Inverse problems in reduced order models of cardiovascular haemodynamics: aspects of data assimilation and heart rate variability

Abstract

Inverse problems in cardiovascular modelling have become increasingly important to assess each patient individually. These problems entail estimation of patient-specific model parameters from uncertain measurements acquired in the clinic. In recent years, the method of data assimilation, especially the unscented Kalman filter, has gained popularity to address computational efficiency and uncertainty consideration in such problems. This work highlights and presents solutions to several challenges of this method pertinent to models of cardiovascular haemodynamics. These include methods to (i) avoid ill-conditioning of the covariance matrix, (ii) handle a variety of measurement types, (iii) include a variety of prior knowledge in the method, and (iv) incorporate measurements acquired at different heart rates, a common situation in the clinic where the patient state differs according to the clinical situation. Results are presented for two patient-specific cases of congenital heart disease. To illustrate and validate data assimilation with measurements at different heart rates, the results are presented on a synthetic dataset and on a patient-specific case with heart valve regurgitation. It is shown that the new method significantly improves the agreement between model predictions and measurements. The developed methods can be readily applied to other pathophysiologies and extended to dynamical systems which exhibit different responses under different sets of known parameters or different sets of inputs (such as forcing/excitation frequencies).

A Lumped Parameter Model to Study Atrioventricular Valve Regurgitation in Stage 1 and Changes Across Stage 2 Surgery in Single Ventricle Patients

“…where T a0 is a material constant, f (l) represents the sarcomere force-length relationship, g(t a ) models the contraction of the chamber, h(v s ) models sarcomere active viscosity. These dependences are described in earlier works by the authors [8], [11] and are unmodified in this study. On the contrary, some changes are here introduced to better model the passive behaviour of the myocardium.…”

“…A one-fibre model [8]- [11] is used to describe the two heart chambers (single ventricle and single atrium) in single ventricle physiology. In this model, the relationship between pressure in the chamber p, volume of the chamber V , fibre stress σ f , and the myocardial wall volume V w is described as σ f /p = (1 + 3V /V w ).…”

“…where T p0 and c p are sarcomere material constants, and compared to the previous works [8]- [11] the additional term to model viscous effects is 1 + ζ(λ) withλ = dλ/dt and ζ(λ) given as…”

“…The valve model is adopted from previous studies [8], [11] to model regurgitation due to incomplete leaflet closure and prolapse. The pressure drop Δp across a valve is described as a function of flow-rate q as follows…”

Closed-Loop Lumped Parameter Modeling of Hemodynamics During Cirrhogenesis in Rats

“…Moreover, in this study, we opted for a minimum number of elements to minimize the parameters to be calibrated. Additional experimental measurements could allow for construction of a more detailed model (e.g including a capacitance in the aorta and in the vena cava), as proposed in [47] or patients with single-ventricle physiology.…”