Nonhomogeneous Strain From Sparse Marker Arrays for Analysis of Transmural Myocardial Mechanics

[+] Author and Article Information
K. Kindberg

Division of Biomedical Modelling and Simulation, Department of Biomedical Engineering, Linköpings Universitet, SE–581 85, Linköping, Swedenkatki@imt.liu.se

M. Karlsson

Division of Biomedical Modelling and Simulation, Department of Biomedical Engineering, Linköpings Universitet, SE–581 85, Linköping, Swedenmatka@imt.liu.se

N. B. Ingels

Department of Cardiothoracic Surgery, School of Medicine,  Stanford University, Stanford, CA 94305, and Laboratory of Cardiovascular Physiology and Biophysics, Research Institute of the Palo Alto Medical Foundation, Palo Alto, CA 94305ingels@stanford.edu

J. C. Criscione

Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843jccriscione@tamu.edu

J Biomech Eng 129(4), 603-610 (Nov 24, 2006) (8 pages) doi:10.1115/1.2746385 History: Received October 25, 2005; Revised November 24, 2006

Background: Knowledge of normal cardiac kinematics is important when attempting to understand the mechanisms that impair the contractile function of the heart during disease. The complex kinematics of the heart can be studied by inserting radiopaque markers in the cardiac wall and study the pumping heart with biplane cineradiography. In order to study the local strain, the bead array was developed where small radiopaque beads are inserted along three columns transmurally in the left ventricle. Method: This paper suggests a straightforward method for strain computation, based on polynomial least-squares fitting and tailored for combined marker and bead array analyses. Results: This polynomial method gives small errors for a realistic bead array on an analytical test case. The method delivers an explicit expression of the Lagrangian strain tensor as a polynomial function of the coordinates of material points in the reference configuration. The method suggested in this paper is validated with analytical strains on a deforming cylinder resembling the heart, compared to a previously suggested finite element method, and applied to in vivo ovine data. The errors in the estimated strain components are shown to remain unchanged on an analytical test case when evaluating the effects of one missing bead. In conclusion, the proposed strain computation method is accurate and robust, with errors smaller or comparable to the current gold standard when applied on an analytical test case.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

The marker (1–13) and bead (15–26) arrays in the ovine left ventricle and the local Cartesian coordinate system (X1,X2,X3). There are three columns of beads: 15–18, 19–22, and 23–26. Further details can be found in (2).

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Figure 2

Cylindrical model of the left ventricle. The cylinder to the left is the undeformed reference configuration of unit height. The cylinder to the right shows the deformed configuration schematically. The deformation is defined by Eq. 13.

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Figure 3

Results from the analytical test case. The transmural strain components in local Cartesian coordinates (X1,X2,X3) for a simulated three-by-four bead array with ∼10mm spacing between the epicardial beads. Dotted: Linear-quadratic polynomial strain, dashed: bilinear-quadratic finite element strain, solid: analytical strain.

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Figure 4

The midwall strain components during one cardiac cycle from end diastole (ED) to ED for one animal, computed with linear-quadratic and linear-cubic polynomials. End systole (ES) is marked.

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Figure 5

Midwall E33 versus left ventricular volume (LVV) for one animal during one cardiac cycle. The strain is computed with a linear-quadratic polynomial.




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