Fig 1.
States and dynamics in the TGP model.
A. The initial state of the TGP model is all-cells-wild-type. Type-I cells (black cells) can arise due to type-I mutations. If the number of type-I cells reaches at least the critical tumor size N, a PA-I tumor developed which cannot regress anymore (black box). We assume that a PA-II tumor occurs as soon as the first type-II cell appears in the system (red box). Solid arrows indicate a single transition whereas dotted arrows indicate several transitions. B. Wild-type cells mutate to type-I cells with probability u and type-I cells mutate to type-II cells with probability v during proliferation. Cell death and proliferation are included as follows. A cell is randomly chosen for cell death and replaced by the offspring of another randomly chosen cell. Spatial aspects are neglected in the model.
Fig 2.
Decomposition of the TGP process.
Assumption Eq (1) implies that no other PA-I mutation occurs if type-I cells are already present in the system. Therefore, u can be set to zero when a single type-I cell emerged. This allows to decompose the process into two sub-processes. First, occurrence of unsuccessful mutants, which go extinct and, second, the occurrence of a successful mutant which leads to absorption in one of the PA states.
Fig 3.
Tumor regression in the TGP model.
A. Partial resection of a PA-I tumor reduces the number of tumor cells to size k which is assumed to be below the critical tumor size N. The residual tumor can regrow, progress or regress based on the same dynamics that led to the primary tumor. Hence, the TGP dynamics with relevant cell number N is utilized to describe the further development of the residual tumor. Regression is achieved if state 0 is reached, i.e. no tumor cells are present anymore. B. The red area indicates the resected part of the diagnosed PA-I tumor. This resection leads to removal of both tumor and wild-type cells. Subsequently, the residual number of tumor cells k competes with other wild-type cells which can lead to regrowth, regression or progression of the residual tumor. We assume that N is the relevant cell number for this competition as in the formation of the primary tumor. This relevant cell number is indicated by the blue circle.
Fig 4.
Parameter estimation from epidemiological data and derivation of the PA-regression-function.
A. In order to estimate the clinically observed fraction of PA-I, we utilize data from the literature. B. The estimated fraction of PA-I cases is interpreted as absorption probability in state N in our model. Eq (3) allows to estimate the corresponding risk coefficient
. C. Substituting the estimated risk coefficient
in Eq (5) determines the PA-regression-function β0.152(ρ) which is plotted in purple. Furthermore, the blue plots indicate the corresponding regression functions for values of γ which are obtained within the standard deviation of
.
Table 1.
Predicted regression probability for cerebellar PA based on the absolute residual tumor size.
Fig 5.
Non-existence of an EOR threshold in PA.
The derived PA-regression-function (purple line) allows to quantitatively predict the regression probability based on the critical tumor size estimated as 9 cm3, see also Table 1. Roughly, one cm3 of resected tumor mass will elevate the chance of regression by 10%. The direct consequence is the non-existence of an EOR threshold implying that any proportion of resected tumor mass will improve prognosis. This stands in contrast to the behavior of the regression function for a fictive high value of the risk coefficient of e.g. γ = 50 (black line).