**Tumour marker as a function of time**

The same as fig. 1C except chanced scale of concentrations U/L. The resulting graph and the exponential graph will in time be nearly indistinguishable, here after approx. 500 days.

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

#### **2.1.6 Basic simulation**

Simulation of concentration data using Microsoft Excel version 2003.

#### **2.1.7 Steady-state**

For each patient, a series of "concentrations" of biomarker was calculated from simulated data as a function of time 'after treatment'. For each 'sample' during *steady-state* conditions, the resulting concentration value (**cij**) is calculated from a chosen '*steady-state*' concentration (**cs-s**) with the addition of a random number (randomi) from a Gaussian distribution multiplied by 'the *steady-state* within-subject biological variation' (CVB), plus a new random number (randomj) from a Gaussian distribution multiplied by the analytical variation (CVA) according to the model (Bliss, 1967):

$$\mathbf{c}\_{\mathbf{i}\mathbf{j}} = \mathbf{c}\_{\mathbf{s}\cdot\mathbf{s}} \text{ } \left[1 + \text{random}\_{\mathbf{i}} \text{ } \text{\textdegree CV}\_{\mathbf{B}} \text{ } \% \text{/} 100 + \text{random}\_{\mathbf{i}} \text{\textdegree CV}\_{\mathbf{A}} \text{ } \% / 100 \right] \text{ }$$

This is performed for 50 samples in series, numbered from 1 to 50 and each corresponding to a specific day of monitoring when sampling is performed every two months (61 days), and further performed for each patient with new random Gaussian numbers for a total of 1000 surrogate patients. The result is a series of random concentration values with a mean close to the chosen value (**cs-s**). Fig. 1A illustrates a steady-state situation, where the fluctuations are based on within- subject biological variation (CVB) plus analytical imprecision variation (CVA).

#### **2.1.8 Tumour growth**

300 Biomarker

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

For each patient, a series of "concentrations" of biomarker was calculated from simulated data as a function of time 'after treatment'. For each 'sample' during *steady-state* conditions, the resulting concentration value (**cij**) is calculated from a chosen '*steady-state*' concentration (**cs-s**) with the addition of a random number (randomi) from a Gaussian distribution multiplied by 'the *steady-state* within-subject biological variation' (CVB), plus a new random number (randomj) from a Gaussian distribution multiplied by the analytical variation (CVA)

**cij** = **cs-s** \* [1 + randomi \* CVB %/100 + randomj \* CVA %/100] This is performed for 50 samples in series, numbered from 1 to 50 and each corresponding to a specific day of monitoring when sampling is performed every two months (61 days), and further performed for each patient with new random Gaussian numbers for a total of 1000 surrogate patients. The result is a series of random concentration values with a mean close to the chosen value (**cs-s**). Fig. 1A illustrates a steady-state situation, where the fluctuations are based on within- subject biological variation (CVB) plus analytical

The same as fig. 1C except chanced scale of concentrations U/L. The resulting graph and the exponential graph will in time be nearly indistinguishable, here after approx. 500 days.

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

Simulation of concentration data using Microsoft Excel version 2003.

**Tumour marker as a function of time**

**Concentration**

**2.1.6 Basic simulation** 

according to the model (Bliss, 1967):

imprecision variation (CVA).

**2.1.7 Steady-state** 

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

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).

Fig. 2. Different rates of tumour growth increase.

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

Computer Simulation Model System for Interpretation

concentration for each algorithm (see e.g. Fig 4).

**2.1.11 Biological variation of tumour growth** 

within the tumour growth is then expressed with:

cut-off concentration.

**2.2.1 Algorithm {1} Barak et al.** 

increased to 97% at 95 U/L.

over time for starting concentrations above 57 U/L.

**2.2 Results** 

algorithm.

al., 1990).

**2.1.10 Varying** *steady-state* **and start (baseline) concentrations** 

and Validation of Algorithms for Monitoring of Cancer Patients by Use of Serial Serum... 303

For each algorithm, a number of steady-state and start concentrations are used (2.38, 4.75, 9.5, 14.3, 19.0, 23.8, 28.5, 38.0, 47.5, 57.0, 66.5, 76.0, 85.5, and 95.0 U/L) where the concentration in the first 'sample' is fixed at 0.1 % below the stated concentration in order to ensure that at least one sample from each 'patient' is below the cut-off threshold. The percentage of positives at a certain time/sample is illustrated as a function of the starting

As defined in the tumour growth situation, the exponential function is added to the steadystate including a biological variation CVB. This exponential function can further be varied by multiplying the concentration due to the exponential function by a random factor times 0.25,

As previously defined, the tumour growth is expressed as an exponential function: 0.95\*eλt, where λ is the slope and t is the time (days or months). The 25% extra biological variation

(1 + randomk\*0.25)\*0.95\*eλ<sup>t</sup>

where randomk is a new random number from a Gaussian distribution, and the start concentration of tumour growth is still expressed by the factor 0.95, when t = 0, i.e. 1% of

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

Two consecutive measurements. The first below and the second above the cut-off (Barak et

In Fig. 4, the fastest tumour growth (λ = 0.0907) is 100 % percentages positive (POS) after two months for all start concentrations (sample 2), whereas the remaining graphs have almost the same development, with POS increasing from 0% at approximately 57 U/L to approximately 50% POS at 95 U/L. After six months (sample 4), the next lower slope ( λ = 0.0346) reaching 100% POS for all start concentrations, whereas the lowest slope (λ = 0.0132) and *steady-state* (λ = 0.0000) slowly increase to approximately 85% POS near the cut-off of 95 U/L. At ten months (sample 6), the slowest tumour growth has separated from the *steadystate* concentrations, increasing from 0 to 100% POS for starting concentrations between 30 and 70 U/L, and false positive (FP) is still zero up to approximately 57 U/L, but has

It is clear from Fig. 4 that true positive (TP) graphs increase with increasing starting concentrations, whereas FP graphs are zero for the low starting concentrations and increase

,

which corresponds to an extra biological variation in tumour growth of 25 %.

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 correspondingly the smaller slopes will show later increases.

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

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

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 tumourproducing biomarkers after surgery, chemotherapy etc.
