Breaking Rank -- A Novel Unscented Kalman Filter for Parameter Estimations of a Lumped-Parameter Cardiovascular Model

TL;DR

The paper addresses the identifiability challenges that arise when estimating a high-dimensional (10-parameter) lumped-parameter cardiovascular model from limited observations, a problem plagued by rank deficiency. It proposes a novel modification of the unscented Kalman filter (UKF) that lengthens the observation window via a Kalman interval $\tau_k$ and expands the effective observation dimension to satisfy $a\ge L$, enabling robust parameter estimation without prior distributions. The authors demonstrate that the modified UKF achieves high accuracy (often $\ge 98\%$) for most parameters across diverse, noise-corrupted synthetic data, outperforming the original UKF which diverges under similar stress. Additionally, the approach supports state estimation of unmeasured variables (e.g., arterial pressure) from LV data, highlighting practical utility for personalisation and digital twin applications in cardiology. Overall, the method offers a robust, gradient-free pathway to personalising high-dimensional cardiovascular models in the presence of noise and limited prior knowledge.

Abstract

We make modifications to the unscented Kalman filter (UKF) which bestow almost complete practical identifiability upon a lumped-parameter cardiovascular model with 10 parameters and 4 output observables - a highly non-linear, stiff problem of clinical significance. The modifications overcome the challenging problems of rank deficiency when applying the UKF to parameter estimation. Rank deficiency usually means only a small subset of parameters can be estimated. Traditionally, pragmatic compromises are made, such as selecting an optimal subset of parameters for estimation and fixing non-influential parameters. Kalman filters are typically used for dynamical state tracking, to facilitate the control u at every time step. However, for the purpose of parameter estimation, this constraint no longer applies. Our modification has transformed the utility of UKF for the parameter estimation purpose, including minimally influential parameters, with excellent robustness (i.e., under severe noise corruption, challenging patho-physiology, and no prior knowledge of parameter distributions). The modified UKF algorithm is robust in recovering almost all parameters to over 98% accuracy, over 90% of the time, with a challenging target data set of 50, 10-parameter samples. We compare this to the original implementation of the UKF algorithm for parameter estimation and demonstrate a significant improvement.

Figures (6)

• Figure 1: Identifiability heatmap for $\geq$ 98% accuracy at 1% noise level ($\sigma_{\text{noise}} = 0.01$). Each number in the heatmap represents the percentage of parameter sets which achieved a minimum of $98\%$ of accuracy at 1% noise, for each parameter, from the available observation combinations.
• Figure 2: Identifiability heatmap for $\geq$ 95% accuracy at 1% noise level ($\sigma_{\text{noise}} = 0.01$). Each number in the heatmap represents the percentage of parameter sets which achieved a minimum of $95\%$ of accuracy at 1% noise, for each parameter, from the available observation combinations.
• Figure 3: Identifiability heatmap for $\geq$ 90% accuracy at 1% noise level ($\sigma_{\text{noise}} = 0.01$). Each number in the heatmap represents the percentage of parameter sets which achieved a minimum of $90\%$ of accuracy at 1% noise, for each parameter, from the available observation combinations.
• Figure 4: The figure shows the comparison of the original (old) and modified (new) UKF implementations on the convergence of the parameter sets over the duration of 100 cycles at 5% observation noise for the full observable set ($p_{lv}$, $p_{sa}$, $p_{sv}$ and $V_{lv}$). The shaded orange and blue regions represent the variance of the parameter estimate. For the new method, with parameters $R_s, C_{sa}$ and $C_{sv}$, there has been covariance collapse, such that the filter has become overconfident on its estimates. To reiterate $\tau$ was a free parameter during tuning, however all target sets had $\tau$ set to 1.
• Figure 5: Identifiability comparison for the 4-observables ($p_{lv}$, $p_{sa}$, $p_{sv}$) and ($V_{lv}$) at $\geq98\%$ accuracy, with varying levels of noise, against the original method with 5% noise. To reiterate $\tau$ was a free parameter during tuning, however all target sets had $\tau$ set to 1.