Integration of Collected Data
(Last Updated February 13, 2006)
The cardiac output measurement provides data similar to that represented by
Figure 1.
To evaluate the denominator
of the flow rate equation (Eq. 1),
Eq. 1
you must integrate the concentration. Several methods are available to do this:
1.
Numerical
Integration by Simpson’s rule:
The region under is
represented by several trapezoidal components, and the areas of the trapezoids
are added. Refer to your calculus book
for more details on this method.
2.
Graphical
Integration:
If you have tracing paper of
uniform weight, you can simply place the tracing paper over the oscilloscope
screen and trace out the curve. Next,
cut out the region under the curve and weigh it on a precise scale. Also weigh a known area of the paper. Assume that your oscilloscope is set to 1
volt per division (vertical axis) and 0.5 seconds per division (horizontal
axis). If an 8 division x 10 division
piece of the paper (80 square divisions) weighs 
grams, and if the piece under the curve weighs
grams, then the
area under the curve is:
- Integrate a Model of
the Curve:
A model was derived for the
two parts of this curve. The rising part
of the curve is represented by:
, Eq. 3
and the falling part of the curve is represented by:
, Eq. 4
where 
. You wish to
determine the appropriate constants from the data provided in a systematic
manner. One method is to transform
equation 3 by taking the logarithm.
.
This equation is in the form
of a linear equation 
, where is 
, 
is 
, and 
is 
. Thus, a linear least
squares fit of the data in the falling part of the curve will provide a value
for from which the time
constant can be directly found. It will
also provide a value of 
. Once the value of 
is known, it can be
substituted back into Eq. 1 for each time value to
obtain several estimates of the constant 
. If the method works,
then the different estimates of should not differ
substantially and they can be averaged to obtain a final estimate of .
Note: The model should fit your data to a
reasonable extent. If you look at your
data and the model and it is clear that you could guess a better fit for the
data, then you have certainly done something wrong in the curve fit.