Q: How do I set initial conditions for a sum of exponentials?

A: We will use two simple Case Studies to explain this. For this example, remember that the relationship between a half-life and the governing exponential is:

1 - Monoexponential decay - bolus injection protocol

This case study will describe how to obtain initial estimates for the coefficient "A" and exponential "a" of the simple monoexponential model

You can plot the data in semi-logarithmic mode using SAAM II. You can either print a copy of the plot to continue this case study, or continue the next steps using the plot that appears on your screen (i.e. you can do the following steps by approximation using what is on the screen as opposed to formally working with a hard copy of the plot).

1. Draw a "best" straight line through the monoexponentially decaying data.
2. The point at which your "best" line intersects the ordinate is an estimate for "A"; estimate A.
3. Calculate the half-time for an arbitrary point on the line. For example, a data value is 2,000 and it occurs at approximately 53 minutes. One half of 2000 is 1000 that occurs at 133 minutes. The half-time, that is, the time it takes to decay from 2000 to 1000, is thus about 80minutes. An estimate for "a" can thus be obtained.
4. Open the parameter dialog box, and enter your initial estimates for "A" and "a".
5. Solve and plot your results. 6. Fit to the data.

2 - Case Study: Biexponential decay - bolus injection protocol

This case study will describe how to obtain initial estimates for the coefficients A1 and A2 and the exponentials a1 and a2 of the biexponential model

You can plot the data in the semi-logarithmic mode using SAAM II. You can either print a copy of the plot to continue this case study, or continue the next steps using the plot that appears on your screen.

1. Draw a "best" straight line (L1) through the monoexponentially decaying tail portion of the data.
2. The point at which this line intersects the ordinate provides an estimate for A2; estimate this value from your plot.
3. Calculate the half-time for an arbitrary point on this line. For example, the point 1600 that occurs at approximately 65 minutes has been chosen. One half of 1600 is 800 that occurs at approximately 135minutes. The half-time is approximately 70 minutes. Now you can obtain an estimate for a2, the smaller of the two exponentials.
4. Draw a line through the initial decaying portion of the data (L2).
5. The point at which this line intersects the ordinate provides an estimate for "A1 +A2". An estimate for A1 can be obtained by subtracting your estimate for A2 from this value. Estimate A1.
6. Calculate the half-time for an arbitrary point on this line. For example, the point 4000 that occurs at approximately 2 minutes has been chosen. One half of 4000 is 2000 that occurs at approximately 22minutes. The half-time is approximately 20 minutes. We can obtain an estimate for a1, the larger of the two exponentials.
7. Open the parameter dialog box, and enter your estimates for A1, A2, and a1and a2.
8. Solve, and plot your results.
9. Fit to the data.

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