Why DeepPumas?

Authors:
Niklas Korsbo, Mohamed Tarek

Machine learning meets scientific modelling

DeepPumas: knowledge + data

Applications

Nonlinear mixed effects (NLME)

Population-level parameters: \(\theta\), \(\Omega\), \(c\), \(\sigma\)

Covariates: \(\text{Age}_i\), \(\text{Weight}_i\)

Random effects: \(\eta_i \sim \mathcal{N}(0, \Omega)\)

Individual derived parameters: \[ \begin{aligned} \text{Ka}_i & = \theta_1 \cdot e^{\eta_{i,1}} + c_1 \cdot \text{Age}_i \\ \text{CL}_i & = \theta_2 \cdot e^{\eta_{i,2}} \\ \text{V}_i & = \theta_3 \cdot e^{\eta_{i,3}} + c_2 \cdot \text{Weight}_i^{c_3} \end{aligned} \]

Dynamics: \[ \begin{aligned} \frac{d[\text{Depot}_i]}{dt} & = - \text{Ka}_i \cdot [\text{Depot}_i] \\ \frac{d[\text{Central}_i]}{dt} & = \text{Ka}_i \cdot [\text{Depot}_i] - \frac{\text{CL}_i}{V_i} \cdot [\text{Central}_i] \end{aligned} \]

Error model: \[ \text{Outcome}_i \sim \mathcal{N}(\text{Central}_i, \text{Central}_i \cdot \sigma) \]

What is a neural network (NN)?

  • Information processing mechanism

  • Loosely based on neurons

  • Mathematically, just a function!

  • NNs are useable anywhere where you’d use a function!

  • Universal approximators!
  • Use data to automatically discover relationships.

DeepNLME

Population-level parameters: \(\theta\), \(\Omega\), \(c\), \(\sigma\)

Covariates: \(\text{Age}_i\), \(\text{Weight}_i\),

Random effects: \(\eta_i \sim \mathcal{N}(0, \Omega)\)

Individual derived parameters: \[ \begin{aligned} \text{Ka}_i & = \theta_1 \cdot e^{\eta_{i,1}} + c_1 \cdot \text{Age}_i + \\ \text{CL}_i & = \theta_2 \cdot e^{\eta_{i,2}} \\ \text{V}_i & = \theta_3 \cdot e^{\eta_{i,3}} + c_2 \cdot \text{Weight}_i^{c_3} + \end{aligned} \]

Dynamics: \[ \begin{aligned} \frac{d[\text{Depot}_i]}{dt} & = - \text{Ka}_i \cdot [\text{Depot}_i] \\ \frac{d[\text{Central}_i]}{dt} & = \text{Ka}_i \cdot [\text{Depot}_i] \, - \end{aligned} \]

Error model: \[ \text{Outcome}_i \sim \mathcal{N}(\text{Central}_i, \text{Central}_i \cdot \sigma) \]

DeepNLME examples

Summary