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title: Untitled

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We derive the expression for the average number of biomarkers received by the $k^{th}$ nanosensor, given by

\begin{align}
\lambda_{k}{(m\tau; t_{0}, x_{T})} &= f_c R_c N_{0} \exp{\bigg(\frac{-d_{k} v}{2 D}\bigg)}\\
&\times \Bigg[a_{0}\quad \!\!\!\!\text{erfc}\bigg(\frac{d_{k}}{\sqrt{4 D (m\tau-t_{0})}}\bigg) \\
&+\frac{a_{0} v d_{k}}{2 \sqrt{\pi} D}\bigg\{\frac{\sqrt{4 D (m\tau-t_{0})}}{
d_{k}}\exp{\bigg(\frac{-d_{k}^2}{4 D (m\tau-t_{0})}\bigg)}-\sqrt{\pi}\quad \!\!\!\!\text{erfc}\bigg(\frac{d_{k}}{\sqrt{4 D (m\tau-t_{0})}}\bigg)\bigg\}\\
&+\frac{a_{1}d_{k}}{4 D b_{1}+d_{k}^{2}}\quad \!\!\!\!\text{erfc}\bigg(\sqrt{\frac{4 D b_{1}+d_{k}^{2}}{4 D (m\tau-t_{0})}}\bigg)\\&+\frac{a_{1}v\sqrt{4 D b_{1}+d_{k}^{2}}}{2\sqrt{\pi}D}\bigg\{\sqrt{\frac{4 D (m\tau-t_{0})}{4 D b_{1}+d_{k}^{2}}}\exp\bigg(-\frac{4 D b_{1}+d_{k}^{2}}{4 D (m\tau-t_{0})}\bigg)\\
&-\sqrt{\pi}\quad \!\!\!\!\text{erfc}\bigg(\sqrt{\frac{4 D b_{1}+d_{k}^{2}}{4 D (m\tau-t_{0})}}\bigg)\bigg\}\Bigg],\tag{1}
\end{align} where $f_c$ is the fraction of the biomarkers secreted by the cancerous cells, $R_c$ is the constant shedding rate with which cancerous cells secrete the biomarkers into the blood vessel, $N_0$ is the initial number of cancer cells at time $t_0$, $d_k$ is the distance of the $k^{th}$ nanosensor from the tumor location, $v$ is the drift velocity, $D$ is the effective diffusion coefficient of the biomarkers in the blood vessel, $\tau$ is the sampling interval, and $a_0$, $a_1$ and $b_1$ are the curve fitting parameters.

The $m^{th}$ sample of the received signal at the $k^{th}$ nanosensor is modelled by 
\begin{equation}
    Y_k \sim\left\{
                \begin{array}{ll}
                  N(\mu_{H},\sigma_{H}^{2}) & :H_{0}\\
                  N(\mu_{H}+\lambda_{k}{(m\tau; t_{0}, x_{T})},\sigma_{H}^{2}) & :H_{1}
                \end{array}
             \right.
\end{equation} where $\mu_{H}$ and $\sigma_{H}^{2}$ are the mean and variance of the noise generated by the healthy cells respectively. The nuisance parameters $t_0$ and $x_T$ exist only for hypothesis $H_1$.      

Then, we solve the following equations to estimate the nuisance parameters $t_{0}$ and $x_{T}$

\begin{equation}
\sum_{m=1}^{M}\Bigg[\bigg(Y_k(m\tau)-\lambda_{k}{(m\tau; t_{0}, x_{T})}-\mu_{H}\bigg)\frac{\partial \lambda_{k}{(m\tau; t_{0}, x_{T})}}{\partial t_{0}}\Bigg]\LARGE{\Bigg\vert_{t_{0} = \hat{t_{0}},~x_{T} = \hat{x_{T}}}} = \large 0 \tag{2}
\end{equation}

\begin{equation}
\sum_{m=1}^{M}\Bigg[\bigg(Y_k(m\tau)-\lambda_{k}{(m\tau; t_{0}, x_{T})}-\mu_{H}\bigg)\frac{\partial \lambda_{k}{(m\tau; t_{0}, x_{T})}}{\partial x_{T}}\Bigg]\LARGE{\Bigg\vert_{t_{0} = \hat{t_{0}},~x_{T} = \hat{x_{T}}}} = \large0 \tag{3}
\end{equation}

Thereafter, we design the detector at the fusion center by assuming that the observations at each nanosensors are independent.

Currently, we are simulating the results.
 


