︠b98ee23d-a522-480b-9969-ccf189067d75i︠ %html This is an adapted version of Section 2.2 in Gregory V. Bard, "Sage for Undergraduates", AMS.

## Biology: Clogged Arteries and Poiseuille’s Law

If an American dies of natural  causes, the probability it is of heart disease is $35.30\%$, of various forms of cancer is $33.94\%$, and of any other cause is $30.75\%$. Considering that heart disease kills roughly a third of us, the realities of clogged arteries and cholesterol should concern us all.

Blood flowing through an artery is governed by the same equation as water flowing in a pipe or oil flowing through a hose. This applies to all such flows of fluids through a cylinder, except for turbulent flow (where the fluid is churning around and tumbling over itself) and compressible flow (such as air flowing in a pneumatic hose). The equation is called Poiseuille’s Law, named for Jean-Lonard Marie Poiseuille (1797-1869), whose dissertation  was applying this mathematics to the human aorta.

In this project, we’re going to explore the relationships between some of the variables in Poiseuille’s Law. We are going to display those interdependencies through tables. In order to create a table, we will use a for-loop with print statements inside.

#### The Background

The formula itself is $$\dot{V} = \left(\Delta P\right)\cdot \frac{\pi r^4}{8\mu L},$$ and regrettably, we do not have time to derive it here. However, I strongly recommend the curious reader to refer to the article “Blood Vessel Branching: Beyond the Standard Calculus Problem”, by Prof. John Adam, published in the Mathematics Magazine, Vol. 84, No. 3, pp. 196-207, by the Mathematical Association of America, in 2011 (it is available on your course site).

Let’s take a moment to identify the meaning of each of those variables.
• The $\dot{V}$ is the volume flow rate, in units such as cubic centimeters per minute. In a major artery, a typical flow might be $100$cm$^3$/min.
• The radius of the artery is $r$, in units such as centimeters. A major artery might be $0.3$cm (or very roughly $\frac18$th of an inch) in diameter.
• The length of the artery is $L$, again in units such as centimeters. A major artery might be $20$cm (or $8$ inches) in length.
• The unit of pressure $P$ (and the pressure difference $\Delta P$) can vary.
• In the metric system, because forces are measured in Newtons, the pressure units then become Newtons per square meter, also called Pascals, i.e., $1$N/m$^2=1$Pa. Using Newtons per square meter is kind of funny because arteries aren’t several square meters in area. Often also the units "bar" and "millibar" (mbar) are used where $1$bar$= 10^5$Pa $= 10^5$N/m$^2$ and thus $1$mbar $= 10$Pa $= 10$N/m$^2$.
• The standard unit in the American system of units is pounds per square inch, because forces are measured in pounds.
• Another unit of pressure used is “atmospheres” where atmospheric pressure is $1$ atmosphere. That’s $101.325$N/m$^2 = 1013.25$mbar (or $14.6959$ pounds per square inch).
• Medicine has been around for a long time. For historical reasons, it is normal to measure blood pressure in a very archaic unit, called "millimeters of Mercury" When you go to the doctor and she tells you that your blood pressure is “$120$ over $80$,” that $120$ means "$120$ millimeters of Mercury", abbreviated $120$mmHg. Just take it as a unit, and consider $120$ to be "normal". If you have $145$mmHg for the first number (called the systolic value, while the second number is the diastolic value), you might be diagnosed as aving "hypertension". A score of $180$mmHg systolic can result in organ damage and is called a "hypertensive emergency".
• Viscosity is a physics variable that you might or might not have seen before, and is denoted $\mu$. It represents how "sticky" or "thick" a fluid might be. Since blood is the fluid in question, and we won’t consider other fluids, we can just take μ to be a constant. The usual unit is a "poise". A typical value for blood might be $\mu = 0.0035$ poise. In comparison, the oil in your car might have $\mu = 0.250$ poise (because it is sticky) and water has $\mu = 0.001$ millipoise (because it is not sticky). Honey is very sticky, and has a viscosity of $\mu=20$ poise. We can’t use the poise as our unit, because we’re using an antique unit for pressure. For us, $$\mu = 4.375359\ldots \times 10^{-8}$$ is just a constant in the formula. The units of viscosity turn out to be "min$\times$mmHg", which is extremely strange, but that is what will make the correct units work out in the formula.

You’re going to create some tables and the cooresponding graphs that can be used to help a physician understand the numerical realities of that quartic relationship in Poiseuille’s Law between arterial radius and blood flow rate. A figure showing the specific relationship should accompany each table with the values from the table emphasized; the Sage documention for 2D plotting might be helpful for this (alternatively, you can always evaluate plot? or list_plot? to show to doc-pages on these functions, although without pictures).

The tables will allow some variables to change, while most variables are locked at "standard values" that are chosen to be very typical. The standard values for our variables here are $\mu = 4.375359 \times 10^{−8}$ for the viscosity, $L = 20$cm for the artery length, $r = 0.3$cm for the arterial radius, $\dot{V} = 100$cm$^3$/min for the blood flow rate, and finally $\Delta P$ is $0.0275$mmHg.

• Your first table should keep our standard values for $\mu$, $L$, $\Delta P$, and compute $\dot{V}$ based on $r$, using Poiseuille’s Law. The $r$ should be computed based upon the percent blockage, from $0\%$ up to $50\%$, in $2\%$ increments. The first column of the table should be the percent blockage, the second column should be the blood flow rate or $\dot{V}$. The accompanying figure should show the blood flow rate in terms of blockage, and also emphasize the values in the table.
Hint: You have to combine a plot with a list_plot here; add axes labels and experiment with colors etc. For an example, see the figure below.
• Your second table should keep our standard values for $\mu$, $L$, and $\dot{V}$. Again, the $r$ should be computed based upon the percent blockage, from $0\%$ up to $50\%$, in $2\%$ increments. This time, you will compute the $\Delta P$ required to achieve the flow rate of $100$cm$^3$/min given the blockage (the $\dot{V}$ given the $\Delta P$), using Poiseuille’s Law. The first column of the table should be the percent blockage, the second column should be the $\Delta P$. The third column should represent $\Delta P$ as the "percent of normal". In other words, $\frac32$ the normal $\Delta P$ should be reported as $150\%$. The accompanying figure should should show this percentage of normal pressure difference as function of blockage, again with the values from the table emphasized.
• The third table should keep our standard values for $\mu$, $L$, and $\Delta P$. The first column should be the "percent normal flow" from $100\%$ down to $25\%$, in steps of $3\%$. The second column should be the flow rate, in cubic centimeters per second, identified by that percentage. The third column should be the radius implied by this, as computed by Poiseuille’s Law. The fourth column should be the percent blockage (In other words, $0.15$cm radius is a $50\%$ blockage because $0.15/0.3 = 0.5$). The accompanying figure should show blockage as function of the percentage of normal flow, again with the values from the table emphasized.

You can check your work by plugging into the original equation, but you might also find it useful to compare your first table to the following, which is given in increments of $6\%$.

$0\%$ blockage implies $99.9617636500122$cm$^3$/min.
$6\%$ blockage implies $78.0450430095128$cm$^3$/min.
$12\%$ blockage implies $59.9466058383290$cm$^3$/min.
$18\%$ blockage implies $45.1948885141475$cm$^3$/min.
$24\%$ blockage implies $33.3494195216211$cm$^3$/min.
$30\%$ blockage implies $24.0008194523679$cm$^3$/min.
$36\%$ blockage implies $16.7708010049720$cm$^3$/min.
$42\%$ blockage implies $11.3121689849831$cm$^3$/min.
$48\%$ blockage implies $7.30882030491648$cm$^3$/min.
$54\%$ blockage implies $4.47574398425329$cm$^3$/min.

In this case, the accompanying figure should look like this: #### Notes

 Based on 2010 data, from the Center for Disease Control and Prevention’s website “FastStats.” This data excludes (as unnatural causes) any homicides, suicides, and accidents.
 Poiseuille, J-L. M., Recherches sur la force du coeur aortique, D.Sc., École Polytechnique, Paris, France. 1828.
︡5363d00b-e6d0-4a31-ae6c-b353e332b2dd︡{"done":true,"html":"\n

\n \nThis is an adapted version of Section 2.2 in Gregory V. Bard, \"Sage for Undergraduates\", AMS.

\n

## \n Biology: Clogged Arteries and Poiseuille’s Law\n

\nIf an American dies of natural  causes, the probability it is of heart disease\nis $35.30\\%$, of various forms of cancer is $33.94\\%$, and of any other cause\nis $30.75\\%$. Considering that heart disease kills roughly a third of us, the\nrealities of clogged arteries and cholesterol should concern us all.

\nBlood flowing through an artery is governed by the same equation as\nwater flowing in a pipe or oil flowing through a hose. This applies to all such\nflows of fluids through a cylinder, except for turbulent flow (where the fluid\nis churning around and tumbling over itself) and compressible flow (such as\nair flowing in a pneumatic hose). The equation is called Poiseuille’s Law,\nnamed for Jean-Lonard Marie Poiseuille (1797-1869), whose dissertation \nwas applying this mathematics to the human aorta.

\nIn this project, we’re going to explore the relationships between some of\nthe variables in Poiseuille’s Law. We are going to display those interdependencies \nthrough tables. In order to create a table, we will use a for-loop with \nprint statements inside.

\n\n

#### \n The Background\n

\nThe formula itself is \n$$\\dot{V} = \\left(\\Delta P\\right)\\cdot \\frac{\\pi r^4}{8\\mu L},$$\nand regrettably, we do not have time to derive it here. However, I strongly\nrecommend the curious reader to refer to the article “Blood Vessel Branching: \nBeyond the Standard Calculus Problem”, by Prof. John Adam, published \nin the Mathematics Magazine, Vol. 84, No. 3, pp. 196-207, by the Mathematical \nAssociation of America, in 2011 (it is available on your course site).

\nLet’s take a moment to identify the meaning of each of those variables.\n
\n
• The $\\dot{V}$ is the volume flow rate, in units such as cubic centimeters per\nminute. In a major artery, a typical flow might be $100$cm$^3$/min.
• \n
• The radius of the artery is $r$, in units such as centimeters. A major\nartery might be $0.3$cm (or very roughly $\\frac18$th of an inch) in diameter.
• \n
• The length of the artery is $L$, again in units such as centimeters.\nA major artery might be $20$cm (or $8$ inches) in length.
• \n
• The unit of pressure $P$ (and the pressure difference $\\Delta P$) can vary.\n
\n
• In the metric system, because forces are\nmeasured in Newtons, the pressure units then become Newtons per\nsquare meter, also called Pascals, i.e., $1$N/m$^2=1$Pa. Using Newtons per square meter is kind of funny\nbecause arteries aren’t several square meters in area. Often also the units \"bar\" and \"millibar\" (mbar) are \nused where $1$bar$= 10^5$Pa $= 10^5$N/m$^2$ and thus $1$mbar $= 10$Pa $= 10$N/m$^2$. \n
• \n
• The standard unit in the American\nsystem of units is pounds per square inch, because forces are measured in pounds.\n
• \n
• Another unit of pressure used is “atmospheres” where atmospheric pressure is\n$1$ atmosphere. That’s $101.325$N/m$^2 = 1013.25$mbar (or $14.6959$ pounds per square inch).\n
• \n
• Medicine has been around for a long time. For historical reasons, it\nis normal to measure blood pressure in a very archaic unit, called\n\"millimeters of Mercury\" When you go to the doctor and she tells\nyou that your blood pressure is “$120$ over $80$,” that $120$ means \"$120$\nmillimeters of Mercury\", abbreviated $120$mmHg. Just take it as\na unit, and consider $120$ to be \"normal\". If you have $145$mmHg\nfor the first number (called the systolic value, while the second number is the diastolic value), \nyou might be diagnosed as aving \"hypertension\". A score of $180$mmHg systolic can result in\norgan damage and is called a \"hypertensive emergency\".
• \n
\n
• \n
• Viscosity is a physics variable that you might or might not have seen\nbefore, and is denoted $\\mu$. It represents how \"sticky\" or \"thick\" a\nfluid might be. Since blood is the fluid in question, and we won’t\nconsider other fluids, we can just take μ to be a constant. The usual\nunit is a \"poise\". A typical value for blood might be $\\mu = 0.0035$\npoise. In comparison, the oil in your car might have $\\mu = 0.250$ poise\n(because it is sticky) and water has $\\mu = 0.001$ millipoise (because it\nis not sticky). Honey is very sticky, and has a viscosity of $\\mu=20$ poise.\nWe can’t use the poise as our unit, because we’re using an antique\nunit for pressure. For us,\n $$\\mu = 4.375359\\ldots \\times 10^{-8}$$\nis just a constant in the formula. The units of viscosity turn out to\nbe \"min$\\times$mmHg\", which is extremely strange, but that is what will\nmake the correct units work out in the formula.\n
• \n

\n

#### \n Your Challenge\n

\nYou’re going to create some tables and the cooresponding graphs that can be used to help\na physician understand the numerical realities of that quartic relationship\nin Poiseuille’s Law between arterial radius and blood flow rate. A figure showing the specific \nrelationship should accompany each table with the values from the table emphasized; the \nSage documention for 2D plotting\nmight be helpful for this (alternatively, you can always evaluate plot? or list_plot? to show \nto doc-pages on these functions, although without pictures).

\nThe tables will allow some variables to change, while most variables\nare locked at \"standard values\" that are chosen to be very typical. The\nstandard values for our variables here are $\\mu = 4.375359 \\times 10^{−8}$ for\nthe viscosity, $L = 20$cm for the artery length, $r = 0.3$cm for the arterial\nradius, $\\dot{V} = 100$cm$^3$/min for the blood flow rate, and\nfinally $\\Delta P$ is $0.0275$mmHg.

\n
\n
• Your first table should keep our standard values for $\\mu$, $L$, $\\Delta P$, and\ncompute $\\dot{V}$ based on $r$, using Poiseuille’s Law. The $r$ should be\ncomputed based upon the percent blockage, from $0\\%$ up to $50\\%$, in\n$2\\%$ increments. The first column of the table should be the percent\nblockage, the second column should be the blood flow rate or $\\dot{V}$. \nThe accompanying figure should show the blood flow rate in terms of blockage,\nand also emphasize the values in the table.
Hint: You have to combine a plot with a list_plot here; \nadd axes labels and experiment with colors etc. For an example, see the figure below. \n
• \n
• Your second table should keep our standard values for $\\mu$, $L$, and $\\dot{V}$.\nAgain, the $r$ should be computed based upon the percent blockage,\nfrom $0\\%$ up to $50\\%$, in $2\\%$ increments. This time, you will compute\nthe $\\Delta P$ required to achieve the flow rate of $100$cm$^3$/min given the\nblockage (the $\\dot{V}$ given the $\\Delta P$), using Poiseuille’s Law. The first\ncolumn of the table should be the percent blockage, the second\ncolumn should be the $\\Delta P$. The third column should represent $\\Delta P$\nas the \"percent of normal\". In other words, $\\frac32$ the normal $\\Delta P$\nshould be reported as $150\\%$. The accompanying figure should should show \nthis percentage of normal pressure difference as function of blockage, again with the values from the \ntable emphasized.\n
• \n
• The third table should keep our standard values for $\\mu$, $L$, and\n$\\Delta P$. The first column should be the \"percent normal flow\" from\n$100\\%$ down to $25\\%$, in steps of $3\\%$. The second column should\nbe the flow rate, in cubic centimeters per second, identified by that\npercentage. The third column should be the radius implied by this,\nas computed by Poiseuille’s Law. The fourth column should be the\npercent blockage (In other words, $0.15$cm radius is a $50\\%$ blockage\nbecause $0.15/0.3 = 0.5$). The accompanying figure should show blockage as function of \nthe percentage of normal flow, again with the values from the table emphasized.\n
• \n

\nYou can check your work by plugging into the original equation, but you\nmight also find it useful to compare your first table to the following, which\nis given in increments of $6\\%$.

\n\n$0\\%$ blockage implies $99.9617636500122$cm$^3$/min.
\n$6\\%$ blockage implies $78.0450430095128$cm$^3$/min.
\n$12\\%$ blockage implies $59.9466058383290$cm$^3$/min.
\n$18\\%$ blockage implies $45.1948885141475$cm$^3$/min.
\n$24\\%$ blockage implies $33.3494195216211$cm$^3$/min.
\n$30\\%$ blockage implies $24.0008194523679$cm$^3$/min.
\n$36\\%$ blockage implies $16.7708010049720$cm$^3$/min.
\n$42\\%$ blockage implies $11.3121689849831$cm$^3$/min.
\n$48\\%$ blockage implies $7.30882030491648$cm$^3$/min.
\n$54\\%$ blockage implies $4.47574398425329$cm$^3$/min.
\n
\n
\nIn this case, the accompanying figure should look like this:

\n \n

\n

#### \nNotes \n

\n Based on 2010 data, from the Center for Disease Control and Prevention’s website “FastStats.” \n This data excludes (as unnatural causes) any homicides, suicides, and accidents.
\n Poiseuille, J-L. M., Recherches sur la force du coeur aortique, D.Sc., École Polytechnique, \n Paris, France. 1828.\n
"} ︠69a41ecc-2aca-4778-bf80-bfb3e471bbf4︠