The Wnt signalling pathway controls cell proliferation and differentiation, and its deregulation is implicated in different diseases including cancer. Learning how to manipulate this pathway could substantially contribute to the development of therapies. We developed a mathematical model describing the initial sequence of events in the Wnt pathway, from ligand binding to β-catenin accumulation, and the effects of inhibitors, such as sFRPs (secreted Frizzled-related proteins) and Dkk (Dickkopf). Model parameters were retrieved from experimental data reported previously. The model was retrospectively validated by accurately predicting the effects of Wnt3a and sFRP1 on β-catenin levels in two independent published experiments (R(2) between 0.63 and 0.91). Prospective validation was obtained by testing the model's accuracy in predicting the effect of Dkk1 on Wnt-induced β-catenin accumulation (R(2)≈0.94). Model simulations under different combinations of sFRP1 and Dkk1 predicted a clear synergistic effect of these two inhibitors on β-catenin accumulation, which may point towards a new treatment avenue. Our model allows precise calculation of the effect of inhibitors applied alone or in combination, and provides a flexible framework for identifying potential targets for intervention in the Wnt signalling pathway.

Colorectal cancer (CRC) is one of the most common causes of cancer-related mortality worldwide. Most cases of deaths result from metastases, assumed to be shed, in many cases, before disease detection. Providing reliable predictions of the metastases' growth pattern may help planning treatment. Available mathematical tumor growth models rely mainly on primary tumor data, and rarely relate to metastases growth. The aim of this work was to explore CRC lung metastases growth patterns. We used data of a metastatic CRC patient, for whom 10 lung metastases were measured while untreated by seven serial computed tomography (CT) scans, during almost 3 years. Three mathematical growth models-Exponential, logistic, and Gompertzian-were fitted to the actual measurements. Goodness of fit of each of the models to actual growth was estimated using different scores. Factors affecting growth pattern were explored: size, location, and primary tumor resection. Exponential growth model demonstrated good fit to data of all metastases. Logistic and Gompertzian growth models, in most cases, were overfitted and hence unreliable. Metastases inception time, calculated by backwards extrapolation of the fitted growth models, was 8-19 years before primary tumor diagnosis date. Three out of ten metastases demonstrated enhanced growth rate shortly after primary tumor resection. Our unique data provide evidence that exponential growth of CRC lung metastases is a legitimate approximation, and encourage focusing research on short-term effects of surgery on metastases growth rate. SIGNIFICANCEProviding reliable predictions of the metastases' growth pattern using mathematical models may help determining the optimal treatment plan that fits a given patient best and maximizes the probability of cure.

BackgroundDespite considerable investigational efforts, no method to overcome the pathogenesis caused by loss of function (LoF) mutations in tumor suppressor genes has been successfully translated to the clinic. The most frequent LoF mutation in human cancers is Adenomatous polyposis coli (APC), causing aberrant activation of the Wnt pathway. In nearly all colon cancer tumors, the APC protein is truncated, but still retains partial binding abilities.Objective & methodsHere, we tested the hypothesis that extracellular inhibitors of the Wnt pathway, although acting upstream of the APC mutation, can restore normal levels of pathway activity in colon cancer cells. To this end, we developed and simulated a mathematical model for the Wnt pathway in different APC mutants, with or without the effects of the extracellular inhibitors, Secreted Frizzled-Related Protein1 (sFRP1) and Dickhopf1 (Dkk1). We compared our model predictions to experimental data in the literature.ResultsOur model accurately predicts T-cell factor (TCF) activity in mutant cells that vary in APC mutation. Model simulations suggest that both sFRP1 and DKK1 can reduce TCF activity in APC1638N/1572T and Apcmin/min mutants, but restoration of normal activity levels is possible only in the former. When applied in combination, synergism between the two inhibitors can reduce their effective doses to one-fourth of the doses required under single inhibitor application. Overall, re-establishment of normal Wnt pathway activity is predicted for every APC mutant in whom TCF activity is increased by up to 11 fold.ConclusionsOur work suggests that extracellular inhibitors can effectively restore normal Wnt pathway activity in APC-truncated cancer cells, even though these LoF mutations occur downstream of the inhibitory action. The insufficient activity of the truncated APC can be quantitatively balanced by the upstream intervention. This new concept of upstream intervention to control the effects of downstream mutations may be considered also for other partial LoF mutations in other signaling pathways.

Most cases of deaths from colorectal cancer (CRC) result from metastases, which are often still undetectable at disease detection time. Even so, in many cases, shedding is assumed to have taken place before that time. The dynamics of metastasis formation and growth are not well-established. This work aims to explore CRC lung metastasis growth rate and dynamics. We analyzed a test case of a metastatic CRC patient with four lung metastases, with data of four serial computed tomography (CT) scans measuring metastasis sizes while untreated. We fitted three mathematical growth models—exponential, logistic, and Gompertzian—to the CT measurements. For each metastasis, a best-fitted model was determined, tumor doubling time (TDT) was assessed, and metastasis inception time was extrapolated. Three of the metastases showed exponential growth, while the fourth showed logistic restraint of the growth. TDT was around 93 days. Predicted metastasis inception time was at least 4–5 years before the primary tumor diagnosis date, though they did not reach detectable sizes until at least 1 year after primary tumor resection. Our results support the exponential growth approximation for most of the metastases, at least for the clinically observed time period. Our analysis shows that metastases can be initiated before the primary tumor is detectable and implies that surgeries accelerate metastasis growth.

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