*This Report Is Produced Under the Supervision of BIOE310 Instructor Prof. Linninger

Design of Spinal Anesthesia

Suraj H. Shah

*This report is produced under the supervision of BIOE310 instructor Prof. Linninger.

1. Abstract

Spinal opioid implementation is an ideal option to assist in solving for chronic pain. While performing under traumatic procedures, anesthesiologists will possibly encounter morphine administration, ideally intrathecally (IT) 3. Doing so, the drug will provide a higher degree of its concentration while having a longer analgesic effect. Intrathecal drug delivery is specialized, proved to show administration of therapeutic molecules within the central nervous system (CNS). The intensity of the drug will depend as it is maneuvered along the different components of the body, such as the kidney, heart and the brain 2.The ideal mechanisms and effects of drug distribution along the cerebrospinal fluid (CSF) filled-spinal canal are misinterpreted. The movement around the CSF filled spaces can possibly predict scattered administrations of each therapeutic drug. Along with this proposal, the movement of fluid around the brain controls the distribution of medications given systemically, into the subarachnoid space or into the brain parenchyma. Systemically-administered medications that penetrate the Blood Brain Barrier (BBB) are rapidly distributed to the compartments of the brain and spinal cord 3. Flow conservation equations will be used in order to describe the direction of incoming and outgoing flow throughout this network; imaging a mechanistic model of the spinal canal to guide the mathematical studies of the effects of pulsatile CSF flow on each opioid.

Keywords: Intrathecal (IT) Delivery, Pulsatile Flow,Cerebrospinal Fluid, Blood Brain Barrier

Shah - 1

2. Introduction

Many variables are needed to be established in order to develop mechanistic models. This project will need to institute objective variations between patients, CSF characteristics, opioid properties, or surgical procedure variables 1, 2, 3.An opioid property that this project will take greatly in consideration is the rate of transfer between each drug in the CSF and the enclosed tissues, such as the spinal cord and plasma membranes 1. The rates will determine the factor of the opioid exhibited during a specific time and the distance of how the drug will be distributed throughout the spinal cord. If performed incompetently, the aspect of performing intrathecal methods can be inaccurate, in some cases hazardous.

As surgeons are operating, many dynamics are implemented while using opioids, affecting the overall distribution of the drugs content. The amount of drug administered depends on the injection process given to the patient3. The location of the injection will also influence the dynamics of the overall administration5. It will demonstrate how much of the biological fluid each compartment will receive. In vivo, it would be difficult to accept precision of controlling the mechanism of the drug. In research, the mechanisms of distribution for drugs administered in the cerebrospinal fluid (CSF) are not well understood 2. Flexible models will be an accurate tool to improve regulation of drug distribution of the important roles of CSF, such as amplitude and frequency for the rapid dispersion after IT administration3. These mechanistic models will predict the distribution of the opioids given experimentally, as it should compute drug distribution in the spinal.

Figure 1. Model describing the circulation of the Cerebrospinal Fluid (CSF), produced by the choroid plexus and deposited into the subarachnoid space and the Central Nervous System (CNS).

Figure 2. Model describing the process of intrathecal drug delivery

Overall, physicians must have a better quantitative and qualitative identification of the specific opioids and adequate doses. A partial route of the results will be mainly used in the spinal site rather than the cerebral analgesic. Some routines and opiates have been used for treating acute and chronic pain, including morphine, fentanyl, alfentanil, and sufentanil 2. Fentanyl, alfentanil, and sufentanil all have extremely detailed pharmacokinetic and pharmacodynamic distinctions 4. By dispersing the drug regionally, the kinetics will slow down, affecting the dosage of the drug. Depending on the pulsations, the flow pattern and the configurations of the spinal cord, a wide range of dosage will need to be taken in considerations 5. Mechanistic models will denote the distribution of these opioids and will guide surgeons in providing an accurate treatment to the patient. The model will also illustrate the absorption rate of the drug, as it binds into the tissue. Different locations that are impacted with the injection will also show a distribution of data of durations. The mechanisms of biodistribution for drugs administered in the CSF will exemplify the opioids being confined in the body, for this research. This experiment will demonstrate the assimilation of the opioids into the tissue, the blood clearance for different injection locations, the number and duration of the injections, and the choice of drug7. A selectivity of these opioids, depending on the procedure of the anesthesiologists, will need to be fully respected in order to receive the full understanding of its uniqueness.

3. Methods

With multiple concentration thresholds, the user can depict mechanistic and biological models to show a distribution of the pulsatile CSF in the human spinal cord.In corresponding to this mechanistic model, the pressure-driven flow, pressures and volumes will be solved for in this experiment. The user can also condition the initial pressure of the spinal cord with a sinusoidal equation, distributing the initial excitation in correspondence to the solved pressure, volumes and flows. Once these variables are functioned properly, they will act physiologically in the body once the drug is administered with a certain concentration. The concentration of the injection and the flow of the injection will be researched in each compartment of the human spinal cord (cervical, thoracic, lumbar, and sacral vertebra). Once the concentration and the flow of the injection are identified, computational analysis of mass transfer will be added. The addition of mass transfer to the reactions will provide the kinetic reaction rate multiplied by the concentration of each compartment of the spinal cord 7.

3.1Introducing the Mechanistic Model

A mechanistic model of a linear system to the sinusoidal or periodic excitations correlating to the drug distribution can be a potential visual. Sinusoidal waves were produced to show the distribution correlating to the different drug (morphine, fentanyl, alfentanil, and sufentanil) concentrations. The mechanistic model contributed from MATLAB exemplifies the components of the human spinal cord (Figure 3 and 4). In figure 4, the MATLAB provided a model that can show the user the deformation of the human spinal cord, as each compartment was enlarged. This biological model can help visual the scatter of each drug from the brain to the end of the spinal cord. The model will demonstrate the cervical vertebrae, thoracic vertebrae, lumbar vertebrae, sacral vertebrae and lastly the brain 8. In order to map out the mechanistic model accurately, concentration equations will be used in order to plot the concentration of each opioid with respect to time at these regions.

Figure 3. Mechanistic model of each vertebra in the Spinal Cord, (cervical, thoracic, lumbar and sacral) bounded by an initial and final pressure.

Figure 4. Mechanistic model of the deformation of each vertebra in the Spinal Cord, (cervical, thoracic, lumbar and sacral) bounded by an initial and final pressure.

3.1Mathematical Equation: Flow

An establishment of flow can take place at each region, showing deformation of volume(1). Compressibility equations will be formulated in order to relate relative change in volume of the CSF as a response to the change in pressure (2). This will help describe the deformation in the flow network. The user will need to solve for concentration in terms for deformation.

3.2Mathematical Equation: Deformation

In equation (1), ∆V/∆t (dV/dt) mathematically analyzes the deformation of volume over a period of time. The differences of flow in between each vertebra will conclude how much volume has been distorted in the CSF filled spinal canal. This compiled equation (2) will introduce the compressibility value, kappa (K), as it will analyze computations of the deformation of the spinal canal.

Knowing from the basic procedure of diffusion, the cervical region of the spine will have a larger flow value only if the system is well mixed. If the CSF goes down, an equal amount must go up, making the spinal cord move up and down. As the volume deforms, (V1 –Vo) the pressure throughout the system should decrease (P1-Po).

3.3Hagen-Poiseuille Law

A pressure change will be marked at each “node,” as it will be mathematically be solved by the Hagen-Poiseuille, prominent by equation (3).

In equation (3) ΔP is the change in pressure, F is the flow, and α is the resistance in the spinal canal. The use of equation (3) will dynamically produce individual constitutive and conservation balance equations of the spinal canal network. In order to predict the patterns for each drug administration in regards to flow, volume and pressure, equation (4) would be declared. The change of each pressure will also be dependent on the kappa value (K), as the change of flow is due to the change of pressure.

3.4Mathematical Equation: Flow

Parameter
(min-1) / Morphine / Alfentanil / Fentanyl / Sufentanil
Kic / 0.37 / 0.170 / 0.0339 / 0.020
Kci / 0.0143 / 0.0236 / 0.0159 / 0.0095
Kplc / 0.0082 / 0.868 / 0.0080 / 0.0131
Kie / 0.0542 / 0.1372 / 0.1078 / 0.0291
Kei / 0.0021 / 0.0063 / 0.0285 / 0.0137
Kplepi / 0.0199 / 0.1088 / 0.0201 / 0.0323

Pressure-driven flows were solved, as volume was also solved by each node in the model. Pressure was solved by determining its derivative in terms of volume. After the correspondence of each variable was solved for, the injection of each drug was placed in the lower vertebrae of the human spinal cord. The concentration of the injection and the flow of the injection were now involved, as both variables were programmed into the volume equations. The flow of the injection was inserted into the volume equations. In order to determine the concentration, equations were written based off the direction of flow, as followed by equation (5). In this equation, F represents the flow, C represents the concentration and V represents the volume in the system, guiding computational analyses of concentration patterns against time. MATLAB functions “max” and “min” determined if a flow was positive or negative, respectively.

3.5Mathematical Equation: Mass Transfer

Lastly, a mass transfer matrix, equation(6) had to be implemented through the rate of kinetics, which was added for eachcompartment of the brain model.As stated before, these models (epidural, vascular, spinal cord and CSF) wereessential fundamentals in order to solve corresponding flows, volumes, pressures and concentrations of each drug. In this equation, C relates to the drug concentration for mass transfer, D is the diffusion coefficient for the drug’s motion; v is the average velocity of transfer throughout the CSF filled spinal canal; k is the kinetic term of each opioid. Overall, equation (6) contributes as a diffusion and convection input; diffusion refers to the transport of mass and convection is refers to the motion of fluid9.

3.6Alternative Equation: Flows

Figure 5. Exhibition of different pharmacokinetic behavior and pharmacodynamic behavior. These parameters were inputted for the “k” values of each opioid being administered 1.

Figure 6.Derivation of physiological parameters for each compartment of the CNS

4. Results

4.1 Results of Morphine

Morphine delivery had a vital concentration of 0.15± 0.01 M in the lumbar vertebrae; making it across the region in 22.0±0.5 seconds in all the models. The cervical region received some quantified intensity at 20.0 ± 0.5 seconds with a concentration of 0.1±0.01 M. Morphine began to show signs of concentration in the thoracic region around 18.0±0.5 seconds with a concentration of 0.13±0.02 M in the models. Obviously, the lower sacral region received close to no signs of morphine patterns in the model, due to it having negligible flow and pressure values. A distorted pattern in the sacral region could have occurred for a few milliseconds after both injections, but immediately leveled off to zero shortly.

4.2 Results of Alfentanil and Sufentanil

When alfentanil administration was placed into the CSF fluid filled spinal canal, the concentration levels in all segments of the mechanistic model were exceptionally the same in regards to morphine delivery. Visually, the biological models of alfentanil distribution shows a time delay in the cervical region, moving across the compartment for 5.0-10.0±0.5 seconds later respectively in the models. Sufentanil drug delivery had the highest concentration levels amongst all opioids being administered into the mechanistic model. Sufentanil had 0.5± 0.01 M in the lumbar vertebrae and had the same time as morphine administration. In the cervical region, the concentration level was the same as morphine administration, but attained at a longer time. The time was delayed to 20.0±0.5 seconds. In the thoracic region, the concentration was also the same as in morphine administration, but with also a delay in time for the epidural model; 7.0±0.5 seconds. Although sufentanil had a higher concentration in one region, the time elapsed between interfaces was deferred for the vascular model.

4.3 Results of Fentanyl

Fentanyl had the smallest concentration amongst all opioids being administered into the mechanistic model. Fentanyl had 0.62± 0.03 M in the lumbar vertebrae; making it across the region in 5.0±0.5 seconds in the epidural model. The cervical region received some quantified administration at 8.0 ± 0.5 seconds with a concentration of 0.8±0.01 M. Fentanyl had an inadequate concentration in the thoracic region around 6.0±0.5 seconds with a concentration of 0.61±0.01 M after both injections were taken place. Similarly to morphine administration, the brain was the last segment of the human spinal cord to receive a concentration quantity. Equally, the lower sacral region received close to no signs of fentanyl pattern in the models. In general, the amount of time delivered in the CSF filled spinal canal was condensed for morphine, as it also acquired higher concentration levels in all segments of the spinal cord. The administration of fentanyl had an inferior criterion; having an extended time frame for delivery, acquiring a smaller amount of concentration in each compartment of the human spinal cord.

4.4 Results of the mechanistic models

These dynamic models analytically demonstrated results of how much concentration was anticipated in the compartments of the spinal cord at specific times for the models. Morphine’s concentration was recorded, as the model represented a drug increase in the brain region against time. The concentration was impacted by the distribution of the intrathecal space. The lumbar vertebrae was still optimized in having the highest concentration level in the human spinal cord. In contrast, the sacral region was represented in having extremely low concentration levels of morphine’s content. This fundamental was similarly exposed in the dynamic results of sufentanil. In contrast, fentanyl and alfentanil had a combination of dynamic concentration levels presented in each compartment. As the distribution of the dynamic system was trajecting towards the lower compartments of the system, the concentration gradually increased in prospect to time. The opioids than gradually decreased as it was dispersed closer to the sacral vertebrae of the human spinal, as the flow and pressure was negligible.The concentration than started to gradually decrease again, as the distribution was projected towards the sacral region. Once the dispersion was closer to the sacral vertebrae, a decrease in concentration emerged.

As seen from these biological models, the flow, volume, and pressure for were highly efficient with IT drug mixing4. In the spinal canal model, the patterns of each opioid were predicted in regards to pulsatile flow.The model simulations discoveredthat fentanylhad a higher concentration level, due to it having a larger area under the curve 2. This high concentration level occurred in the in the lumbar vertebrae. When the injection was placed in the lowest vertebra of the spinal cord; the sacral region, there was a slower clearance and distribution volume; almost to none. Both the cervical and thoracic vertebra, the upper-half segments of the spinal cord received some extent of concentration levels, along with the brain. In general, the mechanistic model of the CSF filled spinal canal proved the anticipated result; the concentration level will be the lowest in the sacral vertebrae and highest in the lumbar vertebrae. The concentration level will eventually start to decay when positioning the injection closer to the brain.

4.5Mechanistic Model: 1

Figure 7. Concentration for each drug dispersion in the compartments of the spinal cord. The dispersion is in the CSF (Model 1)

Figure 8. Concentration for each drug dispersion in the compartments of the spinal cord. The dispersion is in the Spinal Cord(Model 1)

Figure 9. Concentration for each drug dispersion in the compartments of the spinal cord. The dispersion is in the Epidural

Figure 10. Concentration for each drug dispersion in the compartments of the spinal cord. The dispersion is in the Vascular(Model 1)

Figure 11. Volume for each drug dispersion in the compartments of the spinal cord. (Model 1)

Figure 12. Pressure for each drug dispersion in the compartments of the spinal cord. (Model 1)

Figure 13. Flow for each drug dispersion in the compartments of the spinal cord. (Model 1)