Exploring the role of host-tumor interactions in tumor growth and regression

In the last 35 years the characterization of tumor growth using Gompertzian models has led to a new understanding of the spread of tumor cells, and has enabled researchers to develop novel therapeutic strategies to eradicate disease in some cases (see Ciron et al. (2003), Norton et al. (1976), Norton and Simon (1986), and Simon and Norton (2006)). A long standing assertion is that the Gompertzian framework does not allow characterizations of complex biological and clinical phenomena such as tumor regression and dormancy (see Retsky et al. (1998)). We propose a generalized Gompertzian system of delay-differential equations to study host-tumor interaction effects in the absence of external therapy. Our model is a parsimonious extension of the Norton et al. (1976) model: $N(t)$ denotes tumor volume, $G(t)$ represents host-tumor interactions, $N’(t)=K_{1}N(t)G(t)$ and $G’(t)=-K_{2}G(t-\tau ),$ and $\tau \ge 0$ represents time of response of the host to the presence of tumor cells. The first step is to set $G(t)= \exp (\lambda t)$ and study $\lambda = K_{2}\exp (\lambda \tau ).$ Setting $\lambda =\alpha (\tau )+\textrm {i}\beta (\tau ),$ we derive ODEs satisfied by $\alpha (\tau ),$ $\beta (\tau ),$ and prove existence and qualitative properties of infinitely many branches of solutions. Therefore, $G(t)=\sum _{j \in I_{1}} c_{j}\exp \left (\alpha _{j}(\tau )t \right )\cos \left (\beta _{j}(\tau )t\right ) +\sum _{k \in I_{2}} d_{k}\exp \left (\alpha _{k}(\tau )t \right )\sin \left (\beta _{k}(\tau )t\right ),$ where $\exp \left (\alpha _{j}(\tau )t \right )\cos \left (\beta _{j}(\tau )t\right ),$ $\exp \left (\alpha _{k}(\tau )t \right )\sin \left (\beta _{k}(\tau )t\right )$ are eigenfunctions. Substituting $G(t)$ into the $N(t)$ ODE, we: (I) identify an “optimal immunological response” range $\tau >0$ where host-tumor interactions can cause a tumor to remain dormant, or regress from growth state into dormancy, and (II) replicate observed tumor growth in mammograms of 32 breast cancer patients.