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Single-Sample Network Entropy and Early Warning of Sudden Deterioration for Some Complex Diseases
DOI: 10.12677/AAM.2019.81017, PDF, HTML, XML, 下载: 1,227  浏览: 1,671  科研立项经费支持

Abstract: For the early diagnosis of complex diseases, traditional methods require many patient samples to obtain statistical indicators. In real life, people do not go to hospitals for medical examination frequently, so it is difficult to obtain a large number of patients data, and only the result of indi-vidual single sampling has been got, which requires a single sample based algorithm to detect critical points in the development of complex diseases. Based on the theory of dynamic system and dynamic network biomarkers, a new algorithm is developed by using new individual samples combined with local biomolecular networks and the concept of entropy. The algorithm is applied to numerical simulation data and a real disease data. In these applications, the algorithm can detect the critical point in time and accurately. The new algorithm developed in this paper is stable and efficient. It can help medical workers to further understand the dynamic biomolecular mechanism of complex diseases, and can more quickly propose a reasonable treatment plan.

1. 引言

2. 方法

2.1. 单样本网络熵的推导

$Z\left(t+1\right)=f\left(Z\left(t\right);P\right)$ . (1)

${\frac{\partial f\left(Z;{P}_{c}\right)}{\partial Z}|}_{Z=\stackrel{¯}{Z}}$ , (2)

$Y\left(t\right)={S}^{-1}\left(Z\left(t\right)-\stackrel{¯}{Z}\right)$ . (3)

$Y\left(t\right)=\Lambda \left(P\right)Y\left(t-1\right)+\xi \left(t-1\right)$ ,(4)

(4)式中 $\xi =\left({\xi }_{1},{\xi }_{2},\cdots ,{\xi }_{n}\right)$ 是均值为零的微小高斯噪声， ${\xi }_{i}\left(i=1,2,\cdots ,n\right)$ 的标准偏差为 ${\sigma }_{i}\left(i=1,2,\cdots ,n\right)$$\Lambda$ 是关于参数P的对角矩阵。

$Var\left({y}_{i}\right)=\frac{{\sigma }_{i}^{2}}{|1-{\lambda }_{i}^{2}|}$ , (5)

${E}_{i}^{\left(t\right)}\left(\mathcal{X}\right)=-{\sum }_{v}{P}_{i,v}\left(t\right)\mathrm{log}{P}_{i,v}\left(t\right)$ , (6)

${E}^{\left(t\right)}\left(\mathcal{X}\right)=-{\sum }_{u,v}{P}_{u,v}\left(t\right)\mathrm{log}{P}_{u,v}\left(t\right)$ . (7)

$P\left({x}_{{i}_{l}}\left(t\right)=1\right)=P\left(|\Delta {Z}_{{i}_{l}}\left(t\right)|>{d}_{i}\right)={P}_{1}$ , (8)

$P\left({x}_{{i}_{l}}\left(t\right)=0\right)=P\left(|\Delta {Z}_{{i}_{l}}\left(t\right)|\le {d}_{i}\right)={P}_{0}=1-{P}_{1}$ , (9)

$P\left({x}_{{i}_{l}}\left(t\right)=1\right)\to 0$ . (10)

$P\left(\mathcal{X}={A}_{1}\right)\to 1$ , (11)

$P\left(\left\{\cdots ,{x}_{{i}_{l}}=1,\cdots \right\}\right)\le P\left({x}_{{i}_{l}}\left(t\right)=1\right)\to 0$ ，所以局部网络i处于 ${A}_{1}$ 以外的其他状态的概率是趋于0的，

${E}^{\left(t\right)}\left(\mathcal{X}\right)=-{\sum }_{u,v}{P}_{u,v}\left(t\right)\mathrm{log}{P}_{u,v}\left(t\right)\to 0$ . (12)

$sd\left(\Delta {Z}_{{i}_{l}}\right)=sd\left({Z}_{{i}_{l}}\right)$ , (13)

${E}_{i}^{\left(t\right)}\left(\mathcal{X}\right)=-{\sum }_{v=1}^{{2}^{m+1}}{P}_{i,v}\left(t\right)\mathrm{log}{P}_{i,v}\left(t\right)$ , (14)

${\sum }_{v=1}^{{2}^{m+1}}{P}_{i,v}\left(t\right)=1$ . (15)

${G}_{i}^{\left(t\right)}\left(\mathcal{X}\right)=-{\sum }_{v=1}^{{2}^{m+1}}{P}_{i,v}\left(t\right)\mathrm{log}{P}_{i,v}\left(t\right)+\lambda \left({\sum }_{v=1}^{{2}^{m+1}}{P}_{i,v}\left(t\right)-1\right)$ , (16)

${G}_{i}^{\left(t\right)}\left(\mathcal{X}\right)$ 对每个 ${P}_{i,v}$$\lambda$ 求偏导并令其等于0，可得

, (17)

${\sum }_{v=1}^{{2}^{m+1}}{P}_{i,v}\left(t\right)=1$ . (18)

$\lambda =\mathrm{log}\left(\frac{1}{{2}^{m+1}}\right)+1$ , (19)

${E}_{i}^{\left(t\right)}\left(\mathcal{X}\right)=\mathrm{log}\left({2}^{m+1}\right)=m+1$ . (20)

2.2. 探测临界点

1) 针对某种复杂疾病，收集该疾病已有的多个样本，并将该疾病对应的全基因组映射到STRING (STRING数据库是一个搜寻已知蛋白质之间和预测蛋白质之间相互作用的系统)的蛋白质–蛋白质相互作用网络。

2) 针对某个未知状态的样本，将该样本分别与每个已知样本建立状态网络。

3) 局部化全基因组状态网络。

4) 计算每个局部化状态网络的单样本网络熵。

5) 将所有局部状态网络的单样本网络熵排序并取前10%的平均值作为差异评价指标。

3. 主要结果

3.1. 仿真网络

(21)

$\stackrel{¯}{Z}=\left({\stackrel{¯}{z}}_{1},{\stackrel{¯}{z}}_{2},{\stackrel{¯}{z}}_{3},{\stackrel{¯}{z}}_{4},{\stackrel{¯}{z}}_{5},{\stackrel{¯}{z}}_{6},{\stackrel{¯}{z}}_{7},{\stackrel{¯}{z}}_{8},{\stackrel{¯}{z}}_{9},{\stackrel{¯}{z}}_{10},{\stackrel{¯}{z}}_{11}\right)=\left(0,0,0,0,0,0,0,0,0,0,0\right)$ . (22)

$J={\text{e}}^{\Delta t\ast A}$ , (23)

$\left(-\frac{3}{10}\ast |P|,-\frac{1}{2},-\frac{3}{5},-\frac{4}{5},-1,-\frac{6}{5},-\frac{7}{5},-\frac{8}{5},-\frac{9}{5},-\frac{10}{5},-\frac{11}{5}\right)$ , (24)

$S=\left[\begin{array}{ccccccccccc}1& 0& 0& -1& 0& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -1& 0& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 1& 0& -1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& -1& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 1& 0& -1& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0& 1\end{array}\right]$ , (25)

$\Lambda =diag\left({0.7408}^{|P|},0.6065,0.5488,0.4493,0.3679,0.3012,0.2466,0.2019,0.1653,0.14,0.11\right)$ ,(26)

$\Lambda$ 是个对角矩阵。现假设 $\Lambda$ 的特征值 ${\lambda }_{i}\left(i=1,2,\cdots ,11\right)$ 对应的特征向量分别为：

$\left({y}_{1},0,0,0,0,0,0,0,0,0,0\right),\left(0,{y}_{2},0,0,0,0,0,0,0,0,0\right),\cdots ,\left(0,0,0,0,0,0,0,0,0,0,{y}_{11}\right)$ .(27)

$Y\left(k\right)={S}^{-1}\left(Z\left(k\right)-\stackrel{¯}{Z}\right)$ , (28)

Figure 1. Curve: result of algorithm applied to simulation network data

3.2. 真实数据

Figure 2. Curve: result of algorithm applied to data GSE2565

4. 讨论

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