Passing to the limit 2D-1D in a model for metastatic growth

Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...

Tumor progression is a phenomenon where the pre-formed tumor acquires successive mutations to become clinically more aggressive and malignant. In the 1950s, Foulds first described the stepwise progression of cancer cells through successive stages.
Colon cancer is one of the best-documented examples of tumor progression. Early mutation in the APC gene in colon cells causes a small growth on the colon wall called a polyp. With time, this polyp grows into a benign, pre-cancerous tumor. Further...

Metastasis is the spread of cancer cells from the original site to distant locations in the body. Cancer cells can spread via blood vessels (hematogenous) as well as lymph vessels in the body.
Epithelial-to-Mesenchymal Transition
The epithelial-to-mesenchymal transition or EMT is a developmental process commonly observed in wound healing, embryogenesis, and cancer metastasis. EMT is induced by transforming growth factor-beta (TGF-β) or receptor tyrosine kinase (RTK) ligands, which further...

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...