A Quantitative Approach to Blood Pressure (relevance of ...
A Quantitative Approach to Blood Pressure (relevance of mathematics)
• What is Blood Pressure
• How is it Measured
• Relationship to Blood Flow
• Fundamental Mathematics Notions:
o Rates
▪ Definition
• Velocity & Acceleration
o Proportional Reasoning
▪ Arithmetic
• Mr Tall-Mr Short problem
▪ Functional (direct vs inverse proportionality)
• Newton’s Gravitational Law
• Useful Physics Notions:
o Newton’s 2nd Law of Motion
o Pressure
o Density
o Viscosity
• Fluid Flow Equations & Poiseuille’s Law
o Derivation/Explanation
o Connection to Blood Pressure
• Quantitative Analysis of Poiseuille’s Law and (potential) Health Consequences
o Hypertension
▪ Impacts Heart Performance (Heart Attack)
▪ Impacts Blood Vessel Function (clots vs hemorrhage)
o Stroke
▪ Neurological condition due to decreased blood flow to brain
o Poor Vision
▪ Blood vessels in retina hemorrhaging
What is Blood Pressure
Blood Pressure is the force exerted by the blood against the blood vessel walls.
[pic]
It is measured in mercury manometers (mm HG). For example, if the blood pressure is 100 mm HG, it is great enough to lift a column of Mercury (Hg) 100 mm (see illustration below).
How is it Measured
A catheter or cannula (or tube) can be inserted into the blood vessel, and electric transducers can be used for measuring pressure fluctuations (very intrusive to body).
Typically, the auscultatory method is used for clinical determination of blood pressure (this method uses the cuff connected to a sphygmomanometer, and a stethoscope found in doctors’ offices).
Blood pressure readings are usually given as 2 numbers: for example, 110 over 70 (written as 110/70). The first number is the systolic blood pressure reading, and it represents the maximum pressure exerted when the heart contracts. The second number is the diastolic blood pressure reading, and it represents the pressure in the arteries when the heart is at rest.
When blood flow is turbulent (as opposed to smooth, streamlined laminar flow), it produces vibrations in the blood and surrounding tissues that can be heard with a stethoscope. These sounds are called korotkoff sounds.
In adults, the systolic pressure should be less than 120 mmHg, and the diastolic pressure should be less than 80 mmHg.
[NOTE]: Statistical underpinnings above (mean, standard deviation)
• Pre-high blood pressure: systolic pressure consistently 120 to 139, or diastolic 80 to 89
• Stage 1 high blood pressure: systolic pressure consistently 140 to 159, or diastolic 90 to 99
• Stage 2 high blood pressure: systolic pressure consistently 160 or over, or diastolic 100 or over
• Hypotension (blood pressure below normal): may be indicated by a systolic pressure lower than 90, or a pressure 25 mmHg lower than usual
Explanation for measurements of Blood Pressure
The cuff is there to put pressure on the artery in your arm. At first, it is inflated so that the pressure from the outside is greater than the blood pressure - so the blood flow is cut off! Then the air is slowly let out, and the pressure in the cuff drops gradually. At the same time, the nurse puts a stethoscope on your arm and listens for the blood to start flowing. At the moment when the nurse can hear the blood gushing, the blood pressure in the blood pressure cuff (outside pressure) is equal to the blood pressure in the artery, or as you can see on the picture, the two graphs intersect.
[pic]
This story is incomplete, however. Remember that the heart pumps blood by contracting periodically. Blood gushes through the body in waves, and blood pressure also varies like a wave. Therefore, by listening for the first time blood pressure exceeds the outside pressure, we only find the highest value of the blood pressure, also known as the systolic pressure. After that, blood pressure will fluctuate - sometimes going above the outside pressure, and sometimes dipping below it, until finally the blood pressure cuff will lose so much pressure that the blood pressure fluctuations will remain above it, and blood will flow continuously (in the figure, the line intersects with the lowest part of the wave). At this moment we have reached the lowest blood pressure, also known as the diastolic pressure. This is why blood pressure is always given as two numbers, say 120/60 - the first is the peak blood pressure, and the second is the lowest blood pressure.
Relationship of Blood Pressure to Blood Flow
The rate at which fluid flows through a tube is expressed as the volume that passes by a specific point per unit of time. Blood flow is usually reported as either milliliters per minute or liters per minute. The blood flow out of the heart (or equivalently through the aorta) is approximately 5L per minute.
Blood flow (F) through a vessel is proportional to the pressure gradient in that vessel, and inversely proportional to the resistance (R) in the blood itself (such as it’s viscosity). Consider a tube of length l and radius r with a pressure ([pic]) applied at the left opening, and a pressure ([pic]) applied at the right opening.
[pic][pic]
[NOTE]: If [pic]= [pic] then there will be no flow through the tube. If [pic]> [pic] then fluid will flow in the direction from [pic] to [pic] (see diagram above).
Fundamental Mathematics Notions
Walk into math realm…
Rates
A rate is a special kind of ratio, indicating a relationship between two measurements with different units, such as miles to gallons or cents to pounds. The division operator, when dealing with rates, is sometimes expressed as "per". Also, if the units are expressed without abbreviation, the top unit is usually plural while the bottom is singular. For example, (55 miles)/(1 hour) may be written as 55 miles per hour (mph).
Proportionality
A typical definition for proportionality would go “In mathematics and in physics, proportionality is a mathematical relation between two quantities.” There are two different views of this “mathematical relation”; one is based on ratios and the other is based on functions.
Arithmetic Viewpoint
In many textbooks proportionality is expressed as an equality of two ratios:
[pic]
Given the values of any three of the terms, it is possible to solve for the fourth term.
Functional Viewpoint
A scientist has a much different view of proportionality. Given the following equation for the force of gravity (according to Newton):
[pic]
The scientist would say that the force of gravity between two masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses. From this perspective proportionality is a functional relationship between variables in a mathematical equation.
[pic]
[pic]
Useful Physics Notions
Newton’s 2nd Law (Motion)
Intuitively, in order to move an object, you have to exert a force on it. In general, the higher the force, the faster the object moves (movement should invoke the notions of velocity and acceleration in your mind). Stated another way, force is the only item that can cause a change in an object’s velocity (from zero or constant value). When there are changes in an object’s velocity, the object is said to be accelerating (or decelerating). This leads to the following definition:
Force is that which causes acceleration/deceleration in a body
If we denote an object’s acceleration as [pic] and the (resultant) force exerted on it as [pic], then we can express the definition above as:
[pic]
The [pic] symbol is the mathematics to be read as ‘if and only if’ (remember mathematics is a language). This idea is that which Newton’s Second Law expresses. In words it goes as follows: the acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass. This can be written as:
[pic]
or equivalently
[pic]
Pressure
Intuitively, the pressure exerted on an object should be related to the force exerted on the object and the amount of area the force is acting on. It should be noted that when measuring the pressure on an object, the force is assumed to be acting perpendicular to the object’s surface (or ‘normal’ to the surface). Formally, the definition for pressure (with regards to fluids) goes as follows: If [pic] is the magnitude of the normal force on a surface area [pic], then the pressure [pic] is defined as the ratio of force to area:
[pic]
[NOTE]: The pressure in a fluid is not the same at all points. For example a fluid at rest in a container has different pressure values dependent upon the depth considered.
In view of the fact that the pressure in a fluid depends only upon depth, any increase in pressure at the surface must be transmitted to every point in the fluid. This was recognized by the French scientist Blaise Pascal, and is called Pascals’s Law:
A change in pressure applied to an enclosed fluid is transmitted undiminished to every point of the fluid and the walls of the containing vessel
An important application of Pascal’s Law is the hydraulic press (see below). A force [pic] is applied to a small piston of area [pic]. The pressure is transmitted through the fluid to a larger piston of area [pic]. Since the pressure is the same on both sides, it follows that:
[pic]
Therefore the force [pic] is larger than [pic] by the multiplying factor [pic].
Useful Resources (Website Links):
• College- [Physics Cramlets((conversions and algebra)]:
• Conversion Calculator & Table:
• Great Website for Mathematics & Blood Flow (two problems below from this website):
• Two useful problems and prose (from above website):
1) Exercise uses up a lot of energy, which the cells derive from oxidizing glucose. Both glucose and oxygen have to be delivered by the blood. This means that the heart has to work harder to pump more blood through the body. This means it has to beat faster in order to achieve a higher throughput, as described by this equation:
(Blood Flow) = (Heart Rate) X (Stroke Volume)
Heart rate for a human being at rest is about 70 beats/min. During vigorous exercise, heart rate can increase dramatically (the rule of thumb given for maximal heart rate is 220 minus your age). This will result in an increase in blood flow.
Q: Suppose that the heart rate doubled, but, the stroke volume remained constant. How would the flow rate change?
a) Blood flow will double
b) Blood flow will triple
c) Blood flow will halve
d) Blood flow will remain the same
2) It is clear that the higher the pressure exerted by the heart, the faster blood will flow. This is an example of a direct or proportional relationship between two quantities. There is also another factor which controls the blood flow rate, and it is the resistance of the blood vessels to blood flow. This resistance is simply due to the width of the vessels - it's hard to push a lot of blood through a thin tube! Thus, we have an inverse relationship between blood vessel resistance and the blood flow rate - the higher the resistance, the slower the flow rate.
Now let's see if this makes any sense in numbers. The usual pressure difference between the left and right ventricles is about 100 mmHg. The normal cardiac output (the blood flow in the above equation) is about 5 liters/minute. The total peripheral resistance is about 20 (mmHg*min/liters).
Q: Assuming a constant pressure difference, which resistance value will raise blood flow to a rate of 25 L/min?
a) Resistance = 80
b) Resistance = 20
c) Resistance = 4
d) Resistance = 1
[pic][pic]
Conclusion(s):
Although the body can tolerate increased blood pressure for months and even years, eventually the heart may enlarge (a condition called hypertrophy), which is a major factor in heart failure.
Blood pressure is measured in millimeters of mercury (mm Hg). Blood pressure is now categorized as normal, prehypertensive (formerly normal to high-normal), and hypertensive (which is further divided into Stage 1 and 2, according to severity). High blood pressure is generally considered to be a blood pressure reading greater than or equal to 140 mm Hg (systolic) or greater than or equal to 90 mm Hg (diastolic). Blood pressure readings in the prehypertension category (120-139 systolic or 80-89 diastolic) indicate an increased risk for developing hypertension.
High blood pressure, also called hypertension, is, simply, elevated pressure of the blood in the arteries. Hypertension results from two major factors, which can be present independently or together:
• The heart pumps blood with excessive force.
• The body's smaller blood vessels (known as the arterioles) narrow, so that blood flow exerts more pressure against the vessels walls.
It is crucial to "Know Your Numbers" with respect to Blood Pressure. Life-threatening complications can develop over a course of years when hypertension exists. Increased pressure on the inner walls of blood vessels make the vessels less flexible over time and more vulnerable to the buildup of fatty deposits in a process known as atherosclerosis.
Hypertension also forces the heart to work harder to pump adequate blood throughout the body. This extra work causes the muscles of the heart to enlarge, and eventually the enlarged heart becomes inefficient in pumping blood. An enlarged heart may lead to heart failure, in which the heart can not pump enough blood to meet the body's needs.
In some people, the system that regulates blood pressure goes awry: arteries throughout the body stay constricted, driving up the pressure in the larger blood vessels. Sustained high blood pressure - above 140/90 mm Hg, according to most experts - is called hypertension. About 90 percent of all people with high blood pressure have "essential" hypertension - meaning that it has no identifiable cause. In the remaining 10 percent of cases, the elevated blood pressure is due to kidney disease, diabetes, or another disorder.
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