Eindhoven University of Technology. Exam Modeling Cardiac Function (8W160)

Transcription

1 Eindhoven University of Technology department of Biomedical Engineering group Cardiovascular Biomechanics Exam Modeling Cardiac Function (8W160) January 21, 2011, h This exam consists of 6 pages, containing 6 exercises. Answers may be given in English or Dutch. Motivate your answers. 1. The figure below shows left ventricular pressure-volume loops recorded in a sheep: the right-most loop reflects a normal loop, the other loops were obtained upon vena cava constriction (Kind et al, Am J Physiol, 2009). The heart rate was paced at about 80 beats per minute. To model the experiment we use a lumped parameter model of the systemic circulation, with time-varying elastance model of the ventricle, two valves, an arterial system, composed of an aortic impedance and an arterial compliance, a peripheral resistance and a venous system, composed of a venous impedance and a venous compliance. (a) Draw the model. (b) Give the constitutive equation for the left ventricle. Give an estimate of all model parameters. (c) The pv loop in the figure cannot be readily obtained with a time-varying elastance model. Explain the limitations of time-varying elastance model, using your knowledge of cardiac tissue properties. (d) Give the constitutive equations for the peripheral resistance and the arterial compliance. Estimate the parameter values of these elements. (e) As a first extension of the model arterial for the arterial system, an arterial inertance can be added. Give the constitutive equation for this element and explain the physical property it describes.

2 2. The figure below shows a simple electric analog of the cell membrane. In the resting state it holds that V m = 73 mv, E Na = +71 mv, and E K = 89 mv. intracellular J s J Na J K V m C m g Na g K E Na E K intracellular We first consider the resting state, where the stimulus current J s equals 0. (a) Consider Na +. What does the quantity E Na represent? What physical condition determines the value of E Na? (b) What physical condition determines the value of V m? Explain the value of V m in relation to the values of E Na and E K. (c) Discuss the transport of Na + -ions in the resting state. In experiments to elucidate the mechanism underlying the action potential a stimulus current J s is applied to the membrane. (d) Give the relation between the currents J and the membrane potential V m, that should hold at all moments during such an experiment. (e) The voltage-clamp experiment has greatly contributed to our understanding of the mechanism of the action potential. Describe this experiment and a typical result.

3 3. To describe the constitutive behaviour of cardiac muscle in uniaxial loading experiments, a two-element model can be used, consisting of a parallel arrangement of a contractile element CE and a passive element PE: CE PE Phenomenological models for the active and passive element can be used to relate active stress T a and passive stress T p to sarcomere length l s, time after activation t, and sarcomere shortening velocity v s. (a) What variable(s) does T p depend on? Sketch the functional relation(s). Indicate representative values on the figure axes. (b) What variable(s) does T a depend on? Sketch the functional relation(s). Indicate representative values on the figure axes. The one-dimensional preload-afterload experiment has been used to mimic the mechanics of a sarcomere in the cardiac wall during the cardiac cycle. (c) Describe the experiment and sketch the time course of l s, T a and T p during the experiment. (d) Discuss to what extent sarcomere mechanics in this experiment indeed mimics sarcomere mechanics in the cardiac wall during the cardiac cycle.

4 4. In models of left ventricular wall mechanics, fiber orientation in the left ventricular wall has been quantified by two fiber angles. (a) Define the two angles. Illustrate the definition in a figure. (b) Sketch the spatial distribution of the two angles over the left ventricular wall. The figure below (van der Toorn, 2002) shows measurements of the relation between circumferential strain at the inner wall ɛ inner and torsion in the human left ventricle, for healthy volunteers ( Control ) and patients with a stenosed aortic valve ( AVSten ). (c) Describe the deformation mode torsion. Explain how it is related to the structure of the cardiac wall. (d) What meaning has been ascribed to the linear relation between circumferential shortening and torsion in the healthy heart? (e) Explain the difference between the measurement results in the healthy volunteers and the patients.

5 5. Finite element models of left ventricular mechanics have been used to investigate local tissue mechanics during a cardiac cycle. The figure below shows the computed relation between sarcomere length and myofiber stress, both under normal circumstances (dashed line) and during pacing (solid line) of a healthy heart. Timing of activation during pacing is indicated by Early, Mid and Late. (a) Explain the stress-length loop for the normal ventricle. What physical meaning can be attributed to the area enclosed by the loop? (b) Explain the stress-length loop in the early activated region of the paced ventricle. (c) Explain the stress-length loop in the late activated region of the paced ventricle. (d) How would pacing of a healthy heart affect pump function acutely? (e) Discuss two long term adaptation effects of the heart in response to pacing.

6 6. The cardiovascular system is able to adapt to changes in the environment through regulation, growth and remodeling. (a) Illustrate the concept of regulation through the baroreflex response to sudden blood loss. In the one-fiber model of cardiac mechanics fiber stretch ratio λ is related to ventricular cavity volume V c and wall volume V w according to: λ = ( ) 1 + 3(Vc /V w ) 1/ (V 0 /V w ) where V 0 represents the cavity volume in the reference state, to which λ is referred to. This expression was used in a model to describe the end stage of left ventricular adaptation. It was assumed that global function demands, expressed in terms of stroke volume V stroke and ejection pressure p e, were matched to optimal mechanics criteria of the myofibers, expressed by an optimal amount of work w opt per unit of tissue volume per cycle, and an optimal amount of shortening during the ejection phase, expressed by λ opt, by a specific choice of left ventricular cavity and wall volume. (b) Derive an expression for λ opt. (c) Derive an expression for w opt. (d) Assuming that in the healthy human heart adaptation is completed, estimate values of λ opt and w opt. (e) Discuss whether this model would be suitable to describe ventricular adaptation in response to an electrical conduction disorder.