## Error Analysis

Error analysis is a useful tool in the development of an appropriate model-based method. Performance of a thorough error analysis is a critical step in the assessment of the utility of a given method. Papers dedicated solely to error analysis are common in the literature [155, 156, 171, 186-196]. These analyses usually proceed as follows. Choose a particular source of error. Select values for the model parameters and use the model equations to simulate tissue data including this error effect, usually covering a range of effect magnitudes. Then, analyze these simulated measurements with one or more methods, compare the derived parameter estimates to their original values, and determine the magnitude of error that is produced.

Figure 6.11 provides an example of the results of an error analysis. Cerebral blood flow (CBF) measurements with the tracer [15O]water are altered in the presence of errors in correction for the time delay between the measured arterial input function and the actual input to the brain. Using an actual measured input function, tissue time-activity data were simulated Figure 6.11. Example of error analysis - the effect of errors in time delay corrections between the brain and peripheral artery on measurement of cerebral blood flow with [15O]water. A positive time delay means that the tissue data has been shifted forward in time with respect to the arterial input function. The three curves show the percent error in estimated flow, based on data collection periods of 90 sec, 120 sec, and 240 sec. See text for additional details.

Figure 6.11. Example of error analysis - the effect of errors in time delay corrections between the brain and peripheral artery on measurement of cerebral blood flow with [15O]water. A positive time delay means that the tissue data has been shifted forward in time with respect to the arterial input function. The three curves show the percent error in estimated flow, based on data collection periods of 90 sec, 120 sec, and 240 sec. See text for additional details.

over a 4-min period using the model of Eq. 27, with a flow value of 0.5 mL/min/g and a distribution volume of 0.8 mL/g. CBF (K1) was then calculated by direct estimation of the two model parameters for total time intervals of 90, 120, and 240 sec. In each case, the tissue data were shifted with respect to the arterial input function by -3 to +3 sec (a positive shift means that the tissue data have been shifted later in time with respect to the blood data). The figure shows the percent error as a function of time delay. Positive time shifts produce underestimation of blood flow. This error is larger for shorter total acquisition times. This analysis suggests that the effect of time shift errors can be reduced by using longer data-acquisition periods. Even then, errors as large as 10% occur with time shifts of 3 sec, so care should be taken to measure or estimate time delays between tissue and blood data [178,197].

A careful analysis of all the relevant error sources can be used to optimize methodology or to choose one approach over another. For example, various studies have been performed to choose optimal total scanning times and scan schedules [170, 198-201]. Unfortunately, it is difficult to determine the total error of a method based on the independent error analyses of a number of measurements or assumptions. First, error analyses are only as good as their ability to simulate biological reality, i.e., recognizing and analyzing all potential error sources and making appropriate choices for the magnitude of each error term. Even then, many error sources are not independent, i.e., errors in one term affect other terms. Thus, actual errors may be larger or smaller than those predicted from independent error analyses. Therefore, it is best if the ultimate choice of a method can be made by analyzing many studies with a variety of techniques and choosing the approach that has the best reproducibility, the minimum population variability, or the maximum statistical power to extract a particular physiological signal.