Download Intrathecal Opioid Distribution Model

Intrathecal Opioid Distribution Model
Julie C. Wagner
This report is produced under the supervision of BIOE310 instructor Prof. Linninger.
Abstract
The mechanisms of biodistribution for drugs administered intrathecally are not
well understood. This project aims to create a mechanistic model in order to predict the
spread of opioids with given experimental data. The model computes spinal distribution
of the drug based on a Bolus injection, three different flow models derived from the
Navier-Stokes momentum equation, and which one of the four different drugs are being
examined. The injection site occurred at the Lumbar region of the spine and the four
drugs that were tested are Morphine, Alfentanil, Fentanyl, and Sufentanil. The results of
the model based on the data that was used indicated that Alentanil had the lowest
concentration by a considerable margin at each node over time. Morphine, Fentanyl, and
Sufentanil were all very similar but Morphine was the highest while Fentanyl and
Sufentanil were intermediate concentrations when compared to the rest of the opioids.
Introduction
There are several variables that need to be
taken into account in order to develop a
mechanistic model to predict the spread of
opioids intrathecally distributed. The types
of variables that need to be taken into
account can be categorized as either physical
variations between patients, Cerebral Spinal
Fluid
(CSF)
characteristics,
opioid
properties,
or
surgical
procedure
variables1,2,3.
Individual patients have different properties
that can affect drug distribution. The main
patient variation that will be taken into
account is their heart rate. In an experiment
by Hsu et al, it was seen that the heart rate
had a major impact on intrathecal drug
distribution2. The faster the heart rate is, the
more the drug is able to be spread
throughout the system via the CSF2. Heart
pulsations can generate hydrostatic pressure
gradients within the different CSF
compartment which could result in
determining the drug flow throughout the
spinal cord4. Another patient based factor is
the length of their spine. This affects how
far the drug concentration will spread in the
spine based on the rate of drug infusion and
CSF flow1. The closer a vertebra is to the
injection site, the higher the opioid’s
concentration will initially be1.
An important opioid property that affects its
distributed is the rate of transfer between the
drug in the CSF and the surround tissues
(ex: spinal cord, epidural fat, and plasma) 1.
These different rates determine the amount
of the opioid present at a specific time.
Which in turn determines whether the opioid
will be present enough to spread throughout
the spinal cord and seen in different
vertebra.
When performing the surgery using opioids
for the anesthesia, there are certain factors
that the surgeon controls which affect the
drug’s distribution. These factors include the
type of injection (Bolus injection or a
constant injection rate), where the injections
occur, and how many injections are needed3.
The type of injection will affect how much
of the drug is entered into the system and at
what rate it occurs at. In addition, where the
injection occurs will affect which parts of
Wagner - 1
the spinal cord will receive dosage of the
opioid both initially and over time.
Methods
In order to solve the set of differential
equations used, an ODE function was
utilized; this can be seen in Appendix 3. The
following process describes how that was
accomplished.
Brain P1
V1
30
F1
P2
V2
F2
20
P3
Thoracic V3
15
F3
10
Lumbar
P4
V4
F4
5
Sacral
0
Vasculature
Cervical
25
Epidural Space
The mechanistic model demonstrates a
number of different factors that affect opioid
drug distribution. In this simulation four
opioids will be taken into account:
Morphine, Alfentanil, Fentanyl, and
Sufentanil. Previous work of Ummenhofer
et all, shows each opioid concentration will
endure and initial increase at the injection
site then occur a decrease1. Out of each of
the four drugs that will be tested, it is
expected to see that the concentration of
Morphine will be the highest over time,
Fentanyl will have the lowest, while
Alfentanil and Sufentanil will have
intermediate concentrations3.
Spinal Cord Model
35
-5
0
P5
V5
5
10
15
20
30
35
40
Figure1: This model shows the different paths the
species can travel while in the system. Based on the
parameters of this model, each species can flow
between the spinal cord compartments, the epidural
space or to the vasculature. The symbol P describes
the pressure at a given node while the symbol V
describes the volume at that node. Each symbol F
describes the flow between each node which occurs
bidirectionally in the system. The code used to
generate this model can be seen in Appendix 1.
A
Spinal Cord Model
B
Spinal Cord Model with Deformation
35
Brain
C
Spinal Cord Model
P1
35
Brain
V1
35
P1
Brain
V1
30
F1
F1
30
Cervical
Cervical
V2
P1
V1
30
System
The system that will be modeled is based on
the diagram shown in Figure1. It will look at
five different regions: one of which is the
brain while the other four is the Cervical,
Thoracic, Lumbar, and Sacral regions of the
spinal cord. Each species will be injected
into this system through the Lumbar region
of the spine and flow bidirectionally
between each node. In addition the flow will
be traced as it goes from the CSF space to
the spinal cord, epidural space from the
spinal cord, and from the epidural space to
the vasculature. The direction of flow will
be dependent on the pulsation from the heart
beat, which causes the spinal cord will
deform depending whether the heart beat is
in diastole of systole2. The affect that
deformation has on the spinal cord
compartments can be seen in Figure 2.
25
P2
V2
F1
P2
25
Cervical
25
V2
P2
25
F2
F2
F2
20
20
Thoracic
V3
P3
Thoracic
15
V4
V3
Thoracic
P3
V3
F3
F3
P4
10 Lumbar
10 Lumbar
V4
V4
F4
F4
5
F4
V5
5
P5
0
Sacral
Sacral
0
5
10
P4
P4
5
Sacral
P3
15
15
F3
10 Lumbar
20
15
0
0
V5
5
10
V5
P5
0
P5
15
0
5
Figure 2: Above depicts how the spinal cord will
deform based on the pulsations from the heart. A
Shows the spinal in a non-deformed state. B When
the heart beat is in systole the spinal cord deforms
and increases the most at the Cervical node and least,
if at all, at the Sacral node2. C Once the heart beat is
in diastole the deformation returns to the original
state that was seen in A. Therefore this will affect the
flow direction within the system.
Wagner - 2
10
15
Injection
The type of injection used will affect the
results of the mechanistic model created.
This model will show the effects of a Bolus
injection. There will be an injection rate and
inlet of concentration for thirty seconds.
This will occur during the time between 5
and 5.5 minutes of the model. There will be
a total of 50mL of fluid injected into the
system. The injection flow will be reflected
in determining the volume at each node.
Flow
The flow of the system is pressure driven
and can be calculated by the initial pressure
conditions, velocity, and the resistance of
flow at that point. The initial pressure of the
brain is a sinusoidal function with the
frequency of 1.5 Hz, which is approximately
the average heart rate frequency, while all
the other nodes’ initial pressures were set to
zero6. Navier-Stokes momentum equation
was used in order to calculate the flow
between each spinal cord compartment, as
seen in Equation (1)7. U represents the
average fluid velocity, P is the pressure, F is
the frictional term, ρ is the CSF density, t is
time, and x is the distance. The coefficient
of friction used for the flows between each
node can be seen in Table 1.
𝜕𝑈
𝜕𝑡
𝜕
𝑈2
𝑃
+ 𝜕𝑥 ( 2 + 𝜌) = −𝐹
(1)
Coefficients of Friction
Flow 1
0.35879
Flow 2
0.4981
Flow 3
0.7228
Flow 4
0.783081
Table 1: These are the coefficients of friction used in
order to calculate the frictional term in the NavierStokes momentum equation.
Volume
The change in volume is calculated based on
the relationships between the flows in the
system and the flow from the injection. In
addition, at the brain there is more that is
taken into account. For instance, the
production of CSF, the reabsorbation of
CSF, and the volume of fluid based on the
heart pulsations affect the volume at the
brain. These equations are modeled after the
conservation balance equation which
determines that the influx of fluid in one
node must equal the outflow of fluid of the
same node. This is shown below.
𝑑𝑉
= 𝐹𝑖𝑛 − 𝐹𝑜𝑢𝑡
(2)
𝑑𝑡
Pressure
After analytically deriving the change in
volume of the system, the change in pressure
can be calculated. The relationship between
the change in pressure and the change in
volume can be summed up in the continuity
equations below.
∇ ∙ 𝑣⃗ = 0
(3)
Species
In order to calculate the amount of the
opioid at each node, all of the above
information needs to be calculated. This is
dependent on mass transfer which is a
conservation of mass balance equation as
seen in Equation (4).
𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛
𝐴𝑐𝑐 = 𝐹𝑖𝑛 − 𝐹𝑜𝑢𝑡 ± 𝑑𝑒𝑠𝑡𝑟𝑢𝑐𝑡𝑖𝑜𝑛 (4)
Since the flows are bidirectional, a method
was determined to decide which flow and
concentration combination to use at a
specific time. Therefore, for each possible
flow into the node the maximum was taken
between that flow and concentration
relationship and zero. Since the flow can
only occur in one direction, if one of the
flow and concentration relationships where
negative while the other positive using the
above method the positive one would be
used in the calculation. Furthermore, if both
were positive values, then both flows and
concentrations would be taken into account
by adding them together. It should be noted
that during this calculation any flow and
concentration that flowed from the Sacral
region to the brain instead of flowing from
Wagner - 3
the brain to the Sacral region was taken as a
negative value before finding the maximum
between that and zero. This is because when
determining the equations it was all taken as
the flow from the brain towards the Sacral
region as positive while the reverse way as
negative.
The species balance also took into account
the effects of transferring into different
tissues. These parameters are taken into
account in the efflux of fluid from the
compartment.
In addition, the destruction term at the end
will have a different reaction rate value, k,
depending on which opioid is being tested
and for what tissue it is moving in. The
reaction rates that were used can be seen in
Table 21.
Kinetic Terms
Kinetic
Morphine
Alfentanil
Fentanyl
Sufentanil
kic
kci
kplc
kie
kei
kplepi
0.037
0.0143
0.0082
0.0542
0.0021
0.0199
0.17
0.0236
0.0868
0.1372
0.0063
0.0201
0.0339
0.0159
0.0080
0.1078
0.0285
0.1088
0.020
0.0095
0.0131
0.0291
0.0137
0.0323
Table 2: Above shows the kinetic terms for each
opioid that was tested and how the react based on
which tissue they are transferring from. kic and kie
are the rate constants for movement from the
intrathecal space into the spinal cord and epidural
space respectfully. On the other hand, kci and kei
have the opposite movement of their counterparts
above. kplc is the rate constant for movement from
the spinal cord into the plasma and kplepi is
movement from the epidural space into the plasma1.
Models
Three different models were tested. These
models are all under the same conditions
and measured the same results but varied in
how the flow equations were derived from
the Navier-Stokes momentum equation.
Model 1 is the simplest version which only
takes into account the change in pressure
and the frictional term. This can be seen in
Equation (5).
𝜕𝑓
𝜕𝑃
=
−𝐹
(5)
𝜕𝑡
𝜕𝑥
The second and third models both take into
account the CSF densities. In the second
model the flow is derived based on the
velocity and pressure before and after the
flow being calculated, as well as, the
frictional term. This is depicted in Equation
(6) below.
𝜕𝑓𝑖
𝑃
𝑃
= [(𝑈𝑖−1 + 𝑖−1 ) − (𝑈𝑖+1 + 𝑖 )] − 𝐹 (6)
𝜕𝑡
𝜌
𝜌
The third model is similar to the second
except that it takes into account the velocity
and pressure before and the velocity and
pressure of the current flow that is being
calculated. This is seen in Equation (7).
𝜕𝑓𝑖
𝑃
𝑃
= [(𝑈𝑖−1 + 𝑖−1
) − (𝑈𝑖 + 𝜌𝑖 )] − 𝐹 (7)
𝜕𝑡
𝜌
Results
Flow
Before the injection at the 5 minute mark
there is a sinusoidal steady flow between the
compartments. In addition, there is a
noticeable phase shift between the flows
both before and after the injection. Once the
injection takes place there is a large,
instantaneous spike in the flow from the
Lumbar compartment to the Sacral one.
After fifteen minutes, the flow returns to a
periodical wave form. This can be seen in
Figure 3.
Volume
Since the initial conditions for volume of the
brain was determined to be 55 mL, Cervical
23 mL, Thoracic 33 mL, Lumbar 25 mL,
and Sacral 19 mL. Similar to the flow,
Figure 4 shows how before the injection at
the 5 minute mark the volumes periodically
oscillate in the amount of fluid that is held at
that compartment. When the injection occurs
there is a large spike in volume in the brain
that peaks at twice as much volume as
Wagner - 4
before. The other compartments see this
increase as well but at different times. The
Lumbar increases first while the phase shift
has the brain’s volume increasing last. After
fifteen minutes most of the volumes return
to approximately the same as before the
injection. The brain on the other hand is still
decreasing in volume even at the fifty
minute mark
Flow
300
Pressure
f1
f2
f3
f4
200
Pressure
The change in pressure of mmHg over time
of the system can be seen in Figure 5.
Similar to both the volume and flow, there is
a phase shift in the pressure before and after
the Bolus injection. Due to the large amount
of volume in the brain there is a large
amount of pressure within that compartment
as well. Over time there starts to be a
decrease in pressure in each compartment.
700
P1
P2
P3
P4
P5
600
100
500
0
400
300
-100
200
-200
100
-300
0
-400
0
5
10
15
20
25
30
35
40
45
50
Figure 3: Above is the pressure based flow of the
system in mL/s. The flows oscillate with a magnitude
based on the amplitude received from the heart beat
and injection rate. This explains why after 5 minutes
the flow from the Lumbar region of the spine to the
Sacral increases. Over time the flow a dynamic
steady state and no longer has variation in the flow5.
Volume
120
V1
V2
V3
V4
V5
100
80
60
40
20
0
0
5
10
15
20
25
30
35
40
45
50
Figure 4: This represents the change in volume of
each compartment that was modeled over time. The
oscillating volumes are due to the sinusoidal pulse of
the heart. At the five minute mark the injection
begins and is reflected in the volume of reach node.
After the increase in volume there is a steady decline
until the oscillations return to the initial periodic
volume.
-100
0
5
10
15
20
25
30
35
40
45
50
Figure 5: This figure describes the change in
pressure in mmHg over time through each. Until the
opioid was introduced there was a constant periodic
pressure. After the injection was administered, there
is an increase in pressure over time in each
compartment with a steady decline over time.
Species
The amount of species within the system
was traced in the CSF space, spinal cord,
epidural space, and the vasculature for each
opioid. All of this can be seen in Figures 69. Overall, Alfentanil was removed from the
system the fastest, Morphine lasted the
longest and Fentanyl and Sufentanil were
intermediate when compared to the other
opioids. Alfentanil reached a zero
concentration in the CSF space faster than
the other opioids and reached a zero
concentration in the spinal cord as well,
which the other opioids did not. Contrastly,
the opioid that transferred from the epidural
space the fastest was Fentanyl. Lastly,
during this time span each opioid was still
increasing when they transferred to the
vasculature.
Wagner - 5
A
B
Drug Concentration in CSF space for Morphine
2.5
Drug Concentration in CSF space for Alfentanil
1.5
C1
C2
C3
C4
C5
2
1.5
C1
C2
C3
C4
C5
1
1
0.5
0.5
0
0
-0.5
C
0
5
10
15
20
25
30
35
40
45
-0.5
50
D
Drug Concentration in CSF space for Fentanyl
2.5
C1
C2
C3
C4
C5
2
1.5
0
5
10
15
20
25
30
35
40
45
50
Drug Concentration in CSF space for Sufentanil
2
C1
C2
C3
C4
C5
1.5
1
1
0.5
0.5
0
0
-0.5
0
5
10
15
20
25
30
35
40
45
-0.5
50
0
5
10
15
20
25
30
35
40
45
50
Figure 6: These plots are the change in concentration in moles over time in minutes at the CSF space. A
Shows that Morphine lasts the longest within the tissue. B Shows that Alfentanil is removed the fastest
from the tissue. While C Fentanyl and D Sufentanyl are intermediate concentrations.
A
B
Drug Concentration in SC for Morphine
0.4
0.3
0.05
0.1
0
0
5
10
15
20
25
30
35
40
45
D
C1
C2
C3
C4
0.25
0.2
-0.05
50
Drug Concentration in SC for Fentanyl
0.3
C1
C2
C3
C4
0.1
0.2
0
C
Drug Concentration in SC for Alfentanil
0.15
C1
C2
C3
C4
0
5
15
20
25
30
35
40
45
50
40
45
50
Drug Concentration in SC for Sufentanil
0.3
C1
C2
C3
C4
0.25
0.2
0.15
10
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
Figure 7: These plots are the change in concentration in moles over time in minutes at the spinal cord.
Shows that Morphine lasts the longest within the tissue. B Shows that Alfentanil is removed the fastest
from the tissue. While C Fentanyl and D Sufentanyl are intermediate.
Each opioid concentration was also
compared through space and time. This is
done by finding the concentration at each
compartment at the same time. This can be
seen in Figure 10. It can be seen that at the
earliest time is the largest concentration of
each opioid and as time passes the
concentration lowers. Also, there is more of
each opioid seen at the Lumbar and Sacral
nodes than at the Brain, Cervical and
Thoracic. In addition, Alfentanil contains
the lowest amount of concentration at each
time and at each compartment. Similarly,
Morphine has the largest concentration
while Fentanyl and Sufentanil are the
opioids that are in between compared to the
rest. This is consistent with the results from
before.
Wagner - 6
A
B
Drug Concentration in EPI for Morphine
0.6
0.5
0.4
Drug Concentration in EPI for Alfentanil
0.4
C1
C2
C3
C4
C1
C2
C3
C4
0.3
0.2
0.3
0.2
0.1
0.1
0
0
-0.1
0
5
10
C
15
20
25
30
35
40
45
0.4
5
10
20
25
30
35
40
45
50
40
45
50
C1
C2
C3
C4
0.25
0.2
0.3
15
Drug Concentration in EPI for Sufentanil
0.3
C1
C2
C3
C4
0.5
0
D
Drug Concentration in EPI for Fentanyl
0.6
-0.1
50
0.15
0.2
0.1
0.1
0.05
0
0
-0.1
-0.05
0
5
10
15
20
25
30
35
40
45
50
0
5
10
15
20
25
30
35
Figure 8: These plots are the change in concentration in moles over time in minutes at the epidural space.
There are only four concentrations shown here opposed to the five concentrations from Figure 6 because
the brain compartment does not transfer fluid directly into the epidural space. A Shows that Morphine lasts
the longest within the tissue. B Shows that Alfentanil is removed slower from the system and starts to level
off. C Fentanyl is removed the fastest form this tissue and D Sufentanyl is starting to level off and soon
decline like with Alfentanil.
Drug Concentration in VASC for Morphine
0.6
Drug Concentration in VASC for Alfentanil
1
C1
C2
C3
C4
A
0.5
0.4
B
C1
C2
C3
C4
0.8
0.6
0.3
0.4
0.2
0.2
0.1
0
0
-0.1
0
5
10
15
20
25
30
35
40
45
-0.2
50
0
5
10
Drug Concentration in VASC for Fentanyl
1.2
C
1
20
25
30
35
40
45
50
40
45
50
Drug Concentration in VASC for Sufentanil
0.4
C1
C2
C3
C4
0.8
15
D
C1
C2
C3
C4
0.3
0.2
0.6
0.4
0.1
0.2
0
0
-0.2
0
5
10
15
20
25
30
35
40
45
50
-0.1
0
5
10
15
20
25
30
35
Figure 9: These plots are the change in concentration in moles over time in minutes at the vasculature
tissue. There are only four cocentrations shown here opposed to the five concentrations from Figure 6
because the brain compartment does not transfer fluid directly into the vasculature. A Shows that Morphine
lasts the longest within the tissue since it is still increasing at a higher level of concentration for each nod. B
Shows that Alfentanil is removed the fastest from the system and starts to level off at a lower concentration
than the other opioids. C Fentanyl is removed faster form this tissue than Sufentanil. D Sufentanyl is still
increasing like Morphine but at a lesser concenttartion.
Wagner - 7
A
Concentration of Morphine at Each Compartment at a Given Time
Concentration of Fentanyl at Each Compartment at a Given Time
B
2
2
1.5
Concentration
Concentration
1.5
1
1
0.5
0.5
0
Brain
Cervical
Thoracic
Compartment
Lumbar
Concentration of Alfentanil at Each Compartment at a Given Time
C
0
Brain
Sacral
Thoracic
Compartment
Lumbar
Sacral
Concentration of Sufentanil at Each Compartment at a Given Time
D
1.4
Cervical
1.5
1.2
1
Concentration
Concentration
1
0.8
0.6
0.5
0.4
0.2
0
Brain
Cervical
Thoracic
Compartment
Lumbar
0
Brain
Sacral
Cervical
Thoracic
Compartment
Lumbar
Sacral
Figure 10: Each plot shows the concentration at a given compartment of the system. Each line corresponds
to a different time in minutes. This mimics the results seen in previous figures but shows the results
according to position within the system instead of the concentration as a function of time. A shows the
concentration of Morphine. B shows the concentration of Fentanyl. C shows the concentration of Alfentanil
and D shows the concentration of Sufentanil.
Models
Each of the three models were plotted
together in order to compare the effects
of different flow equations on the
system. Figure 11 shows each model
plotted together and compared between
the volume, pressure, flow, and
concentration with Morphine. It can be
A
Model Comparision: Volume
120
seen that Model 1 and Model 2 are
almost identical but vary slightly while
Model 3 has the greatest amount of
differences. Model 3 has the flow from
the Bolus injection return to the original
state earlier than the others and with les
amplitude.
B
Model Comparision: Pressure
5000
4000
100
3000
80
2000
60
1000
40
0
20
0
C
-1000
0
5
10
15
20
25
30
35
40
45
-2000
50
D
Model Comparision: Flow
300
200
5
10
15
20
25
30
35
40
45
50
Model Comparision: Morphine Concentration in the Spinal Cord
3
2.5
100
2
0
1.5
-100
1
-200
0.5
-300
-400
0
0
0
5
10
15
20
25
30
35
40
45
50
-0.5
0
5
10
15
20
25
30
35
40
45
50
Figure 11: Each plot compares the three different models that was tested. The blue line is Model 1, the red
line is Model 2, and the black line is Model 3. Model 3 shows the injection having the least effect on the
system and Models 1 and 2 are almost identical in the effect they take on the system. A compares the
difference in volume, B the difference in pressure, C the difference in flow, and D the difference in
concentration that the different representations of flow have.
Wagner - 8
Discussion
The phase shift that is common between
the flow, volume, and pressure occurs
because of the pulsations from the heart
beat. The flow is first seen in the brain
and then proceeds to the lower
compartments of the spinal cord. Since
there is internal resistance to the fluid
and the length affects the velocity of the
fluid there is a delay in the flow seen
throughout the system. Furthermore,
since the flow of CSF affects the volume
which in turn affects the pressure, the
phase shift is also seen in both of these
parameters.
The flow of CSF in the system increases
when the injection occurs and then
begins to return to its original state. The
reason it does not stay at the amplitude
and frequency that is a direct cause of
the injection is because of the type of
injection that is used. Since a Bolus
injection was tested in this model there
is only an inlet of opioid for thirty
seconds. Afterwards there is no more
influx of fluid into the system allowing it
to return to the original steady state over
time.
There are two possible ways that the
CSF could be dispersed within the brain.
Either it can swell the fluid filled sac that
surrounds the brain or it can be
reabsorbed into the surrounding tissue.
According to the model created the extra
CSF fluid in the brain swells the
surrounding fluid sac. This is depicted in
Figure 4. Since there is a large increase
in volume due to the Bolus injection that
takes a noticeable longer time to start to
return to normal than the other spinal
cord compartments, it can be deduced
that this is not because of reabsorption.
As seen in Figure 1, the brain does not
have an absorption term to the epidural
space and vasculature like the spinal
cord compartments do. The result is that
the extra fluid in the brain is stagnate
until it can be redistributed elsewhere.
The resulting concentrations are similar
to what was expected based on the work
of Ummenhofer et al. The only
discrepancy is found with the opioid
with the least amount of concentration.
Ummenhofer et al stated that Fentanyl
would contain the smallest concentration
over time while it was modeled here as
close to the largest, yet still an
intermediate drug1. The opioid with the
lowest concentration was modeled as
Alfentanil at the end of the model. It is
reasonable that Alfentanil has the least
concentration based on its kinetic terms.
Overall Alfentanil’s kinetics are higher
than the other opioids observed. This
had a great impact on the transfer
between the different tissues. Some key
differences between the two results is
that Ummenhofer did not take into
account the heart rate while this model
did not take into account gravity or shear
stress, both of which affect the NavierStokes momentum equation7. This could
be the reason why the two results were
slightly different.
In order to check the accuracy of the
model an integral balance was
completed on the concentration of each
opioid. The balance summed the
concentrations of each opioid in each of
the different tissues and four different
times of the model. The results can be
seen in Figure 12. Except for the first
time each remaining integral balance is
almost identical for each opioid. This
indicates that the model produced is
accurate in regards to the fact that the
mass of each opioid was conserved.
Wagner - 9
Integral Balance of Morphine
A5
4.2519
4.4468
4.4411
4.5196
B
Integral Balance of Alfentanil
4
3.6808
3.6942
3.7465
21.6087
36.5142
3.3131
4
3
3
2
2
1
1
0
7.0844
14.2867
22.2018
0
37.5641
6.7403
13.9317
Time
C5
Time
Integral Balance of Fentanyl
4.8863
4.8804
4.9663
D
Integral Balance of Sufentanil
4
3.5924
4.2928
3.5483
3.6
21.9485
37.232
3.2189
4
3
3
2
2
1
1
0
6.8632
14.0639
21.8123
36.8201
Time
0
6.8665
14.0834
Time
Figure 12: Integral balances for each opioid are shown. A depicts Morphine, B depicts Alfentanil, C
depicts Fentanyl, and D depicts Sufentanil.
Of the models that were tested, the third
model shows how the effects of a Bolus
injection can occur instantaneously and
leave the system in a short amount of
time. In order to determine if this is the
most accurate model that was
constructed either experimental data
would need to be known in order to
make a comparison or the model should
be tested using a continuous injection to
see if there are any major discrepancies
there.
Conclusion
This model can be used in order to assist
surgeons in deciding which opioid to use
as an Anesthetic during any given
procedure. All they would have to do is
decide on a timeframe for the procedure,
what type of injection will be used, and
approximate the frequency of the
heartbeat. With all of these factors taken
into account, they will be able to analyze
the concentration graphs and make an
educated decision on which opioid to
use.
In order to further the accuracy of this
model, the next step would be to analyze
each vertebra instead of the four regions
of vertebra in the spinal cord. This
would allow for a more precise location
of the opioid concentration in the system
as well as allowing for increased
accuracy in knowing the precise
injection site location.
Wagner - 10
Intellectual Property
Biological and physiological data and some modeling procedures provided to you from Dr.
Linninger’s lab are subject to IRB review procedures and Intellectual property procedures.
Therefore, the use of these data and procedures are limited to the coursework only. Publications
need to be approved and require joint authorship with staff of Dr. Linninger’s lab.
Reference
[1] Ummenhofer, Wolfgang C., et al. "Comparative spinal distribution and clearance
kinetics of intrathecally administered morphine, fentanyl, alfentanil, and
sufentanil." Anesthesiology 92.3 (2000): 739-753.
[2] Hsu, Ying, et al. “The frequency and magnitude of cerebrospinal fluid pulsations
influence intrathecal drug distribution: key factors for interpatient variability.”
Anesthesia & Analgesia 115.2 (2012): 386-394
[3] Shafer, Steven L., and John R. Varvel. "Pharmacokinetics, pharmacodynamics, and
rational opioid selection." Anesthesiology 74.1 (1991): 53-63.
[4] Jeffrey J. Iliff, PhD and Richard D. Penn, MD. “Fluid flow in the brain and the
glymphatic system”
[5] Zhou Xuedong, et al. “Response on Harmonic Excitation Analysis.” (2005)
[6] Berkow, Robert. The Merck Manual of Medical Information. New Jersey: Merck,
1997.
[7] Zagzoule, Mokhtar and Jean-Pierre Marc-Verones. “A Global mathematical model of
the cerebral circulation in man.” J. Biomechanics. 19.12 (1986) 1015-1022
Wagner - 11
Appendix1: System Diagram
C = [7.5,
7.5,
7.5,
7.5,
7.5,
Lx = [7.5,
7.5,
7.5,
7.5,
7.5,
Ly = [32,
24,
16,
8,
0,
26;
18;
10;
2
-6];
7.5;
7.5;
7.5;
7.5
7.5];
28;
20;
12;
4
-4];
%labeling
for i = 1:4
c = C(i,1)-1;
cP = C(i,1)-5;
text(c, C(i,2), ['V',
num2str(i+1)])
cLx = C(i,1)+3;
cLy = C(i,2)+4;
text(cLx, C(i,2)+9, ['P',
num2str(i)])
text(cLx, cLy, ['F',
num2str(i)])
if i == 4
cLx = C(i+1,1)+3;
cLy = C(i+1,2)+8;
text(cLx, cLy+1, ['P',
num2str(i+1)])
end
end
for i = 1:4
line(Lx(i,:), Ly(i,:),
'Color', 'k') %flow line
downward
end
%labeling
text(6, 35, 'Brain')
text(0, C(1,2), 'Cervical')
text(0, C(2,2), 'Thoracic')
text(0, C(3,2), 'Lumbar')
text(0, C(4,2), 'Sacral')
text(6, 33, 'V1')
axis([0 15 0 37])
title('Spinal Cord Model')
Rx = [5; 10; 10; 5; 5];
Ry = [32; 32; 37; 37; 32];
line(Rx, Ry, 'Color', 'k')
%brain rectangle
RLx = [9.5; 15];
RLy = [26; 26; 18; 18; 10; 10;
2; 2];
for i = 2:2:8
line(RLx, RLy(i-1:i), 'Color',
'k') %line to epidural space
end
Bx = [15; 20; 20; 15; 15];
By = [0; 0; 28; 28; 0];
line(Bx, By, 'Color', 'k')
%epidural space box
h = text(17.5,12, 'Epidural
Space');
set(h, 'rotation', 90)
line([20; 25], [18; 18],
'Color', 'k') %line to
vasculature
Bx = [25; 30; 30; 25; 25];
By = [0; 0; 28; 28; 0];
line(Bx, By, 'Color', 'k')
%vasculature box
h = text(27.5,12,'Vasculature');
set(h, 'rotation', 90)
C = [7.5, 26;
7.5, 18;
7.5, 10;
7.5, 2
7.5, -6];
r = [2; 2; 2; 2];
viscircles(C(1:4, :), r,
'Edgecolor', 'k', 'LineWidth',
1) %compartments
axis equal
Wagner - 12
Appendix 2: Deformation
%% normal/ ideal
for i = 1:2:3
subplot(1,3,i)
C = [7.5, 26;
7.5, 18;
7.5, 10;
7.5, 2
7.5, -6];
Lx = [7.5,
7.5,
7.5,
7.5,
7.5,
Ly = [32,
24,
16,
8,
0,
7.5;
7.5;
7.5;
7.5
7.5];
28;
20;
12;
4
-4];
%labeling
for i = 1:4
c = C(i,1)-1;
cP = C(i,1)-5;
text(c, C(i,2), ['V',
num2str(i+1)])
cLx = C(i,1)+3;
cLy = C(i,2)+4;
text(cLx, C(i,2)+8, ['P',
num2str(i)])
text(cLx, cLy, ['F',
num2str(i)])
if i == 4
cLx = C(i+1,1)+3;
cLy = C(i+1,2)+8;
text(cLx, cLy, ['P',
num2str(i+1)])
end
end
%flow lines
for i = 1:4
line(Lx(i,:), Ly(i,:),
'Color', 'k')
end
%labeling
text(6, 35, 'Brain')
text(0, C(1,2), 'Cervical')
text(0, C(2,2), 'Thoracic')
text(0, C(3,2), 'Lumbar')
text(0, C(4,2), 'Sacral')
text(6, 33, 'V1')
axis([0 15 0 37])
title('Spinal Cord Model')
%brain rectangle
Rx = [5; 10; 10; 5; 5];
Ry = [32; 32; 37; 37; 32];
line(Rx, Ry, 'Color', 'k')
C = [7.5, 26;
7.5, 18;
7.5, 10;
7.5, 2
7.5, -6];
r = [2; 2; 2; 2];
%compartments
viscircles(C(1:4, :), r,
'Edgecolor', 'k', 'LineWidth',
1)
axis equal
end
%% deformation
subplot(1,3,2)
C = [7.5, 26;
7.5, 18;
7.5, 10;
7.5, 2
7.5, -6];
Lx = [7.5,
7.5,
7.5,
7.5,
7.5,
7.5;
7.5;
7.5;
7.5
7.5];
Ly = [32, 29.5;
22.5, 21;
15, 12.5;
7.5, 4
0, -4];
%labeling
for i = 1:4
c = C(i,1)-1;
cP = C(i,1)-5;
text(c+0.5, C(i,2), ['V',
num2str(i+1)])
cLx = C(i,1)+3;
cLy = C(i,2)+4;
Wagner - 13
text(cLx+1, C(i,2)+8, ['P',
num2str(i)])
text(cLx+1, cLy, ['F',
num2str(i)])
if i == 4
cLx = C(i+1,1)+4;
cLy = C(i+1,2)+8;
text(cLx, cLy, ['P',
num2str(i+1)])
end
end
for i = 1:4
line(Lx(i,:), Ly(i,:),
'Color', 'k') %flow lines
end
%labeling
text(6.5, 35, 'Brain')
text(0, C(1,2), 'Cervical')
text(0, C(2,2), 'Thoracic')
text(0, C(3,2), 'Lumbar')
text(0, C(4,2), 'Sacral')
text(7, 33, 'V1')
axis([0 15 0 37])
title('Spinal Cord Model with
Deformation')
Rx = [5; 10; 10; 5; 5];
Ry = [32; 32; 37; 37; 32];
line(Rx, Ry, 'Color', 'k')
%brain rectangle
C = [7.5,
7.5,
7.5,
7.5,
7.5,
r = [3.5;
26;
18;
10;
2
-6];
3; 2.5; 2];
%compartments
viscircles(C(1:4, :), r,
'Edgecolor', 'k', 'LineWidth',
1)
Wagner - 14
Appendix 3: Brain ODE
function FP_Main
clear all; close all; clc;
y0 = zeros(30,1);
y0(1:5)=[55;23;33;25;19];
global j
for j = 1:4
[T,Y] = ode45(@brainwaiver
save('model1','Y','T');
,[0 50],y0);
V = Y(:,1:5);
P = .0075*Y(:,6:9);
P0 =V(:,1)/.188;
F = Y(:,10:13);
if j ==1
TM=T;
CM = Y(:,14:18);
CMsc = Y(:,19:22);
CMepi = Y(:,23:26);
CMvas = Y(:,27:30);
end
if j ==2
TA=T;
CA = Y(:,14:18);
CAsc = Y(:,19:22);
CAepi = Y(:,23:26);
CAvas = Y(:,27:30);
end
if j ==3
TF=T;
CF = Y(:,14:18);
CFsc = Y(:,19:22);
CFepi = Y(:,23:26);
CFvas = Y(:,27:30);
end
if j ==4
TS=T;
CS = Y(:,14:18);
CSsc = Y(:,19:22);
CSepi = Y(:,23:26);
CSvas = Y(:,27:30);
end
end
PLOTME(TM,TA,TF,TS, V, P, P0, F, CM, CMsc, CMepi, CMvas, CA, CAsc,
CAepi, CAvas, CF, CFsc, CFepi, CFvas, CS, CSsc, CSepi, CSvas)
Wagner - 15
function dP = brainwaiver(t,Y)
kappa = [.188;.0210;.0174;.0126;.0139]; K = 1./kappa;
E=1;
alfa = [.35879;.49138;.7228;.783081];
Fprod = 0;
reab=6.4e-4;
timei=5;
Pven=0;
A=2;
w=1*pi;
rho = 1.0068;
% Finj = 1;
if (t >=timei && t <=(timei+0.05))
Finj = 1000; %.05sec*1000=50mL
Co=1;
else
Finj = 0;
Co=0;
end
V
P
U
F
=
=
=
=
Y(1:5); CA= .75;
Y(6:9);
Y(10:13);
CA*U;
% Volume
dP(1,1) =
Fprod-F(1)-max(0,((P(1)-Pven)*reab))+A*cos(w*t)*w;
dP(2,1) =
(F(1) - F(2));
dP(3,1) =
(F(2) - F(3));
dP(4,1) =
(F(3) - F(4))+ Finj;
dP(5,1) =
(F(4));
P0 = V(1)*K(1);
% Pressure
dP(6,1) = K(2)*dP(2,1);
dP(7,1) = K(3)*dP(3,1);
dP(8,1) = K(4)*dP(4,1);
dP(9,1) = K(5)*dP(5,1);
%
%
%
%
P2
P3
P4
P5
%Flow Model 1:
dP(10,1) = (P0-P(1))-(U(1)*alfa(1));
dP(11,1) = (P(1)-P(2))-(U(2)*alfa(2));
dP(12,1) = (P(2)-P(3))-(U(3)*alfa(3));
dP(13,1) = (P(3)-P(4))-(U(4)*alfa(4));
%
%
%
%
%
%Flow Model 2:
dP(10,1) = (P0/rho)-((U(2))+P(1)/rho)-(U(1)*alfa(1));
dP(11,1) = ((U(1))+P(1)/rho)-((U(3))+P(2)/rho)-(U(2)*alfa(2));
dP(12,1) = ((U(2))+P(2)/rho)-((U(4))+P(3)/rho)-(U(3)*alfa(3));
dP(13,1) = ((U(3))+P(3)/rho)-(P(4)/rho)-(U(4)*alfa(4));
% % Flow Model 3:
% dP(10,1) = (P0/rho)-((U(1))+P(1)/rho)-(U(1)*alfa(1));
% dP(11,1) = ((U(1))+P(1)/rho)-((U(2))+P(2)/rho)-(U(2)*alfa(2));
Wagner - 16
% dP(12,1) = ((U(2))+P(2)/rho)-((U(3))+P(3)/rho)-(U(3)*alfa(3));
% dP(13,1) = ((U(3))+P(3)/rho)-((U(4))+P(4)/rho)-(U(4)*alfa(4));
C = Y(14:18);
concSCstart=19;%where spinal cord tissue starts
concSCend=22; %where spinal cord tissue ends
Csc = Y(concSCstart:concSCend);
concEPIstart = 23; %where epidural space starts
concEPIend = 26; %where epidural space end
Cepi = Y(concEPIstart:concEPIend);
concVASCstart = 27; %where vasculature starts
concVASCend = 30; %where vasculature ends
Cvas = Y(concVASCstart:concVASCend);
KIC = [0.037 0.170 0.0339 .020]; %%CSF space to SC [MAFS]
KCI = [0.0143 0.0236 0.0159 0.0095]; %%SC to CSF space
KIE = [0.0542 0.1372 0.1078 0.0291]; %%CSF space to Epidural space
KEI = [0.0021 0.0063 0.0285 0.0137]; %%Epidural space to SCF
KPLC = [0.0082 0.868 0.0080 0.0131]; %%SC to Vasclature
KPLEPI = [0.0199 0.0201 0.1088 0.0323]; %Umenhofer
% Concentration
global j
dP(14,1) = (-max(0,C(1)*F(1))+max(0,-F(1)*C(2)))/V(1);%concetration
leaving the brain
dP(15,1) = (-max(0,C(2)*F(2))+max(0,-F(2)*C(3))+max(0,C(1)*F(1))...
-max(0,-F(1)*C(2)))/V(2)-KIC(j)*C(2)+KCI(j)*Csc(1)KIE(j)*C(2)+KEI(j)*Cepi(1);
dP(16,1) = (-max(0,C(3)*F(3))+max(0,-F(3)*C(4))+max(0,C(2)*F(2))...
-max(0,-F(2)*C(3)))/V(3)-KIC(j)*C(3)+KCI(j)*Csc(2)KIE(j)*C(3)+KEI(j)*Cepi(3);
dP(17,1) = (-max(0,C(4)*F(4))+max(0,-F(4)*C(5))+max(0,C(3)*F(3))...
-max(0,-F(3)*C(4))+Finj*Co)/V(4)-KIC(j)*C(4)+KCI(j)*Csc(3)KIE(j)*C(4)+KEI(j)*Cepi(3);
dP(18,1) = (max(0,C(4)*F(4))-max(0,-F(4)*C(5)))/V(5)KIC(j)*C(5)+KCI(j)*Csc(4)...
-KIE(j)*C(5)+KEI(j)*Cepi(4);
for i = concSCstart:concSCend
dP(i,1) = KIC(j)*C(i-17)-KCI(j)*Csc(i-18)-KPLC(j)*Csc(i-18);
end
for i = concEPIstart:concEPIend
dP(i,1) = KIE(j)*C(i-concSCend+1)-KEI(j)*Cepi(i-concSCend)KPLEPI(j)*Cepi(i-concSCend);
end
for i = concVASCstart:concVASCend
dP(i,1) = KPLC(j)*Csc(i-concEPIend)+KPLEPI(j)*Cepi(i-concEPIend);
end
Wagner - 17
function PLOTME(TM, TA,TF,TS, V, P, P0, F, CM, CMsc, CMepi, CMvas, CA,
CAsc, CAepi, CAvas, CF, CFsc, CFepi, CFvas, CS, CSsc, CSepi, CSvas)
%integral balances
for k = 500:500:2000
S_M(k/500) =
sum(CM(k,:))+sum(CMsc(k,:))+sum(CMepi(k,:))+sum(CMvas(k,:));
S_A(k/500) =
sum(CA(k,:))+sum(CAsc(k,:))+sum(CAepi(k,:))+sum(CAvas(k,:));
S_F(k/500) =
sum(CF(k,:))+sum(CFsc(k,:))+sum(CFepi(k,:))+sum(CFvas(k,:));
S_S(k/500) =
sum(CS(k,:))+sum(CSsc(k,:))+sum(CSepi(k,:))+sum(CSvas(k,:));
end
figure
subplot(2,2,1)
bar(S_M)
title('Integral Balance of Morphine')
set(gca, 'XTickLabel', {num2str(TM(500)), num2str(TM(1000)),
num2str(TM(1500)), num2str(TM(2000))})
xlabel('Time')
for i = 1:4
text(i,S_M(i)+.35, num2str(S_M(i)),
'HorizontalAlignment','center','VerticalAlignment','top')
end
subplot(2,2,2)
bar(S_A)
title('Integral Balance of Alfentanil')
set(gca, 'XTickLabel', {num2str(TA(500)), num2str(TA(1000)),
num2str(TA(1500)), num2str(TA(2000))})
xlabel('Time')
for i = 1:4
text(i,S_A(i)+.25, num2str(S_A(i)),
'HorizontalAlignment','center','VerticalAlignment','top')
end
subplot(2,2,3)
bar(S_F)
title('Integral Balance of Fentanyl')
set(gca, 'XTickLabel', {num2str(TF(500)), num2str(TF(1000)),
num2str(TF(1500)), num2str(TF(2000))})
xlabel('Time')
for i = 1:4
text(i,S_F(i)+.35, num2str(S_F(i)),
'HorizontalAlignment','center','VerticalAlignment','top')
end
subplot(2,2,4)
bar(S_S)
title('Integral Balance of Sufentanil')
Wagner - 18
set(gca, 'XTickLabel', {num2str(TS(500)), num2str(TS(1000)),
num2str(TS(1500)), num2str(TS(2000))})
xlabel('Time')
for i = 1:4
text(i,S_S(i)+.25, num2str(S_S(i)),
'HorizontalAlignment','center','VerticalAlignment','top')
end
%Concentration at nodes
node = 1:5;
t = [500; 1000; 1500; 2000];%, floor(length(TS)/4),
floor(length(TS)/2), floor(length(TS))];
figure
subplot(2,2,1)
plot(node, CM(t(1),:), node, CM(t(2),:), node, CM(t(3),:), node,
CM(t(4),:))
title('Concentration of Morphine at Each Compartment at a Given Time')
set(gca, 'XTickLabel', {'Brain',' ', 'Cervical', ' ', 'Thoracic', ' ',
'Lumbar', ' ', 'Sacral'})
xlabel('Compartment')
ylabel('Concentration')
subplot(2,2,2)
plot(node, CF(t(1),:), node, CF(t(2),:), node, CF(t(3),:), node,
CF(t(4),:))
title('Concentration of Fentanyl at Each Compartment at a Given Time')
set(gca, 'XTickLabel', {'Brain',' ', 'Cervical', ' ', 'Thoracic', ' ',
'Lumbar', ' ', 'Sacral'})
xlabel('Compartment')
ylabel('Concentration')
subplot(2,2,3)
plot(node, CA(t(1),:), node, CA(t(2),:), node, CA(t(3),:), node,
CA(t(4),:))
title('Concentration of Alfentanil at Each Compartment at a Given
Time')
set(gca, 'XTickLabel', {'Brain',' ', 'Cervical', ' ', 'Thoracic', ' ',
'Lumbar', ' ', 'Sacral'})
xlabel('Compartment')
ylabel('Concentration')
subplot(2,2,4)
plot(node, CS(t(1),:), node, CS(t(2),:), node, CS(t(3),:), node,
CS(t(4),:))
title('Concentration of Sufentanil at Each Compartment at a Given
Time')
set(gca, 'XTickLabel', {'Brain',' ', 'Cervical', ' ', 'Thoracic', ' ',
'Lumbar', ' ', 'Sacral'})
xlabel('Compartment')
ylabel('Concentration')
%concentration
figure; subplot(2,2,1)
plot(TM,CM); title('Drug Concentration in CSF space for Morphine');
legend('C1' ,'C2', 'C3', 'C4', 'C5');
Wagner - 19
subplot(2,2,2)
plot(TA,CA); title('Drug
legend('C1' ,'C2', 'C3',
subplot(2,2,3)
plot(TF,CF); title('Drug
legend('C1' ,'C2', 'C3',
subplot(2,2,4)
plot(TS,CS); title('Drug
legend('C1' ,'C2', 'C3',
Concentration in CSF space for Alfentanil');
'C4', 'C5');
Concentration in CSF space for Fentanyl');
'C4', 'C5');
Concentration in CSF space for Sufentanil');
'C4', 'C5');
figure; subplot(2,2,1)
plot(TM,CMsc); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,2)
plot(TA,CAsc); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,3)
plot(TF,CFsc); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSsc); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
figure; subplot(2,2,1)
plot(TM,CMepi); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,2)
plot(TA,CAepi); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,3)
plot(TF,CFepi); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSepi); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
figure;subplot(2,2,1)
plot(TM,CMvas); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,2)
plot(TA,CAvas); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,3)
plot(TF,CFvas); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSvas); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
in SC for Morphine');
in SC for Alfentanil');
in SC for Fentanyl');
in SC for Sufentanil');
in EPI for Morphine');
in EPI for Alfentanil');
in EPI for Fentanyl');
in EPI for Sufentanil');
in VASC for Morphine');
in VASC for Alfentanil');
in VASC for Fentanyl');
in VASC for Sufentanil');
figure; plot(TS,V) ;title('Volume'); legend('V1', 'V2', 'V3', 'V4',
'V5');
figure; plot(TS,[P0 P]) ;title('Pressure'); legend('P1', 'P2', 'P3',
'P4', 'P5');
figure; plot(TS,F) ;title('flow'); legend('f1', 'f2', 'f3', 'f4');
Wagner - 20