Figure 1 - uploaded by Pavel Bělík
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The relative arrangements of the regions used in the maximum principle in §4: The case a < α with T + ⊂ S ⊂ D ⊂ T. These four semi-infinite sets are bounded by τ = 0 on the left; τ = ασ 2 2 on the right; and y = 0, y = y 0 (τ ), y = y * (τ ), and y = ˜ y(τ ) on the bottom, respectively (cf. (4.1), (4.5), (2.13), and (4.3)).
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This paper investigates the partial differential equation for the evolving distribution of prostate-specific antigen (PSA) levels following radiotherapy. We also present results on the behavior of moments for the evolving distribution of PSA levels and estimate the probability of long-term treatment success and failure related to values of treatmen...
Contexts in source publication
Context 1
... that D ⊂ T and that the lower part of the boundary of T , the line y = ˜ y(τ ), is tangent to the lower part of the boundary of D, the curve y = y * (τ ) given by (2.14), at the point 0, − αk a + α (cf. Figure 1). Again, by using the method of images and the maximum principle, we find an explicit ...
Context 2
... relative positions of the sets S and D (cf. Figure 1) again make possible the use of the maximum principle to yield another lower bound for the fundamental solution: ...
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Citations
This paper empirically tests a model of stochastic evolutions of prostate-specific antigen (PSA), a trigger for intervention in an early stage prostate cancer surveillance program. It conducts hypothesis testing of the Geometric Browning Motion model based on its attributes of independent increments and linearity of the variance in the increment length versus a wide range of stochastic and deterministic alternatives. These alternatives include the currently accepted deterministic growth model. The paper reports strong empirical evidence in favour of the Geometric Browning Motion model. A model that best describes PSA evolution is a prerequisite to the establishment of decision-making criteria for abandoning active surveillance (i.e. a strategy that involves close monitoring) in early stage prostate cancer. Thus, establishing empirically the type of PSA process is a first step toward the identification of more accurate triggers for abandoning active surveillance and starting treatment while the chances of curing the disease are still high.
Screening for prostate cancer (PC) has led to more cancers being detected at early stages, where active surveillance (AS), a strategy that involves monitoring and intervention when the disease progresses, is an option. Physicians are seeking ways to measure progression of the disease such that AS is abandoned when appropriate. A blood test, prostate-specific antigen (PSA), and the concept of doubling time (PSADT) and PSA kinetics are being used as proxies of disease speed of progression. Studies using these proxies report conflicting results. These studies cast doubts on the current rules for stopping AS and recent research concludes that PSADT and PSA kinetics are unreliable triggers for intervention in an AS program. These findings are consistent with stochastic processes being analyzed as if they were "deterministic" (i.e., current models measure disease progression by PSA's evolution assuming it to be deterministic). A model that best describes PSA evolution is a pre-requisite to the establishment of decision criteria for abandoning AS. This paper suggests modeling PSA evolutions and kinetics as stochastic processes. Consequently, triggers for stopping AS may be different than PSADT and can result in substantially different recommendations, which are likely to have significant impact on patients and the healthcare system.
