Data-Augmented Modeling of Intracranial Pressure

Abstract

Precise management of patients with cerebral diseases often requires intracranial pressure (ICP) monitoring, which is highly invasive and requires a specialized ICU setting. The ability to noninvasively estimate ICP is highly compelling as an alternative to, or screening for, invasive ICP measurement. Most existing approaches for noninvasive ICP estimation aim to build a regression function that maps noninvasive measurements to an ICP estimate using statistical learning techniques. These data-based approaches have met limited success, likely because the amount of training data needed is onerous for this complex applications. In this work, we discuss an alternative strategy that aims to better utilize noninvasive measurement data by leveraging mechanistic understanding of physiology. Specifically, we developed a Bayesian framework that combines a multiscale model of intracranial physiology with noninvasive measurements of cerebral blood flow using transcranial Doppler. Virtual experiments with synthetic data are conducted to verify and analyze the proposed framework. A preliminary clinical application study on two patients is also performed in which we demonstrate the ability of this method to improve ICP prediction.

Appendix A Algorithm: Regularizing Iterative Ensemble Kalman Method

Prior sampling: Use Latin hypercube sampling method to generate the prior state ensemble $\{{\mathbf {x}}_j^{(0)}\}_{j=1}^{N_s}$, where ${\mathbf {x}}_j$ is jth sample of the augmented state, including major arterials’ CBFV and ABP, ICP, and unknown parameters. Let $\rho \in (0, 1)$ and $\tau = 1/\rho$.

For $n = 1 : n_{max}$

• 1. Forward prediction:

  • (a) Evaluate the forward intracranial model with the initial physical state, boundary conditions, and model parameters, which are updated in the last iteration. Namely, the analyzed state ${\hat{\mathbf {x}}}_j^{(n)}$ at iteration step n is propagated through the forward model ${\mathcal {F}}$ at $(n+1)$th iteration,

    $$\begin{aligned} {\mathbf {x}}_j^{(n+1)} = {\mathcal {F}}({\hat{\mathbf {x}}}_j^{n}). \end{aligned}$$

    (14)

  • (b) Obtain the perturbed ensemble of observation data $\{{\mathbf {y}}_j^{(0)}\}_{j=1}^{N_s}$ based on the data uncertainty level $\sigma _d$.

  • (c) Calculate statistical information of predicted state and observation data. We first calculate the sample means of state and data as,

    $$\begin{aligned} {\bar{\mathbf {x}}}^{(n+1)}= & {} \frac{1}{N_s}\sum _{j=1}^{N_s}{\mathbf {x}}^{(n+1)}_j \end{aligned}$$

    (15)
    $$\begin{aligned} {\bar{\mathbf {y}}}^{(n+1)}= & {} \frac{1}{N_s}\sum _{j=1}^{N_s}{\mathbf {y}}^{(n+1)}_j. \end{aligned}$$

    (16)

    The error covariances of the predicted state and observation data can then be obtained,

    $$\begin{aligned} P_m^{(n+1)}= & {} \frac{1}{N_s - 1}\sum _{j=1}^{N} \bigg ({\mathbf {x}}^{(n+1)}_j - {\bar{\mathbf {x}}}) ({\mathbf {x}}^{(n+1)}_j - {\bar{\mathbf {x}}}^{(n+1)} \bigg )^T \end{aligned}$$

    (17)
    $$\begin{aligned} P_d^{(n+1)}= & {} \frac{1}{N_s - 1}\sum _{j=1}^{N}({\mathbf {y}}^{(n+1)}_j - {\bar{\mathbf {y}}}) \bigg ({\mathbf {y}}^{(n+1)}_j - {\bar{\mathbf {y}}}^{(n+1)} \bigg )^T \end{aligned}$$

    (18)

• 2. Regularizing Bayesian analysis:

  (a) Calculate the control variable $\alpha _i^{(n+1)}$ by following sequence,

  $$\begin{aligned} \alpha _i^{(n+1)} = 2^{i+1}\alpha _0, \end{aligned}$$

  (19)

  where an initial guess of $\alpha _0 = 1$ is used in this work. The $\alpha ^{n+1}$ is obtained as $\alpha ^{(n+1)} \equiv \alpha ^{(n+1)}_{N}$, where N is the first integer when the inequality defined by Eq. (13) is satisfied. (b) Compute regularized Kalman gain matrix as,

  $$\begin{aligned} K^{(n+1)} = P_m^{(n+1)}H^T \bigg (HP_m^{(n+1)}H^T + \alpha ^{(n+1)}P_d^{(n+1)} \bigg )^{-1}, \end{aligned}$$

  (20)

  (c) Update each state sample as follows,

  $$\begin{aligned} {\hat{\mathbf {x}}}^{(n+1)} _j= {\mathbf {x}}^{(n+1)} _j + K^{(n+1)} \bigg ({\bar{\mathbf {y}}}^{(n+1)} - H{\mathbf {x}}^{(n+1)} _j \bigg ), \end{aligned}$$

  (21)

• 3. Stopping criteria:

  If

  $$\begin{aligned} \bigg |\bigg|{P_d^{(n+1)}}^{-1/2}({\mathbf {y}}^{(n+1)} - H{\bar{\mathbf {x}}}^{(n+1)}) \bigg |\bigg| \le \tau \sigma _d, \end{aligned}$$

  (22)

  then, stop the iteration. $\sigma _d$ represents noise level of observation data.