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

Date Published: August 19, 2008

Publisher: Public Library of Science

Author(s): Amelie M Lutz, Juergen K Willmann, Frank V Cochran, Pritha Ray, Sanjiv S Gambhir, William Pao

Abstract: BackgroundIncreasing efforts and financial resources are being invested in early cancer detection research. Blood assays detecting tumor biomarkers promise noninvasive and financially reasonable screening for early cancer with high potential of positive impact on patients’ survival and quality of life. For novel tumor biomarkers, the actual tumor detection limits are usually unknown and there have been no studies exploring the tumor burden detection limits of blood tumor biomarkers using mathematical models. Therefore, the purpose of this study was to develop a mathematical model relating blood biomarker levels to tumor burden.Methods and FindingsUsing a linear one-compartment model, the steady state between tumor biomarker secretion into and removal out of the intravascular space was calculated. Two conditions were assumed: (1) the compartment (plasma) is well-mixed and kinetically homogenous; (2) the tumor biomarker consists of a protein that is secreted by tumor cells into the extracellular fluid compartment, and a certain percentage of the secreted protein enters the intravascular space at a continuous rate. The model was applied to two pathophysiologic conditions: tumor biomarker is secreted (1) exclusively by the tumor cells or (2) by both tumor cells and healthy normal cells. To test the model, a sensitivity analysis was performed assuming variable conditions of the model parameters. The model parameters were primed on the basis of literature data for two established and well-studied tumor biomarkers (CA125 and prostate-specific antigen [PSA]). Assuming biomarker secretion by tumor cells only and 10% of the secreted tumor biomarker reaching the plasma, the calculated minimally detectable tumor sizes ranged between 0.11 mm3 and 3,610.14 mm3 for CA125 and between 0.21 mm3 and 131.51 mm3 for PSA. When biomarker secretion by healthy cells and tumor cells was assumed, the calculated tumor sizes leading to positive test results ranged between 116.7 mm3 and 1.52 × 106 mm3 for CA125 and between 27 mm3 and 3.45 × 105 mm3 for PSA. One of the limitations of the study is the absence of quantitative data available in the literature on the secreted tumor biomarker amount per cancer cell in intact whole body animal tumor models or in cancer patients. Additionally, the fraction of secreted tumor biomarkers actually reaching the plasma is unknown. Therefore, we used data from published cell culture experiments to estimate tumor cell biomarker secretion rates and assumed a wide range of secretion rates to account for their potential changes due to field effects of the tumor environment.ConclusionsThis study introduced a linear one-compartment mathematical model that allows estimation of minimal detectable tumor sizes based on blood tumor biomarker assays. Assuming physiological data on CA125 and PSA from the literature, the model predicted detection limits of tumors that were in qualitative agreement with the actual clinical performance of both biomarkers. The model may be helpful in future estimation of minimal detectable tumor sizes for novel proteomic biomarker assays if sufficient physiologic data for the biomarker are available. The model may address the potential and limitations of tumor biomarkers, help prioritize biomarkers, and guide investments into early cancer detection research efforts.

Partial Text: Many cancer types are likely curable by conventional therapies, if detected early enough. Therefore, early detection is a primary objective of cancer research with a high potential of improving both patients’ survival and quality of life. To reach the major goal of these efforts—detection of cancer in early stages when the disease may still be curable— currently two types of cancer early detection tests are most widely studied: blood test(s) and imaging. In both diagnostic fields, tremendous progress has been made during the last decade. Ideally, a less costly highly accurate blood-based diagnostic biomarker test would precede any further imaging or biopsy studies. With emerging new, highly sensitive proteomics test assays there seems to be a nearly limitless potential to detect traces of tumor biomarkers in patient serum, if in fact they exist and are specific enough at early tumor stages [1–3]. The advances in these test assays pose the question: What is a realistic lower detectable tumor size limit for these diagnostic tests? To answer this question for secreted blood biomarkers, a mathematical compartmental model simulating the kinetics of blood biomarkers under varying physiological conditions can be utilized. Many aspects have to be taken into consideration when a model of a new test assay is being developed. We chose the setting of ovarian cancer and prostate cancer to test our model, because they are among the cancer types that meet the profile for early detection. Both cancer types often remain relatively asymptomatic until they are advanced and there is likely significant opportunity to improve survival through early detection of the disease.

For the compartmental model, the following conditions were assumed: (1) the compartment represents the plasma, which is considered well mixed and kinetically homogenous; (2) the tumor biomarker of interest consists of a protein that is secreted by tumor cells into the extracellular fluid compartment, and that a certain percentage of the secreted protein will enter the intravascular space (plasma) at a continuous rate. In the plasma, the protein of interest has a distinct half-life owing to, e.g., degradation by proteases, hepatic metabolism, or in the case of a smaller protein because of filtration by the kidney. Two potential pathophysiologic conditions were assumed: (1) the tumor biomarker is either secreted by the tumor cells only (no background secretion by normal cells) or (2) by tumor cells and to some extent by healthy normal cells (background secretion by healthy cells).

Using a linear one-compartment model, the tumor biomarker level in patient plasma under steady state conditions was described as a function of either tumor biomarker secretion by tumor cells or, if applicable, by healthy normal cells (influx) and tumor biomarker plasma half-life (efflux) (Figure 1).

The aim of this study was to relate serum tumor biomarker levels as detected by a blood assay to detectable tumor sizes, because such a relationship can likely aid in determining the lower limit for detectable tumor sizes. A linear one-compartment model (Figure 1) was used to describe the pharmacokinetics of a protein serum tumor biomarker in blood, including tumor biomarker secretion by tumor cells (and healthy cells, if background biomarker secretion is also present), transfer to the intravascular space, and final degradation and/or removal from the plasma (represented as outflow [Fout] in the model). By priming the model parameters on the basis of the literature data for two established and well-studied tumor biomarkers, CA125 for ovarian cancer and PSA for prostate cancer, the potential detection limits needed for a proteomic blood test with regards to tumor burden were calculated under varying physiological and assay conditions (Tables 1–4) [6–8,14,20–24,26,28–31]. A sensitivity analysis of the model (by varying model parameters over a range of parameter assumptions) resulted in a wide range of minimally detectable tumor sizes depending on the specific chosen conditions. In the ideal case scenario of a novel serum tumor biomarker that is secreted only by tumor cells, depending on the chosen parameters (tumor biomarker secretion rate, its fraction reaching the plasma, and the available blood test assay sensitivity), very small ovarian cancer lesions down to a size of 0.11–7.22 mm3 may be detected with a highly sensitive blood assay. Assuming the same ideal case scenario of exclusive biomarker secretion by tumor cells for prostate cancer, comparably small lesions in a range of 0.21–13.15 mm3 may be detectable even with a clinically available PSA immunoassay (detection limit 0.01 ng/ml). In a clinically more realistic scenario, however, with additional secretion of the serum tumor biomarker by healthy cells, ovarian cancer lesions between 116.7 mm3 and 1.52 × 106 mm3 (1.52 l), roughly corresponding to a size range between 4.893 mm3 and 114.983 mm3, may be detected depending on the chosen physiological and model parameters.



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