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 Small samples of blood are withdrawn at regular intervals from the artery that carries blood directly out of the heart.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 4 8 12 20

Trial II:

 Concentration 0.0141734 0.031606 0.0405562 0.0482163 0.0496632 Time in sec 1 3 5 10 15

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 ).

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

#### MATLAB script for least square approximations by solving a linear over-determined system:

t = - V*d,      d = log(1 – rout c/rin) / rout

Steps in writing the MATLAB script:

1. Define parameters rin = 5 and rout = 100.
2. Define the given set of five data points for t and c.
3. Compute the vectors for d = log(1 – rout c / rin ) / rout.
4. Set up a coefficient matrix A for an over-determined linear system: A V = t.
5. Solve the over-determined linear system for V in the sense of a linear least-square approximation. Display the value for V.
6. Comute the total square error E of the linear least-square approximation. Display the value for E.
7. Plot the data points and the linear least-square approximation on the plane (d,t).
8. Plot the data points and the nonlinear least-square approximation on the plane (t,c).

Exploiting the MATLAB script:

1. Compute the least-square approximations for the two data sets.
2. Which data set is more accurate? Compare the total square error and the graphs of least-square approximations versus the data points.

#### MATLAB script for least square approximations by using MATLAB polynomial functions:

t = a1(-d) + a2,     a1 = V,   a2 = t0,   d = log(1 – rout c/rin) / rout

Steps in writing the MATLAB script:

1. Steps (1)-(3) of the first script.
2. Use "polyfit" to find the linear least square approximation for the data (d,t).
3. Compute the value for V and the total square error E.
4. Use "polyval" to compute and plot the linear least square approximation on the plane (d,t).
5. Plot the data points and the nonlinear least-square approximation on the plane (t,c).

Exploiting the MATLAB script:

1. Compute the least-square approximations for the two data sets for t and c.
2. Which data set is more accurate? Compare the intercepts t0 of the linear least-square approximation on the plane (d,t) with that prescribed by the theoretical formula.
3. Compare graphs of least-square approximations obtained in the first and in the second scripts.

#### QUIZ "Evolution of statistical parameters in a random walk of molecules (diffusion)":

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:

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.