问题¶
描述药物在体内的分布和排除情况
模型假设¶
- 中心室(1)和周边室(2), 容积不变，药物从体外进入中心室, 在二室间相互转移, 从中心室排出体外
- 药物在房室间转移速率及向体外排除速 率, 与该室血药浓度成正比
模型建立¶
- $x_{i}(t)$ 药量
- $c_{i}(t)$ 浓度
- $V_{i}$ 容积
这里$i=1,2$
$f_{0} \sim \text { 给药速率 }$
模型建立 $\quad x_{i}(t)=V_{i} c_{i}(t), i=1,2$ $$ \begin{cases}\dot{c}_{1}(t)=-\left(k_{12}+k_{13}\right) c_{1}+\frac{V_{2}}{V_{1}} k_{21} c_{2}+\frac{f_{0}(t)}{V_{1}} \\ \dot{c}_{2}(t)=\frac{V_{1}}{V_{2}} k_{12} c_{1}-k_{21} c_{2} & \begin{array}{l} \end{array}\end{cases} $$
对应的齐次方程通解： $$ \begin{aligned} &\left\{\begin{array}{l} \overline{c_{1}}(t)=A_{1} e^{-\alpha t}+B_{1} e^{-\beta t} \\ \overline{c_{2}}(t)=A_{2} e^{-\alpha t}+B_{2} e^{-\beta t} \end{array}\right. \\ &\left\{\begin{array}{l} \alpha+\beta=k_{12}+k_{21}+k_{13} \\ \alpha \beta=k_{21} k_{13} \end{array}\right. \end{aligned} $$
几种常见的给药方式¶
快速静脉注射¶
$t=0$ 瞬时注射剂量 $D_{0}$ 的药物进入中心室, 血 药浓度立即为 $D_{0} / V_{1}$
$$ \left\{\begin{array}{l} \dot{c}_{1}(t)=-\left(k_{12}+k_{13}\right) c_{1}+\frac{V_{2}}{V_{1}} k_{21} c_{2}+\frac{f_{0}(t)}{V_{1}} \\ \dot{c}_{2}(t)=\frac{V_{1}}{V_{2}} k_{12} c_{1}-k_{21} c_{2} \end{array}\right. $$$$ f_{0}(t)=0, c_{1}(0)=\frac{D_{0}}{V_{1}}, c_{2}(0)=0 $$$$ \begin{aligned} &c_{1}(t)=\frac{D_{0}}{V_{1}(\beta-\alpha)}\left[\left(k_{21}-\alpha\right) e^{-\alpha t}+\left(\beta-k_{21}\right) e^{-\beta t}\right] \\ &c_{2}(t)=\frac{D_{0} k_{12}}{V_{2}(\beta-\alpha)}\left(e^{-\alpha t}-e^{-\beta t}\right) \quad\left\{\begin{array}{l} \alpha+\beta=k_{12}+k_{21}+k_{13} \\ \alpha \beta=k_{21} k_{13} \end{array}\right. \end{aligned} $$恒速静脉滴注¶
$ 0 \leq t \leq T \text { 药物以速率 } k_{0} \text { 进入中心室 } $
$$ \left\{\begin{array}{l} \dot{c}_{1}(t)=-\left(k_{12}+k_{13}\right) c_{1}+\frac{V_{2}}{V_{1}} k_{21} c_{2}+\frac{f_{0}(t)}{V_{1}} \\ \dot{c}_{2}(t)=\frac{V_{1}}{V_{2}} k_{12} c_{1}-k_{21} c_{2} \quad f_{0}(t)=k_{0}, c_{1}(0)=0, c_{2}(0)=0 \end{array}\right. $$$$ \left\{\begin{array}{l} c_{1}(t)=A_{1} e^{-\alpha t}+B_{1} e^{-\beta t}+\frac{k_{0}}{k_{13} V_{1}}, \quad 0 \leq t \leq T \\ c_{2}(t)=A_{2} e^{-\alpha t}+B_{2} e^{-\beta t}+\frac{k_{12} k_{0}}{k_{21} k_{13} V_{2}}, \quad 0 \leq t \leq T \\ A_{2}=\frac{V_{1}\left(k_{12}+k_{13}-\alpha\right)}{k_{21} V_{2}} A_{1}, B_{2}=\frac{V_{1}\left(k_{12}+k_{13}-\beta\right)}{k_{21} V_{2}} B_{1} \end{array}\right. $$$$ t>T, c_{1}(t) \text { 和 } c_{2}(t) \text { 按指数规律趋于零 } $$口服或肌肉注射¶
$ \text { 相当于药物 }\left(\text { 剂量 } D_{0}\right) \text { 先进入吸收室, 吸收后进入中心室 } $
吸收室药量 $x_{0}(t)$ $$ \left\{\begin{array}{l} \dot{c}_{1}(t)=-\left(k_{12}+k_{13}\right) c_{1}+\frac{V_{2}}{V_{1}} k_{21} c_{2}+\frac{f_{0}(t)}{V_{1}} \\ \dot{c}_{2}(t)=\frac{V_{1}}{V_{2}} k_{12} c_{1}-k_{21} c_{2} \end{array}\right. $$
$$ \left\{\begin{array}{l} \dot{x}_{0}(t)=-k_{01} x_{0} \\ x_{0}(0)=D_{0} \end{array}\right. $$$$ \begin{gathered} x_{0}(t)=D_{0} e^{-k_{01} t} \quad f_{0}(t)=k_{01} x_{0}(t)=D_{0} k_{01} e^{-k_{01} t} \\ c_{1}(t)=A e^{-\alpha t}+B e^{-\beta t}+E e^{-k_{01} t} \\ c_{1}(0)=0, c_{2}(0)=0 \Rightarrow A, B, E \end{gathered} $$参数估计¶
各种给药方式下的 $c_{1}(t), c_{2}(t)$ 取决于参数 $k_{12}, k_{21}, k_{13}, V_{1}, V_{2}$ $t=0$ 快速静脉注射 $D_{0}$, 在 $t_{i}(i=1,2, \ldots n)$ 测得 $c_{1}\left(t_{i}\right)$ $$ c_{1}(t)=\frac{D_{0}}{V_{1}(\beta-\alpha)}\left[\left(k_{21}-\alpha\right) e^{-\alpha t}+\left(\beta-k_{21}\right) e^{-\beta t}\right] $$ 设 $\alpha<\beta, t$ 充分大 $\square c_{1}(t)=\frac{D_{0}\left(k_{21}-\alpha\right)}{V_{1}(\beta-\alpha)} e^{-\alpha t}=A e^{-\alpha t}$ 由较大的 $t_{i}, c_{1}\left(t_{i}\right)$ 用最小二乘法定 $A, \alpha$ $$ \widetilde{c}_{1}(t)=c_{1}(t)-A e^{-\alpha t}=B e^{-\beta t} $$ 由较小的 $t_{i}, \widetilde{C}_{1}\left(t_{i}\right)$ 用最小二乘法定 $B, \beta$
$t \rightarrow \infty, c_{1}, c_{2} \rightarrow 0$ 들 进入中室的药物全部排除 $D_{0}=k_{13} V_{1} \int_{0}^{\infty} c_{1}(t) d t \hookrightarrow D_{0}=k_{13} V_{1}\left(\frac{A}{\alpha}+\frac{B}{\beta}\right)$ $c_{1}(0)=\frac{D_{0}}{V_{1}}=A+B \quad \square k_{13}=\frac{\alpha \beta(A+B)}{\alpha B+\beta A}$ $\left\{\begin{array}{l}\alpha+\beta=k_{12}+k \\ \alpha \beta=k_{21} k_{13}\end{array}\right.$ $k_{21}=\frac{\alpha \beta}{k_{13}}$ $k_{12}=\alpha+\beta-k_{13}-k_{21}$