Application of the method of characteristics for the study of shock waves in models of blood flow in the aorta.

Numerical algorithms are presented for the numerical solution of the one-dimensional model of blood flow in the aorta. The pertinent hyperbolic equations are written using Riemann invariants, which are integrated along the characteristics using two efficient algorithms. Because of the hyperbolic nature of the equations shock waves are to be expected, and their occurrence is discussed.

Active and passive stresses in the myocardium.

A mathematical approach that can be used to calculate the passive stress in the ventricular wall is presented. The active fiber stress (force/unit area) generated by the muscular fibers in the ventricular wall is expressed by means of body force (force/unit volume of the myocardium). It is shown that the total intramyocardial passive stress induced in the passive medium of the myocardium can be expressed as the sum of a passive stress induced by the left ventricular pressure and a passive stress induced by the active fiber stress. Applications to experimental data published in the literature are given. New results are presented that show the relation among those two components of the intramyocardial passive stress. New relations between the intramyocardial passive stress, the slope (elastance) of the pressure-volume relation, and the residual volume are also derived. The results obtained give a better understanding of some aspects of the mechanics of cardiac contraction and can provide a more detailed interpretation of clinical conditions.

Studying the mechanics of left ventricular contraction.

Several interesting aspects of the ESPVR have been discussed in this study, including: a) A possibility to introduce, in an explicit manner, the active force of the myocardium in the formalism describing the PVR of the left ventricle. b) A possibility to express the ventriculo-arterial coupling by using the ratio Emax/eam in a way to distinguish between the normal physiological state and the mildly or severely depressed state of the heart. The possibility of also expressing this coupling by using directly different areas under ESPVR has been indicated. A third method, not discussed here, is to use the windkessel model (see [27, 53]). c) The relationship between oxygen consumption and all the areas under ESPVR. d) A possible mechanism of adaptation to short- or long-term variation in load condition by changing ESPVR in a way to create an SWR (see Fig. 4 and Table 2). e) A possibility to use different areas under ESPVR for clinical diagnostic purposes (see [20]); an example for SWR and SWR/SW is given in Table. 2. f) A possibility of noninvasive clinical application of various results of this study. Item (f) depends on the possibility of noninvasive measurement of the left ventricular pressure, which has been reported by Bourguignon and Wagner [61], and Sato, et al. [62]. Noninvasive measurement of left ventricular dimensions has been discussed by Teichholz, et al. [63], Grassman, et al. [64], Dumesnil et al [65], Dumesnil and Shoucri [66, 67]. Measurement of ESPVR from one loop of the contraction cycle has also been discussed by Takeuchi, et al. [68] and Nakamoto, et al. [69]. Further readings can be found in [70] and [71]. The list of references is not exhaustive, but has been chosen to illustrate various related aspects of the topics discussed.

Ventriculo-arterial coupling and the areas under the end-systolic pressure-volume relation.

Ventriculo-arterial coupling is expressed as the ratio Emax/eam (maximum ventricular elastance/arterial elastance). Different areas under the end-systolic pressure-volume relation (ESPVR) are expressed in terms of Emax/eam. The explicit inclusion of the active force of the myocardium in the mathematical formalism describing the pressure-volume relation (PVR) leads to new insight into the mechanics of left ventricular contraction. Applications to experimental data related to stroke work area SW under ESPVR are discussed and provide further evidence for the consistency of the mathematical formalism used.

Possible clinical applications of the external work reserve of the myocardium.

The concept of external work reserve (EWR) related to the end-systolic pressure-volume relation in the left ventricle and introduced in previous publications is investigated. The potential clinical usefulness of indexes related to EWR as well as to different areas under the end-systolic pressure volume line (ESPVL) is indicated. The possibility of non-invasive clinical application of the results of this study is discussed.

Clinical application of end-systolic pressure-volume relation.

On the basis of a mathematical formalism derived in previous studies, properties of the end-systolic pressure-volume (P-V) relation were analyzed to define indexes that can characterize a normal or failing left ventricle. Careful analysis of different areas under the P-V line can lead to new indexes that describe the performance of the left ventricle. The possibility to distinguish between normal, mildly depressed, or severely depressed left ventricles based on P-V relation is discussed. Implementation of the results for routine clinical work is examined.

Theoretical study related to left ventricular energetics.

This study presents new mathematical relations to link oxygen consumption to different areas under the end-systolic pressure-volume relation (ESPVR). The approach consists of approximating the relation between oxygen consumption and left ventricular pressure by a quadratic polynomial, and then relating the coefficients of the quadratic polynomial to different areas under the ESPVR. The procedure applies to both ejecting contraction and isovolumic contraction. The concept of external energy reserve, EER, is introduced and experimental data that corroborate the derived theoretical results are discussed. The clinical significance of the results obtained enhances the potential possibility of using ESPVR in clinical work.

Pressure-volume relation in the right ventricle.

The concept of body force (force/unit volume of muscle), which has been suggested as an explanation for the mechanical contraction of the left ventricle, is now applied to the right ventricle. The results indicate that the same mathematical formalism can be applied to a description of the pressure-volume relation and the ejection mechanism in both the right and left ventricles.

Pump function of the heart as an optimal control problem.

In order to model the pump function of the heart the left ventricle is represented as an elastic thick-walled cylinder contracting symmetrically. The acceleration is included in the mathematical formalism describing the contraction of the myocardium and optimal control theory is used to solve the differential equation of motion of the cylindrical wall in such a way as to minimize a given performance index. Application of the equations to experimental data published in the literature is discussed. The mathematical formalism presents a new way to study the time variation of the volume ejected from the left ventricle. Methods to quantify the pump function of the heart are suggested.

Non-linear pressure-volume relation in left ventricle.

A thick-walled elastic cylinder contracting symmetrically is used as a model for the myocardium. The active force generated by the myocardium during systolic contraction is represented by body force (force/unit volume of myocardium). A mathematical formalism previously developed and based on large deformation analysis is used to derive a quadratic equation to represent the non-linear pressure volume (P-V) relation in the left ventricle in the Suga-Sagawa model. Experimental application of the results obtained confirms the consistency of the mathematical formalism developed to describe the P-V curve in the Suga-Sagawa model.

The pressure-volume relation in the left ventricle and the pump function of the heart.

The concept of body force (force per unit volume) is introduced to account for the effect of the force generated in the radial direction by the active state of the myocardium in an elastic model of the left ventricle represented as a thick-walled cylinder contracting symmetrically. Experimental evidence for the validity of the model is presented. It is shown how the radial force/unit area developed by the myocardium on its inner surface can be included in the equation of the pressure-volume relation (P-V relation) of the left ventricle according to the Suga-Sagawa model, as well as in the formalism that describes the pump function of the heart.

Quantitative relationships between left ventricular ejection and wall thickening and geometry.

The quantitative relationships that exist between left ventricular (LV) wall shortening, wall thickening, and geometry during LV ejection are not well defined. We used a mathematical model to measure these parameters in 40 patients with various LV geometries studied by echocardiography. As opposed to wall shortening, the percent contribution of wall thickening to LV ejection (% delta Vh) was 25 +/- 2% in normal subjects; in all the patients, it varied from 18 to 45% and was inversely correlated (r = 0.94) to the midwall radius-to-wall thickness ratio (R/h) of the ventricle at end diastole. On the other hand, the ratio of the quantity of blood ejected per unit of LV wall volume magnitude of delta V/V omega magnitude of varied from 0.20 to 1.20 (normal subjects 0.83 +/- 0.11) and was directly correlated (r = 0.94) to R/h; using independent data in the literature, we also found a similar relationship (r = 0.80) between the ratio of quantity of blood ejected per unit of LV mass (magnitude of delta V/M omega magnitude of) and R/h. Patients with presumably abnormal myocardial function did not satisfy the relationship between magnitude of delta V/V omega magnitude of or magnitude of delta V/M omega magnitude of and R/h.(ABSTRACT TRUNCATED AT 250 WORDS)

Theoretical study of pressure-volume relation in left ventricle.

The myocardium is represented as a thick-walled elastic cylinder contracting symmetrically. The concept of body force (force/unit volume) is used to model the force developed by the myocardium in the radial direction during systolic contraction. It is shown that the radial force per unit area developed by the myocardium on its inner surface can be included in the equation describing the pressure-volume relation in the left ventricle. Application of the equations derived to experimental data describing the pressure-volume line in the Suga-Sagawa model is given. The results obtained seem to indicate that the body force developed by the myocardium in a normal ejecting contraction reaches its peak toward the end of the contraction phase and that the peak value is related to the peak isovolumic pressure when a quasistatic approximation of the systolic contraction is considered. Implication of the developed model for future studies in cardiac mechanics is also discussed.

Performance of left ventricle based on pressure-volume relation.

The concept of body force (force per unit volume) is used to derive an expression for the radial force developed by the myocardium (active force) in a model of the left ventricle represented as an elastic thick-walled cylinder contracting symmetrically. This approach leads to a novel equation to describe the pressure-volume relation in the Suga-Sagawa model. New indices to describe the mechanics of the left ventricular contraction are derived. Results tend to demonstrate that the radial active force generated by the myocardium will reach its peak value near end-systole, and that this peak is related to the peak isovolumic pressure. The study was carried out within a quasi-static approximation of the contraction (inertia forces neglected).

The pressure-volume relation and the mechanics of left ventricular contraction.

The left ventricle is represented as an elastic thick-walled cylinder contracting symmetrically. The force generated by the active state of the myocardium in the radial direction is represented by body force (force/unit volume) and is included in the mathematical formalism that describes the contraction of the left ventricle. An equation for the P-V relation in the left ventricle is derived and various applications to study cardiac mechanics are discussed. The results obtained tend to demonstrate that the active force generated by the myocardium during an ejecting contraction reaches its maximum value near the end of the systolic phase, when the slope E of the P-V line reaches its maximum value Em, and that it is related to the peak isovolumic pressure.

Contribution to the theoretical study of the Doppler half-time method for the measurement of the cross-sectional area of the mitral valve.

The acceleration term is included in the expression of the Bernoulli equation and an application of this formalism to the study of the Doppler measured mitral flow velocity is discussed. Based on calculation of the time-to-time variation of the parameters involved, a possible theoretical explanation of the pressure half-time method for the measurement of the cross-sectional area of the mitral valve is given.

Some applications of the P-V relation to the study of left ventricular performance.

There is still controversy as to which characteristics of the pressure-volume relation should be used to define myocardial contractility. In the present study a mathematical model for the left ventricle as a two-dimensional cylinder contracting radially and symmetrically was used to establish a relation between a calculated intramyocardial pressure (Dh) and the P-V relation (PVR) at end-systole. Four new indices are introduced that allow a better assessment of change in inotropic state of the myocardium, namely the calculated intramyocardial pressure (Dh), the calculated resultant pressure across the inner surface of the myocardium (Dh-P) (P = cavity pressure), the work Wt related to the pressure (Dh) and the work Wd related to the pressure (Dh-P). A relation between Wt and Wd and different parts of the area under the PVR is established. Indices derived in this manner from the PVR to study changes in myocardial contractility appear to have a clear physical meaning.