**2.2 Results**

302 Biomarker

example of the resulting graphs from 3 different rates of tumour increases (slopes = λ) and a steady-state situation where λ = 0. When λ is high, the biomarker will increase fast and

For each 'patient', the investigated algorithm is applied in sequential order and when a sample is positive according to the algorithm, it is recorded as a positive biomarker signal (POS). Summing up all the 1000 simulated 'patients', the percentage that are positive in each sample number (same days) is calculated, resulting in a growing graph in a plot of percentage biomarker positive as a function of sample number or day/months. This is

The slopes become steeper for increasing λ-values, which means that the detection of tumour growth is earlier for fast growing tumours, as expected. The POS signals for the *steady-state* situation ( = 0.000) represent false positive signals (in the example in Fig. 3 it is 0% after 600 days, approximately 20 months). In *steady-state,* POS signals will be recorded as false positive because no tumour growth is simulated, and therefore the POS signals cannot be considered true positives. For the three other graphs, the POS signals are recorded as true positives because an exponential tumour growth is simulated. In validation of the different algorithms, the time for 100% POS is important, but from a theoretical point of view, the most interesting variables are the lowest �-values (0.000 and 0.0123), which are the most difficult to distinguish - and at the same time very important for follow-up of tumour-

illustrated in figure 3 for four different values of including zero (= *steady-state*).

**Percentage of POS as a function of time**

0 100 200 300 400 500 600 **Time (days)**

Illustration of the algorithm from Barak et al.{1} considered POS when the concentration first exceeds the cut-off (TPA = 95U/L). Percentage POS as a function of time for four different exponential increases with = 0.0132 (-○-), = 0.0346 (-+-), and = 0.0907 (-x-) according to 0.95\* e\*t U/L. The steady-state

correspondingly the smaller slopes will show later increases.

producing biomarkers after surgery, chemotherapy etc.

0

Fig. 3. Percentage positive patients (POS) as a function of time.

simulation is represented by λ = 0.0000 (-●-).

25

50

**Percentage POS**

75

100

**2.1.9 Testing the algorithms by application to the simulated data** 

Results for each algorithm are presented with illustrations of the characteristics for each algorithm.
