Download Intrathecal Drug Targeting: Distribution of Opioids in the Spinal Canal

Intrathecal Drug Targeting: Distribution of Opioids in the Spinal Canal
Nirav Soni
[email protected]
This report is produced under the supervision of BIOE 310 instructor Professor Linninger
Abstract
Intrathecal opioid application is a method used to treat many diseases, such as chronic pain and spinal cancer1,2,3.
However, the mechanisms of biodistribution for drugs administered in the cerebrospinal fluid (CSF) are not well
understood in academia. This project aims at creating a mechanistic model through three different models of flow to
prove the spread of the drugs in the subarachnoid space is patient specific and depends on the pharmacokinetic
parameters of each opioid4,5. A variety of conservation balance equations will be used in order to describe the
direction of incoming and outgoing flow throughout this network, the pressure drop throughout different spinal
compartments, the change in volume throughout the network, and the concentration of four different opioids at
different positions throughout the intrathecal space. The mechanistic model of the spinal canal will be imaged in order
to guide the mathematical studies of the effects of pulsatile CSF flow on each opioid. The models formed through
MATLAB compute spinal distribution of the opioids, absorption of the drugs into the spinal cord, epidural tissue, and
vasculature, blood clearance for different injection locations, and number and duration of the injections. In order to
illustrate intrathecal drug targeting, plots of flow, volume, pressure, and concentration were created with respect to
a bolus injection for each opioid. When studying the figures produced through each of the three flow models, the
results support the hypothesis by showing the differences in pressure, volume, flow, and concentration by bolus
injections for the different opioids selected.
1. Introduction
Intrathecal drug delivery is a clinical treatment option
for many diseases, including spinal cancer and
chronic pain1,2,3. This process consists of direct drug
injection into the spinal canal where the flow of CSF
allows for a natural drug transport2,3. Because the
medication is delivered directly to the spinal cord, it
can be more effective than taking oral medications,
which must travel through other systems before
reaching the spine.
Recently, researchers at the University of Illinois at
Chicago have discovered the important role of CSF
amplitude and frequency for the rapid dispersion after
intrathecal administration6. The extent of drug
distribution in vivo is highly variable and difficult to
control. Varying CSF pulsatility and heart rate from
patient to patient may lead to different drug
distribution6.
Computations in contemporary
experiments demonstrate that the speed of drug
transport is strongly affected by the frequency and
magnitude of CSF pulsations.
Some opioids used in intrathecal delivery include
morphine, fentanyl, alfentanil, and sufentanil4.
Current studies have sampled the spinal cord, CSF
and epidural space after intrathecal injections of these
drugs in order to characterize the rate and extent of
opioid distribution4. This information can be used to
demonstrate different pharmacokinetic behavior,
which correlates well with their pharmacodynamic
behavior4,5.
When rationalizing choice of drug, computer
simulations in recent studies provide insight into the
comparative pharmacokinetics that can be used to
select the appropriate opioid based on the length of
the procedure, the desired intraoperative opioid
concentration, and the desired time course of
recovery5. Opioid selection requires
acknowledgement of the relationship between the
pharmacokinetic and pharmacodynamics
characteristics of these drugs and the onset of and
recovery from drug effect5.
In order to create a mathematical model, which
represents the illustration below (Figure 1), to predict
the natural intrathecal transport of the opioids,
multiple variables must be accounted for. When
considering spinal distribution of the drug, absorption
rate of the drug into the tissue, blood clearance for
different injection locations, number and duration of
the injections, and choice of drug must be studied.
Figure 1:Intrathecal Injection Illustration. Opioid injection
made in the lumbar region, L2, of the spinal cord 6.
2. Methods
𝐾
Conservation balance equations are essential when
describing the physical world. In order to numerically
illustrate the flow network shown in Figure 2,
algebraic and differential systems of equations were
formulated using the following mathematical
equations:
Three different models of flow were used to solve the
small flow network. The flow network follows the
relationship given by Equations 1, 2, and 3. In order
to establish a standard, the arrival of flow into a node
was noted with positive pressures, whereas the
departure of flow was denoted with negative
pressures.
𝑑𝑈𝑖
𝑑𝑡
𝑑𝑈𝑖
𝑑𝑡
𝑑𝑈𝑖
𝑑𝑡
=[
2
𝑈𝑖−1
=[
2
= 𝑃𝑖 − 𝑃𝑖+1 + 𝛼𝑖 𝑈𝑖
𝑃
+ 𝑖] − [
𝜌
2
𝑈𝑖−1
2
2
𝑈𝑖+1
2
+
𝑈2
𝑃
+ 𝜌𝑖] − [ 2𝑖 +
𝑃𝑖+1
]
𝜌
+ 𝛼𝑖 𝑈𝑖
𝑃𝑖+1
]+
𝜌
= 𝐹𝑖 −𝐹𝑖+1
𝛼𝑖 𝑈𝑖
(2)
(3)
(4)
Compressibility equations, given by Equation 5, were
formulated in order to relate relative change in
volume of the CSF as a response to the change in
pressure. This will help describe the deformation in
the flow network.
𝑑𝑉𝑖
𝑑𝑡
= 𝐾
𝑑𝑃𝑖
𝑑𝑡
(5)
Species balance equations, given by Equation 6, were
used in order to map the concentration of each opioid
with respect to time at the brain, cervical, thoracic,
lumbar, and sacral regions.
𝑉
𝑑𝐶𝑖
𝑑𝑡
=
1
𝛼𝑖−1 (𝑃𝑖−1 −𝑃𝑖 )
−
1
𝛼𝑖 (𝑃𝑖 −𝑃𝑖+1 )
(7)
Using the above information and the attached
equations, pressure-driven flow, pressures and
volumes were solved for. The initial excitation was
represented by a sinusoidal equation so that pressures,
volumes and flows acted physiologically. After the
pressures, volumes and flows were found to function
correctly, drug concentrations were added. Two new
variables were introduced: the concentration of the
injection and the flow of the injection. Then, the mass
transfer was computed using the attached equation.
Finally, reactions were included in the mass transfer
equations. This is the kinetic reaction rate multiplied
by the concentration of the compartment. The values
for the kinetic rates of each drug are shown in Table 1.
(1)
Flow conservation equations, given by Equation 4,
were used in order to describe the direction of
incoming and outgoing flow throughout the
intrathecal space. The incoming flow was denoted
with a positive value whereas the outgoing flow was
given a negative value.
𝑑𝑉𝑖
𝑑𝑡
𝑑𝑃𝑖
𝑑𝑡
= 𝐹𝑖 𝐶𝑖 − 𝐹𝑖+1 𝐶𝑖+1 − 𝑘𝐶𝑖
(6)
In order to simplify computations within Matlab
Code 2 for the purpose of creating a mechanistic
model of the flow network, Equation 7 was derived
using a combination of the previous equations.
Parameter
k ic (min−1 )
k ci (min−1 )
k plc (min−1 )
k ie (min−1 )
k ei (min−1 )
k plepi (min−1 )
M
.037
.0143
.0082
.0542
.0021
.0199
A
.170
.0236
.868
.1078
.0063
.0201
F
.0339
.0159
.008
.1372
.0285
.1088
S
.020
.0095
.0131
.0291
.0137
.0323
Table 1: Values of Pharmacokinetic Parameters of
Intrathecal Opioids (Morphine, Fentanyl, Alfentanil, and
Sufentanil)5.
In order to visualize the intrathecal space through
MATLAB, a flow network was created as a
descriptive model shown in Figure 2. This was done
through Matlab Code 1. The figure is split into 5
compartments/nodes that represent the brain,
cervical, thoracic, lumbar, and sacral regions. They
are connected through 4 faces. Each face has its own
flow and each node contains a pressure, volume, and
concentration value. The model takes into account
the spread of opioids into the CSF, spinal cord,
epidural tissue, and vasculature. Figure 2 was created
through labeling and drawing shapes though built in
MATLAB commands: viscircles, line, and text.
mass transfer equations through the use of kinetic
rates, k, of each drug. In order to compute the results
for both a bolus injection, an if-else loops was utilized
in order to determine a value for injection flow and
injection concentration in the lumbar region based on
the time of injection.
Spinal Cord Model
Brain P1
V1
35
F1
Cervical
25
V2
P2
Thoracic V3
15
P3
F3
Lumbar
10
V4
3. Results
P4
F4
5
V5
Sacral
0
Epidural Space
F2
20
Vasculature
30
-5
0
5
P5
10
15
20
25
30
35
40
Figure 2: Intrathecal Flow Network: Flows (F) 1-4
represented at the faces of the flow network. Pressures (P)
and Volumes (V) 1-5 represented at each node of the
network. The concentration of the opioid spreads to the
epidural tissue and to vasculature
This figure was essential when determining the
equations for the different models of flow, volume,
pressure, and concentration.
In order to solve for pressure-driven flow, volume,
pressure, and concentration, Matlab Code 2 was
created. This was done through the creation of
functions. First off, a matrix y0 was created containing
the initial values for: volume, pressure, morphine
concentration, fentanyl concentration, afentanil
concentration,
and
sufentanil
concentration.
Compression constant and resistance and kinetic
values for were determined through literature5. Initial
pressure was realized to be a sinusoidal function due
to its relationship to the pulsatile flow of CSF in the
spinal canal. A sinusoidal function was used in order
to create a physiologically accurate model. Pressuredriven flows were solved for by the use of three
different flow models represented in Equation 1,
Equation 2, and Equation 3 at each face for its
corresponding flow. Flows were plotted in a bolus
injection. Volume was solved for using Equation 4 for
each corresponding node. Volume was plotted in a
bolus injection. Pressure was solved for through the
use of Equation 5 and the realization that pressure is
the derivative of volume. Pressure was plotted in a
bolus injection. After flow, volume, and pressure was
found, the next step was to solve for drug
concentration. Two new variables were introduced:
the concentration of the injection and the flow of the
injection. The flow of the injection was inserted into
the volume equations. Concentrations were solved for
using the built-in MATLAB function max. Max was
used in order to determine which way flow was going
at a given point. Reactions were then included in the
Plots of flows, volumes, pressures, and
concentrations were produced through the use of
Matlab Code 2. The graphs produced take into
account all four opioids (Morphine, Fentanyl,
Afentanil, and Sufentanil) introduced through a bolus
injection.
3.1 Model 1
Figure 3 illustrates Model 1 flows through each phase
of the flow network for a 50 mL bolus injection of
opioid occurring at 5 seconds. The highest amplitude
of pulsatile motion is seen to occur at the injection
site between the Lumbar and Sacral region (L2).
Figure 3: Flows 1-4 through each face of the flow network
after a bolus injection with respect to time.
Figure 4 shows the volume at each compartment of
the flow network throughout a range of time for a
bolus injection. The volume at the brain (55 mL) is
much larger than any other region. It can be observed
that the volume with respect to time is oscillatory due
to the biological properties of the flow of CSF in the
intrathecal space.
Figure 6: Concentration of each opioid at each spinal
compartment in the CSF space with respect to time.
Figure 4: Volume levels in the brain (55 mL) and spinal
compartments (23 mL, 33 mL, 25 mL, 19 mL) in order of
Cervical, Thoracic, Lumbar, and Sacral.
Figure 5 illustrates the pressures at each compartment
of the intrathecal flow model throughout a range of
time for a bolus injection. It can be seen that the
pressure increases due to the injection at the Lumbar
region (L2).
Figure 7 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the spinal cord
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug throughout time is approaching an equilibrium
concentration.
Figure 7: Concentration of each opioid in different regions
of the spinal cord with respect to time.
Figure 5: Pressure in the brain and each spinal
compartment after a bolus injection with respect to time.
Figure 6 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the CSF space
at each compartment throughout a range of time after
a bolus injection. The injection starts at 5 seconds. It
is seen that at the injection time all of the opioid drug
concentrations are highest in the Lumbar region (L2)
and that they are distributed throughout the
intrathecal space.
Figure 8 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the epidural
tissue throughout a range of time after a bolus
injection. The injection starts at 5 seconds. It is seen
that each drug throughout time is approaching an
equilibrium concentration.
Figure 8: Concentration of each opioid in different regions
of the epidural tissue with respect to time.
Figure 9 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in vasculature
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug within the range of time is increasing and will
approach an equilibrium concentration.
Figure 9: Concentration of each opioid in different regions
of vasculature with respect to time.
3.2 Model 2
Figure 10 illustrates Model 2 flows through each
phase of the flow network for a 50 mL bolus
injection of opioid occurring at 5 seconds. The
highest amplitude of pulsatile motion is seen to occur
at the injection site between the Lumbar and Sacral
region (L2).
Figure 11: Volume levels in the brain (55 mL) and spinal
compartments (23 mL, 33 mL, 25 mL, 19 mL) in order of
Cervical, Thoracic, Lumbar, and Sacral.
Figure 12 illustrates the pressures at each
compartment of the intrathecal flow model
throughout a range of time for a bolus injection. It
can be seen that the pressure increases due to the
injection at the Lumbar region (L2).
Figure 10: Flows 1-4 through each face of the flow network
after a bolus injection with respect to time.
Figure 11 shows the volume at each compartment of
the flow network throughout a range of time for a
bolus injection. The volume at the brain (55 mL) is
much larger than any other region. It can be observed
that the volume with respect to time is oscillatory due
to the biological properties of the flow of CSF in the
intrathecal space.
Figure 12: Pressure in the brain and each spinal
compartment after a bolus injection with respect to time.
Figure 13 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the CSF space
at each compartment throughout a range of time after
a bolus injection. The injection starts at 5 seconds. It
is seen that at the injection time all of the opioid drug
concentrations are highest in the Lumbar region (L2)
and that they are distributed throughout the
intrathecal space.
Figure 13: Concentration of each opioid at each spinal
compartment in the CSF space with respect to time.
Figure 14 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the spinal cord
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug throughout time is approaching an equilibrium
concentration.
Figure 16: Concentration of each opioid in different
regions of vasculature with respect to time.
3.3 Model 3
Figure 17 illustrates Model 3 flows through each
phase of the flow network for a 50 mL bolus
injection of opioid occurring at 5 seconds. The
highest amplitude of pulsatile motion is seen to occur
at the injection site between the Lumbar and Sacral
region (L2).
Figure 14: Concentration of each opioid in different
regions of the spinal cord with respect to time.
Figure 15 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the epidural
tissue throughout a range of time after a bolus
injection. The injection starts at 5 seconds. It is seen
that each drug throughout time is approaching an
equilibrium concentration.
Figure 17: Flows 1-4 through each face of the flow network
after a bolus injection with respect to time.
Figure 18 shows the volume at each compartment of
the flow network throughout a range of time for a
bolus injection. The volume at the brain (55 mL) is
much larger than any other region. It can be observed
that the volume with respect to time is oscillatory due
to the biological properties of the flow of CSF in the
intrathecal space.
Figure 15: Concentration of each opioid in different
regions of the epidural tissue with respect to time.
Figure 16 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in vasculature
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug within the range of time is increasing and will
approach an equilibrium concentration.
Figure 20: Concentration of each opioid at each spinal
compartment in the CSF space with respect to time.
Figure 18: Volume levels in the brain (55 mL) and spinal
compartments (23 mL, 33 mL, 25 mL, 19 mL) in order of
Cervical, Thoracic, Lumbar, and Sacral.
Figure 19 illustrates the pressures at each
compartment of the intrathecal flow model
throughout a range of time for a bolus injection. It
can be seen that the pressure increases due to the
injection at the Lumbar region (L2).
Figure 21 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the spinal cord
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug throughout time is approaching an equilibrium
concentration.
Figure 21: Concentration of each opioid in different
regions of the spinal cord with respect to time.
Figure 19: Pressure in the brain and each spinal
compartment after a bolus injection with respect to time.
Figure 20 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the CSF space
at each compartment throughout a range of time after
a bolus injection. The injection starts at 5 seconds. It
is seen that at the injection time all of the opioid drug
concentrations are highest in the Lumbar region (L2)
and that they are distributed throughout the
intrathecal space.
Figure 22 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in the epidural
tissue throughout a range of time after a bolus
injection. The injection starts at 5 seconds. It is seen
that each drug throughout time is approaching an
equilibrium concentration.
Figure 22: Concentration of each opioid in different
regions of the epidural tissue with respect to time.
Figure 23 represents the concentration of Morphine
(top left), Alfentanil (top right), Fentanyl (bottom
left), and Sufentanil (bottom right) in vasculature
throughout a range of time after a bolus injection.
The injection starts at 5 seconds. It is seen that each
drug within the range of time is increasing and will
approach an equilibrium concentration.
It is seen that the flow at each face from the brain to
the sacral vertebrate decreases significantly
throughout. This information is essential when solving
for pressure and volume within each region of the
intrathecal flow network.
Figure 23: Concentration of each opioid in different
regions of vasculature with respect to time.
3.4 Model Comparison
As shown in the Figure 24, the three different models
of flow are plotted on top of one another in different
colors. Model 1 is blue, Model 2 is red, and Model 3
is black. This is to demonstrate the similarities and
differences between the models. It is seen that
although they follow a similar trend, there are
noticeable differences between each model.
Figure 10: Comparison between the three models of flow
(Model 1= blue, Model 2=red, Model 3= black) for a bolus
injection of Morphine with respect to time.
4. Discussion
In the mechanistic models of the intrathecal flow
network, the effects of each opioid injection within the
system were projected with respect to concentration
with reactions and mass transfer. Initially at the
injection time, the highest concentration occurred in
the in the lumbar vertebrae, due to the injection site
being located at L2. A phase lag was witnessed
between the flows at each phase and the volume,
pressure, and concentration at each compartment. The
compartments with a significantly smaller flow
experienced a slower distribution of CSF volume
which resulted in a lower opioid concentration
throughout a range of time. The compartments with a
significantly higher flow experienced a faster
distribution of CSF volume which resulted in a higher
opioid concentration throughout a range of time. The
brain received a fair amount of the selected opioid due
to the higher level of pulsatile flow in that regions
when compared to the vertebrate.
As the model shows, the pressure occurred to be
highest in the brain and lowest in the sacral region for
a bolus injection. This can be confirmed
mathematically through the use of the three flow
models (Equation 1, 2, and 3). Due to the flow being
highest near the brain, the pressure will be the highest
in the brain. The pressure will be the lowest at the
sacral vertebrate due to the flow being lowest in the
sacral region.
Understandably, the volume of CSF in the brain is the
highest in order for the drug to disperse adequately. It
is given in literature that the initial volume of CSF in
the brain is measured to be about 55 mL while the
individual vertebra of the human spinal cord ranged
from 19 mL to 33 mL7. The figures also show that both
volume and pressure at each compartment in the flow
network have an oscillatory behavior due to the
pulsatile flow of CSF.
For each graph of flow, volume and pressure within
each of the three different flow models, it is evident
that the initial sinusoidal pressure affects each flow,
volume and pressure in every compartment due to
their corresponding sinusoidal qualities. This is
clearly illustrated through the different degree of
oscillations at each node. It is also seen that the
concentrations of the drugs oscillate throughout the
network due to them being affected by the initial
sinusoidal pressure caused by the pulsatile flow of
CSF.
It is shown in Figure 24 that although each of the
three flow models are noticeably different, they
follow the same trend and any one of them can be
used when modeling intrathecal drug distribution.
The dispersion of the concentration of the drug
increased with respect to time. This mechanistic
models demonstrate how the opioids will behave
throughout the intrathecal flow network. Variables can
be changed by researchers to provide a more
applicable model.
5. Conclusion
The computations performed and the mechanistic
models created have demonstrated that there are a
variety of factors that need to be accounted for in
intrathecal drug delivery. It has been determined that
drug distribution is patient specific due to the
resistance being affected by the length and diameter
of the spinal cord along with the viscosity of CSF.
Also, it has been established that drug distribution
depends on the choice of the opioid as a result of the
varying kinetic values of pharmacokinetic
parameters. With the creation of these mechanistic
models, engineers and researchers can now further
study the mechanisms of biodistribution for drugs
administered in the CSF
Acknowledgement
This project was assigned in the Biological Systems Analysis course at the University of Illinois at Chicago in the Fall
semester of 2014 under the instruction of Professor Andreas Linninger. It was completed with the help of Chih-Yang
Hsu and Sebastian Pernal.
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.
References
1. Smith, Thomas J., et al. "Randomized clinical trial of an implantable drug delivery system compared with
2.
3.
4.
5.
6.
7.
comprehensive medical management for refractory cancer pain: impact on pain, drug-related toxicity, and
survival."Journal of Clinical Oncology 20.19 (2002): 4040-4049.
Stearns, Lisa, et al. "Intrathecal drug delivery for the management of cancer pain." J Support Oncol 3
(2005): 399-408.
Smith, Howard S., et al. "Intrathecal drug delivery." Pain Physician 11.2 Suppl (2008): S89-S104.
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.
Shafer, Steven L., and John R. Varvel. "Pharmacokinetics, pharmacodynamics, and rational opioid
selection." Anesthesiology 74.1 (1991): 53-63.
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
Agamanolis, Dimitri P., M.D. “Chapter 14.” Cerebrospinal Fluid. Neuropathology, Jan.-Feb. 2013. Web. 04
Dec. 2014.
Appendix
Matlab Code 1: Brain Model
clear all, close all, clc
%% 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;
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)
%% Full model
figure
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)+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
Matlab Code 2: Brain ODE
function nsonip()
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
V = Y(:,1:5);
P = .0075*Y(:,6:9);
P0 =V(:,1)/.188;
F = Y(:,10:13);
c = ['b';'k';'r'];
plot(T,F)%,c(i))
hold on
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
,[0 50],y0);
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)
function dP = brainwaiver(t,Y)
kappa = [.188;.0210;.0174;.0126;.0139]; K = 1./kappa;
E=1;
alfa = [.35879;.49138;.7228;.783081;1e-1];
Fprod = 0;
reab=6.4e-4;
timei=5;
Pven=0;
A=2;
w=1*pi;
rho = 1.0068;
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;%pi*(mean(V)^(2/3));
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);
%Flow Model 1:
%
%
%
%
P2
P3
P4
P5
dP(10,1)
dP(11,1)
dP(12,1)
dP(13,1)
=
=
=
=
(P0-P(1))-(U(1)*alfa(1));
(P(1)-P(2))-(U(2)*alfa(2));
(P(2)-P(3))-(U(3)*alfa(3));
(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));
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 starts
Csc = Y(concSCstart:concSCend);
concEPIstart = 23;
concEPIend = 26;
Cepi = Y(concEPIstart:concEPIend);
concVASCstart = 27;
concVASCend = 30;
Cvas = Y(concVASCstart:concVASCend);
kicM = 0.037;kicA = 0.170;kicF = 0.0339;kicS = .020;
KIC = [kicM kicA kicF kicS]; %%CSF space to SC
kciM = 0.0143;kciA = 0.0236;kciF = 0.0159;kciS = 0.0095;
KCI = [kciM kciA kciF kciS]; %%SC to CSF space
kplcM = 0.0082;kplcA = 0.868;kplcF = 0.0080;kplcS = 0.0131;
KPLC = [kplcM kplcA kplcF kplcS]; %%SC to Vasclature
kieM = 0.0542;kieA = 0.1372;kieF = 0.1078;kieS = 0.0291;
KIE = [kieM kieA kieF kieS]; %%CSF space to Epidural space
keiM = 0.0021;keiA = 0.0063;keiF = 0.0285;keiS = 0.0137;
KEI = [keiM keiA keiF keiS];
kplepiM = 0.0199;kplepiA = 0.0201;kplepiF = 0.1088;kplepiS = 0.0323;
KPLEPI = [kplepiM kplepiA kplepiF kplepiS]; %Umenhofer
% Concentration
j = 1;
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(iconcSCend);
end
for i = concVASCstart:concVASCend
dP(i,1) = KPLC(j)*Csc(i-concEPIend)+KPLEPI(j)*Cepi(i-concEPIend);
end
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)
%concentration
figure; subplot(2,2,1)
plot(TM,CM); title('Drug
legend('C1' ,'C2', 'C3',
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 Morphine');
'C4', 'C5');
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
,'C2', 'C3', 'C4');
subplot(2,2,2)
plot(TA,CAsc); title('Drug
,'C2', 'C3', 'C4');
subplot(2,2,3)
plot(TF,CFsc); title('Drug
,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSsc); title('Drug
,'C2', 'C3', 'C4');
Concentration in SC for Morphine'); legend('C1'
Concentration in SC for Alfentanil'); legend('C1'
Concentration in SC for Fentanyl'); legend('C1'
Concentration in SC for Sufentanil'); legend('C1'
figure; subplot(2,2,1)
plot(TM,CMepi); title('Drug Concentration in EPI for Morphine'); legend('C1'
,'C2', 'C3', 'C4');
subplot(2,2,2)
plot(TA,CAepi); title('Drug Concentration in EPI for Alfentanil');
legend('C1' ,'C2', 'C3', 'C4');
subplot(2,2,3)
plot(TF,CFepi); title('Drug Concentration in EPI for Fentanyl'); legend('C1'
,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSepi); title('Drug Concentration in EPI for Sufentanil');
legend('C1' ,'C2', 'C3', 'C4');
figure;subplot(2,2,1)
plot(TM,CMvas); title('Drug Concentration
,'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
,'C2', 'C3', 'C4');
subplot(2,2,4)
plot(TS,CSvas); title('Drug Concentration
legend('C1' ,'C2', 'C3', 'C4');
in VASC for Morphine'); legend('C1'
in VASC for Alfentanil');
in VASC for Fentanyl'); legend('C1'
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');