Math of the Heart & the Cardiovascular System
It is said that the heart is similar in size to a fist; in other words, a closed hand, making a fist, represents the size of our heart. It weighs a bit less than one pound.
During its contraction, the impact is felt on the wall of the chest, between ribs five and six. To be more precise, this impact is felt most strongly below the left nipple and approximately 8 cm (about 3 inches) to the left of the symmetric axis. The symmetric axis is the vertical line that divides the human body in two equal parts.
The function of the heart is to supply the proper amount of blood for each part of our body. Some organs will receive more blood than others, depending on the actual function of the organ. It depends as well on the activity performed in a given period of time. For instance, if a person is running or practicing sports, muscles are consuming large amounts of energy and they must receive more blood flow than they do when the person is at rest. During a period of exercise, blood flow will be diverted to muscles from other organs, such as the digestive system. Definitely, the human body is a smart system that can adjust and adapt to different situations.
Cardiac Cycle
There are two phases in a cardiac cycle; a contraction period called systole, and a relaxation period called diastole.
The contraction and relaxation of the heart represents one heartbeat. One heart beat is one cardiac cycle.
Ventricles are relaxed during diastole. This is the moment when ventricular filling happens.
Ventricles contract during systole, propelling blood into the pulmonary and systemic circuits. An average normal cardiac cycle occurs around 70 to 72 times per minute. This number represents the heart rate. Given a pulse rate of 70 to 72, the time for a cardiac cycle is 0.8 second.
The cardiac cycle varies in inverse proportion to the size of a person. This statement is extended as well to all warm-blooded animals. It makes sense that the circulation on a small body will be completed in a small period of time and, therefore, determining more "mini-cycles" per a given period of time than a larger body. The heart of an elephant for example, beats around 25 times per minute. The heart of a mouse beats around 700 times per minute. In general, if the body is small, the consumption of oxygen by the tissues of the animal will be faster in comparison to a larger animal.
Heartbeat Rate Variations
Heartbeat is a bit faster in women than in men. Comparing fetuses, the heart rate of a female fetus is approximately 140 to 145, while for a male fetus is 130 to 135. Heart rate is also influenced by age. At birth the rate is approximately 140 beats per minute. When the individual is three years old, the rate is 100 beats per minute. Youngsters have a rate of 90 beats per minute; adults have it at about 75 beats per minute, while elderly people have it at 70 to 80 beats per minute.
The heartbeat rate is influenced by the posture of our body. Standing up the heartbeat rate is 80; sitting is 70 and when lying down the heartbeat rate is 66. Therefore, some patients are told to lie down when physicians want to slow down the heartbeat rate.
A chemical reaction is always influenced by the variation of temperature. Normally, the chemical reactions tend to be faster when temperature increases. The heartbeat is a function of several chemical reactions and, therefore, it will depend on the temperature of the body. In any chemical reaction the rate is doubled when the temperature increases by 10 oC, but it is reduced by half if the temperature decreases 10 oC. Therefore when a person has a fever, the temperature increases and all chemical processes in our body will occur a bit faster without increasing the physical activity. This will induce our heart to speed up the ejection of blood and therefore, to have a higher heartbeat rate.
It has been demonstrated experimentally that if an animal heart is filled with a hot liquid, the heartbeat rate increases proportionally with the temperature, but up to a maximum point of about 44 oC (111.2 oF). If this point is reached, the heart will stop beating. On the other extreme, if an animal heart is filled with cold liquid, the heartbeat rate decreases and the heart stops beating at about 17 oC (62.6 oF). In order to be able to practice cardiac surgery, the heart is slowed down by surgeons, by an induced hypothermia, which is accomplished by cooling the blood. Hypothermia slows cellular metabolism decreasing the need of oxygen.
Cardiac Output
It is estimated that in a person at rest, the volume of blood ejected in each systole is around 70 to 80 ml. This amount of blood moves from the left ventricle to the aorta. It is called the stroke volume. Similarly, the same amount is forced from the right ventricle to the pulmonary artery. Therefore, the total cardiac output is around 140 to 160 ml. Given a pulse rate of 70 and a stroke volume of 80 ml, the amount of blood that leaves the left ventricle is 70 x 80 ml = 5.6 liters per minute. This is called the cardiac output.
Properly written and considering the transformation of units, we will have the following expression:
While exercising, the cardiac output may be doubled.
The heart has the ability to regulate the cardiac output depending on de activity of an individual. If the demand for oxygen increases, the oxygen supply must also increase. Cardiac output is determined by heart rate, preload, afterload and contractility.
Preload
A way to easily understand the meaning of preload, is to pay attention to the prefix "pre," which means "prior to" or "before." Therefore, preload refers to the amount of blood prior to the load (contraction). While researching for the term "preload," I found several descriptions such as that preload is the end volumetric pressure that stretches the ventricle to its greatest dimension depending on the needed demand. When a person is running, the demand will make the heart beat faster to supply the greater amount of oxygen that is needed.
The volume of the blood is directly proportional to the force used by the ventricle to contract. However, if the volume exceeds the accepted limit, the effectiveness of the contraction will decrease. This is called Starling's Law of the Heart.
Afterload
Afterload is a term that indicates what happens after a chamber of the heart has been loaded with blood. Specifically, the afterload is roughly the pressure that the heart has to push against. For the left ventricle, this is the pressure in the aorta.
Contractility
Contractility is the natural characteristic of the heart to contract during systole. This is due to the fact that the cardiac muscle fibers shorten during systole.
Contractility is important for the circulation of the blood in the system. Poor contractility decreases the ejection of blood with the immediate consequence of reducing stroke volume. In a chain reaction, the cardiac output is reduced, and the rest of the organs will not receive the proper amount of blood needed to function normally.
Rate of Flow
It is common that individuals admitted into the ER room with life-threatening situations are given intravenous fluids. After physicians have taken all the precautions and in order to keep their patients within the normal range of vital signs, they order an infusion of some volume of fluid within a stated amount of time. For example, they might order to administer 1000 ml of saline solution in 8 hours, which is equivalent to 125 ml per hour. These are units of flow. To illustrate the change of units, I will write the following expression:
1000ml/8hours = 125 ml/hour = 3000 ml/24 hours
To calculate the flow rate, it is needed the information about the volume of the fluid to be infused, the calibration of the tubing (also called the drop factor) and the time ordered for the infusion that normally is applied in drops. Drops are units of volume and are represented by gtt. The unit gtt is an abbreviation of the Latin "gutta." In the metric system, 1 drop is equivalent to 0.05 mL or 20 drops per each milliliter. Some old home-cooking-recipes, show 24 drop equals ¼ teaspoon, which according to US definitions, it makes the drop equals to 0.051 milliliters. This number is very close to the metric equivalence.
Blood Flow
The circulation of blood in the arteries is controlled by a balance of blood flow, blood pressure and peripheral vascular resistance. Blood flow (which is measured as a volumetric flow rate) is calculated by multiplying the speed of the blood times the cross-sectional area. Blood flow represents how fast the volume of blood passes through the cross section. The units of blood flow are calculated in milliliters per minute.
Blood Flow = (blood velocity) x (cross-sectional area)
Cross-sectional Area of Vascular Tree
A vascular tree is a set of vessels distributed in an anatomic structure of branches obtained by a continuous division of the vessels. It works as a blood distribution network very similar to the branches on a tree. In the picture below, a simple idea of a division of vessels is presented, although this model does not really represent how the vessels divide.
In this other picture, the model represents a closer idea of the division of vessels, although is not yet the perfect model.
The total cross-sectional area of the vascular tree is a maximum value in the capillaries. This can be demonstrated using a mathematical model of geometric series. The summation of the entire cross sectional areas of the vascular tree will be greater than the initial cross section area of the aorta. The division of the vessels starts right there with the aorta increasing the number of cross sections; and as the division continues, the flow of the blood keeps its circulation towards the tissue. Exactly the reverse occurs starting from capillaries to venules to veins and back to the heart.
Hematocrit
Hematocrit is the volume percent of red cells in the blood. To find the hematocrit of an individual, a blood sample is centrifuged in a tube. Red cells will accumulate at the bottom of the tube; the hematocrit is defined as the height of the red cell column divided by the total height and is usually expressed as a percentage.
Viscosity
Viscosity is a measure of resistance of the fluid to flow. It is also considered "an internal friction." The viscosity of liquids decreases rapidly when the temperature increases. Blood is a liquid composed of plasma and particles such as red cells; its viscosity depends on the viscosity of the plasma and on the hematocrit. Viscosity is studied using ideal liquids, called Newtonian fluids. Although the plasma is considered a Newtonian fluid, the blood is not due to the presence of red cells, which create a non-ideal fluid.
Peripheral Vascular Resistance
The force opposing blood flow is called the Peripheral Vascular Resistance (PVR). The PVR is directly proportional to blood viscosity and to the length of the vessels, but inversely proportional to the diameter of the vessel. With more viscous blood, there is more resistance to flow. With longer vessels, there is more resistance of the blood to flow. With smaller vessels, there is more friction and therefore, more resistance of the blood to move.
Pressure
Pressure is the result of applying a force perpendicularly over a surface area of an object. Mathematically, pressure is defined as the force divided by an area. A force applied on a small area will result in a bigger pressure. A force applied on a larger surface and equally spread on the surface will produce a minimum pressure. If the given area is 1 m 2 and the given force is 1 Newton the pressure will be 1 Newton per square meter, which is a unit called "Pascal." To start with some examples, atmospheric pressure is the pressure that we feel on our head, ears, and the rest of our body. Our body gets used to atmospheric pressure, so that when we go to a place that it is at a different altitude with respect to sea level, our body will feel the difference. There are several units used to measure pressure. We will have the opportunity to explore some of these units, including some conversion factors, before we enter in detail the study of blood pressure.
Units of Pressure
There are many alternate units used to express pressure. I will refer only to those that are related to blood pressure or that are needed to understand this curriculum unit.
One atmosphere is equivalent to the pressure obtained by a column of 760 mm of mercury. Therefore, it is said that one atmosphere equals 760 mmHg (Hg is the chemical symbol for mercury). Since pressure is quantified as force divided by area, the force of the mercury is in reality its own weight. When calculating the pressure, the area of the base where the weight of the mercury is applied gets simplified and disappears from the calculations. The calculations includes the density of the fluid, which is mass divided by volume; the volume of the fluid is calculated as the product of area "A" times the height "h" of the mercury. The mass of mercury is calculated as the product of density "ρ" times volume. Substituting, we find that pressure equals the product of density "ρ" times the height "h" of the given liquid.
This formula illustrates that pressure depends on the height of a liquid, independently from the area, since the area disappears from the calculations as it is shown above.
Pounds per square inch (or PSI) have been for a long time a common way to measure pressure; although recently the International system has switched to Pascal as the unit for pressure. Another unit is the Torr and one Torr is equal to one mmHg.
Blood Pressure
Blood pressure, which is created by the pumping action of the heart, is the force that blood exerts against the walls of the circulatory system. Blood pressure is usually measured in mmHg. Blood pressure measurement typically consists of two readings. One is called the systolic pressure, which reflects the pressure of the heart during systole. The other reading is called diastolic pressure and occurs when the heart is at rest between beats.
Inequalities and Blood Pressure
A normal systolic blood pressure is considered to be no more than 120 mmHg, which it can be written as the inequality: Normal SBP ≤ 120 mmHg
A normal diastolic pressure is considered to be no more than 80 mmHg, or written as an inequality: Normal DBP ≤ 80
The category of pre-hypertension is given when the systolic blood pressure is between 120 and 139 mmHg or the diastolic pressure is between 80 to 89 mmHg. Using inequalities we can express these numbers as:
120 SBP 139 mmHg
80 DBP 89 mmHg
Individuals in pre-hypertension category re treatable, but some steps need to be taken to not reach the next category.
High blood pressure (or hypertension) is the next category after pre-hypertension. It has two stages; stage 1 is defined as a systolic reading between 140 and 159 mmHg and diastolic reading between 90 and 99 mmHg. Using inequalities:
Systolic High Blood Pressure reading 140 ≤ SHBP ≤ 159 mmHg
Diastolic High Blood Pressure reading 90 ≤ DHBP ≤ 99 mmHg
Stage -2 hypertension is defined as a systolic reading of 160 mmHg or higher and a diastolic reading of 100 mmHg or higher. Using inequalities:
Systolic High Blood Pressure reading SHBP ≥ 160 mmHg
Diastolic High Blood Pressure reading DHBP ≥ 100 mmHg
All readings above 120/80 indicate a health risk; the risk increases if the readings increase. Sometimes, the systolic and diastolic readings are not in the same category; in that case, physicians assume that an individual is in the most dangerous of the categories. For example, let's assume that your systolic reading is 160 and your diastolic reading is 80. Because of your systolic reading of 160, you would be in stage-2 hypertension and, because of your diastolic reading of 80 you would be in the normal group. However, when both readings are put together and analyzed, the result will be that you are in the hypertension stage-2, because the most severe one prevails.
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