Intrathecal Opioid Application Is Amethod Used Totreatmany Diseases, Such As Chronic Pain

Abstract

Intrathecal opioid application is amethod used totreatmany 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 anddepends on the pharmacokinetic parameters of each opioid4,5. A variety of conservation balance equationswill 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 CSFallows 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 intrathecaladministration6. 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 accountedfor. 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 cord6.

2. Methods

Conservation balance equations are essential when describing the physical world. In order to numerically illustrate the flow network shown in Figure 2, algebraic and differentialsystems 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.

(1)

(2)

(3)

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.

(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.

(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.

(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.

(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.

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.

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. Pressure-driven 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 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.

3. Results

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 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 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 6: Concentration of each opioid at each spinal compartment in the CSF space with respect to time.

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 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 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 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 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 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 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 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 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 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 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 20: Concentration of each opioid at each spinal compartment in the CSF space with respect to time.

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 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.