A Tissue-Level Electromechanical Model of the Left Ventricle: Application to the Analysis of Intraventricular Pressure

Abstract

The ventricular pressure profile is characteristic of the cardiac contraction progress and is useful to evaluate the cardiac performance. In this contribution, a tissue-level electromechanical model of the left ventricle is proposed, to assist the interpretation of left ventricular pressure waveforms. The left ventricle has been modeled as an ellipsoid composed of twelve mechano-hydraulic sub-systems. The asynchronous contraction of these twelve myocardial segments has been represented in order to reproduce a realistic pressure profiles. To take into account the different energy domains involved, the tissue-level scale and to facilitate the building of a modular model, multiple formalisms have been used: Bond Graph formalism for the mechano-hydraulic aspects and cellular automata for the electrical activation. An experimental protocol has been defined to acquire ventricular pressure signals from three pigs, with different afterload conditions. Evolutionary Algorithms have been used to identify the model parameters in order to minimize the error between experimental and simulated ventricular pressure signals. Simulation results show that the model is able to reproduce experimental ventricular pressure. In addition, electro-mechanical activation times have been determined in the identification process. For example, the maximum electrical activation time is reached, respectively, 96.5, 139.3 and 131.5 ms for the first, second, and third pigs. These preliminary results are encouraging for the application of the model on non-invasive data like ECG, arterial pressure or myocardial strain.

Appendices

Appendix A

1.1 The Bond Graph Formalism

The Bond Graph (BG) formalism is a diagram-based method that is particularly powerful for representation of multi-energy systems, since it is based on the representation of power exchanges; the terminology, the rules, and the construction of Bond Graph models are the same for all energy domains. For example, in the mechanical domain, the effort variable e is the force and the flow variable f is the rate; whereas, in the hydraulic domain, the effort variable e is the pressure and the flow variable f represents flow. The power is the product of the effort and the flow: P = e·f. The elements of the Bond Graph language can be classified as:

1.2 Passive Elements: R, C and I

• Resistive element (R):

The resistive element R is used to describe dissipative phenomena and can represent electrical resistors, dashpots or plugs in fluid lines.

• Capacitive element (C):

The capacitive element C is used to describe energy storage and can represent springs or electrical capacitors.

• Inertial element (I):

The inertial element I is used to model inductance effects in electrical systems and mass or inertia effects in mechanical or fluid systems.

1.3 Active Elements: Se and Sf

An effort source is an element that produces an effort independently of the flow, and a flow source an element that produces flow independently of the effort.

1.4 Junction Elements: 0, 1, TF, GY

• 0 junction:

The 0 junction is characterized by the equality of the efforts on all its links, while the corresponding flows sum up to zero, if power orientations are taken positive toward the junction.

• 1 junction:

1 junction is characterized by the equality of the flows on all its links, and the corresponding efforts sum up to zero with the same power orientations.

• Transformer (TF):

The transformer TF conserves power and transmits the factors of power with scaling defined by the transformer modulus. It can represent an ideal electrical transformer or a mass-less lever.

• Gyrator (GY):

A gyrator establishes a relationship between flow to effort and effort to flow and conserves power. It can represent a mechanical gyroscope or an electrical dc motor.

The Bond Graph formalism can be particularly useful for modelling physiological systems which often include various energy domains. For example, models of the vascular system (LeFèvre and Couteiro 1999; Diaz-Zuccarini 2003) are especially interesting since they take into account different energy phenomena (hydraulic, mechanic, chemical…).

Appendix B

2.1 Parameter Values

Some model Parameters values are not determined by the identification algorithm. They are taken from the literature.

See Table 1

Table 1  

Appendix C

3.1 Identified Parameter Values

See Table 2

Table 2