MATHS PBL by Nurul Afiqah

Roles and responsibilities

rollercoasterdinosaurbiteyou — 2016-12-08 13:58:21 UTC

Cassandra: Leader, Time manager
May: Research
Joanna: Research
Afiqah: Research

Problem

cassnsf — 2016-12-11 08:30:26 UTC

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.

Q1

maythutun2111 — 2017-02-04 11:36:16 UTC

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.

Cardiac output (known as Q) is made up of two components, heart rate (HR) and stroke volume (SV) through a formula as followed: Q = HR x SV. Heart rate refers to the number of heartbeats per minute (bpm) and stroke volume refers to the amount of blood pumped out of the left ventricle with every heartbeat (Ptdirect.com, 2017). Therefore, cardiac output of the heart is the volume of blood pumped by the heart per unit time, that is, the rate of flow into the aorta. Blood from the body enters the right atrium of the heart through the veins and is pumped to the lungs through pulmonary arteries for oxygenation. It then flows back to the left atrium through pulmonary veins and then to the rest of the body through the aorta (Stewart, 2012).

There are a number of clinical methods to measure cardiac output and one of them is the dye-dilution method. It is the method of injecting rapidly a known quantity of a dye at a chosen site of the circulatory system and withdrawing blood at a distal site for determination of a concentration curve of the dye diluted with blood (P, 2017). Dye is usually injected into the right atrium and flows through the heart to the aorta. A probe inserted into the aorta measures the concentration of the dye leaving the heart at equally spaced times over a time interval until the dye has cleared (Stewart, 2012). In recent years, the only dye used has been indo-cyanine green (cardiogreen) which has its absorption maximum in the infrared part of the spectrum.

The dye-dilution method is probably one of the most accurate methods to measure cardiac output during exercise. However, this method does not allow measurement of 'beat to beat' changes and it requires a cardiac output which is stable for approximately 10 seconds during exercise and 30 seconds at rest (P, 2017). It also assumes complete mixing of blood and indicator, with no loss of indicator between place of injection and place of detection. If blood flow is further assumed to be constant, then cardiac output can be found using Stewart-Hamilton equation below:

where ∫c(t) dt is the area under the dilution curve and c(t) is the concentration hof dye at time, t (Geerts, Aarts and Jansen, 2011).

Citations
Ptdirect.com. (2017). Cardiac Output and Blood Pressure — PT Direct. [online] Available at: http://www.ptdirect.com/training-design/anatomy-and-physiology/cardiac-output-and-blood-pressure [Accessed 4 Feb. 2017].
Geerts, B., Aarts, L. and Jansen, J. (2011). Methods in pharmacology: measurement of cardiac output. [online] NCBI. Available at: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3045542/ [Accessed 4 Feb. 2017].
P, L. (2017). The dye dilution method for measurement of cardiac output. - PubMed - NCBI. [online] Ncbi.nlm.nih.gov. Available at: https://www.ncbi.nlm.nih.gov/pubmed/2092991 [Accessed 4 Feb. 2017].
Stewart, J. (2012). Calculus: Early Transcendentals. 7th ed. USA: Cengage Learning, pp.565-566.

Q2

maythutun2111 — 2017-02-04 12:16:57 UTC

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

In the problem given, it is stated that the dye is injected into a main vein near the heart which is then drawn into the right side of the heart. My findings also say that the dye is usually injected into the right atrium.

Both my findings and the problem given state that the dye concentration in blood is measured at the site of aorta.

Through my findings, the dye solution method uses the concentration curve obtained over a period of time in order to determine the cardiac output. The problem given also requires us to use the given graph showing the curve of dye concentration in blood to calculate the cardiac output.

The graph given in the problem shows that the dye concentration was taken at equal intervals of 2 seconds until it was zero concentration at the 30th second. This connects to my findings which state that concentration of the dye leaving the heart is measured at equally spaced times over a time interval until the dye has cleared. Furthermore, the dye dilution method requires a cardiac output which is stable for approximately 30 seconds at rest, thus it connects back to the problem where the measurements were taken for 30 seconds from a healthy, resting patient.

Guys help me see if it's correct idk mannn am I even doing the right thing

Q3

rollercoasterdinosaurbiteyou — 2017-02-05 07:13:41 UTC

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.

Ans:
Use Simpson's Rule to estimate the area under the dye concentration curve. Since the data can be read at intervals of 2 seconds, h=2. Hence, n = 14, where n is the number of even number equal strips of width h. After estimating the y(x) values from the given graph (refer to Figure 1.1), name each y(x) as Y0, Y1,…Yn respectively, where Y0 is the first y(x) value and Yn is the last y(x) value. Sub in the values into the Simpson's Rule formula. Using Simpson's Rule, enables us to find the area under the graph, without having to intergrate y(x) since y(x) has no definite intergral.

Figure 1.1

Q4

rollercoasterdinosaurbiteyou — 2017-02-05 08:55:52 UTC

How does your answer in Q3 enable you to perform the calculation?
The Simpson's Rule formula divides the region under the graph into smaller sections (n), ie 14 strips, and then find the sum of the area of the smaller sections. The greater the n, the better the approximation of the area of the dye concentration curve, Bruce Simmons (2016). Also, using Simpson's Rule helped me calculate the area under the graph of the dye concentration curve since doing the easier method of intergration is not possible.

Citations:
Bruce Simmons, 2016, How does Simpson's Rule work [online]. Available from: http://www.mathwords.com/s/simpsons_rule.htm [Accessed on 5 February 2017]

(placed the solution from qn 3 here)

cassnsf — 2017-02-05 09:19:01 UTC

Q6

cassnsf — 2017-02-05 09:31:00 UTC

In Q3 above, comment on the accuracy of your answer when the data were read at intervals of 1 second instead?
When the data were read at intervals of 1 second, the accuracy of our answer would be better.
At 1s intervals, the width of the strips would be halved and more divisions would be under the curve. Since the coordinates of each point is substituted into Simpson's Rule formula to find the area under the graph, there would be more points substituted. The area disregarded would thus be lesser, making the area under the graph calculated more accurate.

Is the relation represented in the graph above specially named as a function? Explain your answer.
Yes, it is specifically named as a function. A vertical line will not cut the graph more than once. This meets the criteria of a function which is that for each x-value, there is only one corresponding y-value. A horizontal line could cut the graph more than once. However, a function also allows each y-value to have multiple corresponding x-values. Thus, it fits the description of a function and is specifically named as a function.
mathtutor.ac.uk, 2005, Introduction to Functions [online]. Available at: http://www.mathtutor.ac.uk/functions/inroductiontofunctions/text [Accessed 6 February 2017]

Joanna

2017-02-06 01:02:11 UTC

Q5
Calculate the cardiac output in litres of blood per minute.
Cardiac output is the AMOUNT of blood pumped by each side(ventricle) of the heart in ONE minute.
As we inject a known amount of a substance upstream, the change in its concentration downstream is related to the rate of the flow. The flow, or volume over time, in this case, would be the cardiac output. Hence, Cardiac output is calculated by the Stewart-Hamilton Equation, as shown below. Q refers to the cardiac ouput; I refers to the amount of indicator present; and the denominator refers to the integral of indicator concentration over time, simply put, the area under the concentration curve.

I= 10mg
From Q4, the area under concentration curve is calulated to be 95.4mg/Ls
Therefore,
Q=10/95.4
=0.1048218L/s
=6.28930818L/s
=6.29L/min (corrected to 3s.f)

Citation
Alex Yartsev.(2013-2017). The Stewart-Hamilton Equation for measuring cardiac output - Deranged Physiology [online] Derangedphysiology.com. Available at: <http://www.derangedphysiology.com/main/core-topics-intensive-care/haemodynamic-monitoring/Chapter%203.2.1/stewart-hamilton-equation-measuring-cardiac-output> [Accessed: 8 February 2017]

Joanna

2017-02-09 01:48:04 UTC

can i ask. To calculate CO, i just use the formula May posted in her question, but the area under the curve is what Fiq calculated right? Is it that simple or is the question asking something else??

Joanna

2017-02-09 02:02:38 UTC