Practical Use of Regularization in Individualizing a Mathematical Model of Cardiovascular Hemodynamics Using Scarce Data

Abstract

Individualizing physiological models to a patient can enable patient-specific monitoring and treatment in critical care environments. However, this task often presents a unique "practical identifiability" challenge due to the conflict between model complexity and data scarcity. Regularization provides an established framework to cope with this conflict by compensating for data scarcity with prior knowledge. However, regularization has not been widely pursued in individualizing physiological models to facilitate patientspecific critical care. Thus, the goal of this work is to garner potentially generalizable insight into the practical use of regularization in individualizing a complex physiological model using scarce data by investigating its effect in a clinically significant critical care case study of blood volume kinetics and cardiovascular hemodynamics in hemorrhage and circulatory resuscitation. We construct a population-average model as prior knowledge and individualize the physiological model via regularization to illustrate that regularization can be effective in individualizing a physiological model to learn salient individual-specific characteristics (resulting in the goodness of fit to individual-specific data) while restricting unnecessary deviations from the populationaverage model (achieving practical identifiability). We also illustrate that regularization yields parsimonious individualization of only sensitive parameters as well as adequate physiological plausibility and relevance in predicting internal physiological states.

Mathematical model of volume kinetics and renal function after burn injury and resuscitation

“…First, we determined population-average parameter values by fitting our mathematical model simultaneously to all the measurements in all subjects in the dataset using the pooled approach [38]. Second, we estimated all 24 subject-specific parameters on the individual basis by fitting our mathematical model to the measurements associated with each subject, by employing a regularized fitting that intends to minimize the number of parametric deviations from the population-average values [39], [40]. Those subjectspecific parameters exhibiting deviations from population-average values in many sheep were chosen as sensitive subject-specific parameters, which were confirmed via a post-hoc parametric sensitivity analysis.…”

“…Second, we estimated all the 24 subject-specific parameters 𝜃 𝑖 associated with the subject 𝑖 by fitting our mathematical model to all the measurements associated with the subject 𝑖 to minimize the cost function in Eq. ( 38), using a regularized fitting that minimizes the number of parametric deviations from the population-average values [39], [40]: , (38) where 𝜆 𝑝 is the regularization weight and 𝛩 𝑙 is the normalization factor for the 𝑙-th element 𝜃(𝑙) of 𝜃, which is defined so that all the elements in 𝜃 are ranged approximately between 0 and 1 (note that such 𝛩 𝑙 can be estimated by, e.g., solving Eq. ( 38) with 𝜆 𝑝 = 0.05 and setting 𝛩 𝑙 as the inter-individual variability of the corresponding element 𝜃(𝑙)).…”

Credibility Assessment of a Subject-Specific Mathematical Model of Blood Volume Kinetics for Prediction of Physiological Response to Hemorrhagic Shock and Fluid Resuscitation

“…In addition, the post calibration identifiability results may change based on the optimization scheme. While we used a quadratic cost function, use of regularization methods may improve practical identifiability of some of the parameters ( Tivay et al, 2020 ).…”

“…However, there are still open and important questions that need to be answered to determine optimal fluid delivery to subjects in the case of hemorrhage, especially in regard to the type, amount, and timing of fluid delivery to be tailored to specific subjects ( Navarro et al, 2015 ). Several studies have leveraged the utility of subject-specific mathematical models to predict physiological responses such as change in blood volume (BV) or mean arterial pressure (MAP) in response to fluid delivery ( Bighamian et al, 2017 , 2018 ; Tivay et al, 2020 ). Furthermore, studies have been conducted to use these mathematical models to guide autonomous therapy of fluids ( Jin et al, 2019 ).…”

Mathematical Modeling, In-Human Evaluation and Analysis of Volume Kinetics and Kidney Function After Burn Injury and Resuscitation

“…Second, we classified the subject-specific parameters into sensitive and insensitive groups by quantifying and comparing the degree of inter-individual variability associated with all the subject-specific parameters. This task was accomplished by solving the following optimization problem for fitting with regularization [36] on an individual patient basis using a multi-start gradient descent method ("globalsearch" in conjunction with "fmincon") in MATLAB (MathWorks, Natick, MA):…”