Download Konuray, A. O., 2008, Development of Tolerance and Dependence

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
4.2.1. Fundamental Approach and Assumptions ..................................................
26
4.2.2. Description of the Structure........................................................................
28
4.3. Dose Sector ...........................................................................................................
33
4.3.1. Fundamental Approach and Assumptions ..................................................
33
4.3.2. Description of the Structure........................................................................
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 .............................................................................
Figure 6.6. Enzyme and neuroadaptation dynamics with and without fluconazole pre-
62
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 anticonvulsants (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
Ultra-shortacting
Methohexital
(BREVITAL)
I.V.†
HALFLIFE,
HOURS
3-5*
Thiopental
(PENTHOTAL)
I.V., rectal
8-10*
Pentobarbital
(NEMBUTAL)
Oral, I.M. †, I.V.,
rectal
15-50
Secobarbital
(SECONAL)
Oral, I.M., I.V., rectal
15-40
Amobarbital
(AMYTAL)
Oral, I.M., I.V.
10-40
Aprobarbital
(ALURATE)
Butabarbital
(BUTISOL, others)
Oral
14-34
Oral
35-50
Butalbital
Oral
35-88
Mephobarbital
(MEBARAL)
Phenobarbital
(LUMINAL,
others)
Oral
10-70
Oral, I.M., I.V.
80-120
Short-acting
Intermediateacting
Long-acting
†
THERAPEUTIC
USES
Induction and/or
maintenance of
anesthesia
Induction and/or
maintenance of
anesthesia,
preoperative
sedation,
emergency
management of
seizures
Insomnia,
preoperative
sedation,
emergency
management of
seizures
Insomnia,
preoperative
sedation,
emergency
management of
seizures
Insomnia,
preoperative
sedation,
emergency
management of
seizures
Insomnia
Insomnia,
preoperative
sedation
Marketed in
combination with
analgesic agents
Seizure disorders,
daytime sedation
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 receptorion 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 GABAactivated 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 postand 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 presynaptic 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 12-3-1. 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
Re-adaptation rate 8
+
Adaptation rat e
6
+
-
+
Number of
down-regulated
recep tors 7
Chloride current
+
5
-
Enzy me synthesis
rate
+
2
+
Extent of CNS
depression
9
Functionality of
GABA
neurotransmission
4
+
3
M etabolism rat e
+
+
Intensity of
withdrawal syndrome
11
-
-
+
Amount of
phenobarbital in
the body
Rate of phenobarbital
intake
10
1
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
VBraintis
CBraintis
VBraincapil
MBraintis
BraintisTo
Braincapil
FR
DR
CBraincapil
Bplasma
Braincapil
ToBraintis
MBraincapil
QBrain
ArterialToBrain
BrainToVenous
PFat
VFat
QFat
CFat
MFat
ArterialToFat
FatToVenous
QHeart
ArterialToVenous
MArterial
QMuscle
CArterial
VMuscle
MMuscle
ArterialTo
Muscle
VArterial
MKidney
ArterialTo
Kidney
VenousToArterial
CMuscle
QTotal
MuscleTo
Venous
PMuscle
QKidney
Excretion
KidneyToVenous
DaysTreatment
CKidney
VKidney
Kexcr
MGItissue
CGItissue
QGItissue
Intake2
Absorption
VGItissue
GItissue
ToLiver
ArterialToLiver
IndInducByPB
Intake
5
Intake
4
Intake3
Synthesis
CLiver
MLiver
NormKmet
Metabolism
LiverTo
Venous
PLiver
QLiver
Kmet
Figure 4.1. Stock-flow structure of the pharmacokinetics sector
kout
Rin
VLiver
Kabs
PGItissue
<Dose>
HalflifeEnzyme
ReaIInducbyPB
PKidney
Intake1
MGIlumen
ArterialToGItis
MVenous
CVenous
VVenous
EnzymeFactor
Degradation
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 ElMasri 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.
IndInducByPb =
E max * CLiver
EC 50 + CLiver
(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.
1,2
IndInducByPB
1
0,8
0,6
0,4
0,2
0
0
5
10
15
20
25
CLiver (mg/L)
Figure 4.2. Saturability of enzyme induction
The outflow Degradation is given by the following equation.
Degradation = kout * EnzymeFactor
(4.14)
26
where kout=
ln(2)
which has units of min-1.
HalflifeEnzyme
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
4.2.1. Fundamental Approach and Assumptions
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
PB
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 *
C PB
GABAEC50 + C PB
(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
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.
4.2.2. Description of the Structure
The stock-flow diagram of the CNS sector is given in Figure 4.4.
EffPBOnClCur
EffPBOn
Readptn
WithdSignIntensity
<CBraintis>
EffAdptn
OnNorm
ClCur
ClCurWOPB
IndAdptnRate
ReadptnFrac
EffSatur
NormClCur
ClCur
TotalNoRecep
Adaptation
NoDownreg
Recep
Readaptation
RealAdptnRate
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)
Adaptation = RealAdptnRate * EffSatur
(4.19)
where
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)
EffAdptnOnNormClCur = F (NoDownregRecep/TotalNoRecep)
(4.21)
where
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 readaptation 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
4.3.1. Fundamental Approach and Assumptions
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.
4.3.2. Description of the Structure
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
ThresholdSedat
<ClCur>
IncrRate
I1
<DaysTreatment>
Dose
DoseIncr
I2
<Time>
I3
LoadDose
I4
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
PARTITION
FLOW, L/min
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
0
and Dose)
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
MGITissue
6
150
4.5
100
3
mg
mg
MGILumen
200
1.5
50
0
0
0
30
60
90
120 150 180
Time (Minute)
210
240
270
0
300
30
60
90
120
150
180
Time (Minute)
a
270
300
210
240
270
300
210
240
270
300
b
MVenous
MLiver
20
20
15
15
10
10
mg
mg
240
MGITissue : singletablet
MGILumen : singletablet
5
5
0
0
0
30
60
90
120
150
180
Time (Minute)
210
240
270
0
300
30
60
90
120
150
180
Time (Minute)
MVenous : singletablet
MLiver : singletablet
c
d
MFat
MMuscle
100
60
75
45
50
30
25
15
0
0
0
30
60
90
120 150 180
Time (Minute)
210
240
270
0
300
30
60
90
120
150
180
Time (Minute)
MFat : singletablet
MMuscle : singletablet
e
f
MBraintis
2
1.5
mg
mg
210
1
0.5
0
0
30
60
90
120
150
180
Time (Minute)
210
240
270
300
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.
ClCur
4
Dmnl
3
2
1
0
0
30
60
90
120
150
180
Time (Minute)
210
240
270
300
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 downregulated receptors is an insignificant fraction of the total receptor population of 60 billion
(Figure 5.3b).
40
NoDownregRecep
4e-007
1.75
3e-007
Billions
Dmnl
EnzymeFactor
2
1.5
2e-007
1e-007
1.25
0
1
0
30
60
90
120
150 180
Time (Minute)
210
240
270
300
0
30
60
90
120 150 180
Time (Minute)
210
240
270
300
NoDownregRecep : singletablet
EnzymeFactor : 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
Dose
40
mg
35
30
25
20
0
7200
14400
Time (Minute)
Dose : d7nofeedback
21600
28800
Dose : d20nofeedback
a
MBraintis
2
mg
1.5
1
0.5
0
0
18000
36000
Time (Minute)
54000
72000
MBraintis : d7nofeedback
MBraintis : 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
NoDownregRecep
6
1.7
4.5
Billions
Dmnl
EnzymeFactor
2
1.4
1.1
3
1.5
0.8
0
0
18000
36000
Time (Minute)
54000
72000
0
EnzymeFactor : d7nofeedback
EnzymeFactor : d20nofeedback
18000
36000
Time (Minute)
54000
72000
NoDownregRecep : d7nofeedback
NoDownregRecep : d20nofeedback
a
b
ClCur
4
Dmnl
3
2
1
0
0
18000
36000
Time (Minute)
54000
72000
ClCur : d7nofeedback
ClCur : 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
Dose
80
mg
65
50
35
20
0
2520
5040
Time (Minute)
7560
10080
29160
38880
Dose : d7
a
MBraintis
2
mg
1.5
1
0.5
0
0
9720
19440
Time (Minute)
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
NoDownregRecep
4
1.7
3
Billions
Dmnl
EnzymeFactor
2
1.4
1.1
2
1
0.8
0
0
9720
19440
Time (Minute)
29160
38880
0
EnzymeFactor : d7
9720
19440
Time (Minute)
29160
38880
NoDownregRecep : d7
a
b
Intensity of withdrawal signs
0.1
Dmnl
0.075
0.05
0.025
0
8640
16200
23760
Time (Minute)
31320
38880
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
ClCur
4
Dmnl
3
2
1
0
0
7220
14440
Time (Minute)
ClCur : d7
ClCurWOPB : d7
21660
28880
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.
Dose
200
mg
150
100
50
0
0
7200
14400
Time (Minute)
21600
28800
54000
72000
Dose : d20
a
MBraintis
4
mg
3
2
1
0
0
18000
36000
Time (Minute)
MBraintis : d20
b
Figure 5.9. Dose profile (a) and drug amount in the brain (b) in the 20 day drug treatment
followed by abrupt withdrawal
48
NoDownregRecep
20
3
15
Billions
Dmnl
EnzymeFactor
4
2
1
10
5
0
0
0
18000
36000
Time (Minute)
54000
72000
0
EnzymeFactor : d20
18000
36000
Time (Minute)
54000
72000
NoDownregRecep : d20
a
b
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
27360
38520
49680
Time (Minute)
60840
72000
WithdSignIntensity : d20
c
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 current
4
Dmnl
3
2
1
0
0
18000
ClCur : d20
ThresholdSedat : d20
36000
Time (Minute)
54000
72000
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
Dose
200
mg
150
100
50
0
0
21600
43200
Time (Minute)
64800
86400
97200
129600
Dose : d60
a
MBraintis
6
mg
4.5
3
1.5
0
0
32400
64800
Time (Minute)
MBraintis : d60
b
Figure 5.12. Dose profile (a) and drug amount in the brain (b) in the 60 day drug treatment
followed by abrupt withdrawal
51
NoDownregRecep
60
3
45
Billions
Dmnl
EnzymeFactor
4
2
1
30
15
0
0
0
32400
64800
Time (Minute)
97200
129600
0
EnzymeFactor : d60
32400
64800
Time (Minute)
97200
129600
NoDownregRecep : d60
a
b
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
84960
96120
107280
Time (Minute)
118440
129600
WithdSignIntensity : d60
c
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 current
4
Dmnl
3
2
1
0
0
32400
ClCur : d60
ClCurWOPB : d60
64800
Time (Minute)
97200
129600
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).
EnzymeFactor
4
Dmnl
3
2
1
0
0
18000
36000
Time (Minute)
54000
72000
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.
Dose
200
mg
150
100
50
0
0
21600
43200
Time (Minute)
64800
86400
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.
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
84960
96120
107280
Time (Minute)
118440
129600
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
Intensity of withdrawal signs
0.3
0.225
0.225
Dmnl
Dmnl
Intensity of withdrawal signs
0.3
0.15
0.075
0
27360
0.15
0.075
38520
49680
Time (Minute)
WithdSignIntensity : d20
60840
72000
0
84960
96120
107280
Time (Minute)
118440
129600
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)
ClCur = 1 – WithdSignIntensity
(6.2)
it follows that
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 postwithdrawal 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
Dose
400
mg
300
200
100
0
0
21600
43200
Time (Minute)
Dose : d60
64800
86400
Dose : d60epileptic
a
MBraintis
6
mg
4.5
3
1.5
0
0
21240
42480
Time (Minute)
63720
84960
MBraintis : d60
MBraintis : 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
NoDownregRecep
60
3
45
Billions
Dmnl
EnzymeFactor
4
2
1
30
15
0
0
0
21240
42480
Time (Minute)
EnzymeFactor : d60
EnzymeFactor : d60epileptic
63720
84960
0
21240
42480
Time (Minute)
63720
84960
NoDownregRecep : d60
NoDownregRecep : 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
Dmnl
3
2
1
0
0
21240
ClCur : d60
ClCur : d60epileptic
ClCurWOPB : d60
42480
Time (Minute)
63720
84960
ClCurWOPB : d60epileptic
ThresholdSedat : 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 downregulated 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.
Dose
200
mg
150
100
50
0
0
7200
14400
Time (Minute)
Dose : d20fluc
21600
28800
54000
72000
Dose : d20
a
MBraintis
4
mg
3
2
1
0
0
18000
36000
Time (Minute)
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
NoDownregRecep
40
3
30
Billions
Dmnl
EnzymeFactor
4
2
1
20
10
0
0
0
18000
36000
Time (Minute)
54000
72000
0
EnzymeFactor : d20fluc
EnzymeFactor : d20
18000
36000
Time (Minute)
54000
72000
NoDownregRecep : d20fluc
NoDownregRecep : d20
a
b
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
27360
38520
49680
Time (Minute)
60840
72000
WithdSignIntensity : d20fluc
WithdSignIntensity : d20
c
Figure 6.6. Enzyme and neuroadaptation dynamics with and without fluconazole pretreatment. 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 pretreatment. 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 downregulated receptors is the lowest in one-every-two-days scheme (Figure 6.7b).
65
EnzymeFactor
2
Dmnl
1.7
1.4
1.1
0.8
0
18000
36000
Time (Minute)
54000
72000
54000
72000
EnzymeFactor : d20oneeverytwo-nofeedb
EnzymeFactor : d20oneaday-nofeedb
EnzymeFactor : d20twoaday-nofeedb
EnzymeFactor : d20threeaday-nofeedb
a
NoDownregRecep
6
Billions
4.5
3
1.5
0
0
18000
36000
Time (Minute)
NoDownregRecep : d20oneeverytwo-nofeedb
NoDownregRecep : d20oneaday-nofeedb
NoDownregRecep : d20twoaday-nofeedb
NoDownregRecep : 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.
ClCur
4
Dmnl
3
2
1
0
0
18000
ClCur : d20oneeverytwo-nofeedb
ClCur : d20oneaday-nofeedb
36000
Time (Minute)
54000
72000
ClCur : d20twoaday-nofeedb
ClCur : 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
Dose
400
mg
300
200
100
0
0
7200
14400
Time (Minute)
21600
28800
Dose : d20oneeverytwo
Dose : d20oneaday
Dose : d20twoaday
Dose : d20threeaday
Figure 6.9. Difference in the extent of tolerance development w.r.t dosing schemes
(Feedback allowed to increase doses)
NoDownregRecep
40
Billions
30
20
10
0
0
18000
36000
Time (Minute)
54000
72000
NoDownregRecep : d20oneeverytwo
NoDownregRecep : d20oneaday
NoDownregRecep : d20twoaday
NoDownregRecep : d20threeaday
Figure 6.10. Neuroadaptation dynamics for different dosing schemes (Feedback allowed to
increase doses)
68
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
0
18000
36000
Time (Minute)
54000
72000
WithdSignIntensity : d20oneeverytwo
WithdSignIntensity : d20oneaday
WithdSignIntensity : d20twoaday
WithdSignIntensity : 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 current
4
Dmnl
3
2
1
0
0
18000
ClCur : d20oneeverytwo
ClCur : d20oneaday
ClCur : d20twoaday
36000
Time (Minute)
54000
72000
ClCur : d20threeaday
ClCurWOPB : d20oneaday
ThresholdSedat : 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.
EnzymeFactor
4
150
3
Dmnl
mg
MGIlumen
200
100
50
2
1
0
0
0
18000
36000
Time (Minute)
54000
72000
0
MGIlumen : d20w10-2
18000
54000
72000
54000
72000
EnzymeFactor : d20w10-2
a
b
NoDownregRecep
ClCur
20
4
15
3
Dmnl
Billions
36000
Time (Minute)
10
5
2
1
0
0
0
18000
36000
Time (Minute)
NoDownregRecep : d20w10-2
54000
72000
0
18000
36000
Time (Minute)
ClCur : d20w10-2
c
d
Figure 7.1. Dynamics of an unsuccessful withdrawal regimen after partial dependence
72
Intensity of withdrawal signs
0.3
Dmnl
0.225
0.15
0.075
0
41760
49320
56880
Time (Minute)
64440
72000
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
EnzymeFactor
4
150
3
Dmnl
mg
MGIlumen
200
100
50
2
1
0
0
0
21600
43200
Time (Minute)
64800
86400
0
MGIlumen : d20w15-4
21600
43200
Time (Minute)
86400
64800
86400
EnzymeFactor : d20w15-4
a
b
NoDownregRecep
ClCur
4
15
3
Dmnl
20
10
5
2
1
0
0
0
21600
43200
Time (Minute)
64800
86400
NoDownregRecep : d20w15-4
0
21600
43200
Time (Minute)
ClCur : d20w15-4
c
d
Figure 7.3. Dynamics in a successful withdrawal regimen after partial dependence
WithdSignIntensity
0.1
0.075
Dmnl
Billions
64800
0.05
0.025
0
28800
43200
57600
Time (Minute)
72000
86400
WithdSignIntensity : d20w15-4
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
EnzymeFactor
4
150
3
Dmnl
mg
MGIlumen
200
100
50
2
1
0
0
0
36000
72000
Time (Minute)
108000
144000
0
MGIlumen : d60w20-1
36000
72000
Time (Minute)
144000
108000
144000
EnzymeFactor : d60w20-1
a
b
NoDownregRecep
ClCur
60
4
45
3
Dmnl
Billions
108000
30
15
2
1
0
0
0
36000
72000
Time (Minute)
108000
144000
NoDownregRecep : d60w20-1
0
36000
72000
Time (Minute)
ClCur : d60w20-1
c
d
Figure 7.5. Results for an unsuccessful withdrawal regimen after complete dependence
WithdSignIntensity
0.2
Dmnl
0.15
0.1
0.05
0
86400
100800
115200
Time (Minute)
129600
144000
WithdSignIntensity : d60w20-1
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.
EnzymeFactor
4
150
3
Dmnl
mg
MGIlumen
200
100
50
2
1
0
0
0
39600
79200
Time (Minute)
118800
158400
0
MGIlumen : d60w30-1
39600
118800
158400
118800
158400
EnzymeFactor : d60w30-1
a
b
NoDownregRecep
ClCur
60
4
45
3
Dmnl
Billions
79200
Time (Minute)
30
15
2
1
0
0
0
39600
79200
Time (Minute)
NoDownregRecep : d60w30-1
118800
158400
0
39600
79200
Time (Minute)
ClCur : d60w30-1
c
d
Figure 7.7. Results for a gradual withdrawal regimen of 30 days following a 60 day drug
treatment
77
Intensity of withdrawal signs
0.2
Dmnl
0.15
0.1
0.05
0
84960
99720
114480
Time (Minute)
129240
144000
WithdSignIntensity : d60w30-1
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
CBraintis
QBrain
VBraincapil
Eff
PBReadptn
EffPB
WithdSignIntensity
CBraincapil
MBraintis
FR
BraincapilToBraintis
Bplasma
ClCur
BrainToVenous
CFat
QFat
QTotal
VArterial
ArterialToVenous
ArterialTo
Muscle
Kexcr
ArterialTo
Kidney
VGItissue
ArterialToGItis
CMuscle
I3
<Time>
Intake
5
Intake2
Intake Intake3
4
MLiver
CLiver
QLiver
Metabolism
FINAL TIME
DaysTreatment
VLiver
NormKmet
kout
Rin
Intake1
Absorption
GItissue
ToLiver
<Time>
KidneyToVenous
VKidney
MGIlumen
MGItissue
ArterialToLiver
LoadDose
HalflifeEnzyme
Kabs
QGItissue
Dose
DoseIncr
I2
PKidney
MKidney
CKidney
<Time>
I4
I1
<Time>
MuscleTo
Venous
VMuscle
QKidney
PGItissue
RealAdptnRate
ThresholdSedat
MVenous
VenousToArterial
Excretion
CGItissue
IndAdptnRate
FatToVenous
VVenous
CVenous
QMuscle PMuscle
MMuscle
CArterial
NoDownreg
Recep
PFat
QHeart
MArterial
Adaptation
ReadptnFrac
VFat
MFat
ArterialToFat
EffSatur
ClCurWOPB
MBraincapil
ArterialToBrain
TotalNoRecep
NormClCur
VBraintis
BraintisToBraincapil
DR
EffAdptn
OnNorm
ClCur
<TIME STEP> Synthesis
LiverTo
Venous
Enzyme
Factor
PLiver
ReaIInducbyPB
IndInducByPB
Kmet
Figure A.1. Complete stock-flow diagram of the model
Degradation
Readaptation
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-ArterialToLiverArterialToKidney-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-BrainToVenousBraincapilToBraintis,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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