**Tumour marker as a function of time**

Illustration of simulated data for steady-state concentrations (-●-), with a mean concentration 50 U/L, CVB = 24.5% and CVA = 8.5%, and tumour with exponential growth λ = 0.0132 (-■-)),according to 0.95\* eλ\*t U/L. Baseline (starting) concentration for the course is 50 U/L and the cut-off concentration, 95 U/L, (- - - -). Sampling frequency every two months (61 days). The (-○-) graph is the addition result of steadystate concentrations (-●-) plus 0.95\* eλ\*t U/L.

Fig. 1C**.** Steady-state, tumour growth and resulting graphs.

Computer Simulation Model System for Interpretation

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Fig. 2. Different rates of tumour growth increase.

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**2.1.8 Tumour growth** 

and Validation of Algorithms for Monitoring of Cancer Patients by Use of Serial Serum... 301

Based on data from Sölétormos *et al.* (1997, 2000a), increases in biomarkers are shown to be associated with progression of disease and it is found that the concentrations of biomarkers have an exponential relation with time. However, the rate of tumour growth can vary considerably. The rate of increase is expressed as in the exponential function as a factor in the exponent in et or as exp(t). The rate of increase (also called slope) was calculated for TPA in patients Sölétormos *et al.* (2000a) and found as 5th percentile ( = 0.0132), as 50th percentile ( = 0.0346) and as 95th percentile ( = 0.0907). Therefore, in the simulation model tumour growth is described as an exponential increase in the biomarker TPA. The start concentration (t = 0) of the biomarker originating from the tumour is arbitrarily selected as an amount corresponding to a concentration 100 times lower than the cut-off concentration (0.95 U/L). The resulting function of the TPA from the tumour is then expressed as 0.95et or as 0.95 exp(t). See Fig 1B as an example of an exponential tumour growth where

**Tumour marker as a function of time**

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Illustration of simulated data for steady-state concentrations (-●-), with a mean concentration 50 U/L and CVB = 24.5% and CVA = 8.5%., and tumours with exponential growth = 0.0132 (-○-), = 0.0346(-+-) , and = 0.0907 (-x-) according to 0.95\* e\*t U/L. Baseline (starting) concentration for the four courses is 50 U/L and the cut-off concentration, 95 U/L, (- - - -). Sampling frequency every two months (61 days).

As a resulting graph the tumour concentration is now added to the *steady-state* concentration (*steady-state* concentration + initial tumour (0.95 U/L)). An example is illustrated in Fig 1C. At first, the resulting graph has nearly the same concentration as the *steady-state* graph, but after some time TPA products from the tumour take over as the dominating contributor. Thereafter *steady-state* concentrations might be neglected as the resulting graph will be close to the exponential tumour graph (see Fig 1D). This process is repeated until a total of 1000 'patient pathways' are evaluated using the same parameters. In Fig 2 is illustrated an
