Where c is the tumor cell density and D is diffusion coefficient and R is the reaction (proliferation) term. This model describes the density of tumor cells and reflects the net rates of invasion and proliferation for the whole population of tumor cells23. It is known to be highly non-linear providing a unique profile of invasion at the leading edge of the imageable tumor24. The model considers the expansion of tumor detectable edge like a traveling wave that approaches a constant velocity v=2√Dρ 25.
Diffusion takes place due to tumor growth factor concentration gradient and the reaction term describes cell proliferation. Different models are proposed in the literature that mostly differ by the construction of the D tensor and the form of the proliferation term. There are different expressions for R in literature26–29 as exponential30, Logistic31 or Gompertz 32 (table.1).
Table 1. Different expressions of reaction term
Exponential Gompertz Logistic
ρc ρc ln(1⁄c) ρc(1-c)
Equation 2 shows no flux boundary condition, according to which tumor cells do not diffuse toward brain boundaries and ventricles with normal directions η.
2.2 diffusion-reaction-advection models
Two classes of cell motion models are broadly defined: Phenomenological and Mechanical models 33. Phenomenological models assume cells are moving through a combination of chemotaxis, haptotaxis, etc. ignoring mechanical effects33. In contrast, mechanical models focus on the influence of mechanical properties of the tumor and surrounding tissu...
... middle of paper ...
...r of resistance to radiotherapy and some chemotherapeutic drugs66.
Single Photon Emission CT
A single photon emission CT (SPECT) scan provides a blood perfusion map of brain lesions. It helps to differentiate tumors from inflammatory diseases and infections. It is also used to determine if certain tumors are resistant to specific types of targeted therapy. Absolute quantitation of radioactivity concentrations is not as straightforward with SPECT as it is with PET because a SPECT radionuclide, cannot provide the temporal or spatial resolution obtained with PET67.
For more information about imaging modalities, readers are referred to an excellent review by Kurdziel et al.68. It is noticeable that for image-based tumor studies, segmentation and registration are important issues. Very good reviews about these issues are presented by Balafar et al.69 and Bauer et al.70
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