**2.2.7 Algorithm {7} Nicolini et al.**

Three consecutive measurements. The first measured concentration is below the cut-off, the middle measured concentration is above the cut-off and the last is more than 30% higher than the middle value (7).

The algorithm Nicolini et al. {7} can also be compared with algorithm Chan et al. A crossing of cut-off has to be confirmed by a new sample – but this new sample has to be 30% higher compared to the second measurement over cut-off.

This 30% higher concentration for the two last measurements is difficult to fulfil for the two slowest tumour growths for starting concentrations just below the cut-off. The exponential function simulated from tumour growth has to have some time before an increase effect is observed in the results. As a consequence, the graphs decrease with increasing starting concentrations without the possibility of giving a positive signal (POS), because when crossing the cut-off too slowly, there will never be a POS signal. This is illustrated in Fig. 6 where the graphs for sample 10 and sample 13 are almost identical.

penultimate value has to have a significant increase compared to earlier measurements (see below). The more restrictive criteria are shown to give much lower FP signals – even after two years, the percentages of FP results are 24% compared to 91% at algorithm Chan et al. {4} below 95 U/L. Again more restrictive criteria have a 'cost' in regard to detection time here at the slowest slope, which is extended by two months at algorithm Söletormos et al. B {5}. On the other hand these algorithms show only 0% FP results at low start concentrations,

The significant increase, or reference change value (RCV), was introduced by Harris and Yasaka (1983) in order to detect a significant change in consecutive measurements, and was defined as RCV = 1.96\*2½\*CVB, where the 1.96 is the standard deviation from a Gaussian distribution corresponding to a two-tailed probability of 5 %, and 2½ relates to the variation of differences CVDifference = (CVB2 + CVB2)½ or the CVB2 can be substituted by the combination of biological and analytical variation. The calculation in the computer system is the test of the difference between two consecutive measurements (as a percentage) in regard to the

Three consecutive measurements. The first is below cut-off and the next two measurements

This algorithm Molina et al. {6} is comparable with algorithm Chan et al. {4} - a confirmation of crossing cut-off with an extra sample - but at algorithm Molina et al.{6}, there is a

This very restrictive criterion results in 0% FP results – even also near cut-off. This algorithm has the lowest number of FP, i.e. 0%, in comparison to all the other algorithms. And, again, strict restrictions inhibit the ability to detect early tumour progression – algorithm Molina et al. {6} has the longest detection time shared with algorithm Söletormos et al.B {5} and

Three consecutive measurements. The first measured concentration is below the cut-off, the middle measured concentration is above the cut-off and the last is more than 30% higher

The algorithm Nicolini et al. {7} can also be compared with algorithm Chan et al. A crossing of cut-off has to be confirmed by a new sample – but this new sample has to be 30% higher

This 30% higher concentration for the two last measurements is difficult to fulfil for the two slowest tumour growths for starting concentrations just below the cut-off. The exponential function simulated from tumour growth has to have some time before an increase effect is observed in the results. As a consequence, the graphs decrease with increasing starting concentrations without the possibility of giving a positive signal (POS), because when crossing the cut-off too slowly, there will never be a POS signal. This is illustrated in Fig. 6

i.e. below 57 U/L.

**2.2.6 Algorithm {6} Molina et al.** 

doubling of the cut-off (to 190 U/L).

algorithm Nicolini et al {7}.

than the middle value (7).

**2.2.7 Algorithm {7} Nicolini et al.** 

compared to the second measurement over cut-off.

where the graphs for sample 10 and sample 13 are almost identical.

are both over double the cut-off value (Molina et al., 1995).

RCV.

Percentage positive (POS) as a function of starting concentration for algorithm {7} Nicolini et al. after 1 year (sample 7), after 18 months (sample 10), and after two years (sample 13). After sample 8 (14 months) the figures will "freeze" and be identical for the three tumour slopes.

Fig. 6*.* Results from Algorithm {7} Nicolini et al. at different times

Computer Simulation Model System for Interpretation

biomarker in the present model and set-up.

**subject biological variation (CVB)** 

sensitive to lack of variance homogeneity.

{6} - and this is noteworthy: independent of the biomarker.

and Validation of Algorithms for Monitoring of Cancer Patients by Use of Serial Serum... 311

These results indicate that the relative performance of the investigated algorithms for early detection of tumour progression and avoiding FP results – seems to be independent of the

It must be underlined that this statement may only be valid based on general considerations. For example biomarkers with relative low *steady-state* variation combined with high rates of tumour increase may change some of the algorithm performances according to the detection time of progression and percentage of FP signals. In this situation the performance from algorithm Nicolini et al. {7} could be better, because start concentration near cut-off may achieve 100% TP signals within an acceptable timeframe compared with a never ending timeframe in this TPA investigation. Nevertheless, the relative information from the algorithms on performance will still stand. In other words - the best ability to detect tumour progression will often be obtained by using the algorithm from Barak et al. {1} and the best ability to get low FP signals will often be obtained by using the algorithm from Molina et al.

**2.2.9 Performance of the algorithms with impact from extreme values of within-**

values used for the challenging of algorithms in the simulations are selected.

the results from the algorithms based on a 95th percentile where CVB = 48.9%.

An important assumption for calculation of within-subject biological variation as the square root of the mean of the variances from the individual coefficients of variation of reference individuals in projects on biological variation is that these variations are distributed homogeneously. If there is variance homogeneity, this pooled coefficient of variance represents all individuals of the reference group and it is correct to use this pooled CVB in the simulations as a factor for the random Gaussian values. This assumption, however, is not fulfilled for TPA (Sölétormos et al. 2000a), where the range of coefficients of variation goes from 8.5% and 48.9% and represents individual CVB-values, from which the extreme

The results in Table 1 are based on CVB = 24.5% (within-subject biological variation). This value is based on a 50th percentile from an investigation on 127 patients (Söletormos et al., 2000b). Due to the lack of variance homogeneity, we have also investigated the impact on

The results for CVB = 48.9% are listed in Table 2 where it can be seen that the detection times for tumour progression are practically the same as for the 50th percentile of biological variation. Only algorithm Söletormos et al. B {5} shows a 2 months later detection time for a slope of 0.0123. Nearly all algorithms show an increased percentage of false positive signals (the four first algorithms are already close to 100 % for CVB, = 24.5% for the highest start concentrations) with the higher biological variation CVB. Only the algorithm Molina et al. {6} maintains 0% FP results in situations with high biological variations. It should also be noted that the algorithm results from Söletormos et al. A {3} and Söletormos et al. B {5} both markedly increase the number of FP results, when the biological variation, CVB, is high and the start concentration is below cut-off. For Söletormos et al. B {5} this is partly due to the algorithm, where the significant change in the criterion is based on the 50th percentile of biological variation CVB = 24.5% whereas the simulation is based on the much higher extreme CVB = 48.9%. Consequently the use of significant change in the algorithm makes it

It has to be underlined that, in this way, the algorithm Nicolini et al. {7} will never achieve 100% POS with start concentrations near cut-off. The 100% POS will only be fulfilled at start concentrations below 28 U/L, for the two slowest slopes (see footnotes in Table 1)
