Design of Spinal Cord Anesthesia
Arjav Patel
Abstract
Administering drugs through the cerebrospinal fluid (CSF) via the spinal cord is a common method used to treat chronic and postoperative pain. The CSF is the fluid within the brain and spinal cord and is commonly used to deliver drugs directly to the central nervous system (CNS). The fluid dynamics of the CSF has not yet been completely understood therefore, targeted drug delivery is still ongoing in research. After injecting four different opioids into the CSF via the spinal cord, we simulate their distribution to help us understand the pharmacokinetics of the specific opioid. The pharmacokinetics and pharmacodynamics of the injected opioids are studied and the relationship of the two is observed. The pressure-driven flow, volume, and pressure of the CSF, along with the concentration of the four opioids after injectionin the CSF, spinal cord, epidural space, and vascular are studied within this experiment. These simulations help us understand the fluid dynamics of the CSF for improved drug delivery.
Keywords: Cerebrospinal fluid, Flow Dynamics, Opioids, Intrathecal, Pharmacokinetics, Pharmacodynamics, Blood-Brain Barrier
1
1. Introduction
Spinal cord opioid administration is considered to be a strong treatment option for chronic and postoperative pain. This is a common practice that has been used by anesthesiologists for over 20 years4. Clinics often use various types of opioids but the most widespread opioid used is morphine because of its low spinal cord distribution and its longer lasting effects, as opposed to fentanyl, alfentanil, and sufentanil1. The most common technique to administer the opioids is intrathecally, a method of introducing drugs to the body via a spinal cord injection. This is ideal because the drugs can be directly injected into the cerebrospinal fluid (CSF) and a smaller dose of medication can be implemented1. Although this method is used so often, it is still not very well understood because the opioid cannot be targeted once injected. Once the opioid has been introduced into the CSF, the distribution varies from person to person because their pulsatile flow is different5. Using various models and scans, clinicians wish to find a method to better treat patients with intrathecal methods. With the goal of better understanding the CSF and drug distribution within that CSF, the desired results can be achieved.
Cerebrospinal fluid (CSF) is a colorless fluid produced from the choroid plexus found in the brain. The CSF is found in the central nervous system surrounding the brain and spinal cord. It serves to protect the brain and spinal cord from impact and more importantly, it serves to transport nutrients to various areas within the central nervous system5. About 140 mL of CSF is present in the human body. It fills the four ventricles in the brain, the spinal sub-arachnoid, and the cranial sub-arachnoid7. In the last two decades, anesthesiologists have taken advantage of this natural transport system to deliver drugs directly to the brain and spinal cord. It is ideal to use the CSF and administer drug intrathecally because it allowed for drugs to cross the blood-brain barrier (BBB). Unfortunately, as ideal as drug delivery through the CSF seems to be, it is actually not very well understood. The CSF has a complex pulsatile flow that affects the distribution of drugs. There are many variables that affect the fluid dynamics of the CSF, many of which are patient-specific2. The study of pharmacokinetics help clinicians understand the time course of drug distribution and absorption3. The use of pharmacodynamics, referring to the relationship between the concentration of drugs and its effect, is also essential3.
In this experiment, many variables need to be considered, including CSF flow patterns, patient-specific considerations, properties of the injected opioids, and location of intrathecal administration. A compartmental model was created to represent the CNS and locations along the spinal cord. The use of opioids is common due to the unique properties, such as its analgesia effect and its ability to be absorbed by enclosed tissues. Within this experiment, these properties will be tested and observed to give a better understanding as to which of the four opioids is most effective in concentration and distribution. The pressure-driven flow, pressure, and volume will be simulated and compared to one another at all five compartments of our model. We believed that these results will give us a better understanding into the pharmacokinetics of drugs in the CSF and that will lead to more improved and efficient treatment of patients.
Figure 1. Anatomy of the cerebrospinal fluid (CSF) in the central nervous system (CNS). The CSF travels within the cranial and spinal sub-arachnoid space.
2. Methods
To properly simulate the distribution of drugs within the CSF, a compartmental model was created to accurately represent the CSN. The model consists of 5 compartments: Brain, Cervical, Thoracic, Lumbar, and Sacral. Each compartment represents a node in this network system. Each node has a flow from one node to the next and a volume. The initial boundary condition for the incoming pressure is represented by a sinusoidal function. Using the computer program, MATLAB, we ran simulations to predict the drug distributions of the opioids along with visualizations of the CSF pressure-driven flows, pressures, and volume at each compartment. Multiple equations, including conservation balance, constitutive equation, ordinary differential equation along with its derivations, were used to solve for the unknowns in this system and they were input into MATLAB.
Figure 2. The top is a model of the CNS in the compartmental schematic. As you can see, it starts with the brain and continues down the spinal cord, which is represented by 4 compartments: Cervical, Thoracic, Lumbar, and Sacral. The bottom is a model of the deformation of each compartment.
2.1. Flow, Pressure, Volume, and Concentration
This system that representsthe flow networkis given by the following equations. The use of conservation balance, equation (1), and constitutive equations, equation (2), were applied to the compartmental model to solve for pressure-driven flow and pressure.
∑F = 0 (1)
αiFi = ΔP (2)
where F is the flow through each compartment, α is the resistance, and P is the pressure at each compartment. Using equation (1), the flow network can be solved for. The summation of the flows is taken as flow in minus the flow out for each node. Equation (2) represents the Hagen-Poiseuille equation, which gives the pressure drop of the fluid flowing through the specified compartment. It shows the relationship between two physical quantities, flow and pressure. Since the CSF is a Newtonian fluid, the resistance factor can be calculated using equation (3):
(3)
where µ is the dynamic viscosity, L is the length of the compartment, and D is the diameter of the compartment.
After applying mathematical manipulations to equation (2), the flows can be solved for in terms of pressure and resistance. Next, the differential equation of volume over time was applied to the system.
(4)
Equation (4) represents the change in volume over the change in time which equals the change in flow through each compartment of the system. Using equation (4), the compressibility value, K, is introduced and used as follows:
(5)
This equation can be applied to each compartment of the system to solve for the volume. Using equations (1)-(5), the following equation was produced:
(6)
Equation (6) shows the relationship between flow, pressure, and volume. This equation can be solved for the change in pressure by simply dividing out the compressibility value, K. Each of these equations were input into MATLAB and used in the simulations to predict the flows, pressures, and volumes of the CSF at each compartment of the spine.
To solve for the concentration of each of the injected opioids, the following equation was used:
(7)
where Fin and Fout are the flows in and out, respectively, Cin and Cout are the concentrations of the injected opioid coming in and out with the flows, and V is the volume at the respective compartment. The CSF does not only flow in the downward direction but also in the upward direction. To account for this change in direction, the MATLAB function “max” and “min” were applied. These two functions will consider the flow in both the upward and downward directions. Finally, the rate of kinetics was applied to equation (7) as follows:
(8)
where, v is the average velocity, C is the drug concentration for mass transfer, k is the kinetic term, and D is the diffusion coefficient. The kinetic term was applied to each opioid injected into the lumbar region of the spine.
(9)
where u is the velocity and ρ is part of the equation for flow that is reabsorbed. Equation 9 is used in the first model to calculate the flow. This equation is derived from equation 4. It is an alternate method to solving for the flow
(10)
This equation is also used to calculate the flows. This is another alternate method to calculating flow. This equation gives a more accurate calculation. It is derived from equation 4.
Region / Volume (mL) / Deformation (mL) / Alpha / KappaBrain / 55 / 0.08828 / 0.188
Cervical / 23 / 0.04981 / 0.35879 / 0.0210
Thoracic / 33 / 0.03725 / 0.49138 / 0.0174
Lumbar / 25 / 0.02251 / 0.7228 / 0.0126
Sacral / 19 / 0.01896 / 0.783081 / 0.0139
Figure 4. Table shows derived physiological values for the volume, kinetic rates, and kappa used in the equations to solve for concentrations of the injected opioids.
Parameter (min-1) / Morphine / Fentanyl / Alfentanil / Sufentanilkic / 0.037 / 0.0339 / 0.170 / 0.020
kci / 0.0143 / 0.0159 / 0.0236 / 0.0095
kplc / 0.0082 / 0.0080 / 0.868 / 0.0131
kie / 0.0542 / 0.1078 / 0.1372 / 0.0291
kei / 0.0021 / 0.0285 / 0.0063 / 0.0137
kplepi / 0.0199 / 0.1088 / 0.0201 / 0.0323
Figure 5. Table shows parameters of pharmacokinetic and pharmacodynamic behavior
3. Results
After inputting our equations into MATLAB, simulations were accurately run through. We were able to produce graphs and data for the pressure-driven flows, pressures, and volumes of the CSF at each compartment for 3 different models. Along with that, we also produced graphs that illustrate the concentration of the injected opioids over time at each compartment of the system. We were able to do this for the CSF, spinal cord, epidural, and vascular. From figure 6, it can be seen that the pressure-driven flows of the system follows a messy sinusoidal pattern and they are greatest coming from the brain and as it moves further down the spine, they start to decrease in amplitude and frequency. As we follow the pattern with respect to time, we can see that each of the flows converge into a smaller wave with less amplitude and frequency. This pattern is consistent for all 3 models.
We were able to produce a graph for pressure-driven flow, pressure, and volume for the entire system at each compartment. There is a graph to represent each of the three models used to solve for the concentration of the injected drugs. The results show similar patterns and data for each of the three models but the most accurate model is the third one. This is because we used equation 10 to calculate the concentration of each drug at each node rather than taking the average of two node or in between nodes.
Figure 6. Graphs illustrate the pressure-driven flows of our system for all 3 models used in our experiment.
Using the constitutive equaitons, the pressure was simulated in MATLAB. The initial condition for the pressure is a sinusoidal function. As for the other compartments, the pressure decreases as we move further down the spine. The pressure follows this same behavior for all 3 models. Pressures and flows in this system are correlated so they follow the same pattern throughtout the entire simulation. The flows were pressure-driven so they are dependent on the pressure. As the pressure decreases throught the spine, the flow decreases as well. Flow and pressure have a linear relationship. Figure 6 illustrates the pressure at each compartment in the system at specific time points. The equaitons used for these simulations are illustrated by the graphs produced. The pressures and flows resulted as predicted by the system of equations used in the methods section.
Figure 7. Graphs show the pressure of the system at each comparment of the spine for all 3 models.
The volume for the system were simulated at each compartment of the sytem. The initial condition for the volume at the brain was 140mL and 25mL for all other compartments in the system. Figure 5 shows the volume with a wave pattern for all the comparmtents and they are constant all the way through. The frequency and amlpitude of the wave is highest for the volume at the brain and decreases in decending order going down the spinal cord. It follows a very smooth and continuous behavior. This holds ture for both the bous and continuous injections. The equations used to solve for the volume are illustrated in figure 7. Figure 8 illustrates the volume at each compartment in the system at specific time points.
Figure 8. Graph shows the volume of the system at each compartment for each of the 3 models. The values given in figure 4 were used to create these volume graphs.
3.2. Concentration of Injected Opioids
The simulations of the drug concentration after injection of the opioids can be seen in the graphs below. Results show that fentanyl had the highest concentration after injection in the lumbar region of the spine, as can be seen in Figure 10. fentanyl follows the same behavior. For all four opioids that were tested, the concentration in the sacral region was the lowest. The concentration of the opioids seemed to travel upward and the concentration was decreased as it traveled up. Fentanyl had the highest concentration at every compartment of the spine, while alfentanil seemed to have the lowest concentration. The concentration equation, along with the rate of kinetic equation, is illustrated in figures 9-16.
3.2.1 First Model
Using this first model, we used equation 7 and 8 to calculate the concentration of the injected drug. We created a different “envelope” for each compartment and calculated the concentration using that method. The “envelop” was created in between each node.
Figure 9. The four graphs show the concentration of the injected drugs into the CSF at each compartment. These graphs are for the first model.
Figure 10. The four graphs show the concentration of the injected drugs into the spinal cord at each compartment. These graphs are for the first model.
Figure 11. The four graphs show the concentration of the injected drugs in the epidural space at each compartment/. These graphs are for the first model.
Figure 12. The four graphs show the concentration of the injected drugs in the vascular space at each compartment. These graphs are for the first model.
3.2.2 Second Model
In the second model, we used equation 7 and 8 to calculate the concentration of the injected drugs. This time, we created an “envelop” that contained two consecutive nodes along with the flow in between the two. Then the next two were used and so on.
Figure 13. The four graphs show the concentration of the injected drug in the CSF for each compartment. These graphs are for the second model
Figure 14. The four graphs show the concentration of the injected drugs in the spinal cord for each compartment. These graphs are for the second model.
Figure 15. The four graphs show the concentration of the injected drugs in the epidural space for each comparment. These graphs are for the second model.
Figure 16. The four graphs show the concentration of the injected drug in the vascular space for each compartment. These graphs are for the second model.
3.2.3 Third Model
For our final model, we used equation 7 and 8 along with derivaions of equation 9 and 10 to calculate the concentration of the injected drugs. For this, we essentially took each node individually to find the concentration. This is the most accurate calculation used.
Figure 17. The four graphs show the concentration of the injected drug in the CSF for each compartment. These are the graphs for the third model.
Figure 18. The four graphs show the concentration of the injected drug in the spinal cord for each compartment. These graphs are for the third model.
Figure 19. The four graphs show the concentration of the injected drugs in the epidural space for each compartment. These graphs are for the third model.
Figure 20. The four graphs show the concentration of the injected drugs in the vascular space for each compartment.
4. Discussion
We presented the CSF with four different opioids to see the distribution of each and how they were affected by pressure, pulse flow, and volume. Using MATLAB, we simulated the distribution of the opioids and measured the concentration of each at five different locations on the spinal cord. We noticed that the pulsatile flow of the CSF followed sinusoidal waves and it was greatest at the brain and it progressively diminished as it moved down the spinal cord. Very similarly, the pressures within the CSF followed a sinusoidal pattern and they were highest at the brain and decreased in amplitudeas we moved down the spine. From the concentration graphs, we can see that the opioids follow a similar pattern and the concentration is highest at the lumbar region, the injection site. From the graphs, it can be seen that the concentration decreases as it moves in the upward direction through the other compartments of the spine. The concentration at the sacral region is essentially zero for all four injected opioids.