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Cardiac output is the amount of blood that is pumped by the heart per unit time, measured in litres per minute. A normal adult's heart can easily pump 4.7 litres of blood per minute through the blood circulatory system.

A dye is injected into a main vein near the heart in order to measure a patient’s cardiac output without interrupting the flow of blood. The dye is drawn into the right side of the heart, pumped through the lungs and out from the left side of the heart into the aorta, where its concentration can be measured every few seconds as the blood flows past.

The data in the graph on the next page show the response of a healthy, resting patient to an injection of 10 mg of dye. The x-axis values represent time in seconds after the dye injection and the y-axis values represent dye concentration in the blood in mg/litre.

Use the graph shown below to calculate* the patient’s cardiac output in litres of blood per minute.

Graph showing the dye concentration in the blood after injecting the dye

Present your calculations* for the problem in the form of a report, which includes your answers to the following questions.

The questions below will help you to analyse and solve the above problem. Please give as much information as you can to each of the questions.

Q1   You may google or search from books to read about cardiac output. Present your findings in not more than a single A4 page (not more than 400 words). Diagrams may be included in your answer.

Q2   What connections do you see between your findings in Q1 and the problem above presented with given data?

Q3   Name a mathematical method and state the formula to estimate the area under the dye concentration curve. Show how it enables you to get the answer. The data could be read at intervals of 2 seconds.

Q4   How does your answer in Q3 enable you to perform the calculation?

Q5   Calculate the cardiac output in litres of blood per minute.

Q6   (a) In Q3 above, comment on the accuracy of your answer when the data were read at intervals of 1 second instead?

(b) Is the relation represented in the graph above specially named as a function? Explain your answer.

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Q2 What connections do you see between your findings in Q1 and the problem above presented with given data?

From the given data, it is shown that as the time increase, the higher the concentration of the dye and this means that the quantity of the blood increases as the cardiac output is related to the quantity of blood which is being delivered to various parts of the body. Through the findings, a healthy adult heart pumps 5 litres every minute which is similar to the problem which stated that a normal adult's heart can easily pump 4.7 litres of blood per minute.

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Q6  a) The graph will be more accurate if the interval becomes 1 second instead 2 second. This will result in a more accurate result of the cardiac output.

b) In mathematics, the graph of a function f is the collection of all ordered pairs (x, f(x)). If the function input x is a scalar, the graph is a two-dimensional graph, and for a continuous function is a curve.  As you can see from this picture

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Please indicate the contribution from each member of your group.

Q1   You may google or search from books to read about cardiac output. Present your findings in not more than a single A4 page (not more than 400 words). Diagrams may be included in your answer.

ANS: Cardiac output is a term used in cardiac physiology that describes the volume of blood being pumped by the heart, in particular by a left or right ventricle, per unit time. The values can be represented using many physical units, such as dm^3/min and L/min. It is an important indicator of how efficiently the heart can meet the demands of the body because cardiac output is related to the quantity of blood delivered to various parts of the body. A healthy adult's heart pumps 5 L every minute, meaning the average cardiac output is approximately 5 L/min.

Q2   What connections do you see between your findings in Q1 and the problem above presented with given data?

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Each of the X values are paired with another Y value  and since the X values of the graph are paired with a corresponding Y value i can conclude that the graph given is a function

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Q3   In order for us to be able to find out the estimate area of the dye concentration which is the area under the curve, we can use the simpson's rule(1-4-1 approximation) to help us out. The area of the curve would be given as by using simpson's rule we first have to divide the area of the graph into even numbers (n) of equal width strips of width (h). Therefore we can say that the length of each width h = b-a/n

Q4  From the graph,
y0=0 yn=0,y1=0.25,y2=0.75,y3=1.4,y4=2.25 y5=3.6,y6=6.9,y7=8.1,y8=5.8,y9=5.25,y10=4.6,y11=3.4,y12=2.5,y13=2.1,

Using Simpson's rule h= b-a/n

In this case we take the value of h from the graph. From the graph h = 2

Area under the dye concentration curve graph= 2/3(0+0+4(0.25+1.4+3.6+8.1+5.25+3.4+2.1)+2(0.75+2.25+6.9+5.8+4.6+2.5))=94.666667

Q5) Cardiac output in litres of blood per minute = 10/area *60
= 10/94.66667 *60
= 6.3380 litres/minute

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Graph showing the dye concentration in the blood after injecting the dye

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Graph showing the dye concentration in the blood after injecting the dye

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x-axis: time in seconds after the dye injection
y-axis: dye concentration in the blood in mg/litre