ESP Project
Vivian Chow & Thukarasha Sivapatharajah
Differential Equations, Section 00001
Professor Ivan T. Ivanov
May 15 2015
Pharmacokinetics of Adderall
1
Table of Contents
Introduction 2
Theory behind Pharmacology 2
Adderall 4
Pharmacokinetics 6
Situation 1: Blood-Gut Pharmacokinetic Model Using Adderall 7
Situation 2: Contrasting Behaviours of Adderall IR and Adderall XR 11
Situation 3: Adderall’s Effect on a Rhesus Monkey 14
Situation 4: Administering Drugs every 4, 8 & 16 hours 16
Conclusion 19
Works Cited 20
Appendix A 21
Appendix B 23
2
Introduction
Most drugs fall into the categories of stimulants, depressants, opiates and hallucinogens,
where the drugs can be addictive, induce dependency and have different effects on the human
body. Some examples of drugs are Tylenol and Aspirin, as well as Marijuana and Tobacco. In
addition, pharmacologists help expand the medical field by studying drugs through examining
the interactions between chemical substances and living organisms before making drugs
available to the population. Hence, this project will model the pharmacokinetics of a drug by
studying its dissipation and removal rates in the body.
Theory behind Pharmacology
Pharmacology is the science that deals with drugs, their properties, actions and effects in
the body. It involves the sciences of pharmaceutics for the preparation of the drugs, therapeutics
for using drugs to treat diseases and toxicosis for the various side-effects (Magoma).
Pharmacology can be divided into five processes:
1. The pharmaceutical process of drugs – chemical synthesis, formulation and distribution of
drugs.
2. The pharmacokinetic process – the time course of drug concentration in the body via
absorption, distribution, biotransformation and excretion of the drug.
3. The pharmacodynamics process – the mechanism of drug action – interaction of drugs with
the body.
4. The therapeutic process – the clinical response arising from the pharmacodynamics process.
5. The toxicological process – various effects of drugs.
3
Figure 1: Relationships between the five pharmacological processes
*This project will examine mainly the pharmacokinetics of Adderall Extended (XR)*
4
Adderall
History: Shire Pharmaceuticals Group created the Adderall drug in 1996 and has
continued to produce this drug until 2007. After 2007, Teva Pharmaceuticals started to produce
this drug. Presently, two types of Adderall exist: Adderall XR (extended released) and Adderall
IR (immediate release). Originally, the instant released tablet version had been the only type
available. In 2002, Adderall XR began to be manufactured. Adderall XR provides an extended
release by distributing only half the drug immediately within the system whilst releasing the
other half after about four to six hours using the controlled release bead technology.
Usage: This drug is currently prescribed to patients with ADHD (Attention Deficit
Hyperactivity Disorder) or narcolepsy (disorder with the nerves that affects the users control
over sleep and staying awake). Starting from around 2009, more students, mostly college
students started to use the drug without being prescribed during the mid-semester and final
weeks in order to keep themselves awake whilst cramming for exams. Since Adderall is a
stimulant, the student will take it in order to reduce tiredness and increase focus during the
examination period. Similarly to Ritalin, the FDA (Food & Drug Administration (US)) has
labelled this Schedule II drug due to its high risk of abuse, which in consequence can lead to
severe physical and psychological effects.
Composition: This drug is the remake of the former drug known as Obetrol, a
discontinued dieting drug. With slight modification of Obetrol, Adderall was created. Adderall is
composed of two amphetamine stereoisomers salt: 75% dextero-amphetamine (d) and 25% levo-
amphetamine (l). The half-life for dextero-amphetamine is ten hours whilst its other isomer is
thirteen hours. Its given molecular formula is C9H13N. Within this drug, there is one benzene
aromatic (arene) ring linked to an amine group on the side.
5
Figure 2: Molecular Structure
Effects: For children, some side-effects may include vomiting, nausea, fever and
insomnia. For adolescents, some side-effects include weight loss, nervousness, lack of appetite
and insomnia. For adults, some side-effects can include headaches, diarrhea, anxiety attacks,
insomnia, dizziness and dry mouth.
Dosage: The Adderall IR tablets are instructed to be taken up to three tablets daily with a
time gap of four to six hours after the intake of each tablet. But, the total daily intake dose should
be 40mg or more if required. These tablets are available in different doses: 5mg, 7.5mg, 10mg,
15mg, 20mg and 30mg. The Adderall XR is available in capsules of 5mg, 10mg, 15mg, 20mg,
25mg and 30mg. But, the total daily recommended dose is 30mg and is slightly lower than daily
recommended dose of Adderall IR.
Figure 3: Dosages of Adderall XR (left) and Adderall IR (right) available
6
Pharmacokinetics
Bioavailability physiology: “The proportion of a drug or other substance which enters the
circulation when introduced into the body and so is able to have an active effect" (Oxford
English Dictionary).
Within Adderall XR, the active compound is known as amphetamine. The amount of
amphetamine available within the system is dependent on the pH of the gut and the intestines.
Amphetamine has a pKa of about nine to ten and considered a weak base. If the pH within the
system between the gut and intestines is above seven, more of the drug will be absorbed. On the
contrary, if the gastrointestinal pH is below seven, the drug will not be easily absorbed resulting
in lower bioavailability of amphetamine. About less than forty percent of the amphetamine is
transported with the proteins found with the blood around the bloodstream.
The half-life as stated previously of dextroamphetamine ranges from nine to eleven hours
whilst that of levoamphetamine ranges from eleven to fourteen hours. But, the half-life can vary
according to the user’s diet which can in turn affect the pH of their urine. The instant-release
(Adderall IR) reaches its peak concentration of amphetamine within the plasma at about three
hours. Whereas, the extend-release drug after ingestion delays its peak concentration until about
seven hours.
Amphetamine is excreted through the kidney, but the amount that will be excreted is
dependent on the urine pH. An increase in urine basicity will result in a decrease of excretion and
vice-versa.
7
Situation 1: Blood-Gut Pharmacokinetic Model Using Adderall
The pharmacokinetic model using Adderall will illustrate how the drug is consumed and
transferred between the gut and blood. This is examined through a math model of Adderall with
two compartments. What follows are a brief description of the each parameters and the
procedures in solving this situation.
To solve this situation, variation of parameters is used. Although, variation of parameters
is the simplest method to solve for the equations that model the concentration in the blood and
gut, diagonalization (D=T-1
MT) can be used to solve for the nonhomogeneous system since
, where T is the combination of the eigenvectors of M and z is the driving function.
Let Compartment 1 be Gut and Compartment 2 be Blood, then:
c1 = concentration in the gut as a function of time
c2 = concentration in the blood as a function of time
k12 = rate of transfer from blood to gut
k21 = rate of transfer from gut to blood
K = rate constant at which the drug is removed from the blood
ti = initial time (hours)
tf = final time (hours)
V1 = volume of blood in gut (litres)
V2 = volume of blood in body (litres)
Some values are assumed to be zero, for instance, ti, c1(0) and c2(0), while k12 is 0.05 and
k21 is 2.95. This project will analyze the effects of Adderall XR in humans, so V1 is assumed to
be 1L, V2 is 5L and K is (ln2)/8=0.086643397.
8
The rates of the drug circulating in the system are represented by the following equations:
1. 3.
2. 4.
Equation 1 represents what occurs to the drug in compartment 1 (gut), while equation 2
represents what occurs to the drug in compartment 2 (blood). In both equations, the ‘x’ variable
represents the quantity of the drug. Equations 1 and 2 are modified into Equations 3 and 4
repetitively using the relation where x is the amount of Adderall, V is the volume of blood
in the compartments and C is the concentration of Adderall in the compartments. Since the drug
is circulating in the system, a positive equation indicates the drug flowing into the system while a
negative equation indicates the drug flowing out the system.
To find the equations that model the pharmacokinetics of Adderall, meaning to find the
equations that represent the change in concentration over a period of time in the gut and the
blood, combine equations 3 and 4 because that way a matrix is formed, which can be solved
using variation of parameters to get the functions based on the formula where
M= .
Since there is one driving function that represents each dosage, use variation of
parameters , where z is the driving function and C is the constants derived
from the general solution to solve for the equations that demonstrate the concentrations over a
period of time in the gut and blood.
9
Step 1: Eigenvalues of M: (-2.95-λ)(-ln(2)/8+0.05-λ)-(0.25)(0.59) λ1 = -3.0015 & λ2 = -0.0852
Step 2: Eigenvectors of M:
Step 3:
Step 4:
Step 5:
Step 6:
Step 7:
Step 8: Particular solution + General solution
Step 9: Using the initial conditions (t=0 & y=0), the constants can be solved
C1 = 0.5011 and C2 = -0.2220
Step 10: Plug-in the constants and extract the equations
y1 (t) = -0.49081e-3.0015t +
0.01929e-0.0852t
+ 0.51010e-t
y2 (t) = 0.10108e-3.0015t +
0.22114e-0.0852t -
0.32222e-t
10
These functions represent the modeling of one dosage with respect to time and the results
are graphed below. Similar calculations can be done to additional dosages.
* For detailed calculations, refer to the Matlab script attached at the end in Appendix*
Figure 5 represents how the drug spikes quickly in the gut and quickly dissipates while it
takes a longer to spike in the blood as well as a
longer time to leave the system based on figure
6. This is due to the dosage function, represented
in figure 4, which helps the drug spike and then
to decrease. The reason why it spikes in the gut
quicker is because it reaches the gut first before
being transported to the blood where it remains
longer since that is the purpose of the drug.
Figure 5: the concentration of Adderall XR in
the gut over a period of time; y1 (t)
Figure 6: the concentration of Adderall XR in
the blood over a period of time; y2 (t)
Figure 4: represents the function that
represents one dosage (e-t)
11
Situation 2: Contrasting Behaviours of Adderall IR and Adderall XR
There exist two types of Adderall: Adderall IR (immediate release) and Adderall XR
(extended). What is the difference between the usage of Adderall IR (immediate release) and
Adderall XR (extended) in the blood and gut? Well, Adderall XR's half-life is about eight hours.
This value is used in order to calculate the K (constant) because , where T is time in
hours. Therefore, the K is equal to 0.086643397 in Adderall XR and since Adderall IR's half-life
is about half of Adderall XR's, which is about four hours, the K is equal to 0.173286795. As a
result, as the constant value increases, the concentration of the drug in the blood decreases.
The graphs below will compare Adderall XR to Adderall IR. For instance, the
concentration of both drugs in the blood (red) and gut (blue) changing over a period of time will
be compared as well the change of concentration in the gut and blood as one intake six dosages
of either drug will be compared. This is done by solving the system formed using equations #3
and #4 from previous situation with a nonhomogeneous term consisting of six shifts of the
function d(t)=e-t, which corresponds to six dosages through an interval of time, which
corresponds to the half-life of the drug. The figures were generated by the numerical solution of
systems #3 and #4 (stated before) using numerical Runge-Kutta method. See the attached Matlab
file in Appendix for details.
12
Figure 7: the concentration of Adderall XR in the blood
(red) and gut (blue) changing over a period of time
Figure 8: the change of concentration in the gut and
blood as one intake six dosages of Adderall XR
Figure 9: the concentration of Adderall IR in the blood
(red) and gut (blue) changing over a period of time
Figure 10: the change of concentration in the gut
and blood as one intake six dosages of Adderall IR
13
Comparing Drugs’ Concentration in Blood and Gut versus Time - The decay time of
Adderall has changed. Adderall XR provides an extended release, which is why the half-life is
twice that of Adderall IR. Adderall XR's concentration in the blood and gut is close to zero after
about ninety to hundred hours, whereas Adderall IR's concentration in the blood and gut is close
to zero after about seventy to eighty hours. In addition, Adderall XR's blood concentration is
higher than Adderall IR's blood concentration since it increases after each dosage. Thus,
Adderall IR's blood concentration does significantly increase after the third dosage.
Comparing Drugs’ Concentration in Blood versus in Gut - The stable concentration in
the blood is higher in Adderall XR than in Adderall IR based on the graph demonstrating the
limiting cycle. For example, Adderall XR reaches stability at a concentration around 0.22 in the
blood, while Adderall IR reaches stability at a concentration around 0.09 in the blood. On the
other hand, the concentration of both drugs in the gut is about the same since both are around 0.2.
In addition, the transfer of Adderall XR between the gut and blood become stable after the sixth
dosage, but for Adderall IR, the transfer of the drug between the gut and blood become stable after
the third dosage.
14
Situation 3: Adderall’s Effect on a Rhesus Monkey
A human and a rhesus monkey share about 93% of their DNA. Although chimpanzees
share about 98-99%, the rhesus monkey still can be used in order to study human evolution. The
rhesus monkey ancestors are said to have diverged from human ancestors about twenty-five
million years ago, whilst the chimpanzees are said to have diverged about six million years ago.
The genes that scientists have identified as similar are for example, the hair formation gene,
immune response and creation of cell membrane protein.
Question: Rather than studying the effects of Adderall XR in the gut and the blood of an
average human being (~65kg), how about studying the effects of this drug on a rhesus monkey of
similar weight?
The following graphs will help study the differences in administering six pills in eight
hours period in a human and a rhesus monkey.
Figure 11: the concentration of Adderall XR in the blood (red)
and gut (blue) changing over a period of time in a human
Figure 12: the concentration of Adderall XR in
the blood versus the gut in a human
15
Answer: Unlike the human body which carries about 77mL of blood per kilogram, a
monkey's body is composed of 54mL of blood per kilogram. Thus, the rhesus monkey is said to
carry about 3.5L of blood whilst a human carries 5L of blood within their body. For a human,
the amount of blood in the gut is about 1L, which is about 20% of the blood in the entire system.
Assuming that this ratio remains proportional, the blood in the gut of a rhesus monkey is 0.7L.
The concentration of Adderall XR found in the 65kg rhesus monkey’s blood and gut is higher
than that found in 65kg human body. Since, the blood in the rhesus monkey’s system is lower
than a human, the Adderall XR is found in higher concentration in the monkey’s system over a
period of time (hours). The decay time remains the same as Adderall XR found in a human’s
body, but in comparison to the rate that the Adderall XR is transferred between the gut and
blood, it is higher. Within the monkey, the Adderall XR is transferred more rapidly between the
gut and the blood because the drug is found in higher concentration within the blood and gut.
Figure 13: the concentration of Adderall XR in the blood
(red) and gut (blue) changing over a period of time in a
rhesus monkey
Figure 14: the concentration of Adderall XR in
the blood versus the gut in a rhesus monkey
16
Situation 4: Administering Drugs every 4, 8 & 16 hours
Let’s assume that the recommended dosage of Adderall XR for an average human is
about a pill for every eight hours. What happens if this is altered and someone consumes a pill
for every four hours? Or what happens if someone consumes a pill for every sixteen hours?
Based on Figure 15 and Figure 16, it is observed that it takes about sixty hours for the
drugs’ concentration in both the blood and gut to reach zero, and that a steady state cannot be
reached by six intakes.
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (hours)
Concentr
ation
Concentration in Blood and Gut vs. Time
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Gut
Blo
od
Concentration in Blood vs. Gut
Figure 15: the concentration of Adderall XR in
the blood (red) and gut (blue) changing over a
period of time for every four hours
Figure 16: the concentration of Adderall XR in
the blood versus the gut for every four hours
17
Based on Figure 17 and Figure 18, it is observed that it takes about sixty hours for the
drugs’ concentration in both the blood and gut to reach zero, and that a steady state can be
reached by about the third intake.
Figure 17: the concentration of Adderall XR in
the blood (red) and gut (blue) changing over a
period of time for every eight hours
Figure 18: the concentration of Adderall XR in
the blood versus the gut for every eight hours
0 20 40 60 80 100 120 1400
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time (hours)
Concentr
ation
Concentration in Blood and Gut vs. Time
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
0.3
0.35
GutB
lood
Concentration in Blood vs. Gut
18
Based on Figure 19 and Figure 20, it is observed that it takes about sixty hours for the
drugs’ concentration in both the blood and gut to reach zero, and that a steady state can be
reached by about the fifth intake.
Comparing the three observations from above, it is can be concluded that it takes the
same amount of time for the last dosage to get eliminated. Also, the steady state of Adderall XR
is reached when a smaller concentration of the drug is present. This is because having larger
intervals guarantees the body in having more time to eliminate the drug.
Figure 19: the concentration of Adderall XR in
the blood (red) and gut (blue) changing over a
period of time for every sixteen hours
Figure 20: the concentration of Adderall XR in
the blood versus the gut for every sixteen hours
0 0.05 0.1 0.15 0.2 0.250
0.05
0.1
0.15
0.2
0.25
GutB
lood
Concentration in Blood vs. Gut
0 20 40 60 80 100 120 140 160 180 2000
0.05
0.1
0.15
0.2
0.25
Time (hours)
Concentr
ation
Concentration in Blood and Gut vs. Time
19
Conclusion
To conclude, this project examines the pharmacokinetics of Adderall XR in different
situations. First, to model the pharmacokinetics of Adderall XR in the blood and gut, analytical
computations of differential equations that model the effects of the drug is solved using variation
of parameters. The results produce functions that model the concentrations of the drug in the gut
and blood, which using Matlab can be graphed. Then to study the effects of the two types of
Adderall, consumption of Adderall XR by a rhesus monkey and administering Adderall XR at
different time intervals, numerical computation using Runge-Kutta was done. Therefore, with the
help of mathematics, the pharmacokinetics of a drug can be studied.
Figure 21: Rhesus Monkey
20
Works Cited
"Adderall." Wikipedia. The Free Encyclopedia, n.d. Web. 17 Mar. 2015.
<http://en.wikipedia.org/wiki/Adderall#Mechanism_of_action>.
"Adderall." Drugs Forum. RSS, n.d. Web. 17 Mar. 2015. <https://drugs-
forum.com/forum/showwiki.php?title=Adderall>.
Different Mechanisms of Adderall/Dex Ritalin. Different Mechanisms of Adderall/Dex Ritalin,
2002. Web. 17 Mar. 2015. <http://www.dr-bob.org/babble/20020425/msgs/104479.html>.
Magoma, Gabriel. "Chapter 17 Introduction to Biochemical Pharmacology and Drug Discovery."
Intech, 2013. Web. 20 Mar 2015. < http://www.intechopen.com/books/drug-
discovery/introduction-to-biochemical-pharmacology-and-drug-discovery>.
Sherzada, Awista. "An Analysis of ADHD Drugs: Ritalin and Adderall." JCCC Honors Journal
3.1 (2012): n. pag. Web. 20 Mar. 2015.
<http://scholarspace.jccc.edu/cgi/viewcontent.cgi?article=1021&context=honors_journal>.
"Types of Drugs." Castle Craig Hospital, n.d. Web. 20 Mar. 2015.
<http://www.castlecraig.co.uk/resources/drugs/types-drugs>.
21
Appendix A
Purpose: Plot graphs – Concentration in Blood and Gut vs. Time (g1, g2) & Concentration in
Blood vs. Gut (g3)
%This file computes the Pharmacokinetics of Adderall n=5000; ti=0; %Initial Time tf=100; %Final Time h=(tf-ti)/n; int=8; %delay interval between admins. pwexp(0,int) pwexp(8,int) pwexp(16,int) c1=zeros(n+1,1); %Conc. in the gut c2=zeros(n+1,1); %Conc. in the blood T=zeros(n+1,1); k1=zeros(n,1); m1=zeros(n,1); k2=zeros(n,1); m2=zeros(n,1); k3=zeros(n,1); m3=zeros(n,1); k4=zeros(n,1); m4=zeros(n,1); K=0.086643397; %Rate at which drug is removed from the blood k12=0.05; %Rate of transfer from blood to gut k21=2.95; %Rate of transfer from gut to blood V1= 1; %Volume of Gut (Litre) V2= 5; %Volume of Blood in Body (Litre) %d=@(t) exp(-.1*t); %Dosage Function c1(1)=0; %Initial Concentration (gut) c2(1)=0; %Initial Concentration (blood) T(1)=ti; f1=@(x,y,z)-k21*x+(k12*V2/V1)*y+z/V1; f2=@(x,y)(V1*k21/V2)*x-(K+k12)*y;
22
for i=1:n T(i+1)=T(i)+h; k1(i)=f1(c1(i),c2(i),pwexp(T(i),int)); m1(i)=f2(c1(i),c2(i)); k2(i)=f1(c1(i)+h/2*k1(i),c2(i)+h/2*m1(i),pwexp(T(i)+h/2,int)); m2(i)=f2(c1(i)+h/2*k1(i),c2(i)+h/2*m1(i)); k3(i)=f1(c1(i)+h/2*k2(i),c2(i)+h/2*m2(i),pwexp(T(i)+h/2,int)); m3(i)=f2(c1(i)+h/2*k2(i),c2(i)+h/2*m2(i)); k4(i)=f1(c1(i)+h/2*k3(i),c2(i)+h/2*m3(i),pwexp(T(i+1),int)); m4(i)=f2(c1(i)+h/2*k3(i),c2(i)+h/2*m3(i)); c1(i+1)=c1(i)+h/6*(k1(i)+2*k2(i)+2*k3(i)+k4(i)); c2(i+1)=c2(i)+h/6*(m1(i)+2*m2(i)+2*m3(i)+m4(i)); end; hold on %g1=plot (T,c1); %set(g1,'Color', 'blue') %Gut %g2=plot(T,c2); %set(g2,'Color', 'red') %Blood %g3=plot(c1,c2); %Concentration of Blood(Y) vs. Gut(X) %This file computes six dosages function y=pwexp(x,L) if x<0 y = 0; elseif 0<= x & x< L y = exp(-x); elseif L <= x & x < 2*L y = exp(-x) + exp(-(x-L)); elseif 2*L<= x & x < 3*L y = exp(-x) + exp(-(x-L)) + exp(-(x-2*L)); elseif 3*L<= x & x < 4*L y = exp(-x) + exp(-(x-L)) + exp(-(x-2*L)) + exp(-(x-3*L)); elseif 4*L<= x & x < 5*L y = exp(-x) + exp(-(x-L)) + exp(-(x-2*L)) + exp(-(x-3*L)) + exp(-(x-4*L)); elseif 5*L<= x & x < 6*L y = exp(-x) + exp(-(x-L)) + exp(-(x-2*L)) + exp(-(x-3*L)) + exp(-(x-4*L)) + exp(-(x-5*L)); else y = exp(-x) + exp(-(x-L)) + exp(-(x-2*L)) + exp(-(x-3*L)) + exp(-(x-4*L)) + exp(-(x-5*L))+exp(-(x-6*L)); end
23
Appendix B
Purpose: Calculations for solving the modelling functions (y1, y2)
A = [ -2.95 0.25; 0.59 -(log(2)/8+0.05) ] [u,d] = eig(A) syms t psi = [ u(1,1)*exp(d(1,1)*t) u(1,2)*exp(d(2,2)*t); u(2,1)*exp(d(1,1)*t) u(2,2)*exp(d(2,2)*t)] B= vpa(psi) C=vpa(det(B)) psiinv=[ vpa(psi(2,2)/C) vpa(-psi(1,2)/C); vpa(-psi(2,1)/C) vpa(psi(1,1)/C)] D=[(exp(-t));0] E=[psiinv*D] F=vpa(int(E)) G=((B*F)) I=sym('c', [2,1]) J=(B*I) K=vpa(G+J) L=subs(K,t,0) M=solve(L) N=structfun(@subs,M) O=vpa(B*N) P=G+O Q=P(1,1) R=P(2,1) %S=ezplot('0.019297817272772538653612807409388*exp(-0.085157152820060616482678028660303*t) - 0.49080803451864529017037186452461*exp(-3.0014862447499326414401821239153*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) + 0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)',[0,100]) %axis([0 35 0 0.2]) %set (S,'Color', 'cyan') %Gut %T=ezplot('0.10107945036184128645676776432014*exp(-3.0014862447499326414401821239153*t) + 0.22114085512035159915629924495257*exp(-0.085157152820060616482678028660303*t) - 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)',[0,100]) %axis([0 60 0 0.18]) %set (T,'Color', 'magenta') %Blood
24
>> Matrix
A =
-2.9500 0.2500
0.5900 -0.1366
u =
-0.9794 -0.0869
0.2017 -0.9962
d =
-3.0015 0
0 -0.0852
psi =
[-(275689237860359*exp(-(6758746166706733*t)/2251799813685248))/281474976710656, -(6264286587770691*exp(-(3068109773666859*t)/36028797018963968))/72057594037927936]
[(7267432230605219*exp(-(6758746166706733*t)/2251799813685248))/36028797018963968, -(280409322244689*exp(-(3068109773666859*t)/36028797018963968))/281474976710656]
B =
[-0.97944492644458236441096232738346*exp(-3.0014862447499326414401821239153*t), -0.086934440032419721400280820944317*exp(-0.085157152820060616482678028660303*t)]
[0.20171176480802185948526528136426*exp(-3.0014862447499326414401821239153*t), -0.99621403480188419621299544814974*exp(-0.085157152820060616482678028660303*t)]
C =
0.99327248136112860935235845641696*exp(-3.0014862447499326414401821239153*t)*exp(-0.085157152820060616482678028660303*t)
25
psiinv =
[-1.0029614768313369922617508162919*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-0.085157152820060616482678028660303*t), 0.087523254357444108298198470388676*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-0.085157152820060616482678028660303*t)]
[-0.2030779756745164397037544430927*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t), -0.98607878988291542359632593836738*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)]
D =
exp(-t)
0
E =
-1.0029614768313369922617508162919*exp(-t)*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-0.085157152820060616482678028660303*t)
-0.2030779756745164397037544430927*exp(-t)*exp(0.085157152820060616482678028660303*t)*exp(3.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
F =
-0.50110835358583632664196507975923*exp(2.0014862447499326414401821239153*t)
0.2219812684774407675961537190039*exp(-0.9148428471799393835173219713397*t)
G =
0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t)
26
- 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
I =
c1
c2
J =
- 0.97944492644458236441096232738346*c1*exp(-3.0014862447499326414401821239153*t) - 0.086934440032419721400280820944317*c2*exp(-0.085157152820060616482678028660303*t)
0.20171176480802185948526528136426*c1*exp(-3.0014862447499326414401821239153*t) - 0.99621403480188419621299544814974*c2*exp(-0.085157152820060616482678028660303*t)
K =
0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t) - 0.97944492644458236441096232738346*c1*exp(-3.0014862447499326414401821239153*t) - 0.086934440032419721400280820944317*c2*exp(-0.085157152820060616482678028660303*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t)
0.20171176480802185948526528136426*c1*exp(-3.0014862447499326414401821239153*t) - 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.99621403480188419621299544814974*c2*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
27
L =
0.47151021724587269561496562374043 - 0.086934440032419721400280820944317*c2 - 0.97944492644458236441096232738346*c1
0.20171176480802185948526528136426*c1 - 0.99621403480188419621299544814974*c2 - 0.32222030548219285066065275669906
M =
c1: [1x1 sym]
c2: [1x1 sym]
N =
0.5011
-0.2220
O =
0.019297817272772538653612807409388*exp(-0.085157152820060616482678028660303*t) - 0.49080803451864529017037186452461*exp(-3.0014862447499326414401821239153*t)
0.10107945036184128645676776432014*exp(-3.0014862447499326414401821239153*t) + 0.22114085512035159915629924495257*exp(-0.085157152820060616482678028660303*t)
P =
0.019297817272772538653612807409388*exp(-0.085157152820060616482678028660303*t) - 0.49080803451864529017037186452461*exp(-3.0014862447499326414401821239153*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) + 0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
0.10107945036184128645676776432014*exp(-3.0014862447499326414401821239153*t) + 0.22114085512035159915629924495257*exp(-0.085157152820060616482678028660303*t) - 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
28
Q =
0.019297817272772538653612807409388*exp(-0.085157152820060616482678028660303*t) - 0.49080803451864529017037186452461*exp(-3.0014862447499326414401821239153*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) + 0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
R =
0.10107945036184128645676776432014*exp(-3.0014862447499326414401821239153*t) + 0.22114085512035159915629924495257*exp(-0.085157152820060616482678028660303*t) - 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t)
S = 0.019297817272772538653612807409388*exp(-0.085157152820060616482678028660303*t) - 0.49080803451864529017037186452461*exp(-3.0014862447499326414401821239153*t) - 0.019297817272772536644261338575425*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) + 0.49080803451864523225922696231585*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t) ; ([0 35 0 0.2]) %Gut T = 0.10107945036184128645676776432014*exp(-3.0014862447499326414401821239153*t) + 0.22114085512035159915629924495257*exp(-0.085157152820060616482678028660303*t) - 0.22114085512035157613039451311464*exp(-0.9148428471799393835173219713397*t)*exp(-0.085157152820060616482678028660303*t) - 0.10107945036184127453025824358442*exp(2.0014862447499326414401821239153*t)*exp(-3.0014862447499326414401821239153*t) ; ([0 60 0 0.18]) %Blood