# A Photolysis-Assist Molecular Communication for Tumor Biosensing
## 3 Receiver Model  
In general, the symbol sequence is denoted by \(a_{i}\in\{0,1\}\), \(\{i=0,1,2,\ldots\}\). During the same interval \(T_{b}\), the impulse with the number of molecules \(Q\) are emitted into the diffusion channel for the symbol \"\(1\)\", denoted \(\delta(t)\), while symbol \"\(0\)\" is denoted by emitting a light impulse. The transmitted signal \(s(t)\) at the transmitter is  
\[s(t)=Q\sum_{i=0}^{\infty}a_{i}\delta(t-iT_{b}), \tag{11}\]  
Subsequently, the received signal at the receiver is  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]  
When noise is taken into account, the received signal can be expressed as  
\[z(t)=y(t)+n(t), \tag{13}\]  
and \(n(t)\) denotes the additive white Gaussian noise, which follows a normal distribution. When \(n\) symbols \"\(1\)\" are transmitted, then received signal can be expressed as  
\[y(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b})\quad(a_{i}=1), \tag{14}\]  
Thus, the current signal observed by receiver in pure diffusion scenario is denoted as:  
\[y(t)=\frac{QR}{\sqrt{4\pi Dt}}\exp\left(-\frac{d^{2}}{4 Dt}\right), \tag{15}\]  
The concentration throughout the receiver radius is assumed to be uniform and equal [40]. The photolysis reaction always happens after the symbols \"\(1\)\" are transmitted, thus, the signal observed by receiver in photolysis scenario when the previous symbols are \"\(1\)\" of \(n\) is denoted as:  
\[y(t)=nQ(\sum_{i=0}^{n}\frac{R}{\sqrt{4\pi D(t-iT)}}\cdot\exp(-\frac{d^{2}}{4D (t-iT)}-J(t-nT))), \tag{16}\]  
The mean expectation and standard deviation for symbol \"\(0\)\" and symbol \"\(1\)\" in the \(i\)th symbol interval are represented by \(\mu_{0}\), and \(\sigma_{0}\), \(\mu_{1}\) and \(\sigma_{1}\), respectively. For symbol \"\(0\)\", the mean expectation is  
\[\mu_{0}=\mu_{ISI0}+\mu_{n}, \tag{17}\]  
and the variance is  
\[\sigma_{0}^{2}=\frac{\mu_{ISI0}+\mu_{n}}{R}, \tag{18}\]  
For symbol \"\(1\)\", the mean expectation is  
\[\mu_{1}=\mu_{s}+\mu_{ISI1}+\mu_{n}, \tag{19}\]  
and the variance is  
\[\sigma_{1}^{2}=\frac{\mu_{s}+\mu_{ISI1}+\mu_{n}}{R}, \tag{20}\]The optimal detection threshold can be calculated by the maximum posterior criterion, the expression is (21). When \(z\) is above the threshold value, the received signal is judged as \"1\"; otherwise, the received signal is judged as \"0\". \[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]0  
When symbol \"1\" sent is misjudged as \"0\", the conditional probability function is  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]1  
When symbol \"0\" sent is misjudged as \"1\", the conditional probability function is  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]2  
The error probability, denoted by \(P_{e}\), is expressed as:  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]3 \[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]4  
Molecules diffused from the transmitter to the receiver meanwhile molecules degrade over time. The probability that a molecule transmitted in slot \(i\in\{1,2,\ldots,n\}\) arrives in slot \(n\) can be expressed as  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]5  
where \(m=n-i\), \(f(t)\) is the probability density function (pdf) of first-arrival time of the molecule [41], in the one dimensional domain assumed in this paper, \(f(t)\) is expressed  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]6  
We thus describe the probabilities of successful bit transmission as follows:  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]7  
\[y(t)=s(t)*h(t)=\sum_{i=0}^{n}a_{i}h(t-iT_{b}), \tag{12}\]8  
respectively, the false probability is \(1-Pn\) and \(1-Qn\). Equations (29) and (30) are derived in [42], for \(n=1\), \(Q_{1}=q_{0}\), and \(P_{1}=1\), that is, communicating a '0' is always successful.