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rutgerfick committed Feb 23, 2018
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"source": [
"Panagiotaki et al. (2014) proposed a multi-compartment model called VERDICT to characterize the composition of tumorous tissues. VERDICT models the diffusion in tumor cells, the extra-cellular space and surrounding bloodvessels as a restricted Sphere, an isotropic Gaussian Ball and a Stick compartment, respectively. VERDICT's design is as follows:\n",
"\\begin{equation}\n",
" E_{\\textrm{VERDICT}}= \\underbrace{f_{\\textrm{Tumor}}\\overbrace{E(D|\\lambda_{\\textrm{intra}})}^{\\textrm{Sphere}}}_{\\textrm{Tumor Cells}}\n",
" + \\underbrace{f_{\\textrm{extra}}\\overbrace{E(\\cdot|\\lambda_{iso})}^{\\textrm{Ball}}}_{\\textrm{Hindered Extra-Cellular}}+\\underbrace{f_{blood}\\overbrace{E(\\lambda_\\parallel, \\boldsymbol{\\mu})}^{\\textrm{Stick}}}_{\\textrm{Vascular}}\n",
" E_{\\textrm{VERDICT}}= \\underbrace{f_{\\textrm{Tumor}}\\overbrace{E(D|\\lambda_{\\textrm{intra}})}^{\\textrm{Sphere}}}_{\\textrm{Tumor Cells}} + \\underbrace{f_{\\textrm{extra}}\\overbrace{E(\\cdot|\\lambda_{iso})}^{\\textrm{Ball}}}_{\\textrm{Hindered Extra-Cellular}}+\\underbrace{f_{blood}\\overbrace{E(\\lambda_\\parallel, \\boldsymbol{\\mu})}^{\\textrm{Stick}}}_{\\textrm{Vascular}}\n",
"\\end{equation}\n",
"where $D$ is the sphere's diameter. VERDICT uses the Gaussian Phase approximation to model the sphere *(Balinov et al. 1993)*, which accounts to changes in gradient pulse duration $\\delta$ and separation $\\Delta$. Furthermore, some particular parameter constraints are imposed:\n",
"- The intra-cellular (intra-spherical) diffusivity is fixed to 0.9e-9 m^2/s\n",
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