Density Driven Spread of Anesthesia in Cerebrospinal Fluid
Density-driven flow in porous media, spinal anesthesia, drug diffusion, sensory-motor blockage
BEE 4530 - Computer-Aided Engineering: Applications to Biological Processes
Rajesh Bollapragada, Victor Chen, Priyanka Khan, Michael Pierides©
May 2017
Table of Contents
1.0 Executive Summary
2
2.0 Introduction
2.1 Background of Spinal Anesthesia Spread
2.2 Current Research and Literature Review
3
3
4
3.0 Problem Statement
5
4.0 Design Objectives
6
5.0 Problem Schematic
6
6.0 Methods
6.1 Variables
6.2 Governing Equations
6.3 Boundary Conditions
6.4 Initial Conditions
6.5 Implementation of Density-Driven Diffusion
6.6 Material Properties
8
8
8
9
10
10
10
7.0 Results and Discussion
7.1 Qualitative Description
7.2 Validation
7.3 Sensitivity Analysis
7.4 Objective Function
12
12
14
15
15
8.0 Conclusion and Design Recommendations
18
Appendix
19
References
22
1
1.0 Executive Summary
Intrathecal administration of anesthesia is performed for purposes of sensory and motor nerve blockage
during surgeries to the abdomen and lower extremities. The spread of anesthesia is primarily based on the
injection velocity, concentration, and density of the drug and determines which spinal nerves are blocked
and the duration for which the block is maintained. As abdominal surgeries differ in the nerves that require
blockage and the duration the block is required, the properties of the anesthesia, such as density,
concentration, and injection velocity must be made unique for a given surgery to produce the optimal
blockage for that procedure.
Anesthesiologists refer to the density ratio of the anesthesia to the cerebrospinal fluid as the baricity.
Hyperbaric anesthesia, which is more dense than cerebrospinal fluid, tends to diffuse downward in the
direction of gravity. Hypobaric anesthesia, which is less dense than cerebrospinal fluid, tends to diffuse
upward, against gravity. The baricity of anesthesia can be increased by the addition of dextrose and
decreased by the addition of distilled water. While the baricity of the anesthesia affects its spread and the
nerves that are blocked, the concentration of the drug affects the duration of blockage. In many cases, the
degradation rate of intrathecal anesthesia is proportional to the concentration of the drug in the spinal canal
and a higher concentration of the drug would increase the duration of blockage. However, the spread and
dosage of anesthesia must be carefully monitored throughout the procedure as slight fluctuations in these
parameters can put the patient at risk of cardiac complications such as hypotension, bradycardia, and
potentially death.
The density-driven spread and duration of blockage by the anesthesia bupivacaine were modelled with
respect to its baricity, concentration, and injection pressure. The results demonstrated that hyperbaric
anesthesia tended to diffuse in the direction of gravity while hypobaric anesthesia tended to diffuse against
gravity. An objective function was also created that measured the harm the anesthesia did to the patient.
The safe range of anesthesia concentrations was found to be 2.5 to 7.5 mg/mL. The optimum injection
pressure was found to be 915,279 Pa, which is comparable to the pressure of injection. However, while
baricity had a great effect on the spread of the drug, it appeared to have no noticeable effect on the block
provided by anesthesia.
Keywords: Density-driven flow in porous media, spinal anesthesia, drug diffusion, sensory-motor
blockage
2
2.0 Introduction
2.1 Background of Spinal Anesthesia Spread
Spinal anesthesia is a widely used anesthetic technique that provides complete sensory and motor block
during surgeries to the abdomen, legs, and lower extremities. The anesthesia is injected into the
cerebrospinal fluid through a long thin needle that is inserted between the L4 and L5 vertebrae [1]. The
cerebrospinal fluid is contained within the intrathecal space, a region enclosed by protective membrane
tissues that line the central canal of the vertebral column, where it bathes the tissues of the brain and spinal
cord, and circulates nutrients for the central nervous system.
The spread of anesthesia in the cerebrospinal fluid determines which spinal nerves are blocked [2]. The
spinal nerves extend from the left and right side of the spinal cord at each intervertebral space and transmit
signals between the central nervous system and the rest of the body. Each spinal nerve is composed of a
dorsal root nerve and a ventral root nerve. Spinal anesthesia blocks neural transmission at the root nerves
and causes a numbing sensation called a block. The strength of the block depends on the concentration of
anesthesia at the insertion of the root nerves to the spinal cord and the duration of contact [3]. The
effectiveness of the block can be determined by studying the sensory responses at the dermatomes, areas of
skin that are innervated by the spinal nerves (Figure 1.1). Each pair of spinal nerves is responsible for
innervating a different dermatome. The adequacy of a block is judged by a lack of response from a pinprick
at each of the dermatomes [4]. Once the block has been deemed adequate, the surgery can proceed.
Figure 1.1: Diagram of spinal nerves and corresponding dermatomes [5]
The spread of anesthesia is determined by the interaction of several physical factors related to the
cerebrospinal fluid, anesthesia and injection technique. When the anesthesia enters the intrathecal space, it
initially spreads due to the weak circulatory forces within the cerebrospinal fluid [6]. However, the spread
of the anesthesia is ultimately determined by its baricity, the density ratio of the anesthesia to the
cerebrospinal fluid [3]. Hyperbaric anesthesia has a baricity greater than one and tends to diffuse downward
in the direction of gravity. Hypobaric anesthesia has a baricity less than one and tends to diffuse against
3
gravity. Baricity of the anesthesia can be increased by mixing with dextrose or decreased by mixing with
distilled water [7]. The spread of the anesthesia is also slightly influenced by the velocity with which it is
injected. Evidence has shown that the range of the spread becomes greater as the injection velocity is
increased [8]. The concentration of the anesthesia also has a slight effect on its spread as higher concentrated
doses tend to result in greater spreads of the anesthesia [9]. However, the effect concentration has on spread
is eclipsed by the effects of baricity and injection velocity. Instead, the concentration of the anesthesia has
a far greater effect on the duration of the block as higher concentrations tend to produce longer periods of
block [3].
Patient positioning during the injection procedure is another important factor that is considered when
administering spinal anesthesia [6]. In most procedures, anesthesia is administered with the patient in the
sitting position to maximize the width of the intervertebral space and harness the effects of gravity on the
spread of the drug [3]. In this position, the patient arches the back forwards with chin tucked to the chest
such that the arch of the spine resembles the shape of a “C” (Figure 1.2). After the anesthesia has been
delivered, the patient is asked to remain in the sitting position for a period of five to ten minutes [10]. The
anesthesia settles during this period and the patient is repositioned to either the supine or prone position to
rest horizontally facing up or down respectively on the operator bed. Surgery is thereafter commenced.
Figure 1.2: Position of patient and angle of spine during spinal anesthesia administration [11].
4
2.2 Current Research and Literature Review
Several clinical studies have been conducted to address how the parameters of baricity, drug concentration,
and injection velocity affect the spread of anesthesia in the cerebrospinal fluid. However, some of these
studies have shown inconsistent results regarding the effects of these parameters on the spread of anesthesia.
A clinical study conducted by the University of Rochester School of Medicine showed no difference in the
spreads of anesthesia with respect to hyperbaric and hypobaric anesthesia [12]. However, studies have more
frequently shown that hyperbaric anesthesia tends to diffuse in the direction of gravity and hypobaric
anesthesia against gravity [13] [14]. In regard to injection velocity, the study conducted by Tuonimen M.
et. al illustrated that slower injection velocity produced a higher and more predictable spread of spinal
anesthesia [15].
However, Schwagmaier et. al observed no direct relationship between injection velocity and spread of
spinal anesthesia [16]. Similarly, there has not been unanimity regarding the effects of concentration on the
spread of anesthesia in spinal fluid. Some clinical studies showed no significant difference while others
showed that a higher anesthesia concentration resulted in a higher level of block [12] [9]. Current research
has, therefore, demonstrated that several variables affect the spread of anesthesia in cerebrospinal fluid in
unpredictable manners. Although the effects of baricity and slow injection of anesthesia are thoroughly
understood, the techniques that produce predictable results are not universally practiced.
3.0 Problem Statement
As surgeries that involve spinal anesthesia differ in the nerves that require blockage and the duration for
which blockage is required, anesthesia must be made unique with respect to each surgery. Parameters that
the anesthesiologist can modify include anesthesia baricity, concentration, injection velocity, and the patient
position during injection. Erroneous determination of these characteristics can put the patient in risk of
cardiac complications such as bradycardia and hypotension, which appear to result from an excess
concentration of drug or the drug travelling too high up the spinal canal [5]. In extreme cases, these
complications can put the patient at risk of severe nerve damage or even death.
As a result, it is necessary to thoroughly monitor how the characteristics of the injected anesthesia such as
the baricity, drug concentration, and injection velocity affect the spread and duration of the block. However,
clinical studies have shown that the effects of these parameters on anesthesia spread are complicated and
difficult to predict. Therefore, the goal of this project is to use mathematical modelling to make the spread
of anesthesia in cerebrospinal fluid as predictable as possible for anesthesiologists.
5
4.0 Design Objectives
The primary goal of the project was to use finite element modelling to accurately predict the spread of
spinal anesthesia with respect to various factors. This goal was divided into following objectives:
1. Model the density driven spread of anesthesia in spinal fluid by simulating realistic surgical
injection physics.
2. Analyze the effects of factors such as baricity, anesthesia concentration, and injection velocity on
the spread of spinal anesthesia, factors that are controlled by the anesthesiologist.
3. Use sensitivity analysis to study the effects of spinal fluid properties on the spread of spinal
anesthesia. These factors are not controlled by the anesthesiologist and add variability to anesthesia
administration during surgery.
4. Optimize the characteristics of anesthesia to provide an appropriate lower lumbar block for a
duration of 5 hours. Studies have shown that regional anesthesia concentrations above 73 μg/mL
result in adequate blocks. [17]
5. Prevent anesthesia from exiting the model domain or going above the minimum toxic
concentration, 1060 μg/mL, to prevent hypotension and bradycardia.
5.0 Problem Schematic
A 3D model was created on COMSOL Multiphysics® that simulates the injection anesthesia into the spinal
fluid (Figure 2.1). The model approximates the shape of the spine in the sitting position. Data regarding the
curvature of the spine in the sitting position was obtained from a study that used mathematical models to
reconstruct the shape of the spine in various postures such as the “C-shaped” sitting position. After plotting
the curvature data on the COMSOL model builder, a torus structure was fitted to the data points to represent
the spinal canal. Half of the torus was removed by using a block to allow for visualization of the interior of
the spinal column. Furthermore, this halved the number of domain elements to reduce computation. The
spinal canal was then partitioned into domains that indicate the positions of the vertebrae and the
intervertebral spaces. A cylindrical element was then fused to the torus between the domains of the L4 and
L5 vertebrae to represent the inserted needle.
6
Figure 2.1: Schematic of spinal cord. The schematic shows the dimensions of the spinal cord, domains of the
vertebrae, the boundary conditions at the inlet and outlet, and the no flux conditions along the canal.
Our study modelled the spread of anesthesia in the cerebrospinal fluid of the spinal canal using the reacting
flow in porous media physics interface. The spinal canal was modelled as porous media because it is jointly
occupied by both solid nerve and liquid spinal fluid elements that interweave to create a porous network.
In order to simulate the injection, the needle element was given a fluid flow inlet condition and a
concentration inflow condition. The injection was modelled using a Gaussian pulse pressure condition to
create a laminar inflow of the anesthesia into the spinal canal. The degradation of the anesthesia was
modelled using a first order reaction. To ensure that that the fluid within the domain stayed incompressible,
outlet and outflow boundary conditions were applied to the top of the domain. The model does not account
for vertebrae above the T8 so the outlet condition effectively states that the fluid exiting the top domain is
flowing into the rest of the spinal canal.
7
6.0 Methods
The injection and diffusion physics of the model were established using the following variables, governing
equations, boundary and initial conditions, and parameters. Furthermore, parameter relationships were
defined to accommodate for density-driven flow.
6.1 Variables
Table 1: Variables used in this model
Symbol
Unit
Name
⍴
kg/m3
Density of Spinal Fluid and Drug
(Refer to section 6.6)
u
m/s
Velocity
t
s
Time
P
Pa [kg/m·s2]
Pressure
μ
Pa·s [kg/m·s]
Dynamic Viscosity
I
-
Identity Matrix
g
m/s2
Gravitational Acceleration
c
kg/m3
Concentration of Drug
Dc
m2/s
Diffusivity of Drug
κ
m2
Permeability
�P
-
Porosity
k’’
1/s
Degradation Rate
6.2 Governing Equations
The fluid flow of the anesthetic injectate through the syringe is modelled using the Navier-Stokes equations:
Conservation of momentum:
𝜌(
∂u
∂t
+ 𝑢 ∙ ∇𝑢) = −∇𝑝 + ∇ ∙ (𝜇(∇𝑢 + (∇𝑢)𝑇 ) −
2
3
𝜇(∇ ∙ 𝑢)𝐼) + 𝜌𝑔𝑧
(1)
Conservation of mass:
𝜕𝜌
𝜕𝑡
+ ∇ ∙ (𝜌𝑢) = 0
(2)
8
The Gaussian pulse pressure condition was used to describe the pressure boundary condition at the site of
the injection. The range of the function is between the maximum injection pressure and the natural pressure
of the spinal canal at the side of needle insertion. The curve illustrating the Gaussian pressure condition is
shown in Figure 6.1.
Gaussian pulse pressure condition:
𝑝(𝑡) = (𝑃𝑖𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛 − 𝑃𝑠𝑝𝑖𝑛𝑎𝑙 𝑐𝑜𝑟𝑑 ) ∙ e−(
(𝑡−3.5)2
)
12.5
+ 𝑃𝑠𝑝𝑖𝑛𝑎𝑙 𝑐𝑜𝑟𝑑
(3)
Figure 6.1: Gaussian pulse injection pressure over time.
The value of this function ranges between the maximum injection pressure and the natural spinal canal pressure at
the site of insertion
The cerebrospinal fluid flow and the spread of anesthesia through the spinal column was modelled using
the diffusion equation and Brinkman equation. These equations describe reactive fluid flow through a
porous media, which the environment of the intrathecal space in the spinal canal is modeled as:
Transport of diluted species:
∂c
∂t
+ ∇ ∙ (𝑐𝑢) = ∇ ∙ (𝐷𝑐 ∙ ∇c) − 𝑘′′𝑐
(4)
Brinkman equation:
𝜌
𝜖𝑝
(
𝜕𝑢
𝜕𝑡
𝑢
𝜇
𝜖𝑝
𝜖𝑝
+ (𝑢 ∙ ∇ )) = ∇ ∙ [−𝑝𝐼 +
(∇𝑢 + (∇𝑢)𝑇 ) −
9
2𝜇
𝜖𝑝
𝜇
(∇ ∙ 𝑢)𝐼] − ( ) 𝑢 + 𝜌𝑔𝑧
𝜅
(5)
6.3 Boundary Conditions
An inlet boundary condition is applied at the end of the needle for the injectate to flow in and an inflow
boundary conditions is applied at the end of the needle for the drug dissolved in the injectate. Outlet and
outflow boundary conditions are applied at the top of the entire domain to maintain conservation of mass
and to ensure that the fluid in the spinal column remains incompressible. Because the model only contains
the portion of the spinal column between L5 and T8, there exists more vertebrae above the top of the model
for which fluid flows into. All the other faces on the domain have zero flux boundary conditions, which do
not permit fluid flow through. Furthermore, there is a symmetry boundary condition on the flat face of the
domain where the spinal column was cut in half. There is a constant pressure boundary condition at the top
of the spinal column which is 103000. [17]
6.4 Initial Conditions
There is no concentration of drug in the spinal column or the needle at zero seconds. Furthermore, the initial
fluid velocities in the domain are zero. A constant pressure value was established throughout the model to
study the effects of density driven flow.
6.5 Implementation of Density-Driven Diffusion
The local density of the spinal fluid, ⍴ , was defined as a linear interpolation between the density of the
drug and the density of the spinal fluid. This results in the effective density of the spinal fluid at any time
and is represented by the equation:
Density variation equation:
𝜌 = 𝜌𝑓𝑙𝑢𝑖𝑑 +
𝜌𝑑𝑟𝑢𝑔 − 𝜌𝑓𝑙𝑢𝑖𝑑
𝑐𝑑𝑟𝑢𝑔 − 𝑐𝑓𝑙𝑢𝑖𝑑
× 𝑐(𝑡)
(6)
Thus when ⍴ drug is less than ⍴ fluid, ⍴ becomes increasingly less than ⍴ fluid with respect to increased
concentration, causing the anesthesia to rise. However, when ⍴ drug is greater that ⍴ fluid, ⍴ becomes
increasingly greater than ⍴ fluid, causing the anesthesia to sink with respect to gravity.
10
6.6 Material Properties
Table 2: Input parameters used in this model
Parameter
Symbol
Value
Unit
Source
Pressure at top of spinal canal
P
103000
Pa
[17]
Pressure at bottom of spinal canal
P
103000
Pa
[17]
Density of anesthesia
ρsa
0.9983-1.032
g/cm3
[18] [19]
Viscosity of anesthesia
μa
0.009795
g/cm·s
[20]
Diffusion coefficient of anesthesia
Dc
6.71·10-10
m2/s
[21]
Degradation rate of anesthesia
R
0.00005501168
s-1
[22] [23]
Density of cerebrospinal fluid
ρcf
1000.59
kg/m3
[24]
Dynamic viscosity of cerebrospinal
fluid
μa
1.003*10-3
Pa·s
[25]
Effective porosity of cerebrospinal
fluid
𝜖𝑝
0.3
Effective permeability of cerebrospinal
fluid
𝜅
0.7 * 10-8
m2
This value was selected to match
the clinical data. Sensitivity
analysis was performed on this
parameter
[25]
Concentration range of injected
anesthesia
ca
2-10
kg/m3
[26]
Inflow pressure of syringe
ps
1220372
Pa
[27]
11
7.0 Results and Discussion
7.1 Qualitative Description
After the 3D model was run for 120 minutes, the typical duration of a related surgery, concentration profiles
of the anesthesia obtained. Figures 7.1 and 7.2 show the concentrations of hypobaric and hyperbaric
anesthesia in the spinal canal between the T8 and L5 vertebrae at various times between 3 seconds and 40
seconds. The hyperbaric model clearly shows concentrated drug settling below the point of injection (Figure
7.2), whereas the hypobaric model sees the drug flow upward out of the domain (Figure 7.1). The hypobaric
plot seems more diffuse, whereas the hypobaric plot is more concentrated toward the bottom of the domain
over the period measured. The color legends have been normalized to represent the same range of
concentration for both the hypobaric and hyperbaric plots.
Figure 7.1: Concentration profile of hypobaric anesthesia at 3, 5, 10, 20, 30, and 40 seconds.
12
Figure 7.2: Concentration profile of hyperbaric anesthesia at 3, 5, 10, 20, 30, and 40 seconds.
13
7.2 Validation
The results from a clinical study that measured anesthesia concentration in the spinal fluid over time were
used to validate the COMSOL model in this paper [28]. The clinical data was plotted against the
computational data from the model to obtain Figure 7.3. The model solution and the experimental data
illustrated the same general degradation trend. While the model solution initially differed from the
experimental solution, the model solution eventually converged toward the experimental solution.
Figure 7.3: Validation of Anesthesia Concentration in Spinal Fluid over Time.
The experimental solutions [28] show a higher concentration at each time point, but both the experimental solution
and model data have the same overall trends. At later times during the procedure, the model solution converged
toward the experimental solution.
14
7.3 Sensitivity Analysis
By performing a sensitivity analysis, the effects of parameters that are not controlled by the anesthesiologist
on the spread of the anesthesia were determined (Figure 7.4). In the sensitivity analysis, the average
concentration was measured at 10 seconds, with a 10% increase and decrease in diffusivity, porosity,
permeability, reaction rate, and spinal fluid density. The sensitivity analysis showed that other than spinal
fluid density, a 10% change in any of the other immutable parameters had a negligible effect on the results.
Even though spinal fluid density was the only parameter that seemed to make a significant difference on
the model, it is also a highly-documented parameter value which does not vary significantly between
patients.
Figure 7.4: Sensitivity Analysis with a 10% increase or decrease in shown parameters.
Changes in spinal fluid density affected the results, while the other parameters tested did not significantly change the
results.
7.4 Objective Function
The objective function was designed to measure the amount of harm the anesthesia would do to the patient.
An anesthesia concentration of 0.076 kg/m^3 is the minimum effective concentration to provide adequate
sensory block, so any concentration lower than that in the spinal will result in pain for the patient.
Conversely, an anesthesia concentration of 1.06 kg/m^3 is the minimum toxic concentration at which there
is a risk of nerve damage to the patient. Because of these two physiological constraints, the first expression
of the objective function taxes concentrations above and below the thresholds determined from literature
[29]. The nerve damage is weighted higher than the inadequate blockage because whereas pain is
temporary, nerve damage is irreversible. The final term of the objective function penalizes any anesthesia
that travels above the top of the domain (past the T8 vertebrae). This term was weighted less highly than
the other two terms in the function because the model only includes the vertebrae between T8 and L5 while
nervous system damage occurs if the anesthesia rises above the T4 vertebrae [30].
15
The objective function was applied to three parameters, injection pressure, density, and concentration,
because these parameters can be manipulated by the anesthesiologist. Seven different values for each of the
parameters were used in the COMSOL model, keeping an even distribution of the parameters between the
maximum and minimum values of each parameter found in literature (Table 2). Concentration data was
taken from each parameter’s model for each vertebra and the loss function was applied to the
concentrations. Loss function plots were produced to find minima for each parameter:
𝐿(𝑐𝑖 (𝑡)) = ∑𝑛𝑖 20 [(𝑐𝑖 (𝑡) < 0.076
10 ∑𝑜𝑢𝑡𝑙𝑒𝑡 𝑐(𝑡)
𝑘𝑔
𝑚3
) × (0.076
𝑘𝑔
𝑚3
− 𝑐𝑖 (𝑡))] + 25 [(𝑐𝑖 (𝑡) > 1.06
𝑘𝑔
𝑚3
) × (𝑐𝑖 (𝑡) − 1.06
(7)
Figure 7.5: Concentration Objective Loss function.
The loss function was minimized at a concentration of 2.5 kg/m3.
Figure 7.6: Density Objective Loss function.
The loss function did not vary significantly with changes in density.
16
𝑘𝑔
𝑚3
)] +
Figure 7.7: Pressure Objective Loss function.
The loss function was minimized at a pressure of 915,279 Pa, a value close to those used in surgery [27].
The objective function plots provide insight on the effects of various parameters on drug effectiveness and
potential damage to the body. The loss function was minimized at a concentration of 2.5 kg/m3 and a
pressure of 915,279 Pa, which is comparable to the pressure of injection (Figures 7.5, 7.7) [27]. The
anesthetic density however appeared to have had little effect on the loss function as shown in Figure 7.6,
despite its importance in predicting the flow pattern.
17
8.0 Conclusion and Design Recommendations
This model seeks to provide anesthesiologists with an accurate method of determining which locations in
the body will have sensory blocks based on an anesthesiologist’s given parameter values. The densitydriven spread plots, validation, and objective function results demonstrated that the model provided
accurate results and met the design objectives of our problem.
Although this model predicts the spread of anesthesia throughout the spinal column with high precision,
the model can be made more accurate. Improvements could be made by including a more precise spinal
cord geometry, the effect of temperature, and the posture of the patient during the surgery. A more precise
spinal cord geometry would match better the curvature and anatomy of the spinal canal, potentially using
CT scan data for realistic measurements. A future model could also include the entire length of the spinal
cord. This would reduce physical approximation error in the model. The effect of temperature is another
parameter that could affect spread of anesthesia. In most cases, the anesthesia is colder than the body, so
changing anesthesia temperatures would affect its spread [31]. The posture of the patient is another factor
that could be accounted for in future studies, because a patient is laid down into the supine or prone position
for their surgery following 5 to 10 minutes in the sitting position. In this model, patient posture was
considered negligible under the assumption that the anesthesia had settled in the spinal canal after ten
minutes and that changing the patient’s posture would not lead to different results [10].
18
Appendix
Appendix A: Mathematical Statement of the Problem
To generate a concentration profile, a physics defined mesh was used (Figure A.1).
Figure A.1: Mesh plot of model with magnifications along wall of spinal canal and needle. The mesh is finer along
the wall of the spinal canal and the needle itself.
19
Appendix B: Mesh Convergence Analysis
Upon performing a mesh convergence, the plot converged at 49,995 degrees of freedom. This was used as
a baseline for degrees of freedom in determining the final mesh, which ultimately had 83,920 degrees of
freedom. A significantly finer mesh allowed for a more accurate computation and incorporation of the fine
dimensions of the injection needle.
Mesh Convergence Analysis at 10 Seconds
Concentration (kg/m3)
0.3
0.295
0.29
0.285
0.28
0.275
0.27
40000
42000
44000
46000
48000
50000
52000
54000
56000
Degrees of Freedom
Figure A.2: Mesh Convergence Analysis at 10 Seconds.
The mesh was shown to converge at 49,995 degrees of freedom.
Appendix C: Solution Strategy
The iterative solver used to solve the governing equations was the general minimal residual method solver
GMRES. The time stepping used was default physics based time stepper which automatically adjusted the
time step to keep the error within the acceptable tolerances. The relative tolerance used was 0.01 and the
absolute tolerance used was 0.0005. An average computation required 2.51 GB of memory and took
approximately 30 min to compute.
20
Appendix D: Glossary
Baricity - Relative density of anesthesia to spinal fluid
Bradycardia
- Condition of slow heart rate
Dermatome
- Region of skin innervated by a spinal nerve
Hyperbaric
- Pertaining to anesthesia that is more dense than spinal fluid
Hypobaric
- Pertaining to anesthesia that is less dense than spinal fluid
Hypotension - Condition of low blood pressure
Prone position - Posture in which patient is lying horizontally face down
Sitting position - Posture in which the patient is sitting hunched over with spine in the shape of a “C”
Supine position - Posture in which patient in lying horizontally face up
21
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