
MBW:Gravitational Effects on Blood FlowFrom MathBioAuthor: Tracey Morland This paper extends Womersley's model (see MBW:Womersley Arterial Flow) on arterial blood flow to include the effects of gravity on arterial flow in the aorta. In the same manner as Womersley's model, this model assumes a Newtonian fluid and laminar flow in a rigid tube. In addition, gravitational variances on the moon and are Jupiter are compared to the results on earth. There is very little quantitative information regarding how much gravity effects blood flow, so the goal of this research is to give an idea of this magnitude in a simplified model. If the effects of gravity are known quantitatively on blood flow velocity and volumetric flow rate, this could assist in space mission planning and aid in the understanding of long term effects on astronauts in space. ContentsOverviewBiological ContextThe biological context is the same as in MBW:Womersley Arterial Flow as we are analyzing the arterial blood flow velocity. The extension in this paper considers an inclined artery at some angle relative to the earth (see Figure (2)). AssumptionsThere are several assumptions that are made in this model that will greatly influence the result. This model assumes a Newtonian fluid (noncompressible), laminar flow (no turbulence), and a rigid tube (the tube has no elasticity). In reality, blood is a nonNewtonian fluid ^{[1]} can have turbulence, particular under conditions of high flow ^{[2]}, and the vessel has elasticity. HistoryIn most research on arterial flow velocity, the effects of gravity are ignored and frequently cited as “neglible.” There are several studies that do consider the effects of gravity. However, I have not encountered any research on extending Womersley's model to include gravitational effects. Even in the papers that do consider gravity, most of them are studying venous flow or blood flow in the arteries where gravitational effects are most profound. See ^{[3]} for one such example. In ^{[4]}, arterial pulse transit time is calculated using a simple forumla based on Bernoulli's equation. Experimental evidence is aquired to validate the model. However, none of the complexities of the NavierStokes equations are considered. There have been other research conducting simulations of the gravitational effects on arterial flow. Notatble among this research is ^{[5]} which uses the Navierstokes equations to model blood flow, and derives the boundary conditions from similarities between artierial flow and electric circuits. There model is significantly more complex than Womersley's, however, since they define blood as a nonNewtonian fluid, they consider the elastic properties of the artery, and they also analyze the flow in bifurcations of the arteries. The data they collected focuses on the effects of gravity on the change in diameter of the blood vessel instead of the effect on blood flow velocity. Mathematical BackgroundMost of the mathematics in this paper will look familar from MBW:Womersley Arterial Flow. We will take a more indepth look at Fourier series, though, and discuss how the coefficients are derived. The Navierstokes equations will become more important in this paper as we will utilize an additional gravitational term in these equations and determine how this term gets propogated throughout the model. Furthermore, we will continue to use Bessel functions as they arise in the Womersley model. Last, the notation and parameters defined in MBW:Womersley Arterial Flow will also be referenced in this paper. Brief SynopsisMathematics Used:  partial differential equations, Fourier series and analysis, classical physics (forces and gravity), partial derivatives, use of the Jacobian, vector gradients, and linear systems. Type of model: This project uses the  NavierStokes partial differential equations to study the fluid dynamics of blood under the effects of gravity. Biological system: The biological system studied in this project is the flow of blood in humans. The NavierStokes equations theoretically model Netwonian fluids perfectly. In the summary of Womersley's model, we made several simplications to the model allowing us to ignore many of the terms in these equations. In Womersley's paper, he considers a cylindrical tube that is parallel with the Earth and therefore the gravitational effect is zero (see the figure below). When considering gravity our model looks like the model in Figure 2. In Figure 1, there is no gravity acting in the direction of flow. Now in Figure 2, gravity has an x component and a y component. Let's look at this in more detail. Relative to the x and y axis, we have: Therefore we have the following relationships: (1) Therefore we have, (2) Just like in traditional Womersley flow, we are assuming that flow in the y direction is zero, and there is no flow in the z direction, so all of its derivatives are zero. From the continuity equation we have: (3) And since we assumed that it follows that: (4) Thus, the NavierStokes equations simplify to: (5) In order to compare our model to Womersley's, let's convert to cylindrical coordinates. Then our equation for motion becomes: (6) Last, if we have the fluid velocity is changing periodically, then we have a term on the left hand side. The final equation for motion is therefore: (7) Notice that this equation is exactly the equation for Womersley flow with an additional gravitational term. As a validity check of the model we see that if then we get precisely Womersley's formula. The same method will be used to find an analytical solution to this equation, namely, rewriting and as periodic functions to remove the timedependence from Equation (5). Before we can do this, however, we must determine how to “absorb” the constant term, into the equation. Otherwise, there will be a remaining timedependent term when we ultimately divide both sides by (see MBW:Womersley Arterial Flow for more details). Analyzing the Effect of the Gravity Term  Approach #1The pressure gradient function for a heart beat is periodic, and therfore can be represented as a fourier series. The general form is: (8) where is the freqency, or in this case the beats per minute. Note that this unit is usually in radians/second. Recall that's Womersley flow velocity is defined as follows: (9) where is defined as (10) For the initial velocity, we are assuming Poiseuille flow, so (11) The gravitational effect is added to the pressure gradient function, as shown in Equation (7). Notice that the only constant term in Equation (8) is . Therefore, since is a constant, it only gets added to the term . And since this term only appears in the initial mean pressure gradient term, Equation (9) becomes: (12) Gravitational Effects on Flow RateSimilarly, only the initial flow rate, or volumetric flow, term gets effected by gravity. Recall that the Womersley flow rate is defined as: (13) Since all the terms will not be effected by adding a constant, only the term will be changed. Again, since we are assuming Poiseulle flow for the initial flow rate, we have: (14) Therefore the final rate is then: (15) ResultsTo see the effect that including gravity has on flow velocity and flow rate, I calculated the these quantities using the extended equations. An initial hypotheses was that gravity would effect the flow velocity and flow rate, but I was unsure of the magnitude of the change. Pressure Gradient and Parameters for Human AortaIn order to analyze the effects of gravity, we need to define a pressure gradient function for the human aorta. Later on, we will use a different pressure function for analysis of flow in a dog's femoral artery. The pressure gradient function, , we will use for the human aorta is from Womersley's paper. The coefficients for the Fourier series are as follows. Note that the units have been converted from mmHg/cm to Pa/m. These coefficients are derived in a rather complicated way. First, a heart monitor is used to collect the data, then a FastFourier Transform (FFT) on a computer is used to convert the signal to a function.
From the above plot of human blood pressure, the pressure drop in the arteries (the focus of this model) is around 15 mmHg. Assuming the blood travels along a tube 1m in length, which is a reasonable arm length, this equates to a mean pressure gradient of 15 mmHg/m, or 1999.8355 Pa/m. Thus, the value for the mean pressure gradient is approximated to be Pa/m. Other constants needed for this problem are listed in the table below.
The graph of the pressure gradient function is seen in Figure 4 below. Flow Velocity and Flow Rate with No GravityTo have a comparison to the flow velocity and rate when gravity is present, we plot both the velocity and volumetric flow with no gravity. A comparison between the velocity at the center of the artery (r = 0 cm) and midradius (r = .00075 cm) is shown in the Figure (5). The velocity is highest at the center of the artery and slowest near the boundary, which supports Womersley's result that the fastest flow occurs at the center of the artery. Also, notice that the peak velocity for the midradius occurs slightly before the peak velocity at the center. This is a result that is also verified by Womersley. Figure (6) shows the volumetric flow rate over time with no gravitational effects compared to the pressure gradient function. One of the main results from Womersley's model is seen here, as the pressure gradient spike occurs before the peak blood flow. Physiologically, this means that the pulse wave travels through the body first, with the blood flow following shortly after (a little more than 0.1 seconds in this case). Changes in Flow Velocity on EarthThe flow velocity is calculated using Equation (13) for various artery positions: , , , and . Just as expected, the flow velocity is highest in position (when the artery is pointed downward toward the earth), and lowest for (when the artery is pointed upward). A simplistic way to visualize this is to put your arm down at your side, or in position , and then point your arm upwards at position . The results obtained are graphed in Figure (7). All of the velocities peak at the center of the tube. These velocity profiles are calculated at time 0.15 seconds, approximately the time when the velocity peaks. Changes in Velocity in Various Gravitational FieldsTo determine the magnitude of the effect that gravity has on blood flow velocity, the velocity is calculated in both microgravity and on Jupiter. The gravitational fields are as follows:
The various flow velocities are shown in Figure (8). The table below lists the velocities at the various arm positions.
Gravitational Effect on Flow RateFlow rate was calculated using Equation (16) using the same positions. Just like for flow velocity, the flow rate was calculated using the gravity of Earth, Jupiter, and the Moon. The results are shown in Figure (9) below. .The table below shows the changes in flow rates at various gravities.
Analyzing the Effect of the Gravity Term  Approach #2The difference in this approach is that instead of combing the gravity term with the pressure gradient term from the start, this term is left as and propogated through the solution. So from Equation (7), we use the same approach we did in MBW:Womersley Arterial Flow and first look at the homogenous Bessel function.
(16) where . And we get the expected form: (17) The difference is now when we consider the nonhomogenous form, we get that (18) After substituting in these constants, we get that: (19) Also, using the same approach for flow rate, we get:
(20)
ResultsWe are using the same pressure gradient function as defined in Approach #1, and also the initial velocity and flow plots will be the same before considering gravity. We see a very similar velocity profile plot at various radii in the tube. As before, these velocity profiles are calculated at time 0.15 seconds, approximately the time when the velocity peaks.
Flow velocities on Jupiter, Earth and MoonAgain we are analyzing the effect of gravity on the arm positions , , , and . Using Equation (19) the flow velocities were calculated for the gravity on Jupiter, Earth and the Moon at these positions. The results are shown in the figure below. .
Flow Rates on Jupiter, the Moon and EarthJust like in Approach #1, we plot the various flow rates due to the different gravitational fields. The results are very similar.
Likewise, we also list the velocities at the various positions in the table below.
AnalysisBlood Flow Velocity with GravityAnalysis of Approach #1On Earth, the velocity increased 1.087 m/s between the supine (horizontal) position, and the vertical position with the arm held down at the side. The velocity decreased 1.086 m/s from the supine position when holding the arm overhead. Therefore, the change in velocity due to posture change is the same magnitude whether the flow is in the direction gravity or away. Also, in Figure (8) we see that the highest velocity occurs at position , and steadily decreases as the arm is rotated to the lowest velocity at . This confirms what we expect intuitively that the velocity would steadily decrease when rotating the arm from position (arm down at your side) to position (arm overhead). The gravitational effect on blood velocity using the moon's gravitational field is much less than those on the earth, which is what we would expect from the model. The velocity from the supine position to holding the arm overhead increased only 0.181 m/s. On Jupiter, we would expect a much greater gravitational effect on the flow velocity. This is exactly what we see from the model. The flow velocity increases from 1.004 m/s in the horizontal position to 3.113 m/s with the arm head down at the side. This difference in velocity on earth is about 50% of the amount for Jupiter. Interestingly, the amount the velocity decreased between the horizontal position and having the arm overhead is only about 0.8537 m/s for Jupiter. It would seem like the difference in flow velocity between holding the arm overhead to holding the arm down at the side should be similar, like it was for Earth. However, this model predicts a much greater change when the arm is held down at the side. Analysis of Approach #2The results using this approach are very similar to the results in Approach #1. The change in velocity when the arm is horizontal to when the arm is down at one's side is 1.275 m/s, or about 15% greater than we saw in Approach #1 but still on the same order of magnitude. However, in Approach #2 we see a much smaller decrease in velocity when holding the arm overhead. This difference is only about 0.6168 m/s or about 57% of what we saw in Approach #1. It seems like the difference in velocities from the horizontal position should be the same between position and position . For the moon we see very similar results to Approach #1. The amount that the velocity increases when the arm is down at the side is 0.211 m/s, and the amount the velocity decreases when the arm is held overhead is 0.212 m/s. For Jupiter, we get an increase of 2.975 m/s when the arm is held down, and a decrease of 4.055 m/s when the arm is overhead. So in the approach we see a much steeper decrease in velocity when the arm is overhead.
Volumetric Flow with GravityAnalysis of Approach #1A much larger change in seen when analyzing the flow rates. On Earth we see that the flow rate increases 5.12 cc/s when the arm is held down at the side, and a decrease of 5.12 cc/s when the arm is overhead. This is an increase of 0.31 liters of blood flow per minute in the artery. On the moon we see a much smaller increase in flow rate, around 0.854 cc/s, but a much larger decrease of around 2.6 cc/s when the arm is overhead. On Jupiter we see the most dramatic change which is what we expect intuitively. There is an increase and decrease in flow rate of about 12 cc/s which rougly equates to a change of 0.75 liters/min of blood flow. Analysis of Approach #2When using this approach, we see a greater increase in the flow rate. On the Earth, the flow rate when the arm is down at the side is 10.85 cc/s, whereas in Approach #1 it was 9.924 cc/s. However, there is a much smaller decrease when the arm is overhead. In this approach the flow rate decreases only 2.817 cc/s, whereas in Approach #1 it decreased 5.12 cc/s. Similar results are seen for Jupiter (a larger increase and a smaller amount of decrease in flow rate). However, on the Moon the amount of change between the overhead position and having the arm down at the side remains almost the same, at around 1.0 cc/s. Comparison Between Approach #1 and Approach #2Here we juxtapose the results of both approaches to see how they differ. VelocityBelow, the flow velocities are plotted for Earth, Jupiter, and the Moon for both approaches. Flow RateBelow, the flow rates are plotted for Earth, Jupiter, and the Moon for both approaches. DiscussionOne thing we need to verify is that the values for the flow velocities and flow rates are of the correct magnitude for arterial flow. I was unable to find any data that gives a quantitative change in flow velocity due to gravity. However, in ^{[8]} for a healthy subject the velocity is given as 100 cm/s. Before considering gravity, our models predicted a velocity of 1.004 m/s, so our value is accurate. In ^{[4]} they conduct an experiment to calculate the difference in pulse transit time due to gravitational effects. The subjects in their study are sitting in chair and then lying on their back. They find that the pulse traveling through a large artery of length 1.05 meter travels about 45.4 ms faster in the sitting position (somewhat equivalent to our position). Using approach #2 of our model, we get an increase of 1.275 m/s. For an artery of length 1.05 m, this equates to a difference in blood flow time of 585 ms. As a comparison, during exercise the flow rate increases approximately 4.13 cc/s from rest to 8090\% of maximum heart rate. This is data is taken from ^{[9]}, and depicts the flow rate in the common carotid artery. Note that the change in flow rate during exercise is similar to the change in flow rate on earth between positions and horizontal, which is a difference of 5.12 cc/s. Both approaches add a constant term to the pressure gradient, either increasing it or decreasing it based on gravity. This is precisely what the body does to compensate for gravitational effects. Blood pressure increases when you stand up, and decreases when you lie down. This is exactly what this model does by adding the term to the velocity solution. Last, elasticity of the blood vessels would effect blood flow, which is not considered in this paper. Future research could examine the difference in gravitational effects by extending the model to account for changes in the radius. Another area of extension would be to determine the effect of including bifurcations in the model, and how gravity influences the flow dynamics at these regions. Matlab CodeThe following code was used to calculate the velocity at zero gravity. Similar code was used for each angle position, as well as for calculating the velocity on Jupiter and the Moon (by varying g). Note that only two harmonics were used (even though we have fourier coefficients for six) because this seemed to give the most accurate curve. r = 0.0075; R = 0.0015; an = [.5915*i.5006 .1824*i+.4825 .17842*i+.1237 .03999*i.02679 .001683*i.0379 .0215*i.00187]; lambda = [2224.412*1i^1.5 3145.7935*i^1.5 3852.7944*i^1.5 4448.824*i^1.5 4973.936*i^1.5 5448.674*i^1.5]; u0 = (1/(4*mu))*A0*(r^2R^2); u1 = real( (an(1) + (g*sin(0))/(1i*6*pi)) * (1besselj(0,lambda(1)*r)/besselj(0,lambda(1)*R)) *exp(6*pi*t*i)); u2 = real( (an(2) + (g*sin(0))/(1i*6*pi*2)) * (1besselj(0,lambda(2)*r)/besselj(0,lambda(2)*R)) *exp(2*6*pi*t*i)); u = (u0+u1 + u2); The following code was used to calculate the flow rates for various angles, and for various gravitational fields. q0 = (1/(8*mu))*A0*pi*R^4; alpha = [R*sqrt(w/v) R*sqrt(2*w/v) R*sqrt(3*w/v) R*sqrt(4*w/v) R*sqrt(5*w/v) R*sqrt(6*w/v)]; q1 = pi*R^2*real( (an(1) + (g*sin(0))/(1i*6*pi)) * (1(2*alpha(1)*1i^1.5*besselj(1,alpha(1)*1i^1.5))/(1i^3*alpha(1)^2*besselj(0,alpha(1)*1i^1.5))) *exp(6*pi*t*i)); q2 = pi*R^2*real( (an(2) + (g*sin(0))/(1i*6*pi*2)) * (1(2*alpha(2)*1i^1.5*besselj(1,alpha(2)*1i^1.5))/(1i^3*alpha(2)^2*besselj(0,alpha(2)*1i^1.5))) *exp(2*6*pi*t*i)); q = (q0+q1 + q2); Recent CitationWomersley's work, although seminal, was completed many years ago. However, it still sees important uses. In a recent paper by Achille and Humphrey^{[10]}, it is again cited and used as a part of a larger computational system attempting to detect aneurisms in the brain. Achille and Humphrey are attempting to model several brain processes in order to predict possible aneurisms, in a clinical environment, based on patient data. Their work focus on the Willis circle of arteries that deliver blood to the brain. It is in this context that the Womersley's work presented here is used. The velocity of the blood arriving at the brain is required because the amount of blood involved is required in the predictions. It is important to note that Achille and Humphrey use Womersley's work as input to other problems, such as fluidsolid interaction models. References
