Mathematical models can be useful tools in exploring population disease trends over time and can be used to gain insight into the fundamental mechanisms of cancer development. In this paper, we provide a systematic comparison between the exact and the approximate solutions for estimating the length of time between the biological initiation of cancer and diagnosis through the development of a Weibull-like survival model. A total of 1,608,484 malignant primary cancers were used in the analysis using cancer incidence data obtained from the National Cancer Institute’s Surveillance, Epidemiology, and End Results (SEER) program. We find that the approximate solution provides a reliable comparison of the latency periods for different types of cancer and has no significant effect on the estimation accuracy, which differs from the exact solution by 0% to 11.3%. Thirty-five of the 44 cancers in this analysis were found to progress silently for 10 years or longer prior to detection representing 89% of the patients in this analysis. The results of this analysis differentiate cancer types that progress undetected over a period of years to identify new opportunities for early detection which increases the likelihood of successful treatment and alleviates the ever-growing cancer burden. 1. Introduction Cancer is the second leading cause of death in the United States and across the world [1]. It is estimated that 13 million Americans are currently living with cancer and 40.8 percent of men and women can expect to be diagnosed with cancer at some point in their lifetime [2]. In addition to the devastating effects on patients and their families, the economic costs of cancer are enormous, both in terms of direct medical-care resources for its treatment and in the loss of human capital due to early mortality [3]. According to the National Institutes of Health, cancer costs the United States an estimated $263.8 billion in medical costs and lost productivity in 2010 and the cost of cancer care is expected to escalate more rapidly in the near future as more expensive targeted treatments are adopted as standards of care [4]. To estimate the period between the biological initiation of cancer and the medical diagnosis, we utilized the popular two-parameter Weibull distribution as our framework in order to develop the approximate and exact parameter solutions. The Weibull distribution has been used to describe the mechanisms of cancer development in previous research. In contrast to the memoryless exponential distribution which assumes a constant failure rate, the shape of the
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