pharmacokinetics of adderall - vanier...

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)


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


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


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


the blood (red) and gut (blue) changing over a

period of time for every four hours


the blood versus the gut for every four hours

17


drugs’ concentration in both the blood and gut to reach zero, and that a steady state can be

reached by about the third intake.



period of time for every eight hours


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


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


18


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.



period of time for every sixteen hours


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


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


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

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