DEVELOPMENT OF TOLERANCE AND DEPENDENCE IN BARBITURATE … · development of tolerance and dependence in barbiturate use: ... development of tolerance and dependence in barbiturate

DEVELOPMENT OF TOLERANCE AND DEPENDENCE IN BARBITURATE USE:

A SYSTEMS MODELING APPROACH

by

Ali Osman Konuray

B.S., Chemical Engineering, Istanbul Technical University, 2005

Submitted to the Institute for Graduate Studies in

Science and Engineering in partial fulfillment of

the requirements for the degree of

Master of Science

Graduate Program in Industrial Engineering

Bo�aziçi University

2008

ii

DEVELOPMENT OF TOLERANCE AND DEPENDENCE IN BARBITURATE USE:

A SYSTEMS MODELING APPROACH

APPROVED BY:

Prof. Yaman Barlas …………………

(Thesis Supervisor)

Assist. Prof. Aybek Korugan ………………...

Assoc. Prof. Cengizhan Öztürk ………………...

DATE OF APPROVAL: 22.09.2008

iii

ACKNOWLEDGEMENTS

I am deeply grateful to Professor Yaman Barlas, my thesis supervisor, for being a

great example of an enthusiastic scientist. Without him, I would never indulge the field of

System Dynamics which seemed, at first, very distinct from my scientific background. His

contribution to my studies in recent years is invaluable.

I would like to thank Assist. Prof. Aybek Korugan and Assoc. Prof. Cengizhan

Öztürk for taking part in my thesis jury and providing valuable feedback.

I would like to thank Ceyhun Eksin and Genco Fas for their company, during and

after intense academic moments. I would also like to thank members of SESDYN Research

Group for their support and friendship, and all the bright people in the department for

contributing to my academic development.

I would like to thank Süheyla Ayar for sharing her life with me in the last couple of

years.

I would like to express my deepest gratitude to my mother Gülsün Konuray for

inspiring me with her artistic personality. Her wisdom is my guiding light. Lastly, I would

like to thank my late father Dr. M. Mehmet Konuray for installing in me an unfailing

respect for science.

iv

ABSTRACT DEVELOPMENT OF TOLERANCE AND DEPENDENCE IN

BARBITURATE USE: A SYSTEMS MODELING APPROACH

A system dynamics model is constructed to study the development of tolerance and

dependence to phenobarbital in prolonged use. Phenobarbital is a sedative barbiturate drug

whose target of action is the brain. Although its use has decreased over the years,

phenobarbital is still being prescribed to many patients. As a side effect, phenobarbital

enhances the synthesis of its own metabolic enzymes in the liver. This enzyme induction

problem causes increased tolerance to phenobarbital over time. Moreover, the brain adapts

to the presence of the drug and its sensitivity decreases with time. The resulting decrease in

drug effectiveness urges the drug user to increase the dose. A feedback loop results, as the

increased dose in turn leads to more metabolic induction and neuroadaptation. Furthermore,

the brain’s adaptation to the drug plays a major role in rendering the user dependent on the

drug hence complicating withdrawal from the drug. Because adaptive changes persist even

after drug intake stops, upon abrupt discontinuation to the drug, the user experiences

unwanted rebound effects.

The model incorporates phenobarbital absorption, distribution, metabolism, and

elimination processes with enzyme induction and neuroadaptation related structures. We

start with validating the model by assuming a normal person. We then consider three

scenarios: An epilepsy patient, a normal person taking an enzyme inhibitor drug

concurrently with phenobarbital, and a normal person adopting different dosing schemes.

We finally search for dosing regimens that facilitate gradual withdrawal from the drug so

that rebound effects are avoided. Results show that an epilepsy patient is more prone to

developing tolerance and dependence. Also, it is shown that concurrent intake of an

enzyme inhibitor drug weakens rebound effects after sudden discontinuation since

phenobarbital is cleared slower. Experiments with dosing frequencies show that the patient

is more prone to tolerance and dependence development if dosing frequency is decreased.

Finally, experiments confirm that in order to withdraw from the drug safely, doses should

be reduced gradually.

v

ÖZET

BARB�TURAT KULLANIMINDA TOLERANS VE BA�IMLILIK OLU�UMU: B�R S�STEM MODELLEMES�

Sürekli fenobarbital kullanımında tolerans ve ba�ımlılık olu�umunu ara�tırmak için

bir sistem dinami�i modeli kurulmu�tur. Fenobarbital, beyni etkileyen sedatif

(sakinle�tirici) bir ilaçtır. Geçmi� yıllara kıyasla kullanımı azalmı� olmasına ra�men bir

çok insan halen fenobarbital kullanmaktadır. Fenobarbital bir yan etki olarak kendini

metabolize eden karaci�er enzimlerinin sayısını arttırır. Bu enzim artı�ı ilaca tolerans

olu�umuna neden olur. Bunun yanında, zamanla beyin ilaca adapte olur ve dolayısıyla ilaca

kar�ı hassasiyeti azalır. Bu iki faktör, ilacın etkinli�ini azalttı�ından kullanıcının aynı

etkiyi hissedebilmesi için dozu arttırması gerekir. Artan dozlar metabolizma ve

nöroadaptasyon etkilerini güçlendirerek kısır bir geri bildirim döngüsü olu�turur.

Nöroadaptasyon, kullanıcıyı ilaca ba�ımlı kılarak ilacın bırakılmasını zorla�tırır. �laç alımı

kesilmesine ra�men adaptif de�i�imler hemen yokolmaz ve dolayısıyla kullanıcı ilacı

bıraktıktan kısa bir süre sonra yoksunluk sendromu ya�ar.

Kurulan model, fenobarbital ilacının emilimi, da�ılımı, metabolizması ve atılımı

süreçlerini içermektedir. Enzim artı�ı ve nöroadaptasyon mekanizmaları da modele

eklenmi�tir. Tezde öncelikle normal bir insan ele alınmakta ve model empirik veriler

kullanılarak gerçeklenmektedir. Bunun ardından, bir epilepsi hastasının, bir enzim

inhibitörüyle birlikte fenobarbital kullanan bir insanın, ve normal bir insanın uyguladı�ı

farklı doz uygulamalarının modellendi�i üç ayrı senaryo incelenmi�tir. Son olarak

yoksunluk sendromunu engelleyebilecek doz stratejileri ile deneyler yapılmı�tır. Sonuçlar

epilepsi hastalarının tolerans ve ba�ımlılık geli�imine daha hassas olduklarını göstermi�tir.

Di�er taraftan, fenobarbital ile beraber enzim inhibitörü bir ilaç alınırsa, fenobarbital

vücuttan daha yava� atılmakta, dolayısıyla da fenobarbital alımı aniden kesildi�inde ortaya

çıkan yoksunluk sendromunun �iddeti daha az olmaktadır. Farklı doz stratejileriyle yapılan

deneylerde, doz alım sıklı�ı azaldıkça tolerans ve ba�ımlılık geli�iminin hızlandı�ı

görülmü�tür. Son olarak, yoksunluk sendromundan kaçınmak için, dozun kademeli bir

�ekilde azaltılması gerekti�i gösterilmi�tir.

vi

TABLE OF CONTENTS

ACKNOWLEDGEMENTS.............................................................................................. iii

ABSTRACT...................................................................................................................... iv

ÖZET ................................................................................................................................. v

LIST OF FIGURES ........................................................................................................ viii

LIST OF TABLES............................................................................................................ xi

LIST OF ABBREVIATIONS.......................................................................................... xii

1. INTRODUCTION ......................................................................................................... 1

1.1. Neurotransmission in the Central Nervous System ................................................ 4

1.2. Definition of Pharmacokinetics .............................................................................. 7

1.3. Pharmacokinetics of Barbiturates ........................................................................... 7

1.4. Action mechanism of barbiturates .......................................................................... 9

1.5. Development of Tolerance and Dependence to Barbiturates ............................... 11

2. RESEARCH OBJECTIVE AND DYNAMIC HYPOTHESIS ................................... 16

3. METHODOLOGY ...................................................................................................... 19

4. MODEL DESCRIPTION ............................................................................................ 20

4.1. Pharmacokinetics Sector....................................................................................... 20

4.1.1. Fundamental Approach and Assumptions .................................................. 20

4.1.2. Description of the Structure........................................................................ 20

4.2. Central Nervous System Sector ............................................................................ 26



4.3. Dose Sector ........................................................................................................... 33



4.4. Model Parameters ................................................................................................. 35

5. VALIDATION OF THE MODEL .............................................................................. 37

5.1. Simulation Results ................................................................................................ 37

5.1.1. Single Dose ................................................................................................. 37

5.1.2. Continuous Drug Intake with Constant Dose ............................................. 40

vii

5.1.3. Continuous Drug Intake with Dose Increase as a Result of Feedback ....... 42

5.1.3.1. Drug Treatment for Seven Days .................................................... 43

5.1.3.2. Drug Treatment for 20 Days.......................................................... 47

5.1.3.3. Drug Treatment for 60 Days.......................................................... 49

5.2. Model Validity Discussion ................................................................................... 52

6. SCENARIO ANALYSES............................................................................................ 56

6.1. Epilepsy Patient .................................................................................................... 56

6.2. Co-administration of a Drug That Causes Enzyme Inhibition ............................. 60

6.3. Different Dosing Frequencies ............................................................................... 64

7. ANALYSIS OF WITHDRAWAL POLICIES ............................................................ 70

7.1. Withdrawal after 20 days of treatment ................................................................. 70

7.1.1. An unsuccessful regimen ............................................................................ 70

7.1.2. A successful regimen .................................................................................. 72

7.2. Withdrawal after 60 days of treatment ................................................................. 74

7.2.1. An unsuccessful regimen ............................................................................ 74

7.2.2. A Successful Regimen ................................................................................ 76

8. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS................................. 78

APPENDIX. EQUATIONS OF THE MODEL............................................................... 80

REFERENCES ................................................................................................................ 89

viii

LIST OF FIGURES

Figure 1.1. Relative safety of barbiturates and benzodiazepines....................................... 1

Figure 1.2. Frequency of barbiturate use among twelfth grade high school students ....... 2

Figure 1.3. Different types of synapses (top). A schematic representation of

neurotransmission (bottom) ............................................................................ 5

Figure 1.4. Steps in excitatory and inhibitory neurotransmission ..................................... 6

Figure 1.5. Blood plasma concentration – time data for a single IV dose of thiopental ... 8

Figure 1.6. GABAA receptor-chloride channel complex ................................................... 9

Figure 1.7. Pre- and post-synaptic neuroinhibition by barbiturates................................. 10

Figure 1.8. Proposed mechanism of enzyme induction by phenobarbital ....................... 12

Figure 1.9. Administered doses of PB ............................................................................. 13

Figure 1.10. Change in the intensity of rebound behavior with time .............................. 14

Figure 1.11. The Himmelsbach hypothesis ..................................................................... 15

Figure 2.1. Causal loop diagram for tolerance and dependence development ................ 17

Figure 4.1. Stock-flow structure of the pharmacokinetics sector .................................... 21

Figure 4.2. Saturability of enzyme induction .................................................................. 25

Figure 4.3. Concentration-response data for phenobarbital............................................. 27

Figure 4.4. Stock-flow structure of the CNS sector......................................................... 28

Figure 4.5. Graphical function for IndAdptnRate ............................................................ 29

Figure 4.6. Graphical function for EffSatur ..................................................................... 30

Figure 4.7. Graphical function for EffAdptnOnNormClCur ............................................ 31

Figure 4.8. Graphical function for EffPBOnReadptn ...................................................... 32

Figure 4.9. Stock-flow structure for the Dose Sector ...................................................... 34

Figure 5.1. Absorption and distribution of a single dose................................................. 38

Figure 5.2. Increasing chloride current in the brain after a single dose........................... 39

Figure 5.3. Dynamics of enzyme induction and neuroadaptation for a single dose ........ 40

Figure 5.4. Constant doses (a) and drug profiles in the brain (b) in both a seven day

and a 20 day treatment. ................................................................................. 41

Figure 5.5. Enzyme induction and neuroadaptation and the resulting chloride current

profile when the user takes constant doses (for seven and 20 days)............. 42

ix

Figure 5.6. Dose profile (a) and drug amount in the brain (b) in the seven day drug

treatment followed by abrupt withdrawal..................................................... 44

Figure 5.7. Enzyme and neuroadaptation dynamics in the seven day drug treatment

followed by abrupt withdrawal ..................................................................... 45

Figure 5.8. Behavior of chloride current in the seven day drug treatment ...................... 46

Figure 5.9. Dose profile (a) and drug amount in the brain (b) in the 20 day drug

treatment followed by abrupt withdrawal ..................................................... 47

Figure 5.10. Enzyme and neuroadaptation dynamics in the 20 day drug treatment

followed by abrupt withdrawal ................................................................... 48

Figure 5.11. Behavior of chloride current in the 20 day drug treatment ......................... 49

Figure 5.12. Dose profile (a) and drug amount in the brain (b) in the 60 day drug

treatment followed by abrupt withdrawal ................................................... 50

Figure 5.13. Enzyme and neuroadaptation dynamics in the 60 day drug treatment

followed by abrupt withdrawal ................................................................... 51

Figure 5.14. Behavior of chloride current in the 60 day drug treatment ......................... 52

Figure 5.15. Progression of enzyme induction in 20 days of continuous PB use............ 53

Figure 5.16. Comparison of tolerance dynamics generated by the model (a) against

real data (b) from Gay et al (1983). ............................................................ 53

Figure 5.17. Tolerance and dependence indicators for 60 days of continuous PB

intake, (a) Model output, (b) Real data. ...................................................... 54

Figure 5.18. Differences in withdrawal dynamics between a partially dependent

(20 day user) and a completely dependent (60 day user). ........................... 55

Figure 6.1. Dose profiles (a) and drug profiles in brain tissue (b) of both a healthy and

an epileptic individual in 20 days of continuous PB use .............................. 58

Figure 6.2. Enzyme and neuroadaptation dynamics in both a healthy and an epileptic

individual taking PB for the last 60 days ...................................................... 59

Figure 6.3. Chloride current in a healthy and an epileptic individual) ............................ 59

Figure 6.4. Flurbiprofen average clearance as influenced by fluconazole

pre-treatment ................................................................................................. 61

Figure 6.5. Dose profiles (a) and drug amounts in the brain (b) with and without

fluconazole pre-treatment ............................................................................. 62

Figure 6.6. Enzyme and neuroadaptation dynamics with and without fluconazole pre-

x

treatment ....................................................................................................... 63

Figure 6.7. Enzyme and neuroadaptation dynamics in different dosing schemes (No

feedback to increase the doses)..................................................................... 65

Figure 6.8. Comparative behavior of chloride current (No feedback to increase the

doses) ............................................................................................................ 66

Figure 6.9. Difference in the extent of tolerance development w.r.t dosing schemes

(Feedback allowed to increase doses)........................................................... 67

Figure 6.10. Neuroadaptation dynamics for different dosing schemes (Feedback

allowed to increase doses) ........................................................................... 67

Figure 6.11. Dependence dynamics for different dosing schemes (Feedback allowed

to increase doses) ........................................................................................ 68

Figure 6.12. Behavior of chloride current in different dosing schemes (Feedback

allowed to increase doses) ........................................................................... 69

Figure 7.1. Dynamics of an unsuccessful withdrawal regimen after partial

dependence.................................................................................................... 71

Figure 7.2. Severity of withdrawal signs after an unsuccessful dosing strategy in

partial dependence ......................................................................................... 72

Figure 7.3. Dynamics in a successful withdrawal regimen after partial dependence ...... 73

Figure 7.4. Severity of withdrawal signs after a successful dosing strategy in partial

dependence..................................................................................................... 73

Figure 7.5. Results for an unsuccessful withdrawal regimen after complete

dependence..................................................................................................... 75

Figure 7.6. Severity of withdrawal signs after an unsuccessful dosing strategy in

complete dependence..................................................................................... 75

Figure 7.7. Results for a gradual withdrawal regimen of 30 days following a 60 day

drug treatment ................................................................................................ 76

Figure 7.8. Severity of withdrawal signs after a successful dosing strategy in complete

dependence..................................................................................................... 77

xi

LIST OF TABLES Table 1.1. Classification and properties of barbiturates .................................................... 3

Table 4.1. Main pharmacokinetic parameters used in the model .................................... 36

Table 4.2. Other pharmacokinetic parameters ................................................................. 36

Table 5.1. Initial values for the stocks ............................................................................. 37

xii

LIST OF ABBREVIATIONS

Adptn Adaptation

Arterial Arterial Blood

Braincapil Brain Capillary

Braintis Brain Tissue

C Concentration (of phenobarbital in)

ClCur Chloride Current

ClCurWOP B Chloride Current Without Phenobarbital

CNS Central Nervous System

CYP Cytochrome P

Eff Effect

GABA Gamma Amino Butyric Acid

GI Gastrointestinal

Ind Indicated

Induc Induction

M Amount (of phenobarbital in)

Norm Normal

P Tissue-Blood Partitition Coefficient (of phenobarbital in)

PB Phenobarbital

Q Blood Volumetric Flow Rate (through)

Readptn Re-adaptation

Real Realized

ThresholdSedat Sedation Threshold

V Volume (of)

Venous Venous Blood

1

1. INTRODUCTION

Barbiturates are classified as central nervous system (CNS) depressants. They act

generally on the CNS. In low doses, they cause sedation and as the dose is increased, the

user experiences hypnosis (i.e. sleep). Further increase in the dose results in anesthesia and

finally coma. Overdose of barbiturates causes severe respiratory depression and may lead

to death. For instance, Jimi Hendrix, the famous rock artist, died of barbiturate overdose in

the year 1970.

Because of having high abuse potential, they are being replaced by the safer

benzodiazepines. Figure 1.1 gives an idea about the relative safety of barbiturates and

benzodiazepines.

Figure 1.1. Relative safety of barbiturates and benzodiazepines (Katzung 2004)

The dose-effect relationship of barbiturates is rather linear and lethal overdoses are

more likely. On the other hand, this relationship is saturable for benzodiazepines. At high

doses, as the dose is further increased, CNS depression stays almost constant. This enables

a wider margin of safety.

2

Despite their high abuse potential, barbiturates are still being used as anti-

convulsants (i.e. anti-epileptic drugs), intravenous anesthetics, and death inducing agents

(Hardman and Limbird, 2001). Furthermore, a lot of people still use barbiturates for

sedation or to fall asleep. Alarmingly, a statistical study revealed that the frequency of

barbiturate use among twelfth grade high school students in the U.S. has increased slightly

over the last few years (See Figure 1.2 below).

Figure 1.2. Frequency of barbiturate use among twelfth grade high school students (From

http://www.monitoringthefuture.org)

Barbiturates are classified with respect to their onset and duration of action. However,

the action mechanism is the same for all barbiturates. Different barbiturate classes are

tabulated in Table 1.1.

3

Table 1.1. Classification and properties of barbiturates (Hardman and Limbird, 2001)

CLASS COMPOUND

(TRADE NAMES) ROUTES OF ADMINISTRATION

HALF-LIFE, HOURS

THERAPEUTIC USES

Methohexital (BREVITAL)

I.V.† 3-5* Induction and/or maintenance of anesthesia

Ultra-short- acting

Thiopental (PENTHOTAL)

I.V., rectal 8-10* Induction and/or maintenance of anesthesia, preoperative sedation, emergency management of seizures

Pentobarbital (NEMBUTAL)

Oral, I.M. †, I.V., rectal

15-50 Insomnia, preoperative sedation, emergency management of seizures

Short-acting

Secobarbital (SECONAL)

Oral, I.M., I.V., rectal 15-40 Insomnia, preoperative sedation, emergency management of seizures

Amobarbital (AMYTAL)

Oral, I.M., I.V. 10-40 Insomnia, preoperative sedation, emergency management of seizures

Aprobarbital (ALURATE)

Oral 14-34 Insomnia

Butabarbital (BUTISOL, others)

Oral 35-50 Insomnia, preoperative sedation

Intermediate-acting

Butalbital Oral 35-88 Marketed in combination with analgesic agents

Mephobarbital (MEBARAL)

Oral 10-70 Seizure disorders, daytime sedation

Long-acting

Phenobarbital (LUMINAL, others)

Oral, I.M., I.V. 80-120 Seizure disorders, status epilepticus, daytime sedation

† I.M.: intramuscular injection, I.V.: intravenous administration

* Value represents terminal half-life due to metabolism by liver; redistribution following intravenous administration produces effects lasting only a few minutes

4

Other than the therapeutic uses mentioned in Table 1.1, some barbiturates have had

different uses. For example, other than its common use as an inducer of anesthesia, the

ultra-short acting thiopental is used in large doses in the United States to execute prisoners

on death row. In lower doses, it is sometimes used as a truth serum. The drug does not

itself force people to tell the truth, but is thought to make subjects more likely to be caught

off guard when questioned (Stevens and Bannon, 2007).

Barbiturate use can cause dependence. This dependence may be psychological in the

initial stages of barbiturate treatment. However, as treatment continues, tolerance and then

physical dependence develops. As people develop tolerance for barbiturates, they may

need more of the drug to get the desired effect. This can lead to an overdose. As Weil and

Rosen (2004) point out in From Chocolate to Morphine, “People who get in the habit of

taking sleeping pills every night to fall asleep might start out with one a night, progress to

two, and then graduate to four to get the same effect. One night the dose they need to fall

asleep might also be the dose that stops their breathing." Overdoses occur because

tolerance to the lethal effects of the drug is less than tolerance to its therapeutic effects (e.g.

sedation). In physical dependence, the user experiences difficulties in stopping drug

treatment. Upon discontinuation of the drug, the user experiences a withdrawal syndrome

in which he/she goes through a state of rebound hyperexcitability manifested as excessive

nightmarish dreaming, restlessness, irritability, and convulsions (Liska, 2001).

Although their use is decreasing, mechanism of action of barbiturates is just recently

being clarified. Before reviewing the mechanism, it would be useful to briefly overview

first the subject of neurotransmission and then pharmacokinetics.

1.1. Neurotransmission in the Central Nervous System

Neurotransmission means the communication of nerve cells (i.e. neurons). This is

accomplished by billions of interconnected neurons. The point where two neurons meet is

called a synapse. Different types of synapses exist and these are shown in Figure 1.3.

5

Figure 1.3. Different types of synapses (top). A schematic representation of

neurotransmission (bottom) (From http://www.answers.com/topic/synapse?cat=health)

The message between two neurons is conveyed through synapses via substances

called neurotransmitters. Neurotransmitters are stored in specialized sacs (i.e. vesicles)

inside the presynaptic nerve endings (i.e. nerve terminals). When a reversal of electrical

charge is experienced in the nerve terminal, the vesicles translocate and bind to the

neuronal membrane. This process is called docking. The reversal of charge is called the

action potential. It is accomplished through an influx of sodium ions and efflux of

potassium ions through specialized ion channels located on the axon of the presynaptic

neuron. This depolarization is conveyed to the nerve ending and causes ion channels to

open and allow an influx of calcium. The influx of calcium ions induces the release of the

neurotransmitter to the synaptic cleft by exocytosis of the docked vesicles. The

neurotransmitter then travels to the postsynaptic neuron and binds to specific receptor

proteins on its membrane and changes the membrane electrical potential. If the

neurotransmitter is excitatory, an influx of sodium ions to the postsynaptic neuron causes

depolarization and this initiates an action potential in the neuron. However, if the

6

neurotransmitter is inhibitory, an influx of chloride and potassium ions occurs which

hyperpolarizes the membrane and thus an action potential is inhibited (Hardman and

Limbird, 2001). In figure 1.4, inhibitory and excitatory neurotransmission are summarized.

Figure 1.4. Steps in excitatory and inhibitory neurotransmission (Hardman and Limbird,

2001)

The most widespread excitatory and inhibitory transmitters in the CNS are glutamate

and gamma-aminobutyric acid (GABA), respectively (Powis and Bunn, 1995). As

mentioned previously, there exist receptors on neuronal membranes that are specialized to

bind neurotransmitters. Each receptor is specialized to bind a specific type of

neurotransmitter. Furthermore, there are many sub-types of a receptor for a specific

7

neurotransmitter and functions of each of these subunits are modulated by different

mechanisms (Hardman and Limbird, 2001).

1.2. Definition of Pharmacokinetics

There are several phases before an administered drug causes a response. After

administration, the drug goes through many phases during which it may lose effectiveness.

After oral administration, the drug must dissolve in stomach fluids, and it must be absorbed

from the gastrointestinal tract. Once absorbed, it is directly transported to the liver via the

hepatic portal vein. The metabolism in liver at this stage is referred to as first-pass

metabolism. In drug development, it is aimed to design drugs that have little first-pass

metabolism since it has a negative impact on drug efficacy. Furthermore, a drug may also

undergo elimination in different regions such as the gastrointestinal wall which too is an

undesired property. After first-pass metabolism, the remaining drug enters blood

circulation and reaches the target organ. There, it binds its receptor to exert its effect.

While in blood circulation, the drug is transported to the liver once more and it undergoes

further elimination. Also while in circulation, it may bind to blood plasma proteins or

tissues of different organs. Once bound, a drug molecule is ineffective. This process of

drug delivery in the body is referred to as pharmacokinetics.

1.3. Pharmacokinetics of Barbiturates

Most barbiturates are rapidly absorbed into the blood following oral intake. The most

important factor that plays a role in the entrance of a barbiturate into the brain is its lipid

solubility. To exemplify the differences in pharmacokinetic profiles of barbiturates, we

consider two barbiturates: ultra-short acting thiopental and long acting phenobarbital (See

Table 1.1).

Due to its high lipid solubility, the ultra-short acting thiopental has a very rapid onset

of effects in the CNS. In comparison, the long acting phenobarbital has low lipid solubility

and thus penetrates into the brain slower.

8

In order to be cleared from the body, barbiturates must be transformed into more

water-soluble forms so that they can be filtered in the kidneys. Only insignificant quantities

(less than 1per cent) of thiopental are excreted unchanged in the urine. Unlike thiopental,

20 to 30 percent of the administered dose of phenobarbital is excreted unchanged.

The elimination-half life of phenobarbital is 4 to 5 days. For thiopental, the situation

is much more complex. Upon intravenous administration, thiopental rapidly penetrates into

the brain due to its very high lipid solubility and if the dose is sufficient, produces loss of

consciousness in one circulation time. The blood plasma-brain equilibrium is reached in

less than a minute. After that, thiopental diffuses out of the brain and out of other tissues

that receive high blood supply and is redistributed to all the remaining less perfused tissues

such as muscle and fat. It is because of this rapid redistribution that a single dose of

thiopental is very short acting (Katzung, 2004). The redistribution phenomenon causes the

half-life of thiopental to be time dependent. Initially, the fall in plasma concentration is

very rapid corresponding to a half-life of less than ten minutes. It is denoted as t1/2α in

Figure 1.5 below. After redistribution to less perfused areas, the fall of concentration slows

down. The half-life increases to more than ten hours. This half-life is denoted as t1/2β in the

figure.

Figure 1.5. Blood plasma concentration – time data for a single IV dose of thiopental

(From http://www.accessmedicine.com/popup.aspx?aID=414128&print=yes)

9

1.4. Action mechanism of barbiturates

It was shown that barbiturates exert their CNS-depressant effects by both

potentiating the inhibitory effects of GABA and suppressing excitatory effects of

glutamate. However, suppression of excitatory neurotransmission does not contribute to

their sedative/hypnotic effects (Powis and Bunn, 1995; Joo et al., 1999).

At low to moderate concentrations, barbiturates bind to the GABAA receptor. The

GABAA receptor is a sub-type of GABA receptors which is classified as a ligand-gated ion

channel meaning that the binding of a ligand (a molecule) to the receptor causes the ion

channel to open. The GABAA receptor is composed of different sub-units. The distribution

of these sub-units in the CNS is widespread and heterogeneous and this heterogeneity has

yet to be fully defined (Hardman and Limbird, 2001). Schematically, the GABAA receptor-

ion channel complex is as in Figure 1.6.

Figure 1.6. GABAA receptor-chloride channel complex. There are five binding sites

(subunits) on the complex (From http://www.ifcc.org)

By binding to its specific site, barbiturates enhance the inhibitory chloride ion

currents mediated by GABA. Essentially, barbiturates increase the time for which GABA-

activated channels are open. At higher concentrations, they activate the chloride channels

even in the absence of GABA. This action is regarded as postsynaptic inhibition. In

10

addition to postsynaptic effects, barbiturates induce GABA-mediated presynaptic

inhibition as well. This takes place in axo-axonic synapses (See Figure 1.3). GABA

released from the ending of the inhibitory neuron binds to GABA receptors on the terminal

of the excitatory neuron and causes a modest depolarization which decreases excitatory

neurotransmitter release. It was also shown that especially at higher concentrations,

barbiturates directly suppress excitatory transmission mediated by glutamate. The post-

and pre-synaptic inhibition effects of barbiturates are shown in Figure 1.7.

Figure 1.7. Pre- and post-synaptic neuroinhibition by barbiturates (Powis and Bunn, 1995)

Also, at anesthetic concentrations, barbiturates inhibit calcium influx to the pre-

synaptic nerve ending and thus reduce transmitter release. In addition to these, barbiturates

reduce axonal conduction through ion channels and thus increase the threshold for

electrical excitability and decrease the rate of rise of the action potential. However, these

11

effects are realized at very high concentrations which are practically irrelevant (Powis and

Bunn, 1995).

1.5. Development of Tolerance and Dependence to Barbiturates

Barbiturates have been shown to cause the phenomenon of enzyme induction. In the

liver, there exists a system of enzymes that are responsible for converting many

endogenous and exogenous substances into active and/or inactive forms. The so-called

cytochrome P450 family of enzymes constitutes the majority of the enzyme population in

the liver (Hardman and Limbird, 2001). By convention, cytochrome enzymes have the

prefix CYP. The CYP enzymes catalyze various destructive reactions such as oxidation.

The inducing effect of barbiturates causes more enzymes to be synthesized and thus a

faster metabolism of the substrates of these enzymes. When the set of substrates include

the drug itself, this is called autoinduction. In time, a tolerance to the barbiturate occurs

and higher doses are required to induce the same drug effect. Among barbiturates,

phenobarbital (will be denoted by PB hereafter) is the most potent inducer of CYP2C

subfamily of enzymes. Since PB itself is mostly metabolized by this subfamily of enzymes

(Tanaka, 1999), it has autoinduction properties. This was also reported by Magnusson

(2007).

Induction of enzymes by PB in rats is studied by Magnusson et al. (2006). Their

purpose is to characterize the magnitude, time course, and specificity of PB mediated

enzyme induction, and to develop an integrated pharmacokinetic model that represents the

change in the activities of different CYP enzymes. In another study, Raucy et al. (2002)

work with human liver cells in vitro to investigate the extent of induction of CYP2C

enzymes by several inducers including PB.

The mechanism of induction is not fully understood. Nevertheless, there is progress.

A variety of drugs and xenobiotics cause enzyme induction and each is believed to have its

own mechanism. It is believed that inside liver cells, there exist several receptors that

respond to different types of chemicals. These receptors are called nuclear receptors. An

excellent review on the topic is provided by Handschin and Meyer (2003). It is believed

12

that upon exposure to the chemical to which it is sensitive, these nuclear receptors

translocate to the nucleus of the cell and bind to specific regions on the DNA molecule and

modulate protein synthesis. PB is believed to activate the CAR (Constitutively Active

Receptor) type of nuclear receptors. The proposed mechanism of enzyme induction by PB

in a liver cell is shown in Figure 1.8.

Figure 1.8. Proposed mechanism of enzyme induction by phenobarbital (Simplified from

Zelko and Negishi, 2000). PB: Phenobarbital, HSPs: Heat Shock Proteins, CAR:

Constitutively Active Receptor, RXR: Retinoid X Receptor

Upon exposure to PB, the heat shock proteins dissociate from the CAR receptor by a

dephosphorylation reaction. The true PB target in this event is not known. Upon liberation,

the CAR receptor enters the nucleus and it is activated by a phosphorylation reaction. The

CAR receptor then heterodimerizes (i.e. combines with another molecule of different

13

structure) with the RXR type of receptor and finally binds to a specific area on the DNA.

The binding eventually leads to an increased rate of enzyme synthesis. The increased rate

of enzyme synthesis in turn leads to faster metabolism of the drug and thus a tolerance

develops to the effects of the drug.

Development of tolerance to PB during chronic treatment is studied on rats by Gay et

al. (1983). Their aim is to quantify the development of tolerance. They give rats two daily

doses of PB so as to achieve the same level of CNS depression with each dose. They show

that the doses show an increasing trend (Figure 1.9). This is also referred to as

pharmacokinetic tolerance.

Figure 1.9. Administered doses of PB. Half-filled circles are morning doses, filled circles

are total daily doses (sum of morning and evening doses). All doses result in the same level

of CNS depression (Gay et al., 1983)

Physical dependence to barbiturates develops over a time period of weeks to months

as opposed to pharmacokinetic tolerance which peaks in a few days to a week (Hardman

and Limbird, 2001). The major cause of physical dependence is brain’s adaptation to the

drug. This adaptation is called neuroadaptation. Physical dependence renders withdrawal

from barbiturates difficult. Upon withdrawal, a dependent barbiturate user experiences

rebound effects such as seizures and is urged to continue the drug.

14

In a research, although PB enhances inhibitory neurotransmission by increasing the

rate of GABA binding, it is shown that after rats were treated with PB for a long time, they

show decreased GABA binding. It is believed that this is due to an adaptive response by

the rats which results in desensitized or down-regulated GABAA receptors (Ito et al., 1996).

This down-regulation decreases chloride flow through the channel and thus inhibitory

neurotransmission weakens.

In the same study by Gay et al., it is shown that upon abrupt cessation of PB

treatment, rats experience rebound effects such as ear twitches, tremor, and tail erection.

However, these withdrawal syndromes weaken with time as shown in Figure 1.10.

Figure 1.10. Change in the intensity of rebound behavior with time. Rats are observed

twice daily for withdrawal signs following abrupt termination of 35 days of PB treatment

(Gay et al., 1983)

It is believed that all chemicals promoting inhibitory neurotransmission trigger

similar mechanisms of neuroadaptation. The mechanisms of neuroadaptation induced by

chronic ethanol use that lead to tolerance and dependence are studied by many researchers

(Brailowsky and Garcia, 1999; Finn and Crabbe, 1997; Kokka et al., 1993; Littleton, 1998).

Similar to barbiturates, ethanol acutely promotes the inhibitory effects of the

15

neurotransmitter GABA by increasing chloride ion flow through the GABAA channel. It is

being speculated that, as an adaptation, the receptor-channel complex counteracts this

effect by changing the composition of its subunits, and thus reducing the chloride flow.

Unintended effects such as hallucinations or seizures occur upon withdrawal from chronic

ethanol exposure. This is called alcohol withdrawal syndrome. It is believed that the

adaptive changes on GABAA receptors and calcium ion channels persist during alcohol

withdrawal and contribute to the withdrawal syndrome. This suggestion is in agreement

with the Himmelsbach hypothesis which illustrates the development of tolerance and

dependence. Schematically, the hypothesis is as in Figure 1.11.

Figure 1.11. The Himmelsbach hypothesis (Littleton, 1998)

The hypothesis can be applied as well to the barbiturate case since it involves similar

neuroadaptative changes and it is shown that a disrupt discontinuation of barbiturate use

results in rebound hyperexcitability, characterized by excessive nightmarish dreaming,

restlessness, irritability and convulsions. It is generally suggested that barbiturate dosage

must be reduced gradually to avoid these unwanted effects (Liska, 2001).

16

2. RESEARCH OBJECTIVE AND DYNAMIC HYPOTHESIS

This thesis focuses on phenobarbital (PB) use. Other than its use as a sedative drug, it

is also an anti-epileptic drug of choice. Continuous use of PB unfolds interesting dynamics

that are likely to be counter-intuitive and thus require careful research.

Prolonged use of PB enhances liver enzymes in a few days so that the rate of

metabolism approximately doubles (enzyme induction). As the drug is continued, the body

tries to counteract the increase in inhibitory neurotransmission by down-regulating the

GABAA receptors. This neuroadaptation is much slower than enzyme induction. Peak of

neuroadaptation is reached after several weeks and the number of down-regulated

receptors comes to stagnation. Down-regulated receptors reduce the efficiency of

inhibitory transmission and together with enzyme induction, they decrease the efficacy of

the drug. The decreased efficacy urges the drug user to increase the doses. Upon abrupt

withdrawal, the drug is cleared much rapidly but the reduced efficiency in inhibitory

neurotransmission persists. This disrupts the normal activity of the CNS since excitatory

neurotransmission is not balanced by inhibitory neurotransmission, which is manifested by

a chloride current lower than normal. The result is a withdrawal syndrome. Nevertheless,

as re-adaptation commences with decreasing levels of the drug, the physiology gradually

returns to normal and withdrawal syndrome ceases. The causal loop structure is given in

Figure 2.1.

There are three negative feedback loops in the system. The first one is related to the

development of pharmacokinetic tolerance as a result of enzyme induction. The loop is 1-

2-3-1. Sustained levels of PB in the body lead to a higher rate of enzyme synthesis. This

leads to a faster PB metabolism and thus the amount of drug in the body decreases.

17

Amount ofphenobarbital in

the body

Extent of CNSdepression

Metabolism rate

Functionality ofGABA

neurotransmission

Number ofdown-regulated

recep tors

Enzy me synthesisrate

Adaptation rate

Re-adaptation rate

+

+

-

+-

+

+

Chloride current+

+

+

-

+

-

Intensity ofwithdrawal syndrome

-

Rate of phenobarbitalintake

-

+

1

2

3

4

5

6

7

8

9

10

11

Figure 2.1. Causal loop diagram for tolerance and dependence development

The second negative feedback loop is related to neuroadaptive changes in the brain.

The loop is 4-5-6-7-4. The primary effect of PB is to increase chloride current which leads

to the depression of the CNS. If treatment is continued, continuous potentiation of chloride

current is counteracted by desensitization of GABAA receptors. This weakens the

inhibitory neurotransmission system.

There is a third negative feedback loop which is a consequence of the two

aforementioned loops. The decrease in inhibitory neurotransmission as a result of increased

metabolism and desensitized receptors leads to less CNS depression. This urges the drug

user to increase the administered dose, which leads to stronger inhibitory

neurotransmission. The loop is 4-5-9-10-1-4. This loop is operational only at later phases

when the functionality of inhibitory neurotransmission is weakened.

When all three loops are operational, they result in positive feedbacks that lead to

continuously increasing doses. The most potent positive feedback is through

18

neuroadaptation rather than enzyme induction and it is 4-5-9-10-1-4-5-6-7-4. Verbally, as

functionality of inhibitory neurotransmission weakens as a result of neuroadaptation, the

user compensates by increasing the dose which leads to further neuroadaptation and thus

inhibitory neurotransmission is further weakened.

The aim of this research is to build a simulation model that represents a regular PB

user taking into account the two related aspects: Enzyme induction and neuroadaptation.

Tolerance development will be traced by monitoring the dose increase decisions of the user.

To provide insight on dependence development, the situation after withdrawal will also be

studied. In addition to a hypothetical healthy person who takes PB for sedation, we will

study three other cases: An epilepsy patient, a person taking another drug that interacts

with PB, and a normal person employing dfferent dosing frequencies. These different

scenario analyses will improve our insights on prolonged PB use. Finally for the

hypothetical healthy person, a feasible dosing scheme during withdrawal will be

investigated, so that the unwanted rebound effects are avoided.

19

3. METHODOLOGY

In Section 2, we have defined a medical problem that involves several interdependent

variables and feedback relationships. Indeed, this is a rather complex system: Human body

exposed to an exogenous chemical. To capture the long-term dynamics, one has to study

the system as a whole rather than focusing one at a time on individual elements of the

system. By creating a mathematical model of the system and defining accurately the

relationships, one can unfold the behavior of the system in the long term. System

Dynamics (SD) methodology is most suitable for this task.

In general, SD is a simulation modeling methodology for studying and managing

complex feedback systems. SD models contain sets of differential/difference equations

which when solved simultaneously, produce certain dynamics of behavior. The focus is on

pattern prediction rather than point prediction, unlike the “black box” statistical models. As

a result, SD models are descriptive in the sense that they explain the direct causalities

(rather than drawing correlations) that give birth to the dynamic behavior of interest. SD

can be applied to all sorts of systems (e.g. businesses, medical systems, socio-economic

systems) that contain complex feedback relationships. The methodology first identifies a

problem, then develops a dynamic hypothesis explaining the causes of the problem, builds

a computer simulation model of the system with regard to the root of the problem,

validates the model against structural and behavioral information seen in the real world,

suggests policies to address the problem and implements the solution. The process is not

purely sequential since one usually finds him/herself visiting some previous steps and

revising decisions (Sterman, 2000; Barlas, 2002; Forrester, 1961).

20

4. MODEL DESCRIPTION

4.1. Pharmacokinetics Sector

4.1.1. Fundamental Approach and Assumptions

This sector models the absorption, distribution, metabolism, and excretion phases.

We model each organ separately and calculate the amount of drug in each organ at a given

instance. We regard only the organs and tissues that are large in volume and those that

receive high blood supply. These are brain, lungs, heart, muscle tissue, fat tissue, kidney,

gastrointestinal tissue, and liver. We model the organs as stock variables, each stock

representing the amount of drug accumulated in that organ. Blood is divided into two parts:

Arterial blood and venous blood. Arterial blood flows into organs whereas venous blood

flows out of organs.

Preliminary simulation runs revealed that lungs and heart do not decouple arterial

and venous blood phenobarbital (PB) content. That is, for both lungs and heart, the amount

of PB entering the organ is practically equal to the amount leaving the organ at a given

time. There is no significant drug uptake into or drug elimination in these tissues.

Therefore, we do not model lungs and heart as stock variables. We only include the flow of

drug from venous blood to arterial blood through the lungs and the flow of drug from

arterial blood to venous blood through the heart.

Another assumption is that within an organ, the drug is distributed uniformly, so that

the concentration of drug inside the organ is equal to the concentration of drug in the blood

that flows out of the organ. Finally, in all our simulation experiments, we assume oral

administration of PB in the form of tablets.

4.1.2. Description of the Structure

The stock-flow structure of the pharmacokinetics sector is given in Figure 4.1

21

MBraintis

MBraincapil

MArterial

MMuscle

MVenous

MKidney

MGItissue

MGIlumen

MLiver

EnzymeFactor

ArterialToBrain

ArterialToMuscle

MuscleToVenous

ArterialToKidney

ArterialToGItis

ArterialToLiver

BrainToVenous

KidneyToVenous

Absorption

LiverToVenous

BraincapilToBraintis

QBrain

CBraincapil

FRBplasma

CBraintisVBraintis

DR

QTotal

CVenous

CMuscleQMuscle

PMuscle

CArterial

QKidney

PKidney

CKidney

QHeart

QGItissue

PGItissue

CGItissue Kabs

QLiver

CLiver

NormKmet

PLiver

Rin kout

Metabolism

GItissueToLiver

IndInducByPB

Synthesis Degradation

BraintisToBraincapil

Intake1

Kmet

Intake2

ReaIInducbyPB

Intake3Intake

4

VBraincapil

VMuscle

VKidney

VGItissue

VLiver

VVenous

VArterial

Kexcr

DaysTreatment HalflifeEnzyme

Intake5

Excretion

MFatArterialToFat FatToVenous

QFatCFat

PFatVFat

VenousToArterial

ArterialToVenous

<Dose>

Figure 4.1. Stock-flow structure of the pharmacokinetics sector

22

As mentioned before, the stocks represent the amounts of drug in different organs.

The flows represent the amounts flowing in blood. In modeling absorption and distribution,

the assumptions used by El-Masri and Portier (1998) were utilized. Absorption is assumed

to follow a first-order rate equation. Its equation is given below.

Absorption = Kabs * MGIlumen (4.1)

where Kabs (min-1) is the absorption constant, and MGIlumen (mg) is the amount of drug

present in the gastrointestinal lumen.

To calculate concentrations, we divide the amounts to volumes. For example, the

concentration of PB in brain tissue is given by Equation 4.2 below.

CBraintis = MBraintis / VBraintis (4.2)

where Mbraintis (mg) is the amount of drug in brain tissue, and VBraintis (L) is the

volume of brain tissue.

The amounts flowing via arterial blood into all organs are assumed to be flow-limited.

To exemplify, the rate of PB transfer from arterial blood to brain is given in Equation 4.3

below.

ArterialtToBrain = CArterial * QBrain (4.3)

where CArterial (mg/L) is the concentration of the drug in arterial blood and QBrain

(L/min) is the rate of blood flow through the brain.

The outflows of all organs except liver and brain are formulated considering that

only unbound drug can flow out of the organ into venous blood. For example, the rate of

PB flow from kidney to venous blood is given by Equation 4.4.

KidneyToVenous = CKidney * QKidney / PKidney (4.4)

23

where QKidney is the rate of blood flow through the kidney; CKidney is the concentration

and PKidney is the tissue-blood partition coefficient in the kidney. The tissue-blood

partition coefficient in an organ is simply the equilibrium ratio of the concentration of drug

in the blood (mobile) to the concentration of drug bound to tissue (immobile) in that organ.

The liver is perfused by both the arterial blood and also by the blood coming from GI

tissue via the hepatic portal vein. Therefore, its outflow towards venous blood is

LiverToVenous = CLiver * (QLiver + QGItissue) / PLiver (4.5)

where CLiver (mg/L) is the concentration of drug in the liver, QLiver (L/min) is the blood

flow rate through the liver, PLiver is the tissue-blood partition coefficient in the liver, and

Qgi (L/min) is the blood flow rate through the GI tissue.

The brain is divided into two parts: Blood (in capillaries) and tissue. Blood in the

brain is denoted by the stock “Brain capillary”. The amount of drug flowing from the brain

into the venous blood is simply QBrain*CBraincapil where QBraincapil (L/min) is the

blood flow rate through the brain and CBraincapil (mg/L) is the concentration of the drug

in brain capillaries. The equations for the flow of drug between brain capillary and brain

tissue are derived by El-Masri and Portier (1998). We copy these equations in our

formulations for BraincapilToBraintis and BraintisToBraincapil as follows.

BraincapilToBraintis =VBraintis * DR * Cbraincapil / (1+Bplasma) (4.6)

BraintisToBraincapil =VBraintis * DR * CBraintis * FR 4.7)

BraincapilToBraintis (mg/min) is the amount of drug diffusing from brain capillaries

into brain tissue, BraintisToBraincapil (mg/min) is the amount of drug diffusing out of

brain tissue to the capillaries, VBraintis is the volume of brain tissue (ml), DR is the

diffusion rate constant (min-1), CBraincapil (mg/L) is the concentration of drug in brain

capillary, CBraintis is the concentration of the drug in brain tissue, Bplasma is the bound

fraction of drug in red blood cells, and FR is the ratio of free to tissue concentrations of the

24

drug. The values for blood flow rates, organ volumes used in calculating concentrations,

partition coefficients, bound fractions and rate parameters are taken from the paper by El-

Masri and Portier (1998) and are given in Tables 4.1 and 4.2 (and in the Appendix together

with all the equations of the model).

Urinary excretion was assumed to be a first-order rate process. It is given in Equation

4.8.

Excretion = Kexcr * MKidney (4.8)

In modeling metabolism rate (mg/min), we use the following equation.

Metabolism = CLiver* Kmet (4.9)

As a matter of fact, Kmet is a function of CLiver. This functional relationship

underlies the process of enzyme induction. To clarify, we start with the equation for Kmet

given below.

Kmet = NormKmet * EnzymeFactor (4.10)

NormKmet (L/min) is a constant and EnzymeFactor is modeled as a stock variable

(See Figure 4.1). Initially, it equals 1, and its inflow and outflow are equal to each other. Its

differential equation is given below.

d(EnzymeFactor) / dt = Synthesis – Degradation (4.11)

As drug concentration in the liver increases, the inflow Synthesis also increases. The

following equation holds for Synthesis.

Synthesis = Rin *(1+RealInducByPB) (4.12)

Rin is the synthesis rate of the enzyme when no drug is present. RealInducByPB is a

smoothed version of IndInducByPb, the latter being a saturable function defined by

25

Equation 4.13. We assume a smoothing time of 2 days. The reason for the delay is that

enzyme induction is a process of protein synthesis involving several genetic processes such

as transcription of genes, mRNA synthesis, etc. which take time.

IndInducByPbCLiverECCLiverE

+=

50

max * (4.13)

Emax is the maximal induction effect and EC50 (mg/L) is the concentration of the drug

that causes half the maximal effect. This function is linear in CLiver for small values of

CLiver since when CLiver << EC50, IndInducByPB ≅ Emax * CLiver. On the other hand,

when CLiver is large so that CLiver >> EC50, IndInducByPb ≅ Emax, thus the function

becomes constant (i.e. the function saturates). We plot the function in Figure 4.2 to clarify

further. Emax and EC50 are as given in Table 4.2.

0

0,2

0,4

0,6

0,8

1

1,2

0 5 10 15 20 25

CLiver (mg/L)

IndI

nduc

ByP

B

Figure 4.2. Saturability of enzyme induction

The outflow Degradation is given by the following equation.

Degradation = kout * EnzymeFactor (4.14)

26

where kout=zymeHalflifeEn

)2ln( which has units of min-1.

To establish a baseline situation, initially (i.e. when no drug is present) we set

EnzymeFactor = 1, and we also set Rin = kout. We assume an enzyme half-life of 2 days

regarding the information in the literature that half-lives of CYP enzymes range between 1

to 6 days (Michalets, 1998).

Other mathematical aspects of the model will be explained where relevant. For

numerical values of model parameters, refer to Appendix.

4.2. Central Nervous System Sector


As mentioned in Section 1.4, for sedation, the major effect of barbiturates is their

potentiation of inhibitory neurotransmission mediated by GABA. The effect on excitatory

neurotransmission is seen in relatively high concentrations of the barbiturate. Therefore,

the model does not take into account the effects on excitatory neurotransmission. Any

adaptation that may be experienced in glutamate (excitatory neurotransmitter) receptors is

also not taken into account.

In the literature, there is no data regarding the quantitative relationship between PB

concentration and GABA mediated inhibitory neurotransmission in humans. However,

there are animal data. In their research, Ffrench-Mullen et al. (1993) use in vitro assays

from pig brains and derive concentration-response functions for several drugs including PB

by measuring peak chloride currents with special equipment. Their results are given in

Figure 4.3. To use this data, we assume that pigs and humans respond equally to PB

treatment.

27

Figure 4.3. Concentration-response data for phenobarbital (Ffrench-Mullen et al., 1993).

The concentration-response relationship of phenobarbital given in the figure can be

represented by the following equation.

Response = Pmax * PB

PB

CGABAECC

+50

(4.15)

where Pmax is the maximum chloride current increase percentage and GABAEC50 is

the concentration of PB that causes half the maximal response. The numerical values of

these parameters are 600 per cent and 2.79 mg/L, respectively.

Neuroadaptation rate is dictated by the extent of inhibitory neurotransmission.

However, as the number of desensitized receptors grows very large, neuroadaptation

saturates. This is presumably achieved by a decreased adaptation rate rather than by an

opposing re-adaptation process. Nevertheless, re-adaptive mechanisms are operational

once the drug concentration drops below a certain threshold.

In the human CNS, there exist around 100 billion neurons and GABA receptors can

be found amongst 60-80 per cent of all neurons. Furthermore, the GABAA subtype of

PB

28

GABA receptors is claimed to be present in ubiquitous amounts (Birnir, 2008). These facts

led to the assumption that there are approximately 60 billion GABAA receptors.


The stock-flow diagram of the CNS sector is given in Figure 4.4.

EffPBOnClCur

ClCur

NormClCur

IndAdptnRate

EffAdptnOnNorm

ClCur

NoDownregRecep

Adaptation ReadaptationClCurWOPB

EffPBOnReadptn

WithdSignIntensity TotalNoRecep

EffSatur ReadptnFrac

RealAdptnRate

<CBraintis>

Figure 4.4. Stock-flow structure of the CNS sector

EffPBOnClCur is simply the concentration-response function. It is a proxy for how

PB presence affects the central nervous system. PB presence not only affects chloride

currents, it also influences the rate of re-adaptation. Unless PB concentration in the CNS is

below a certain level, re-adaptation is inhibited. The concentration-response function for

PB was given in Equation 4.15. We use this function for EffPBOnClCur.

EffPBOnClCur is used in the following equation.

ClCur = NormClCur*(1+EffPBOnClCur/100) (4.16)

It was not possible to find numerical data on chloride currents in the human brain.

Thus, we model the chloride current variables as multiplicative factors. We assume that

when no drug is present in the body,

29

ClCur= NormClCur = ClCurWOPB = 1 (4.17)

We define the number of down-regulated GABAA receptors as a stock variable called

NoDownregRecep having units of billions. It is an indicator of the extent of brain’s

adaptation to the drug. Its differential equation is given below.

d (NoDownregRecep) / dt = Adaptation – Readaptation (4.18)

where

Adaptation = RealAdptnRate * EffSatur (4.19)

Adaptation involves several steps at the cellular level which delay the desensitization

of GABAA receptors. Therefore, we model RealAdptnRate as a third order smoothing of

IndAdptnRate. The smoothing time is assumed to be 15 days. The rate of adaptation is

assumed to be proportional to the discrepancy between a base chloride current (without PB)

and the actual chloride current. IndAdptnRate is therefore defined as a function of ClCur /

ClCurWOPB and is given in Figure 4.5.

EffSatur, as the name implies, slows down neuroadaptation as the number of

desensitized receptors approach the total number of receptors. It is therefore defined as a

function of NoDownregRecep / TotalNoRecep and is given in Figure 4.6.

Figure 4.5. Graphical function for IndAdptnRate. Abscissa is ClCur / ClCurWOPB

30

Figure 4.6. Graphical function for EffSatur. Abscissa is NoDownregRecep / TotalNoRecep

As can be seen, the saturation effect is operational after 80 per cent of the receptor

population is down-regulated.

Since adaptation modifies brain physiology, normal chloride current is affected.

NormClCur = ClCurWOPB* EffAdptnOnNormClCur (4.20)

where

EffAdptnOnNormClCur = F (NoDownregRecep/TotalNoRecep) (4.21)

and F is assumed to be a decreasing function given in Figure 4.7.

31

Figure 4.7. Graphical function for EffAdptnOnNormClCur. Abscissa is NoDownregRecep /

TotalNoRecep

As can be seen, when all receptors are down-regulated, physiology becomes such

that chloride current is 30 per cent less than that in a healthy person.

In modeling the re-adaptation process, we use the following equation.

Readaptation = EffPBOnReadptn* ReadptnFrac * NoDownregRecep

(4.22)

We assume that there is a critical concentration of the drug above which no re-

adaptation can occur. This is captured by EffPBOnReadptn which is given in Figure 4.8.

32

Figure 4.8. Graphical function for EffPBOnReadptn. Abscissa is EffPBOnClCur

Figure 4.8 implies that when PB concentration in the brain is such that when the

concentration-response function (i.e. EffPBOnClCur) indicates less than a 70 per cent

potentiation of the chloride current, re-adaptation can commence.

Finally, to see the intensity of withdrawal signs, we define a variable called

WithdSignIntensity which is merely a shifted and inverted version of ClCur and is given

below.

WithdSignIntensity = - (ClCur – 1) (4.23)

The variable is only meaningful after drug treatment stops. We assume that when

ClCur drops below its base value of 1, WithdSignIntensity becomes greater than 0 implying

that inhibitory neurotransmission is compromised. Given that WithdSignIntensity is greater

than 0, the larger it is, the less the inhibitory neurotransmission and the more likely the

outburst of a withdrawal syndrome. To interpret this variable, we will first establish

reference values that imply insignificant and significant withdrawal signs. This will be

clarified in Section 5.

33

4.3. Dose Sector


PB depresses the activity of the CNS by enhancing chloride currents. We assume that

the PB user is content as long as his/her level of CNS depression corresponds to a chloride

current that is 250 per cent higher than normal. This value was found by regarding the

expected concentration of PB in the venous blood. The therapeutic range of PB plasma

levels (sampled from venous blood) is 10-40 mg/L. Since sedation is the primary effect

and the user is assumed to take the drug for this purpose, the venous blood concentration is

expected to be around 10 mg/L. Hence the threshold was calibrated to yield our

expectation.

We assume the user is urged to take a larger dose at the next dosing time if the

chloride current drops below the sedation threshold. That is, if he/she had taken the single

daily dose, say, 2 hours before he/she has realized the reduced effectiveness, he/she waits

until the next day to increase the dose. Therefore, the frequency of dosing does not change.

We also assume constant dose increments.


We model the amount of administered dose as a stock which only has a single inflow.

The stock-flow structure is given in Figure 4.9.

The single differential equation in this sector is given below.

d(Dose)/dt = DoseIncr (4.24)

where

DoseIncr=11*12*13*14*IncrRate (4.25)

34

DoseDoseIncr

<Time>

I2

ThresholdSedat

I1

I3I4

LoadDose

IncrRate

<ClCur>

<DaysTreatment>

Figure 4.9. Stock-flow structure for the Dose Sector

IncrRate is equal to 10 mg/min. The variables I1, I2, I3 and I4 are 0,1 binary

indicator variables. We want the dose dynamics to be operational only after the initial dose

is effective. The variable I3 serves this purpose and is given below.

I3 = IF THEN ELSE(Time>1440, 1, 0 ) (4.26)

That is, if Time is later than 1440 minutes, I3 = 1, and I3 = 0 otherwise.

The purpose of the variable I2 is to stop dose increase decisions after the end of PB

treatment. It is given in Equation 4.27.

I2 = IF THEN ELSE (Time<DaysTreatment, 1, 0) (4.27)

During drug treatment, I1 helps start the inflow when chloride current is below the

sedation threshold and stop it when the threshold is exceeded.

I1 = IF THEN ELSE (ClCur<ThresholdSedat, 1, 0) (4.28)

where Sedation threshold is 2.5 as explained and dose incr rate is calibrated to give

1/3 mg/min.

35

To model constant dose increments, we use I4. For example, in one-a-day dosing and

for a constant increment of 30 mg, I4 is as in Equation 4.29.

I4 = IF THEN ELSE (MODULO(Time,1440)>=1437, 1, 0) (4.29)

Since the inflow DoseIncr, when it is open, equals 10 mg/min, after 3 minutes of

inflow, 30 mg accumulate in the stock Dose. Additionally, since the inflow opens before a

day is over, the dose increase decision can be implemented at the beginning of the next day.

4.4. Model Parameters

The model represents the actual physical structure of a human being. For the

pharmacokinetic sector, we use data from a pharmacokinetic modeling study by El-Masri

and Portier (1998). Rate parameters are assumed regarding the variations between the three

human subjects who had participated in their work. Note that rate parameters naturally

vary among humans. The parameters we use in the pharmacokinetic sector are given in

Tables 4.1 and 4.2.

36

Table 4.1. Main pharmacokinetic parameters used in the model

ORGAN VOLUME, L BLOOD

FLOW, L/min

PARTITION

COEFFICIENT,

Dimensionless

GI tissue 1.19 0.9 1

Liver 1.925 0.235† 2.25

Kidney 0.308 0.875 2.05

Fat 16.394 0.26 1

Muscle 28 1.67 1.12

Brain capillary 0.0447 0.57

Brain tissue 1.3553

Arterial blood 1.556*

Venous blood 3.811*

Heart 0.2

Lungs 4.475†† *: Blood volume †: Contribution of the hepatic portal vein is not included

††: Sum of all flows

Table 4.2. Other pharmacokinetic parameters

PARAMETER NUMERICAL

VALUE

Kabs 0.02 min-1

NormKmet 0.00314 L/min

DR 0.02

FR 1.75

Bplasma 0.438

Emax 1.15

EC50 1 mg/L

In the CNS sector, we mostly use graphical functions which were explained in detail

in Section 4.2.2.

37

5. VALIDATION OF THE MODEL

In this section, we present evidences of the model’s validity with respect to the real

system. We first present the simulation results under basic assumptions. We then introduce

new assumptions and shed light on dynamics generated by different structures in the model.

Having presented all relevant results, we then draw comparisons between model outputs

and real data and present discussions.

5.1. Simulation Results

In these structural validation runs, we assume that the user employs one-a-day dosing.

He/she is assumed to continue with 30 mg tablets when allowed after an initial loading

dose of 180 mg (i.e. first dose). We use the initial conditions given in Table 5.1 for the

stocks.

Table 5.1. Initial values for the stocks

STOCK INITIAL VALUE

All (except MGIlumen

and Dose)

0

MGIlumen 180

Dose 30

EnzymeFactor 1

NoDownregRecep 0

5.1.1. Single Dose

To observe the initial pharmacokinetic processes such as absorption from the

gastrointestinal lumen, distribution to organs and tissues, and elimination, we give the

results for the first 300 minutes (5 hours) after the loading dose of 180 mg only. We

display only the most informative stocks for this run.

38

MGILumen200

150

100

50

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MGILumen : singletablet

MGITissue6

4.5

3

1.5

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MGITissue : singletablet a b

MLiver20

15

10

5

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MLiver : singletablet

MVenous20

15

10

5

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MVenous : singletablet c d

MMuscle100

75

50

25

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MMuscle : singletablet

MFat60

45

30

15

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

MFat : singletablet e f

MBraintis2

1.5

1

0.5

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

mg

MBraintis : singletablet g

Figure 5.1. Absorption and distribution of a single dose

39

After diffusing from the gastrointestinal lumen into the gastrointestinal tissue, the

drug does not stay here and it is immediately distributed to various organs, its first

destination being the liver. The sharp increase in liver PB content during the first 15

minutes confirms this (Figure 5.1c). From Figure 5.1e, we note that about half the amount

administered is distributed to muscle tissue. This is expected since muscle tissue

constitutes 40 per cent of total body volume and receives approximately 35 per cent of

total blood supply. The amount of PB accumulated in fat is also large (Figure 5.1f). Similar

to muscle tissue, fat constitutes a large percentage of total body volume. As can be seen

from Figure 5.1g, the amount of drug in the target site (i.e. brain tissue) reaches a plateau

in 3 hours. Although, it is only a small fraction of the amount administered, its effect is not

insignificant. This can be seen in the following figure.

ClCur4

3

2

1

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

Dm

nl

ClCur : singletablet

Figure 5.2. Increasing chloride current in the brain after a single dose

It can be seen that chloride current (i.e. inhibitory neurotransmission) has more than

doubled. As expected, no enzyme induction or neuroadaptation took place in such a short

time. Enzyme amount stays at the undrugged level (Figure 5.3a). The number of down-

regulated receptors is an insignificant fraction of the total receptor population of 60 billion

(Figure 5.3b).

40

EnzymeFactor2

1.75

1.5

1.25

10 30 60 90 120 150 180 210 240 270 300

Time (Minute)

Dm

nl

EnzymeFactor : singletablet

NoDownregRecep4e-007

3e-007

2e-007

1e-007

00 30 60 90 120 150 180 210 240 270 300

Time (Minute)

Bill

ions

NoDownregRecep : singletablet a b

Figure 5.3. Dynamics of enzyme induction and neuroadaptation for a single dose

5.1.2. Continuous Drug Intake with Constant Dose

In this section, we give the results of simulation experiments in which we assume a

regular user of PB. We use the terms “drug use”, “drug intake”, and “drug treatment”

interchangeably in the following sections. Recall that our dynamic hypothesis defends that

the user would be urged to increase the doses as tolerance develops to the effects of the

drug. In this section, however, we assume that the user is not urged and takes constant

doses of 30 mg after the loading dose. We therefore show the failure of constant doses to

maintain a constant level of sedation. We comparatively study two scenarios to show

different levels of tolerance and dependence development: Intake for seven days and intake

for 20 days. The results are as follows.

41

Dose40

35

30

25

200 7200 14400 21600 28800

Time (Minute)

mg

Dose : d7nofeedback Dose : d20nofeedback

a

MBraintis2

1.5

1

0.5

00 18000 36000 54000 72000

Time (Minute)

mg

MBraintis : d7nofeedbackMBraintis : d20nofeedback

b

Figure 5.4. Constant doses (a) and drug profiles in the brain (b) in both a seven day and a

20 day treatment.

42

EnzymeFactor2

1.7

1.4

1.1

0.80 18000 36000 54000 72000

Time (Minute)

Dm

nl

EnzymeFactor : d7nofeedbackEnzymeFactor : d20nofeedback

NoDownregRecep6

4.5

3

1.5

00 18000 36000 54000 72000

Time (Minute)

Bill

ions

NoDownregRecep : d7nofeedbackNoDownregRecep : d20nofeedback

a b

ClCur4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

ClCur : d7nofeedbackClCur : d20nofeedback

c

Figure 5.5. Enzyme induction and neuroadaptation and the resulting chloride current profile when the user takes constant doses (for seven and 20 days).

The constant 30 mg doses can be seen in Figure 5.4a. Although the extent of enzyme

induction is the same in seven days and 20 days (Figure 5.5a), neuroadaptation progresses

much slower (Figure 5.5b). As a result of enhanced metabolism, the amount of PB in the

brain decreases constantly (Figure 5.4b). Although in the 20 day treatment the amount in

the brain approaches a steady state, chloride current continues to fall as can be seen in

Figure 5.5c. These results demonstrate that to maintain sedation, the doses must be

increased. Starting from the following section, we incorporate this feedback loop into our

analyses.

5.1.3. Continuous Drug Intake with Dose Increase as a Result of Feedback

We consider the following drug treatment durations all of which end with abrupt

withdrawal: 7 days, 20 days, and 60 days. In all drug treatments, the user is assumed to

43

start with 30 mg tablets after the loading dose of 180 mg. As time elapses, the user would

increase the dose in constant increments to compensate the reduced effectiveness. To

model the daily drug administration process, we use a pulse function. The inflow named

Intake1 of the stock MGIlumen is given in Equation 5.1.

Intake1 = (Dose/TIME STEP)*PULSE TRAIN( 1440, TIME STEP, 1440,

DaysTreatment*1440+TIME STEP)

(5.1)

Dose is a stock variable previously explained in detail in Section 4.3, and

DaysTreatment is simply the number of days of PB administration. The first term in

parentheses is the pulse amplitude. The function PULSE TRAIN is a built-in function in

Vensim whose arguments are start time of pulse, pulse duration, pulse repeat time, and

final time of pulse, respectively.

5.1.3.1. Drug Treatment for Seven Days

To see the situation after withdrawal as well, we set the final time to 27 days (38,880

minutes), and DaysTreatment to 6. Recall that at time zero, the stock MGIlumen contains

the loading dose. The tablets are administered starting from the second day (i.e.

Time=1440) and for four days. The sum is seven days of drug treatment. We obtain the

following dynamics for the key variables. We first present the dose profile (excluding the

loading dose) and the profile for the amount of drug in the brain.

44

Dose80

65

50

35

200 2520 5040 7560 10080

Time (Minute)

mg

Dose : d7

a

MBraintis2

1.5

1

0.5

00 9720 19440 29160 38880

Time (Minute)

mg

MBraintis : d7

b

Figure 5.6. Dose profile (a) and drug amount in the brain (b) in the seven day drug

treatment followed by abrupt withdrawal

45

EnzymeFactor2

1.7

1.4

1.1

0.80 9720 19440 29160 38880

Time (Minute)

Dm

nl

EnzymeFactor : d7

NoDownregRecep4

3

2

1

00 9720 19440 29160 38880

Time (Minute)

Bill

ions

NoDownregRecep : d7 a b

Intensity of withdrawal signs0.1

0.075

0.05

0.025

08640 16200 23760 31320 38880

Time (Minute)

Dm

nl

WithdSignIntensity : d7 c

Figure 5.7. Enzyme and neuroadaptation dynamics in the seven day drug treatment

followed by abrupt withdrawal

Looking at figure 5.6a we see that the user increases the third dose. This is because

chloride current drops below the threshold as can be seen in Figure 5.8. By doubling the

dose, the user avoids a decrease below the threshold.

46

ClCur4

3

2

1

00 7220 14440 21660 28880

Time (Minute)

Dm

nl

ClCur : d7ClCurWOPB : d7

ThresholdSedat : d7

Figure 5.8. Behavior of chloride current in the seven day drug treatment

In Figure 5.7a, it is interesting to note that although drug treatment stops on the

seventh day (8640 minutes), enzyme induction continues its progress until around the ninth

day (12,000 minutes). Furthermore, there is an onset of enzyme induction. This inertia is

due to genetic processes related to enhanced synthesis of enzymes such as gene

transcription, mRNA synthesis, etc. which take time. Nevertheless during drug treatment,

EnzymeFactor approaches 2, implying that induction is almost complete (Recall that at

maximal induction, rate of metabolism doubles).

The inertia in neuroadaptation is more significant. Observe from Figure 5.7b that

although drug intake stops, similar to enzyme induction, down-regulation continues its

progress six more days (i.e. the curve peaks around the 13th day). However, only a very

small fraction of total receptor population is down-regulated implying that dependence has

not yet developed. We therefore assume that the peak intensity in Figure 5.7c is

insignificant and thus establish a reference. Hereafter, we regard any peak intensity below

0.025 as insignificant. The reports in literature stating that dependence to barbiturates

develops in several weeks also support the validity of our assumption.

47

5.1.3.2. Drug Treatment for 20 Days

Following are the results for 20 days of 30 mg one-a-day doses ending with abrupt

withdrawal.

Dose200

150

100

50

00 7200 14400 21600 28800

Time (Minute)

mg

Dose : d20

a

MBraintis4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

mg

MBraintis : d20

b

Figure 5.9. Dose profile (a) and drug amount in the brain (b) in the 20 day drug treatment


48

EnzymeFactor4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

EnzymeFactor : d20

NoDownregRecep20

15

10

5

00 18000 36000 54000 72000

Time (Minute)

Bill

ions



0.225

0.15

0.075

027360 38520 49680 60840 72000

Time (Minute)

Dm

nl


Figure 5.10. Enzyme and neuroadaptation dynamics in the 20 day drug treatment followed

by abrupt withdrawal

The inertia in enzyme and neuroadaptation dynamics is again evident. The onset of

enzyme induction is shorter than that of receptor down-regulation. Figure 5.10a shows that

in a few days, enzyme induction peaks and although intake stops on the 20th day, fast

metabolism persists six more days (until the 36,000th minute).

As can be seen in Figure 5.10b, more than a quarter of the receptor population is

down-regulated. This weakens inhibitory neurotransmission by decreasing normal chloride

current (See the variable named NormClCure in Section 4.2.2). Together with fast

metabolism this reduces the effectiveness of the drug, urging the user to increase the dose

several times (Figure 5.9a). The decrease in drug effectiveness is so severe that the final

dose is five times the initial dose. Looking at Figure 5.11 below, we conclude that the dose

increase decisions are justified since chloride current is maintained above the threshold

with a few insignificant undershoots throughout 20 days.

49

Chloride current4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

ClCur : d20ThresholdSedat : d20

ClCurWOPB : d20

Figure 5.11. Behavior of chloride current in the 20 day drug treatment

The peak intensity of withdrawal signs is around 0.1 as can be seen in Figure 5.10c.

As mentioned in Section 1.5, clinical research suggests that dependence to barbiturates

develops in a few weeks. This suggestion and our outputs showing that more than a quarter

of total receptors are desensitized in 20 days lead to the conclusion that the user has

become dependent-at least partially-to PB and thus upon abrupt discontinuation, he/she

would experience a significant withdrawal syndrome. In Figure 5.10c, the peak intensity of

withdrawal signs is around 0.1. Accordingly, hereafter we shall regard any withdrawal sign

intensity above 0.1 as severe. We now have two reference points to help us assess the

significance of withdrawal signs in further simulation experiments. Finally, the delay in the

outburst of the withdrawal syndrome is due to the long half-life (despite enhanced

metabolism) of PB.

5.1.3.3. Drug Treatment for 60 Days

We set the final time to 90 days (129,600 minutes) and DaysTreatment to 59 days.

We obtain the following results.

50

Dose200

150

100

50

00 21600 43200 64800 86400

Time (Minute)

mg

Dose : d60

a

MBraintis6

4.5

3

1.5

00 32400 64800 97200 129600

Time (Minute)

mg

MBraintis : d60

b

Figure 5.12. Dose profile (a) and drug amount in the brain (b) in the 60 day drug treatment


51

EnzymeFactor4

3

2

1

00 32400 64800 97200 129600

Time (Minute)

Dm

nl

EnzymeFactor : d60

NoDownregRecep60

45

30

15

00 32400 64800 97200 129600

Time (Minute)

Bill

ions



0.225

0.15

0.075

084960 96120 107280 118440 129600

Time (Minute)

Dm

nl


Figure 5.13. Enzyme and neuroadaptation dynamics in the 60 day drug treatment followed

by abrupt withdrawal

Elevated enzyme levels persist even after drug administration stops as was the case

in the shorter treatment durations studied previously. We see from Figure 5.13b that in 60

days, practically all receptor population is desensitized implying that the user has been

rendered completely dependent. Around the 45th day, desensitization saturates. Tolerance,

on the other hand, is almost complete after the user increases the dose to 150 mg at the 19th

dose. Further dose increase is a month later (45th day).

Abrupt withdrawal causes a severe withdrawal syndrome as can be verified from

Figure 5.13c. The peak intensity is nearly twice our reference of significance. In Figure

5.14 below, we present the chloride current profile. The elevated dosages are efficient in

maintaining the desired sedation level.

52

Chloride current4

3

2

1

00 32400 64800 97200 129600

Time (Minute)

Dm

nl

ClCur : d60ClCurWOPB : d60

ThresholdSedat : d60

Figure 5.14. Behavior of chloride current in the 60 day drug treatment

5.2. Model Validity Discussion

A point-by-point match with real data is not a major goal, and it is not realistic in SD

models (Forrester, 1961; Barlas, 1996; Sterman 2000). The crucial idea is to capture the

behavior patterns. We thus draw our comparisons according to this approach.

The validity of the pharmacokinetic sector is established since the same structures

were used in a previous study and a good fit with real data has been shown (El-Masri and

Portier, 1998). Urinary excretion was also included in this model and it is assumed to

follow first-order kinetics. In the literature, it is reported that 24 per cent of administered

PB is excreted unchanged (Engasser et al., 1981). It was a straightforward issue to

calibrate the rate constant using this information.

Regarding enzyme induction, it is reported in the literature that the rate of

metabolism doubles at maximal induction and this peak occurs in days to weeks. Parameter

calibrations were done using this information. We repeat the enzyme induction related

model outputs for the 20 day treatment case in Figure 5.15. As can be seen, metabolism

53

doubles in approximately 2 weeks (18,000 minutes). This is in good agreement with

literature (Hardman and Limbird, 2001).

EnzymeFactor4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

EnzymeFactor : d20

Figure 5.15. Progression of enzyme induction in 20 days of continuous PB use

Since no quantitative human data regarding tolerance and dependence development

are available in the literature, we use data from studies on animal models such as the one

by Gay et al. (1983). Our assumptions are fairly similar to theirs. Similar to our model,

they target a constant sedation level in rats while adjusting doses. They administer PB

orally to rats for 35 days and observe that tolerance development is complete after the first

ten days. To compare, in Figure 5.16 we display their daily dosing history together with

our model outputs for the 60 day drug treatment case.

Dose200

150

100

50

00 21600 43200 64800 86400

Time (Minute)

mg

Dose : d60 a b

Figure 5.16. Comparison of tolerance dynamics generated by the model (a) against real

data (b) from Gay et al (1983).

54

Similar to the findings by Gay et al, our drug user increases the doses most

aggressively in the first few weeks in order to reach a constant (desired) sedation level.

Afterwards, relatively constant doses are enough to sustain the constant sedation level. To

confirm this constant level of sedation, refer to Figure 5.14 (See the first 60 days). A good

pattern match is thus observed between the model’s dose output and experimental dose

data as seen in Figure 5.16a and b, so as to sustain a desired sedation level.

Gay et al. also monitor rats for withdrawal signs after abrupt discontinuation to the

drug. They quantify the intensity of withdrawal signs which occur a few days after

discontinuation. They also observe that the signs attenuate as time elapses. Although their

proxy for the intensity of withdrawal signs is different from ours (i.e. their proxy is

behavioral outcomes like ear twitches; ours is chloride currents), they indicate the same:

the more intense the behavioral sign (the lower the chloride current), the more severe is the

withdrawal syndrome. We compare our results in Figure 5.17 below. As can be seen, a

sharp-boom-then-decay behavior is well captured by our model.


0.225

0.15

0.075

084960 96120 107280 118440 129600

Time (Minute)

Dm

nl

WithdSignIntensity : d60 a b

Figure 5.17. Tolerance and dependence indicators for 60 days of continuous PB intake, (a)

Model output, (b) Real data.

Gay et al. also observe that although tolerance development is almost complete in the

ten day group, the rats withdrawn from PB after 35 days of continuous administration

experience more intense withdrawal signs. Our 20 day case is analogous to their ten day

group. In Figure 5.18 below, we show that the difference in withdrawal signs of the 20 day

drug user and the 60 day user is captured by our model.

55


0.225

0.15

0.075

027360 38520 49680 60840 72000

Time (Minute)

Dm

nl

WithdSignIntensity : d20


0.225

0.15

0.075

084960 96120 107280 118440 129600

Time (Minute)

Dm

nl

WithdSignIntensity : d60 a b

Figure 5.18. Differences in withdrawal dynamics between a partially dependent (20 day

user) and a completely dependent (60 day user).

56

6. SCENARIO ANALYSES

6.1. Epilepsy Patient

In epilepsy, the essential balance between excitatory and inhibitory

neurotransmission is disrupted. There are various types of epilepsy. The major

classification regards the extent to which the brain is affected. According to this

classification, a seizure may either be general or partial, the former meaning that all parts

of the brain are affected while the latter means that the disease starts in one lobe of the

brain only. The causes of epilepsy are difficult to dissect. It may be due to anything that

damages the brain such as an infection involving the brain or a head injury. It may even be

due to incomplete maturation of the brain (Chappell and Crawford, 2001).

As mentioned previously, this thesis focuses on inhibitory neurotransmission

mediated by the neurotransmitter GABA. Therefore, in this scenario we consider a form of

epilepsy called juvenile myoclonic epilepsy that is caused by decreased inhibitory

neurotransmission as a result of mutations in GABAA receptors (Kapur, 2003). As the

name implies, this type of seizure is experienced by young people (in juvenile time of their

life) and it is characterized by sudden, jerky or shock-like contractions usually in arms and

legs (Chappell and Crawford, 2001).

All epileptic seizures are tried to be controlled by anti-epileptic drugs. These drugs

do not cure epilepsy (From http://www.epilepsy.org). There are numerous antiepileptic

drugs and several different prescriptions depending on the type of seizure. Sometimes, it is

not possible to control a form of seizure with a single drug in which case multi-drug

treatment is employed (Chappell and Crawford, 2001). We assume a form of juvenile

myoclonic epilepsy that can be controlled with phenobarbital (PB).

Recall that our proxy for inhibitory neurotransmission was the chloride current in the

brain and the undrugged state was defined by a constant named ClCurWOPB and we

assumed that its value was 1 in a healthy individual (See Section 4.2.2). Since an epilepsy

57

patient is suffering from decreased inhibitory neurotransmission, we have to set this to a

lower value. We may assume that a myoclonic seizure is comparable to a state of rebound

hyperexcitability. Therefore in assigning the value of ClCurWOPB for an epilepsy patient,

we regard our previous assumption that in severe rebound hyperexcitability, value of the

model variable named WithdSignIntensity was at least 0.1. Since

WithdSignIntensity = -(ClCur-1) (6.1)

it follows that

ClCur = 1 – WithdSignIntensity (6.2)

Therefore, in a state of rebound excitability

Cl current < 1 – 0.1 = 0.9 (6.3)

As a result, in our epilepsy patient, we assume that Cl current without PB is equal to

0.9. This is an average value. In reality, we expect that between seizures, this value is close

to 1. However, being more realistic and assuming a time dependent profile to this variable

would not contibute to the quality of our analysis.

In order to make sound comparisons, we assume that the epilepsy patient not only

wants to control seizures, but also desires sedation similar to the healthy person we have

studied in Section 5. That is, ThresholdSedat is assumed to be the same for the epilepsy

patient. We assume the same daily doses of 30 mg.

Epilepsy patients sometimes continue drug treatment for a life time. Usually,

withdrawal is not an issue for epilepsy patients. Therefore, we do not study post-

withdrawal dynamics in this scenario. Instead, we comparatively present the dynamics in

the first 60 days of PB use by a healthy person and an epilepsy patient to provide insights

on the differences in development of tolerance and dependence in a disease state.

58

Dose400

300

200

100

00 21600 43200 64800 86400

Time (Minute)

mg

Dose : d60 Dose : d60epileptic

a

MBraintis6

4.5

3

1.5

00 21240 42480 63720 84960

Time (Minute)

mg

MBraintis : d60MBraintis : d60epileptic

b

Figure 6.1. Dose profiles (a) and drug profiles in brain tissue (b) of both a healthy and an

epileptic individual in 20 days of continuous PB use.

59

EnzymeFactor4

3

2

1

00 21240 42480 63720 84960

Time (Minute)

Dm

nl

EnzymeFactor : d60EnzymeFactor : d60epileptic

NoDownregRecep60

45

30

15

00 21240 42480 63720 84960

Time (Minute)

Bill

ions

NoDownregRecep : d60NoDownregRecep : d60epileptic

Figure 6.2. Enzyme and neuroadaptation dynamics in both a healthy and an epileptic

individual taking PB for the last 60 days.

Chloride current

4

3

2

1

00 21240 42480 63720 84960

Time (Minute)

Dm

nl

ClCur : d60ClCur : d60epilepticClCurWOPB : d60

ClCurWOPB : d60epilepticThresholdSedat : d60

Figure 6.3. Chloride current in a healthy and an epileptic individual (Undrugged levels of

chloride current are also given)

Since the undrugged level of chloride current in an epilepsy patient is lower as can be

seen in Figure 6.3 (gray colored line), the discrepancy between the desired level of chloride

current and normal chloride current is larger and thus more drug is necessary to sustain

sedation. This can be verified by looking at Figure 6.1. The amount of drug administered

and thus the amount of drug in the brain is higher in the epileptic individual.

60

Enzyme induction profile is not different in an epilepsy patient as can be seen in

Figure 6.2a. This is because the PB concentration in the liver is at saturation in both cases

and thus enzyme synthesis accelerates at maximum rate.

In the CNS, the rate of neuroadaptation is alarmingly higher in the epileptic

individual due to elevated doses. In about 35 days, the number of downregulated receptors

approaches the total receptor population of 60 billion (Figure 6.2b). This shows that an

epilepsy patient becomes dependent earlier than a healthy person. When compared to the

epilepsy patient, in 35 days, slightly more than half the receptor population is down-

regulated in a healthy person.

As mentioned previously, dependence is not a major concern in an epilepsy since the

patient is not expected to discontinue the drug. However, this scenario shows the increased

susceptibility of an epilepsy patient to rebound effects had he/she discontinued the drug

contrary to a doctor’s advice.

6.2. Co-administration of a Drug That Causes Enzyme Inhibition

In this scenario, we study a possible drug-drug interaction. Most drug-drug

interactions are due to the effects of drugs on liver enzymes (i.e. CYP enzymes). The CYP

enzymes are either inhibited or induced by drugs leading to altered metabolism of the

substrates of (chemicals that are metabolised by) these enzymes. Usually, the drugs

themselves are also substrates of these enzymes and thus pharmacokinetics of a drug may

vary considerably if administered together with another drug. To illustrate, suppose that

drug A is taken together with drug B which is an inhibitor of a CYP enzyme. Suppose also

that drug A is a substrate of this CYP enzyme. This would lead to a slower metabolism of

drug A and a normal dose of drug A might actually be fatal. Therefore, in multi-drug

treatment, levels of drugs must be carefully monitored to avoid unwanted results.

There may be infinitely many forms of drug-drug interactions. In this scenario, we

assume that our hypothetical person has been taking fluconazole, an anti-fungal drug,

61

before starting PB treatment. Fluconazole has been shown to be an inhibitor of PB

metabolizing enzymes (Venkatakrishnan, 2000).

To investigate the extent of enzyme inhibition by fluconazole, Kumar et al. (2008)

use flurbiprofen as a substrate of the inhibited enzyme. They study three groups of subjects.

To the first group, they administer flurbiprofen only. To the second group, they administer

flurbiprofen after pre-treatment with 200 mg fluconazole for 7 days. Finally to the third

group, they administer flurbiprofen after pre-treatment with 400 mg of fluconazole for 7

days. They monitor flurbiprofen clearance in all groups. Their averages are plotted in

Figure 6.4.

Figure 6.4. Flurbiprofen average clearance as influenced by fluconazole pre-treatment.

Values are given as median + 25th percentile (Kumar et al., 2008).

Observe from the figure that a seven day pre-treatment with 200 mg fluconazole

halves the rate of metabolism of flurbiprofen. Clearing rate drops from 1.6 L/hr to 0.8 L/hr.

Although there is no comprehensive clinical study on PB-fluconazole interaction, it

is reported in the literature that when co-administered with PB, fluconazole leads to

increased PB levels via inhibition of enzymes similar to the flurbiprofen case. Since both

62

PB and flurbiprofen are substrates of the same enyzme, we may argue that extent of

inhibition will be similar for both drugs. Here we assume that before starting PB treatment,

the user has been taking 200 mg doses of fluconazole for the past seven days. Therefore,

by the start of treatment, metabolism rate of PB is assumed to be half the normal rate (i.e.

initially the model variable EnzymeFactor is equal to 0.5). However, enzyme induction

still occurs and in a few days the metabolism rate is doubled (i.e. EnzymeFactor becomes

approximately 1). We assume that fluconazole has no effect on any other part of the system.

Assuming that PB treatment duration is 20 days, we get the following results.

Dose200

150

100

50

00 7200 14400 21600 28800

Time (Minute)

mg

Dose : d20fluc Dose : d20 a

MBraintis4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

mg

MBraintis : d20fluc MBraintis : d20 b

Figure 6.5. Dose profiles (a) and drug amounts in the brain (b) with and without

fluconazole pre-treatment. PB is taken for 20 days and is discontinued abruptly

63

EnzymeFactor4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

EnzymeFactor : d20flucEnzymeFactor : d20

NoDownregRecep40

30

20

10

00 18000 36000 54000 72000

Time (Minute)

Bill

ions

NoDownregRecep : d20flucNoDownregRecep : d20

a b


0.225

0.15

0.075

027360 38520 49680 60840 72000

Time (Minute)

Dm

nl

WithdSignIntensity : d20flucWithdSignIntensity : d20

c

Figure 6.6. Enzyme and neuroadaptation dynamics with and without fluconazole pre-

treatment. PB treatment duration is 60 days ending with abrupt discontinuation.

Figure 6.5a shows that inhibition of metabolism has slowed down the progression of

tolerance. Since PB is cleared much slower, a milder increase in dose is enough to yield the

same level of sedation. Compared to a total PB dose of 2040 mg, pre-treatment with

fluconazole necessitates only 1320 mg. In Figure 6.5b, we see that the PB amount in the

brain is not increased significantly and thus there is no toxicity concern.

Neuroadaptation and withdrawal dynamics are rather interesting. Observing Figure

6.6b, we see that neuroadaptation has progressed more severely after fluconazole pre-

treatment. Interestingly, upon withdrawal, the intensity of rebound effects is much lower.

This can be explained as follows: Although the number of down-regulated receptors is

larger when fluconazole is administered prior to PB, phenobarbital is cleared slower and

thus there is more time for re-adaptive mechanisms to restore brain physiology. This can

64

be verified by observing Figure 6.6c and noting that the withdrawal syndrome is not only

lighter, but also the outburst of the syndrome is later in the fluconazole pre-treatment case.

6.3. Different Dosing Frequencies

In all preceding simulation runs, we assume the drug user employs one-a-day dosing.

One may reasonably suspect that in terms of tolerance and dependence, employing

different dosing schemes could yield different results. Therefore in this section we

compare four different dosing schemes in which we vary the initial doses and dosing

frequencies. We experiment with four dosing schemes: One tablet every two days, one

tablet every day (our basic assumption), two tablets every day, and finally three tablets

every day. The initial dose and dose increment are the same for each case and these are 60,

30, 15, and 10 mg for tablets taken one-every-two, one-a-day, two-a-day, and three-a-day,

respectively. Had the doses not increased, the average daily doses would be the same in all

dosing schemes. We comparatively show tolerance and dependence dynamics together

with the behavior of chloride current. Treatment duration is assumed to be 20 days

followed by abrupt discontinuation.

As in Section 5, we first disengage feedback and assume constant doses. As shown in

Figure 6.7a, without the feedback, the behavior of EnzymeFactor does not change

significantly as a function of dosing scheme. On the other hand, the peak number of down-

regulated receptors is the lowest in one-every-two-days scheme (Figure 6.7b).

65

EnzymeFactor2

1.7

1.4

1.1

0.80 18000 36000 54000 72000

Time (Minute)

Dm

nl

EnzymeFactor : d20oneeverytwo-nofeedbEnzymeFactor : d20oneaday-nofeedbEnzymeFactor : d20twoaday-nofeedbEnzymeFactor : d20threeaday-nofeedb

a

NoDownregRecep6

4.5

3

1.5

00 18000 36000 54000 72000

Time (Minute)

Bill

ions

NoDownregRecep : d20oneeverytwo-nofeedbNoDownregRecep : d20oneaday-nofeedbNoDownregRecep : d20twoaday-nofeedbNoDownregRecep : d20threeaday-nofeedb

b

Figure 6.7. Enzyme and neuroadaptation dynamics in different dosing schemes (No

feedback to increase the doses)

Observing the comparative behavior of chloride current given in Figure 6.8, we see

that in one-every-two-days scheme, the average chloride current in the first few days of

66

treatment (when the chloride current is relatively high) is lower in comparison to the other

schemes. The rate of neuroadaptation is more sensitive to chloride current at these levels as

can be verified from our effect formulation given in Figure 4.5. Therefore, the overall rate

of neuroadaptation in this scheme is lower in comparison to the other three schemes. This

results in a lower peak in the number of down-regulated receptors.

ClCur4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

ClCur : d20oneeverytwo-nofeedbClCur : d20oneaday-nofeedb

ClCur : d20twoaday-nofeedbClCur : d20threeaday-nofeedb

Figure 6.8. Comparative behavior of chloride current (No feedback to increase the doses)

When the feedback loop is operational, the picture changes drastically. Similar to our

base case, we assume that the feedback process is operational once the first dose is taken

(the dose after the loading dose). In Figure 6.9, we give the dynamics of tolerance

development in all dosing schemes.

67

Dose400

300

200

100

00 7200 14400 21600 28800

Time (Minute)

mg

Dose : d20oneeverytwoDose : d20oneadayDose : d20twoadayDose : d20threeaday

Figure 6.9. Difference in the extent of tolerance development w.r.t dosing schemes (Feedback allowed to increase doses)

NoDownregRecep40

30

20

10

00 18000 36000 54000 72000

Time (Minute)

Bill

ions

NoDownregRecep : d20oneeverytwoNoDownregRecep : d20oneadayNoDownregRecep : d20twoadayNoDownregRecep : d20threeaday

Figure 6.10. Neuroadaptation dynamics for different dosing schemes (Feedback allowed to increase doses)

68


0.225

0.15

0.075

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

WithdSignIntensity : d20oneeverytwoWithdSignIntensity : d20oneadayWithdSignIntensity : d20twoadayWithdSignIntensity : d20threeaday

Figure 6.11. Dependence dynamics for different dosing schemes (Feedback allowed to increase doses)

It turns out that both tolerance and dependence development is less when frequency

of doses is increased. In three-a-day dosing, the total amount of drug administered in 20

days is 1690 mg. As the frequency is decreased, this total amount increases. In the extreme

case where the user takes one tablet every two days, the total amount administered is 2460

mg which is approximately 50 per cent more than the three-a-day case. Additionally, in

one-every-two-days dosing, the peak number of down-regulated receptors is 23 billion

whereas in three-a-day dosing, the peak is 13 billion (Figure 6.10). As anticipated from this,

the severity of rebound effects is most potent in one-every-two-days dosing (Figure 6.11).

Finally, the amplitude of oscillations in chloride current is less in frequent dosing. This

outcome is in favor of homeostasis: The body prefers stability. The comparative behavior

of chloride current is given in Figure 6.12 below. In one-every-two-days scheme, the

enormous overshoots of chloride current increase the rate of neuroadaptation: Chloride

current sometimes exceeds 3.5 (Recall that the threshold is 2.5). The extent of

neuroadaptation is proxied by the area between the chloride current and the sedation

threshold.

69

Chloride current4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

ClCur : d20oneeverytwoClCur : d20oneadayClCur : d20twoaday

ClCur : d20threeadayClCurWOPB : d20oneadayThresholdSedat : d20oneaday

Figure 6.12. Behavior of chloride current in different dosing schemes (Feedback allowed to increase doses)

We may confidently argue that the most appropriate dosing scheme is three-a-day

dosing. Although further increases in dosing frequency could prove better, such high

frequencies would not be practical since the user would have to remember too often taking

a tablet.

70

7. ANALYSIS OF WITHDRAWAL POLICIES

It is shown in the preceding sections that abrupt withdrawal results in an unwanted

withdrawal syndrome. This suggests that the dose should be reduced gradually. During the

withdrawal period, the decision variables are dosing times, dosing amounts and duration of

the withdrawal regimen. The best policy would be the one that causes very few or no

withdrawal signs with a minimum total amount of administered PB. In this section, we

demonstrate both unsuccessful and successful withdrawal dosing regimens after one-a-day

dosing for both 20 and 60 days. We assume a healthy user taking PB one-a-day for

sedation as in Section 5. It is anticipated that withdrawal would be easier after the 20 day

treatment since the drug user would not be totally dependent on the drug as was shown in

section 5.1.2.2. On the other hand, we have shown in section 5.1.2.3 that the user reaches a

maximum tolerance and dependence level in the 60 day treatment and this would

complicate withdrawal.

In hypothesizing effective withdrawal regimens, we use intuition and therefore start

with relatively good regimens. Simulation experiments were conducted as follows: We

start with an initial guess and we check, at the end of the regimen, whether the user

experiences rebound effects. If this is the case, we prolong the regimen and/or modify the

doses until we observe no withdrawal syndrome. In summary, by improving upon our

previous postulations, we try to come up with regimens that help avoid a withdrawal

syndrome. For each case of drug treatment duration, we present first an unsuccessful

postulation. Then we discuss necessary modifications that lead to a successful regimen.

7.1. Withdrawal after 20 days of treatment

7.1.1. An unsuccessful regimen

Since the half-life of PB is long, when drug intake is stopped on the 20th day, the

drug stays in the body and is still effective. Trials show that the chloride current stays

above the base value (i.e. 1) for at least seven days after the last dose. Thus, we wait for

71

seven days before starting the withdrawal regimen. This regimen lasts for ten days: After

taking no tablet in the first seven days, the user is supposed to take one-tenth of the last

dose (dose on the 20th day) for the following three days and then discontinue. The

dynamics that result are given in Figures 7.1 and 7.2.

MGIlumen200

150

100

50

00 18000 36000 54000 72000

Time (Minute)

mg

MGIlumen : d20w10-2

EnzymeFactor4

3

2

1

00 18000 36000 54000 72000

Time (Minute)D

mnl

EnzymeFactor : d20w10-2 a b

NoDownregRecep20

15

10

5

00 18000 36000 54000 72000

Time (Minute)

Bill

ions

NoDownregRecep : d20w10-2

ClCur4

3

2

1

00 18000 36000 54000 72000

Time (Minute)

Dm

nl

ClCur : d20w10-2 c d

Figure 7.1. Dynamics of an unsuccessful withdrawal regimen after partial dependence

72


0.225

0.15

0.075

041760 49320 56880 64440 72000

Time (Minute)

Dm

nl

WithdSignIntensity : d20w10-2

Figure 7.2. Severity of withdrawal signs after an unsuccessful dosing strategy in partial

dependence

Looking at Figure 7.1b, we see that although enzyme levels are being restored during

withdrawal doses, the duration of the withdrawal regimen falls short complete this

restoration. EnzymeFactor is more than 1.5 at the time of complete withdrawal. Figure 7.1c

shows that down-regulated receptors merely stop increasing and re-adaptive mechanisms

are not operational at all. The result is a severe withdrawal syndrome as can be seen in

Figure 7.1d (chloride current undershoots 1) and more clearly in Figure 7.2.

7.1.2. A successful regimen

The failure of the ten day withdrawal period suggests a longer withdrawal period

with decreased dosages. After trial-and-error, we come up with the following 15 day

regimen: We administer one-fiftheenth of the final dose between days 27 and 31; and we

administer one-twentieth of the final dose between days 32 and 35. The following

dynamics result.

73

MGIlumen200

150

100

50

00 21600 43200 64800 86400

Time (Minute)

mg

MGIlumen : d20w15-4

EnzymeFactor4

3

2

1

00 21600 43200 64800 86400

Time (Minute)

Dm

nl


NoDownregRecep20

15

10

5

00 21600 43200 64800 86400

Time (Minute)

Bill

ions


ClCur4

3

2

1

00 21600 43200 64800 86400

Time (Minute)

Dm

nl


Figure 7.3. Dynamics in a successful withdrawal regimen after partial dependence

WithdSignIntensity0.1

0.075

0.05

0.025

028800 43200 57600 72000 86400

Time (Minute)

Dm

nl


Figure 7.4. Severity of withdrawal signs after a successful dosing strategy in partial

dependence

74

Our anticipation turned out correct. The duration of withdrawal is now long enough

so as to facilitate complete recovery of down-regulated receptors (Figure 7.3c). Although

the metabolism is still 50 per cent higher than normal (Figure 7.3b, around the 35th day),

complete withdrawal does not lead to a significant withdrawal syndrome as can be seen

from Figure 7.4. Observe that the peak intensity of withdrawal signs is well below the

0.025 reference. This suggests that the contribution of enzyme induction to development of

dependence is minor. In fact, this is reported in the literature as well. This result is thus an

additional clue of our model’s validity.

7.2. Withdrawal after 60 days of treatment

7.2.1. An unsuccessful regimen

We now experiment with withdrawal regimens after 60 days of continuous PB use

after which the user becomes completely dependent on the drug. As a first trial, we

propose a 20 day regimen as follows: We wait seven days before administering reduced

doses and after that, between days 67 and 80, we administer one-fifteenth of the final dose.

The following dynamics result.

75

MGIlumen200

150

100

50

00 36000 72000 108000 144000

Time (Minute)

mg

MGIlumen : d60w20-1

EnzymeFactor4

3

2

1

00 36000 72000 108000 144000

Time (Minute)

Dm

nl


NoDownregRecep60

45

30

15

00 36000 72000 108000 144000

Time (Minute)

Bill

ions


ClCur4

3

2

1

00 36000 72000 108000 144000

Time (Minute)

Dm

nl


Figure 7.5. Results for an unsuccessful withdrawal regimen after complete dependence

WithdSignIntensity

0.2

0.15

0.1

0.05

086400 100800 115200 129600 144000

Time (Minute)

Dm

nl


Figure 7.6. Severity of withdrawal signs after an unsuccessful dosing strategy in complete

dependence

Although chloride current is maintained in an appropriate range during the regimen

so that both the down-regulated receptors and elevated enzyme levels are decreased

76

(Figures 7.5b and 7.5c), re-adaptation is partial because the duration of withdrawal falls

short. The result is a severe withdrawal syndrome as can be seen in Figure 7.6.

7.2.2. A Successful Regimen

We prolong the duration of withdrawal to 30 days. Since the dosage in the previous

regimen was shown to be appropriate, the regimen in the first 20 days is exactly the same

as in 7.2.1. We then assume that the user is supposed to take one-twentieth of the final dose

for the following ten days (i.e. between days 81 and 90). The following dynamics are

observed.

MGIlumen200

150

100

50

00 39600 79200 118800 158400

Time (Minute)

mg

MGIlumen : d60w30-1

EnzymeFactor4

3

2

1

00 39600 79200 118800 158400

Time (Minute)

Dm

nl


NoDownregRecep60

45

30

15

00 39600 79200 118800 158400

Time (Minute)

Bill

ions


ClCur4

3

2

1

00 39600 79200 118800 158400

Time (Minute)

Dm

nl


Figure 7.7. Results for a gradual withdrawal regimen of 30 days following a 60 day drug

treatment

77


0.15

0.1

0.05

084960 99720 114480 129240 144000

Time (Minute)

Dm

nl


Figure 7.8. Severity of withdrawal signs after a successful dosing strategy in complete

dependence

As anticipated, prolonging the last phase of the regimen cured the failure. The drug

user experiences no rebound effects (Figure 7.8). The duration of drug intake is long

enough so that almost all down-regulated receptors are restored by the end of the 90th day

(Figure 7.7c).

78

8. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

Although being replaced by safer drugs, a lot of people still use phenobarbital (PB)

regularly for sedation or against sleep disorders. As a side effect, phenobarbital enhances

the synthesis of its own metabolic enzymes in the liver. This enzyme induction problem

causes increased tolerance to phenobarbital over time. Moreover, the brain adapts to the

presence of the drug and its sensitivity decreases with time (neuroadaptation). The system

dynamics model constructed in this thesis is a representation of prolonged barbiturate use,

including phenobarbital absorption, distribution, metabolism, and elimination processes

with enzyme induction and neuroadaptation related structures.

The validity of the model is first demonstrated using available experimental data and

other qualitative information. The model is used as an experimental platform to study

various scenarios, including an epileptic patient, potential drug interactions and alternative

dosing schemes to minimize withdrawal syndromes. Adaptive changes in the body as a

response to drug use result in drug tolerance, dependence and eventually withdrawal

syndromes. The situation is further complicated in an epilepsy patient. We show that an

epilepsy patient is more prone to development of barbiturate tolerance and dependence.

Possible drug-drug interactions should also be taken into account if PB is being taken

together with other chemicals. In this thesis, we study a drug-drug interaction involving

only the liver. However, in epilepsy, several drugs may be prescribed and concurrent

intake of these drugs would involve more complex dynamics especially in the central

nervous system (CNS). Additionally, the consumption of alcohol while taking barbiturates

has well-known synergistic and thus lethal effects. As such, these could be subjects of

future study.

The model provides an experimental platform to test different dosing schemes and

dose adjustment policies in prolonged use. We experiment with different dosing

frequencies and show that the more frequent the doses, the better it is in terms of tolerance

and dependence development. However, we neglect the possible impracticality of frequent

doses. Since we explicitly model drug content in arterial blood, the simulation model could

79

also be used to simulate clinical settings such as constant intravenous infusion where the

infusion of drug is more continuous contrary to cases studied here.

Parallel to literature reports, we show that when a dependent user abruptly

discontinues PB use, harmful rebound effects are experienced. To avoid, the doses should

be reduced gradually. We have proposed relatively efficient withdrawal regimens for both

partial and complete dependence cases. As anticipated, a longer period of withdrawal is

necessary in complete dependence cases. It is shown that he duration of the withdrawal

period should be at least half the actual treatment duration in all cases. The method of

search for feasible withdrawal regimens was intuitive. A more systematic approach could

prove useful in the future. It may be interesting to define the problem as an optimization

problem where it is tried to minimize both the duration of the withdrawal period and the

amount of doses while keeping the resulting intensity of withdrawal signs at minimum.

The model does not take into account neuroadaptation dynamics in the excitatory

neurotransmission system. It is likely that when inhibitory neurotransmission is potentiated

by PB, besides desensitizing inhibitory receptors to counteract potentiation,

neuroadaptation could up-regulate excitatory neurotransmission as well. Including this

mechanism in the future versions of the model could enhance its realism.

Finally, the model is built using rather generic structures and generic assumptions.

This is especially true for the pharmacokinetic sector. The parameters can be modified so

that a different CNS-active drug can be modeled as well. Receptor down-regulation is also

a rather common mechanism of neuroadaptation. Therefore, the parameters in the CNS

sector of the model can be modified to capture the dynamics of a different drug that causes

receptor down-regulation.

80

APPENDIX. EQUATIONS OF THE MODEL

The complete stock-flow structure of the model is given in Figure A.1.

81

MBraintis

MBraincapil

MArterial

MMuscle

MVenous

MKidney

MGItissueMGIlumen

MLiver

EnzymeFactor

ArterialToBrain

ArterialToMuscle MuscleTo

Venous

ArterialToKidney

ArterialToGItis

ArterialToLiver

BrainToVenous

KidneyToVenous

Absorption

LiverToVenous

BraincapilToBraintis

QBrain

CBraincapil

FR

Bplasma

CBraintis

VBraintis

DR

QTotal CVenous

CMuscle

QMuscle PMuscle

CArterial

QKidney

PKidney

CKidney

QHeart

QGItissue

PGItissue

CGItissue Kabs

QLiver CLiver

NormKmet

PLiver

Rin kout

Metabolism

GItissueToLiver

IndInducByPB

Synthesis Degradation

BraintisToBraincapil

Intake1

Kmet

EffPB

ClCur

NormClCur

IndAdptnRate

EffAdptnOnNorm

ClCur

NoDownregRecep

Adaptation ReadaptationClCurWOPB

Intake2

<Time>

EffPBReadptn

ReaIInducbyPB

Intake3

WithdSignIntensity

Intake4

VBraincapil

VMuscle

VKidney

VGItissue

VLiver

VVenous

VArterialKexcr

DoseDoseIncr<Time>

DaysTreatment

I2

HalflifeEnzyme

ThresholdSedat

TotalNoRecep

EffSatur

Intake5

<TIME STEP>

Excretion

MFatArterialToFat FatToVenousQFat

CFat

PFat

VFat

VenousToArterial

ArterialToVenous

ReadptnFrac

RealAdptnRate

I1

I3

<Time>I4

<Time>

LoadDose

FINAL TIME

Figure A.1. Complete stock-flow diagram of the model

82

Equations of the model are given below for one-a-day treatment lasting for 20 days

and ending with abrupt withdrawal.

Dose=INTEG (DoseIncr,30)

I4=IF THEN ELSE(MODULO(Time,1440)>=1437,1,0)

LoadDose=180

MGIlumen=INTEG (Intake2+Intake3+Intake4+Intake5+Intake1-Absorption, LoadDose)

DoseIncr=I1*I2*I3*I4*10

I3=IF THEN ELSE(Time>1440, 1 ,0 )

I1=IF THEN ELSE(ClCur<ThresholdSedat, 1 , 0 )

Adaptation=EffSatur*RealAdptnRate

RealAdptnRate=SMOOTH3(IndAdptnRate , 15*1440 )/15

Readaptation=EffPBOnReadptn*ReadptnFrac*NoDownregRecep

ReadptnFrac=0.000325

VenousToArterial=QTotal*CVenous

MVenous=INTEG (ArterialToVenous+BrainToVenous+FatToVenous+KidneyToVenous+

LiverToVenous+MuscleToVenous-VenousToArterial,0)

Excretion=MKidney*Kexcr

83

MArterial= INTEG (VenousToArterial-ArterialToMuscle-ArterialToLiver-

ArterialToKidney-ArterialToGItis-ArterialToFat-ArterialToBrain-ArterialToVenous,

0)

ArterialToVenous=CArterial*QHeart

MFat= INTEG (ArterialToFat-FatToVenous,0)

FatToVenous=QFat*CFat/PFat

ArterialToFat=QFat*CArterial

VFat=16.394

PFat=1

CFat=MFat/VFat

QFat=0.26

Intake3=0

Intake4=0

Intake5=0

MKidney= INTEG (ArterialToKidney-KidneyToVenous-Excretion,0)

TotalNoRecep=60

84

EffAdptnOnNormClCur= WITH LOOKUP (NoDownregRecep/TotalNoRecep,

([(0,0.6)-(1,1)],(0,1),(0.0825688,0.829825),(0.100917,0.807018),

(0.131498,0.784211),(0.153333,0.77193),(0.186544,0.764912),

(0.266055,0.750877),(0.33945,0.742105),(1,0.7) ))

EffSatur= WITH LOOKUP (NoDownregRecep/TotalNoRecep,

([(0.8,0)-(1,1)],(0,1),(0.8,1),(0.8263,0.969298),(0.849541,0.907895),

(0.877676,0.758772),(0.899083,0.605263),(0.933945,0.179825),

(0.944342,0.100877),(0.95841,0.0351),(0.975535,0),(1,0) ))

ThresholdSedat=2.5

Intake2=0

kout=LN(2)/HalflifeEnzyme

HalflifeEnzyme=2880

Rin=LN(2)/(HalflifeEnzyme)

Intake1=(Dose/TIME STEP)*PULSE TRAIN( 1440, TIME STEP,

1440,DaysTreatment*1440+TIME STEP)

DaysTreatment=19

I2=IF THEN ELSE(Time<DaysTreatment*1440,1,0 )

CLiver=MLiver/VLiver

CMuscle=MMuscle/VMuscle

CVenous=MVenous/VVenous

85

CArterial=MArterial/VArterial

CBraincapil=MBraincapil/VBraincapil

CBraintis=MBraintis/VBraintis

CGItissue=MGItissue/VGItissue

CKidney=MKidney/VKidney

Kexcr=0.0035

VKidney=0.308

VArterial=1.556

VBraincapil=0.0447

VMuscle=28

VGItissue=1.19

VLiver=1.925

VVenous=3.811

MBraincapil= INTEG (ArterialToBrain+BraintisToBraincapil-BrainToVenous-

BraincapilToBraintis,0)

EffPBOnClCur=CBraintis*600/(2.79+CBraintis)

WithdSignIntensity=-(ClCur-1)

86

NormClCur=ClCurWOPB*EffAdptnOnNormClCur

ReaIInducbyPB=SMOOTH3(IndInducByPB, 2*1440)

Synthesis=Rin*(1+ReaIInducbyPB)

EffPBOnReadptn= WITH LOOKUP ( EffPBOnClCur,

([(0,0)-(800,1)],(0,1),(10.7034,0.938596),(18.9602,0.850877),(25.9939,0.714912),

(42.5076,0.232456),(46.1774,0.144737),(52.9052,0.0614035),(60.367,0.0175),(70,0)

,(600,0) ))

IndAdptnRate= WITH LOOKUP (ClCur/ClCurWOPB,

([(0,0)-(6,0.04)],(0,0),(1.5,0),(1.88991,0.0008635),(2.04587,0.00185025),

(2.20183,0.003455),(2.37156,0.0061675),(2.52294,0.009375),(2.66972,0.0133225),

(2.78899,0.0173925),(3,0.028125),(3.04465,0.03),(3.13211,0.0317544),

(3.23976,0.0331579),(3.40061,0.0342105),(3.61468,0.035),(4,0.035614),(6,0.03561)

))

ClCur=NormClCur*(1+EffPBOnClCur/100)

ClCurWOPB=1

NoDownregRecep= INTEG (Adaptation-Readaptation,0)

Kmet=NormKmet*EnzymeFactor

Metabolism=CLiver*Kmet

MBraintis= INTEG (BraincapilToBraintis-BraintisToBraincapil,0)

BraincapilToBraintis=VBraintis*DR*CBraincapil/(1+Bplasma)

BraintisToBraincapil=VBraintis*DR*CBraintis*FR

87

Absorption=MGIlumen*Kabs

ArterialToBrain=QBrain*CArterial

ArterialToGItis=CArterial*QGItissue

ArterialToKidney=CArterial*QKidney

ArterialToLiver=CArterial*QLiver

ArterialToMuscle=QMuscle*CArterial

Bplasma=0.438

BrainToVenous=QBrain*CBraincapil

Degradation=(EnzymeFactor)*kout

DR=0.02

EnzymeFactor= INTEG (Synthesis-Degradation,1)

FR=1.75

MGItissue= INTEG (Absorption+ArterialToGItis-GItissueToLiver,

GItissueToLiver=QGItissue*CGItissue/PGItissue

Kabs=0.02

KidneyToVenous=QKidney*CKidney/PKidney

NormKmet=3.14/1000

88

MLiver= INTEG (ArterialToLiver+GItissueToLiver-LiverToVenous-Metabolism, 0)

LiverToVenous=(QLiver+QGItissue)*CLiver/PLiver

MuscleToVenous=QMuscle*CMuscle/PMuscle

MMuscle= INTEG (ArterialToMuscle-MuscleToVenous,0)

PGItissue=1

PKidney=2.05

PLiver=2.25

PMuscle=1.12

QBrain=0.57

QGItissue=0.9

QHeart=0.2

QKidney=0.875

QLiver=0.235

QMuscle=1.67

QTotal=4.475

IndInducByPB=1.15*CLiver/(1+CLiver)

VBraintis=1.3553

89

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