LAB 7: VOLUME OF A HEART AND LEAST SQUARE APPROXIMATION
Mathematics:
Experimental and numerical data do not generally fit to theoretical dependences because of experimental and numerical errors. If the error of data detection is sufficient small, data points are expected to resemble a theoretical curve. This assumption enables us to approximate parameters of a theoretical formula by computing the least-square approximation between a theoretical curve and real (experimental and numerical) data points. If the theoretical curve is simply a polynomial, the least-square approximation is a polynomial approximation. In many modelling problems, the theoretical curve is however expressed by exponential, logarithmic, power, and trigonometric functions, i.e. the least-square approximation is a nonlinear approximation. This project exploits the least-square approximation with an exponential function for an applicational modelling problem in medical science.
Consider a cardiology research unit that performs an experiment in which a dye is injected at a constant rate into the vein that empties directly into the heart. Small samples of blood are withdrawn at regular intervals from the artery that carries blood directly out of the heart. Concentration of the dye in the heart's blood was found in two trials as a function of time:
Trial I:
Concentration |
0.0243291 |
0.0368201 |
0.0465258 |
0.0480842 |
0.0499363 |
Time in sec |
2.0 |
4.0 |
8.0 |
12.0 |
20.0 |
Trial II:
Concentration |
0.0141734 |
0.031606 |
0.0405562 |
0.0482163 |
0.0496632 |
Time in sec |
1.0 |
3.0 |
5.0 |
10.0 |
15.0 |
The mathematical model for concentration of dye in the heart c=c(t) is:
c(t) = rin/rout
(1 – exp(-routt/V))
where V is the volumne of the heart in mL,
rin is the total amount of dye pumped in the heart per
second, and rout is the total amount of blood pubped
out of the heart per second. We assume rin = 5
mL/sec and rout = 100mL/sec. In
order to determine V, we have a set of five data points (tk,ck)
and fixed values for parameters rin and
rout. By regrouping terms of the formula for c(t) and
taking a logarithm, we reduce the formula for c(t) to a linear
formula for t = t(c):
t = - V*log(1 – rout c/rin) / rout
The linear least-square approximation can now be used to
find a value for V from the two sets of five data points for t
versus d = log(1 – rout c / rin ).
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The first data set
results in the least square approximation for the volume of a heart: V = 311.25 ml with the total square error E = 4.09. The second data set
results in the least square approximation V = 299.99 ml with
the total square error E = 0.00. Therefore, the second data set is more
accurate for the theoretical dependence c = c(t).
Objectives:
·
understand
the computational algorithm for nonlinear least-square approximations
·
exploit
linear least-square approximations by solving a linear over-determined system
·
exploit
MATLAB polynomial functions and least-square approximations
t = - V*d, d = log(1 – rout c/rin)
/ rout
Steps
in writing the MATLAB script:
Exploiting
the MATLAB script:
t = a1(-d) + a2, a1 = V, a2 = t0, d = log(1 – rout c/rin)
/ rout
Steps
in writing the MATLAB script:
Exploiting
the MATLAB script:
Consider
the same problem as in Script #2 of Lab. # 2, i.e. the random
walk of N molecules after n collisions. The
diffusion theory predicts that = 0 and 2 = D n, where D is
diffusion coefficient, is the mean and 2 is the variance:
= xk(n),
2 = ( xk(n) - )2
Evolution
of the mean and the variance2 in time n depends on the
number of molecules N in a statistical ensemble, as well as on
each particular random data realization. A typical pattern for evolution of the
mean and the variance2 for 10 different ensembles
with different values of N is shown here:
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The
coefficient of diffusion D can be computed as the slope of the
least-square approximation of the curve2 in time n. The basic
script with computations of and 2 is attached. Continue the script and
compute the coefficient of diffusion for an ensemble of N = 500 molecules.