You are using a browser not supported by the Wolfram Cloud
Supported browsers include recent versions of Chrome, Edge, Firefox and Safari.
I understand and wish to continue anyway »
WOLFRAM NOTEBOOK
Modeling Gas Exchange in the LungsFrancisco Veloso Aires
Modeling Gas Exchange in the LungsFrancisco Veloso Aires
Abstract
Abstract
In this experiment through deriving a discrete-time dynamical system, the breathing of a simplified lung was modeled. The experiment solicited the concentration of a chemical in the lung after a breath. The main goal of the experiment is to understand and discover if the usage of the program Wolfram Cloud can be done to provide models of bio-medicine elements and materials. Using the program allows for the graphing of points, creating equations for the amount of air and chemicals being inhaled and exhaled, but most of all it allows for the equilibrium point to be discovered. At the end of the experiment, it was found that the program is valuable and compatible with doing medicine models. The results showed that in trial 1, the equilibrium concentration of the unknown chemical in the lungs is 9.550%, of the ambient concentration, after exactly 33 breaths and on. In trial 2, the equilibrium concentration of the unknown chemical in the lungs is 24.09%, of the ambient concentration, after 25 breaths. In trial 3, the equilibrium concentration of the unknown chemical in the lungs is 7.000%, of the ambient concentration, after 19 breaths.
Introduction
Introduction
The Wolfram Cloud is a very complete mathematical website and program. This website allows for mathematical coding and the creation of notebooks with projects, and code. In addition, it has a utility that allows mathematical problem-solving, like a calculator. It allows one to get help for a mathematical problem and understand the problem better. The program side of the Wolfram Cloud is very useful to create models of math-related processes, in this case, the gas exchange in the lungs, but this can be done for blood exchange in the heart, or other bio-medicine topics. This also allows for the creation of vast systems and functions, just by using math and code. being able to do all these types of models allows for a better understanding of each, and to get better and more accurate and precise results. In a model of the gas exchange, through the program, it is possible to gain an accurate number for the equilibrium point and how the gas in the lungs acts, through the creation of a discrete-time dynamical system. For these equations the values that are needed are the volume of the lungs, the amount of air being inhaled and exhaled, the amount of the chemical being investigated, and how much of that chemical gets absorbed by the blood. In the average human, many of these values are constant, for example, the volume of the lungs is usually going to be the same for the average human. According to Lumen: Biology for Majors II, "Tidal volume (TV) measures the amount of air that is inspired and expired during a normal breath. On average, this volume is around one-half liter, which is a little less than the capacity of a 20-ounce drink bottle" ("Breathing Capacity", para. 4). This is the amount of air that is inhaled in a regular breath, which allows the calculation of the ratio between the air inhaled and the volume of the lungs. Likewise, the blood also takes in chemicals and transports them through the body. The blood usually absorbs the oxygen, and transports it to the muscles, this is because when exercising the muscles require more oxygen to perform better, so the blood becomes the provider. According to InformedHealth.org, "By picking up or releasing hydrogen they help to keep the pH of the blood stable; when they release nitrogen the blood vessels expand, and blood pressure falls" ("What does blood do?", para. 7). By having the lungs providing the different chemicals to the blood, our body is able to become healthier, but this is only a factor when the chemicals are positive, when they are negative they destroy our body since they enter the bloodstream through absorption. This type of program is a good way to study smokers, how badly their lungs work, and how the bad chemicals inhale interact. Using this program it is possible to study individual chemicals and learn about them.
Methods
Methods
1. Create an account on the Wolfram Cloud website, which is the program used to construct the project.2. Understand and study what a discrete-time dynamical system is. 3. Using the book "Modeling the Dynamics of Life", read chapter 10.9, and understand how the system is incorporated in the modeling of the lungs, when exhaling and inhaling with the absorption of the chemical.4. Using the equations learned, create a Recurrence Table with the equation, showing the values of the concentration of the chemical in the lungs, and also the equilibrium point.5. Write down the equation being used: f[x_]=(1-q)(1-a)x+q γ, and set parameters for the values: q,a,γ. 6. Then name the function and nest it, using [x[n_]=Nest[f,1,n]].7. Once that is done create a table with the values calculated from the equation: Table[{n,N[x[n]]},{n,1,35}]//TableForm.8. Calculate the equilibrium concentration of the unknown chemical using the equation: h[x_]=(q γ)/(1-(1-q)(1-a)).9. Then graph the table formed with the initial equation and the function of the equilibrium.10. Once both equations have been graphed, name them and group them together with the text Show[{graph1,graph2}].11. Then use cob webbing to visually show where the equilibrium point is located at.
Data
Data
Functions being used:c_t+1=f(c_t) Without Absorption: f(c_t)=(1-q)c_t+qγ With Absorption: f(c_t)=(1-q)(1-a)c_t+qγ Variables used:c_t = concentration of the unknown chemical after every breath. (mmmol/L).a = ratio of how much of the chemical was absorbed by the blood. q = W/V, ratio of the amount exhaled over volume of the lung. (W= L, V=L).γ = ambient concentration of unknown chemical. (mmmol/L).x = number of breaths.Trial 1. Equilibrium concentration of Unknown Chemical in the lungs:Parameter used:q= 0.2 γ= 0.21 mmmol/La= 0.3
General Function
In[]:=
Clear[x]q=0.2γ=0.21a=0.3f[x_]=(1-q)(1-a)x+qγ
Out[]=
0.2
Out[]=
0.21
Out[]=
0.3
Out[]=
0.042+0.56x
In[]:=
x[n_]=Nest[f,1,n]
Out[]=
Nest[f,1,n]
In[]:=
list1=Table[{n,N[x[n]]},{n,1,35}];
In[]:=
Table[{n,N[x[n]]},{n,1,35}]//TableForm
Out[]//TableForm=
1 | 0.602 |
2 | 0.37912 |
3 | 0.254307 |
4 | 0.184412 |
5 | 0.145271 |
6 | 0.123352 |
7 | 0.111077 |
8 | 0.104203 |
9 | 0.100354 |
10 | 0.0981981 |
11 | 0.0969909 |
12 | 0.0963149 |
13 | 0.0959364 |
14 | 0.0957244 |
15 | 0.0956056 |
16 | 0.0955392 |
17 | 0.0955019 |
18 | 0.0954811 |
19 | 0.0954694 |
20 | 0.0954629 |
21 | 0.0954592 |
22 | 0.0954572 |
23 | 0.095456 |
24 | 0.0954554 |
25 | 0.095455 |
26 | 0.0954548 |
27 | 0.0954547 |
28 | 0.0954546 |
29 | 0.0954546 |
30 | 0.0954546 |
31 | 0.0954546 |
32 | 0.0954546 |
33 | 0.0954545 |
34 | 0.0954545 |
35 | 0.0954545 |
Equilibrium Equation
In[]:=
h[x_]=(qγ)/(1-(1-q)(1-a))
Out[]=
0.0954545
Graphing
In[]:=
p5=Plot[f[x],{x,0,25},PlotStyleBlack];p6=Plot[y=x,{x,0,25},PlotStyleBlack];Show[{p1,p2},PlotRange{{0,0.2},{0.0,0.15}}]
Out[]=
Graphing Cobwebbing
Graphing Absorbtion
Parameter used:
q= 0.2
γ= 0.53 mmmol/L
a= 0.3
q= 0.2
γ= 0.53 mmmol/L
a= 0.3
General Function
Equilibrium Equation
Graphing
Graphing Absorption
Parameter used:
q= 0.2
γ= 0.21 mmmol/L
a= 0.5
q= 0.2
γ= 0.21 mmmol/L
a= 0.5
General Function
Equilibrium Equation
Graphing
Graphing Cobwebbing
Graphing Absorption