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Research Article

# Cancer Screening: A Mathematical Model Relating Secreted Blood Biomarker Levels to Tumor Sizes

• Affiliation: Department of Radiology, Stanford University School of Medicine, Stanford, California, United States of America

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• Affiliation: Department of Radiology, Stanford University School of Medicine, Stanford, California, United States of America

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• Affiliation: Department of Radiology, Stanford University School of Medicine, Stanford, California, United States of America

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• Affiliation: Department of Radiology, Stanford University School of Medicine, Stanford, California, United States of America

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• To whom correspondence should be addressed. E-mail: sgambhir@stanford.edu

Affiliations: Department of Radiology, Stanford University School of Medicine, Stanford, California, United States of America, Department of Bioengineering, the Bio-X Program, Stanford University School of Medicine, Stanford, California, United States of America

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• Published: August 19, 2008
• DOI: 10.1371/journal.pmed.0050170

### Fundamental problems with the mathematical model published by Lutz et al.

#### Posted by plosmedicine on 16 Jun 2009 at 22:03 GMT

Author: George D Swanson
Position: Member
Institution: California State University
e-mail: dswanson@csuchico.edu

The model developed in this paper is fundamentally flawed and leads to erroneous conclusions. In Figure 1, Fout as indicated in the figure is labeled “efffux of the tumor biomarker by elimination” in units of ng/ml/h. This is clearly wrong since the units must be h-1 to be consistent with equations (Figure 1.). However, of a more serious nature, the differential equation (Figure 1) violates mass balance. The correct mass balance equation should be:

d(qplasma Vplasma)/dt = (IN VN + IT VT) – qplasma Qe

where V symbolizes the respective volumes in ml and Qe is the elimination flow in ml/h with IN and IT the influx of biomarker in ng/ml/h. The parameter, qplasma, represents the tumor biomarker level in plasma (ng/ml). The solution to a step increase in influx (from zero) is given by

qplasma = (IN VN + IT VT) / Qe (1-exp[-(Qe / Vplasma) t]

qplasma = (IN VN + IT VT) / Qe (1-exp[-(t/tau)]

where tau is the time constant (Vplasma / Qe). The steady state value is given by

qstsplasma = (IN VN + IT VT) / Qe.

Note if all of the volumes are equal (Vplasma = VN = VT), then

qstsplasma = (IN + IT) / (Qe/ Vplasma) = (IN + IT) / tau-1) = (IN + IT) / ln(2)/t1/2).

This is the equation given in the manuscript (Figure 1.). The problem then is that the paper overestimates the concentration of qstsplasma by as much as 1000 (assuming the volumes associated with the influx are at least 1000 times smaller than the plasma volume). This means that the smallest tumor size is a 1000 times larger than indicated in the paper.

No competing interests declared.

### RE: Fundamental problems with the mathematical model published by Lutz et al.

#### AmelieLutz replied to plosmedicine on 22 Oct 2009 at 23:47 GMT

Yes, elimination rate constant Fout should have units of h-1; this is a typographical error.

In Fig. 1, model parameters and state variables were published with units of concentration (U/ml or ng/ml) instead of the conventional mass units (U or ng). While this is mathematically valid, the manuscript’s results were actually calculated based on mass units. Therefore, incorporation of the volumes of distribution (as mentioned in the previous post) into the calculations is not an issue here. The published calculations and conclusions of the manuscript are still correct. To clarify this, we have updated Fig. 1 so that all state variables and model parameters are described in terms of mass units. We also note that model secretion rates IN and IT were based on published mass values, which were assessed by measuring total mass of biomarker secreted by a specified number of cells over a given time period (Refs 6, 29, 30). To account for potential influences of the tumor microenvironment on the biomarker secretion rates we chose a wide range of secretion rates for the sensitivity analysis of the model.

REVISION OF FIG. 1 EQUATIONS

Equation A

d(qplasma(t))/dt = (IN + IT) – Fout qplasma(t)

Equation B

qplasma(t) = ((IN + IT) / Fout) (1 – exp[-Fout t])

Equation C

qplasma,SS = (IN + IT) / Fout = (IN + IT) / (ln 2 / t1/2)

No competing interests declared.

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