Background. Improved understanding of human breast cancer growth rates may have many clinical applications. Previous reports have used small numbers of patients and assumed an exponential growth rate. Methods. The exponential equation and the most commonly used decelerating growth equations, the Gompertz equation and seven generalized forms of the logistic equation, were fitted to mammographic measurements of primary breast cancer using the least squares method. An average of 3.4 observations was made in 113 patients, whereas two measurements were made in another 335 patients. Tumors were assumed to originate as a single cell with the lethal tumor volume assumed to be 240 cells. Results. All decelerating equations tested provided a better fit than the exponential, whereas a form of the logistic equation provided the best fit to the data. Limitations in the number of tumor measurements, the assumption of maximal tumor size, and biases inherent in the method of data collection are reviewed. These observations suggest families of curves that characterize breast cancer growth during the early period of clinical observation. Conclusions. Breast cancer growth in the early clinical period was modeled by a form of the logistic equation. The exponential equation fit the data least well.
Part I of this study [Spratt JS, Meyer JS, Spratt JA: J Surg Oncol 60:137-146, 1995] reviewed the early reports of investigators, predominantly mathematical biologists and statisticians considering the mathematical laws that would describe the growth of a neoplasm. Included were cytokinetic measurements of the mitotic index, thymidine labeling index, bromodeoxy-uridine labeling index, and the relation of these indices to the potential tumor volume doubling time. The actual doubling time of benign and malignant colonic neoplasms were reported. This second part provides the cumulative observations on the actual doubling times of pulmonary metastases, primary pulmonary cancers, skeletal sarcomas, melanomas, a chemodectoma, tumors of maxillary antrum, testicular cancers, prostate cancer, and the relation between the accumulation of multiple primary cancers and growth rates. The most complete data set is for breast cancer concluding that the cancer growth curve is a decelerating curve with great natural variance. Understanding of the rates of growth of human cancers is essential for understanding the spectrum of cancer behavior observed clinically.
The purpose of this article is to consolidate data collected from a variety of sources that have permitted calculations of the rates of growth of human neoplasms. These sources include Fischel State Cancer Hospital (Columbia, MO); Mallinckrodt Institute of Radiology, (St. Louis, MO); Roentgen Diagnostic Institute, Allmanna Sjukhuset (Malmo, Sweden); University of Louisville (Louisville, Kentucky); University of Heidelberg (Heidelberg, Germany); and St. Luke's Hospital (St. Louis, MO). Included in the data are laboratory measurements of cell replication rates. All gross measurements were made either on imaging studies or with a centimeter scale for surface or palpable neoplasms. Data have been reported for breast and pulmonary cancers and metastases of many types, melanomas, skeletal sarcomas, benign and malignant colonic neoplasms, and isolated cases of less frequent neoplasms. Related cytokinetic measurements by tritriated thymidine labelling, bromodeoxyuridine labelling, S-phase fraction from DNA flow cytometric analysis, and mitotic indices are discussed. The various mathematical formulae applicable to the analysis of the collected data and the determination of rates and patterns of growth are included. Also considered are the clinical implications of these data and the importance of ever better knowledge on the cytokinetics of human cancer. Prior studies on the evolution of insight into this field are cited and discussed. The authors conclude that a more accurate quantification of the growth rates of human cancer is essential for understanding the biological variance of human cancers seen clinically.
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