On mining latent treatment patterns from electronic medical records

Abstract

Clinical pathway (CP) analysis plays an important role in health-care management in ensuring specialized, standardized, normalized and sophisticated therapy procedures for individual patients. Recently, with the rapid development of hospital information systems, a large volume of electronic medical records (EMRs) has been produced, which provides a comprehensive source for CP analysis. In this paper, we are concerned with the problem of utilizing the heterogeneous EMRs to assist CP analysis and improvement. More specifically, we develop a probabilistic topic model to link patient features and treatment behaviors together to mine treatment patterns hidden in EMRs. Discovered treatment patterns, as actionable knowledge representing the best practice for most patients in most time of their treatment processes, form the backbone of CPs, and can be exploited to help physicians better understand their specialty and learn from previous experiences for CP analysis and improvement. Experimental results on a real collection of 985 EMRs collected from a Chinese hospital show that the proposed approach can effectively identify meaningful treatment patterns from EMRs.

Appendix

In this appendix, we give the derivation of Eq. (11)

$$\begin{aligned}&P(\mathbf{f}, \mathbf{v}, \mathbf{a}, \mathbf{t}, \mathbf{z}|\alpha ,\beta , \gamma , \eta , \iota , \mathbf{g}, \mathbf{k}, \mathbf{x}, \mathbf{y})\nonumber \\&\quad = P(\mathbf{z}|\alpha ) P(\mathbf{f}|\mathbf{z},\eta ) P(\mathbf{v}|\mathbf{z}, \mathbf{f}, \iota , \mathbf{g}, \mathbf{k}, \mathbf{x}, \mathbf{y} ) P(\mathbf{a}|\mathbf{z},\beta ) P(\mathbf{t}|\mathbf{z},\mathbf{a},\gamma ) \end{aligned}$$

(27)

For $P(\mathbf{z}|\alpha )$, we have

$$\begin{aligned} P(\mathbf{z}|\alpha )&= \int P(\mathbf{z}|\varTheta )P(\theta |\alpha ) d\varTheta \nonumber \\&= \int \prod _{d=1}^{D} \left( \prod _{i=1}^{N_d} P(z_{d,i}|\theta _d)P(\theta _d|\alpha )\right) d\varTheta \nonumber \\&= \int \prod _{d=1}^{D} \prod _{z=1}^Z \theta _{d,z}^{C_{d,z}} \prod _{d=1}^{D} \left( \frac{\varGamma (Z \alpha )}{\varGamma (\alpha )^Z} \prod _{z=1}^Z\theta _{d,z}^{\alpha - 1} \right) d\varTheta \nonumber \\&\propto \prod _{d=1}^{D} \frac{\prod _{z=1}^Z \varGamma (C_{d,z}+\alpha )}{\varGamma (\sum _{z=1}^Z (C_{d,z} + \alpha ))} \end{aligned}$$

(28)

For $P(\mathbf{f}|\mathbf{z},\eta )$, we have

$$\begin{aligned} P(\mathbf{f}|\mathbf{z},\eta )&= \int P(\mathbf{f}|\varPsi ,\mathbf{z})P(\varPsi |\eta )d\varPsi \nonumber \\&= \int \prod _{d=1}^{D} \prod _{i=1}^{N_d^f} P(f|\psi _{z_{d_i}}) \prod _{z=1}^Z P(\psi _{z_{d_i}}|\eta )d\varPsi \nonumber \\&= \int \prod _{z=1}^Z \prod _{f=1}^{F} \psi _{z,f}^{C_{z,f}} \prod _{z=1}^Z \left( \frac{\varGamma ({F} \beta )}{\varGamma (\beta )^F}\prod _{f=1}^{F}\psi _{z,f}^{\beta -1}\right) d\varPsi \nonumber \\&\propto \prod _{z=1}^Z \frac{\prod _{f=1}^{F} \varGamma (C_{z,f}+\beta )}{\varGamma (\sum _{f=1}^{F}(C_{z,f}+\beta ))} \end{aligned}$$

(29)

For $P(\mathbf{v}|\mathbf{z}, \mathbf{f}, \iota , \mathbf{g}, \mathbf{k}, \mathbf{x}, \mathbf{y} )$, we have

$$\begin{aligned}&P(\mathbf{v}|\mathbf{z}, \mathbf{f}, \iota , \mathbf{g}, \mathbf{k}, \mathbf{x}, \mathbf{y} )\nonumber \\&\quad = \int P(\mathbf{v}|\mathbf{z},\mathbf{f},\varDelta )P(\varDelta |\iota ) d\varDelta \prod _{z=1}^Z \prod _{f=1}^F P(V_{z,f}|\mathbf{g}, \mathbf{k}, \mathbf{x}, \mathbf{y})\nonumber \\&\quad = \left\{ \begin{array}{lll} \int \prod _{d=1}^{D} \prod _{i=1}^{N_d^f} P(v_i|\delta _{z,f}) \prod _{z=1}^Z \prod _{f=1}^{F}P(\delta _{z,f}|\iota )d\varDelta &{} : &{} f \,\hbox {is a categorical feature}\\ \prod _{z=1}^Z \prod _{f=1}^F \int \int \prod _{v\in \mathbf{v}_{z,f}} P(v|\mu _{z,f}, \lambda _{z,f}^{-1})&{}\\ P(\mu _{z,f}|g_f, (k_f \lambda _{z,f})^{-1}) P(\lambda _{z,f}|x_i, y_i) d\mu _{z,f} d\lambda _{z,f}&{}:&{} f \,\hbox {is a numerical feature} \end{array} \right. \nonumber \\&\quad = \left\{ \begin{array}{lll} \int \prod _{z=1}^Z \prod _{f=1}^{F} \big (\prod _{v=1}^{V_{z,f}} \delta _{z,f,v}^{C_{z,f,v}} \frac{\varGamma (\iota V_{z,f} )}{\varGamma (\iota )^{V_{z,f}}}\prod _{v=1}^{V_{z,f}}\delta _{z,f,v}^{\iota -1}\big )d\varDelta &{}:&{} f \,\hbox {is a categorical feature}\\ \prod _{z=1}^Z \prod _{f=1}^F (2\pi )^{-\frac{C_{z,f}}{2}} \frac{\varGamma (x_{z,f})}{\varGamma (x_f)}\frac{{y_{f}}^{x_f}}{{y_{z,f}}^{x_{z,f}}} (\frac{k_f}{k_{z,f}})^{\frac{1}{2}}&{}:&{} f \,\hbox {is a numerical feature} \end{array} \right. \nonumber \\&\quad \propto \left\{ \begin{array}{lll} \prod _{z=1}^Z \prod _{f=1}^{F} \frac{\prod _{v=1}^{V_{z,f}}\varGamma ( C_{z,f,v}+\iota )}{\varGamma (\sum _{v=1}^{V_{z,f}} (C_{z,f,v} + \iota ))} &{}:&{} f \,\hbox {is a categorical feature}\\ \prod _{z=1}^Z \prod _{f=1}^F (2\pi )^{-\frac{C_{z,f}}{2}} \frac{\varGamma (x_{z,f})}{\varGamma (x_f)}\frac{{y_{f}}^{x_f}}{{y_{z,f}}^{x_{z,f}}} (\frac{k_f}{k_{z,f}})^{\frac{1}{2}}&{}:&{} f \,\hbox {is a numerical feature} \end{array} \right. \end{aligned}$$

(30)

For $P(\mathbf{a}|\mathbf{z},\beta )$, we have

$$\begin{aligned} P(\mathbf{a}|\mathbf{z},\beta )&= \int P(\mathbf{a}|\varPhi ,\mathbf{z})P(\varPhi |\beta )d\varPhi \nonumber \\&= \int \prod _{d=1}^{D} \prod _{i=1}^{N_d^e} P(e_i.a|\phi _{z_i}) \prod _{z=1}^Z P(\phi _z|\beta )d\varPhi \nonumber \\&= \int \prod _{z=1}^Z \prod _{a=1}^{A} \phi _{z,a}^{C_{z,a}} \prod _{z=1}^Z \left( \frac{\varGamma (A \beta )}{\varGamma (\beta )^A}\prod _{a=1}^{A}\phi _{z,a}^{\beta -1}\right) d\varPhi \nonumber \\&\propto \prod _{z=1}^Z \frac{\prod _{a=1}^{A} \varGamma (C_{z,a}+\beta )}{\varGamma (\sum _{a=1}^{A}(C_{z,a}+\beta ))} \end{aligned}$$

(31)

For $P(\mathbf{t}|\mathbf{z},\mathbf{a},\gamma )$, we have:

$$\begin{aligned} P(\mathbf{t}|\mathbf{z},\mathbf{a},\gamma )&= \int P(\mathbf{t}|\mathbf{z},\mathbf{a},\varXi )P(\varXi |\gamma ) d\varXi \nonumber \\&= \int \prod _{d=1}^{D} \prod _{i=1}^{N_d^e} P(e_i.t|\xi _{z_i,e_i.a}) \prod _{z=1}^Z \prod _{a=1}^{A}P(\xi _{z,a}|\gamma )d\varXi \nonumber \\&= \int \prod _{z=1}^Z \prod _{a=1}^{A} \left( \prod _{t=1}^{T} \xi _{z,a,t}^{C_{z,a,t}} \frac{\varGamma (T\gamma )}{\varGamma (\gamma )^T}\prod _{t=1}^{T}\xi _{z,a,t}^{\gamma -1}\right) d\varXi \nonumber \\&\propto \prod _{z=1}^Z \prod _{a=1}^{A} \frac{\prod _{t=1}^{T}\varGamma ( C_{z,a,t}+\gamma )}{\varGamma (\sum _{t=1}^{T}(C_{z,a,t} + \gamma ))} \end{aligned}$$

(32)

Substituting Eqs. (28)–(32) into Eq. (27), and using the chain rule and $\varGamma (\alpha ) \!=\! (\alpha -1)\varGamma (\alpha -1)$, we can obtain the conditional probability conveniently,

$$\begin{aligned}&P(z_{d,i} = z| \mathbf{f}, \mathbf{v}, \mathbf{a}, \mathbf{t},\mathbf{z}_{d}^{-i},\alpha , \beta , \gamma , \eta , \iota , \mathbf{g},\mathbf{k},\mathbf{x},\mathbf{y}) \nonumber \\&\quad = \left\{ \begin{array}{lll} \frac{P(z_{d,i}, f_{d,i}, v_{d,i}| \mathbf{f}_{d}^{-i}, \mathbf{v}_d^{-i}, \mathbf{z}_d^{-i}, \alpha , \eta , \iota , \mathbf{g},\mathbf{k},\mathbf{x},\mathbf{y})}{P(f_{d,i}, v_{d,i}| \mathbf{f}_{d}^{-i}, \mathbf{v}_d^{-i}, \mathbf{z}_d^{-i}, \alpha , \eta , \iota , \mathbf{g},\mathbf{k},\mathbf{x},\mathbf{y})} &{}:&{} i \,\hbox {is a patient feature} \\ \frac{P(z_{d, i}, p_{d,i}. a_{d, i}, t_{d, i}| \mathbf{a}_{d}^{-i}, \mathbf{t}_{d}^{-i}, \mathbf{z}_{d}^{-i}, \alpha , \beta , \gamma )}{P( a_{d, i}, t_{d, i}| \mathbf{a}_d^{-i}, \mathbf{t}_d^{-i}, \mathbf{z}_{d}^{-i}, \alpha , \beta , \gamma )} &{}:&{} i \,\text {is a clinical event}\end{array} \right. \nonumber \\&\quad \propto \left\{ \begin{array}{lll} \frac{P(\mathbf{z}, \mathbf{f}, \mathbf{v}|\alpha , \eta , \iota , \mathbf{g},\mathbf{k},\mathbf{x},\mathbf{y})}{P(\mathbf{z}_{d}^{-i}, \mathbf{f}_{d}^{-i}, \mathbf{v}_{d}^{-i}|\alpha , \eta , \iota , \mathbf{g},\mathbf{k},\mathbf{x},\mathbf{y})}&{}:&{} i \,\hbox {is a patient feature} \\ \frac{P(\mathbf{z}, \mathbf{a}, \mathbf{t}|\alpha , \beta , \gamma )}{P(\mathbf{z}_d^{-i}, \mathbf{a}_d^{-i}, \mathbf{t}_d^{-i}|\alpha , \beta , \gamma )} &{}:&{} i \,\text {is a clinical event} \end{array} \right. \nonumber \\&\quad \propto \frac{C_{z,d}^{-i} + \alpha }{C_{d,*}^{-i}+ \alpha Z}\cdot \left\{ \begin{array}{lll} \frac{C_{z,f_i}^{-i} + \eta }{C_{z,*}^{-i} + \eta F} \cdot \left\{ \begin{array}{lll} \frac{C_{z,f_i,v_i}^{-i} + \iota }{C_{z,f_i,*}^{-i} + V_{z,f_i} \iota } &{}:&{} f_i \,\hbox {is a categorical feature}\\ \frac{\varGamma (x_{z,f_i})}{\varGamma (x_{z,f_i}^{-i})} \cdot \frac{{y_{z,f_i}^{-i}}^{x_{z,f_i}^{-i}}}{{y_{z,f_i}}^{x_{z,f_i}}} \cdot (\frac{k_{z,f_i}^{-i}}{k_{z,f_i}})^{\frac{1}{2}}&{}:&{} f_{i} \,\hbox {is a numerical feature} \end{array} \right. &{}:&{} i \,\hbox {is a patient feature} \\ \frac{C_{z,a}^{-i} + \beta }{C_{z,*}^{-i} + A \beta } \cdot \frac{C_{z,a,t}^{-i} + \gamma }{C_{z,a,*}^{-i} + T \gamma } &{}:&{} i \,\text {is a clinical event} \end{array} \right. \nonumber \\ \end{aligned}$$

(33)