.
Model Construction
E = energy
S = surface area
m = mass
W = weight of the animal
r = pulse rate
V = volume of the animal
b = volume of blood pumped by the heart in one stroke.
Model Construction (Continued)
From assumption A, we can assume that the energy required to maintain the body temperature of a mammal E is proportional to its body surface area S, i.e. \( E \propto S \)
Since blood flow through the lungs provides animals oxygen, larger animals have larger lungs and require more oxygen. Therefore, we assume that the amount of energy available is proportional to the blood flow b through the lungs, i.e \( E \propto b \)
Also, we assume that the least amount of blood needed to circulate, the amount of available energy will equal the amount of energy used to maintain the body temperature. This implies \( b \propto S \)
Model Construction (Continued)
Since larger animals have larger hearts and larger lungs, we can assume the mammals are geometrically similar. Thus, \( S \propto V^{2/3} \)
where V is the volume of the animal.
Since \( V \propto m \propto W \) where m is the mass of the animal and W is the weight of the animal, we have \( S \propto W^{2/3} \). This implies \( b \propto W^{2/3} \).
Also, since blood flow is determined by the total amount of blood pumped by the heart per unit time (minutes), and every time the heart beats the blood is pumped into the arteries, we have \( b \propto rV \propto rW \) where r is the pulse rate and Vh is heart volume. From \( b \propto W^{2/3} \) and \( b \propto rW \), we have \( W^{2/3} \propto rW \) which is \( r \propto W^{{-1/3}} \).
Methodology
First we will be modeling using Geometric Similarity given our power curve model function
We will be using Least squares criterion to fit our curve model of the form \( y = Ax^n \) where n is fixed to a given collection of data points. In our case, our curve is \( r = Aw^{-1/3} \) . Therefore we will use the least squares criterion to estimate A with n = -1/3.
In addition we will use the Chebyshev criterion to find the bounds.
Least Squares Criterion Methodology
\[ f(x) = ax^n \] requires minimization of:
\[ S = \sum_{i=1}^{m} [y_i - f(x_i)]^2 \]
\[ S = \sum_{i=1}^{m} [y_i - ax^n]^2 \]
Solving for a yields
\[ a = \frac {{\sum xi^{2} y_i}} {\sum xi^2n} \]
Body Weight vs Pulse Rate
Least Squares Criterion Model Results
mammals body_weight pulse_rate pulse_rate_fit
1 Vespergo pipistrellas 4 660 864.06
2 Mouse 25 670 469.08
3 Rat 200 420 234.54
4 Guinea pig 300 300 204.89
5 Rabbit 2000 205 108.86
6 Little dog 5000 120 80.21
7 Big dog 30000 85 44.14
8 Sheep 50000 70 37.23
9 Man 70000 72 33.28
10 Horse 450000 38 17.90
11 Ox 500000 40 17.28
12 Elephant 3000000 48 9.51
Least Squares Criterion Model Results
Chebyshev Criterion Model Results
Given N points (x_1, y_1),(x_2, y_2),(x_3, y_3), .., (x_n, y_n), to fit the model y=cx using Chebyshev criterion, we need to:
\[ E = max_i|yi - c(xi)| \]
Goal: to minimize r
Chebyshev LP Model Statement
r - ( 660 - c*4 ) >= 0, r + ( 660 - c*4 ) >= 0
r - ( 670 - c*25 ) >= 0, r + ( 670 - c*25 ) >= 0
r - ( 420 - c*200 ) >= 0, r + ( 420 - c*200 ) >= 0
r - ( 300 - c*300 ) >= 0, r + ( 300 - c*300 ) >= 0
r - ( 205 - c*2000 ) >= 0, r + ( 205 - c*2000 ) >= 0
r - ( 120 - c*5000 ) >= 0, r + ( 120 - c*5000 ) >= 0
r - ( 85 - c*30000 ) >= 0, r + ( 85 - c*30000 ) >= 0
r - ( 70 - c*50000 ) >= 0, r + ( 70 - c*50000 ) >= 0
r - ( 72 - c*70000 ) >= 0, r + ( 72 - c*70000 ) >= 0
r - ( 38 - c*450000 ) >= 0, r + ( 38 - c*450000 ) >= 0
r - ( 40 - c*500000 ) >= 0, r + ( 40 - c*500000 ) >= 0
r - ( 48 - c*3000000 ) >= 0, r + ( 48 - c*3000000 ) >= 0
Chebyshev Criterion Geometric Interpretation
Conclusion:
For the majority of the mammals, the model makes good predictions in both least Square and Geometric Similarity, and support the biology that larger the animals slower pulse rate. However, Elephant is bigger than all the animals but it has pulse rate slightly higher than Ox and Horse. So, apart from the body weight, there is something else that affects the pulse rate. Chebysev has a large error which is not acceptable.