Download Exam Cardiac Modeling (8W160)

Eindhoven University of Technology — department of Biomedical Engineering
Exam Cardiac Modeling (8W160)
February 1, 2007, 14.00 – 17.00 h
This exam consists of 4 pages, containing 6 exercises.
Answers may be given in English or Dutch. Motivate your answers.
1. The figure below shows left ventricular pressure-volume (pV) loops recorded in a patient
during vena cava constriction (from Steendijk et al, 2004).
(a) The behaviour of the left ventricle can be described with a time-varying elastance
model. Give the mathematical formulation of the model, and estimate the values
of the model parameters.
(b) The patient, in which the pV-loops were measured, received a cardiac pacemaker.
How would a successful pacing treatment be reflected in
(1) pV-loops, acquired after the intervention and
(2) in the associated parameters in the time-varying elastance model?
The time-varying elastance heart model may be coupled to a lumped parameter model
of blood flow through the vascular system, formulated in terms of pressure p and flow
q. Based on the Navier-Stokes equation
¶
µ
∂~v
~ v = −∇p
~ + η∇
~ 2~v
+ ~v · ∇~
ρ
∂t
three basic elements of such a model can be defined.
(c) Give the constitutive equations for each of these three elements, and discuss their
relation to the Navier-Stokes equation.
(d) One other important element in a lumped parameter model of the circulation is
not related to the Navier-Stokes equation. Give the constitutive equation for this
element, and describe its physiological meaning.
1
Exam Cardiac Modeling (8W160) — February 1, 2007
2. The quantities in the table below characterise the resting state of the cell membrane of
a normal nerve cell. ci and ce represent intracellular and extracellular concentrations,
P represents permeability, and E and Vr represent the Nernst and the resting potential,
respectively.
Na+
K+
ci
[mM/l]
10
140
ce
[mM/l]
142
5
P/PK
[-]
0.04
1
E
[mV]
+71
−89
Vr
[mV]
−69
(a) Consider Na+ . What does the quantity E represent? What physical processes
determine the value of E?
(b) What does the quantity Vr represent? What physical processes determine the value
of Vr ? Explain the value of Vr on the basis of the other quantities in the table.
(c) What mechanism gives the nerve cell the possibility to generate an action potential?
(d) 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. 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
Models for the active and passive element relate active stress Ta and passive stress Tp
to sarcomere length ls , time after activation t, and sarcomere shortening velocity vs .
(a) What variable(s) does Tp depend on? Sketch the functional relation(s), and give
representative values of the quantities involved.
(b) What variable(s) does Ta depend on? Sketch the functional relation(s), and give
representative values of the quantities involved.
(c) In isometric uniaxial loading experiments, total muscle length is kept constant.
Still, mechanical inhomogeneity may occur due to tissue damage at the sites where
the tissue is clamped. How would you model this experimental imperfection? And
what would be the local tissue response in the experiment?
Exam Cardiac Modeling (8W160) — February 1, 2007
4. Left ventricular wall mechanics has been analysed using a geometrically simplified
model, consisting of a large number of nested thinwalled cylinders. In the model, deformation of the left ventricle was described by the deformation modes shown in the figure
below.
reference state
volume change
shape change
torsion
(a) The deformation modes are related to the fiber structure of the left ventricular
wall. Define the two angles, used to characterise this fiber structure.
(b) Only one of the two angles, mentioned in (a), can be accounted for in the nested
cylinder model. Sketch the transmural distribution of this angle.
(c) Discuss how the deformation modes are caused by the typical fiber structure, described by the fiber angle mentioned in (b).
(d) The nested cylinder model was used to demonstrate the possibility of homogeneous
shortening of the myofibers across the left ventricular wall. Explain the proposed
mechanism.
5. Local tissue mechanics during a cardiac cycle can be described by the time course of
fiber stress σf and sarcomere length ls .
(a) Sketch the time course of σf and ls during the four phases of the cardiac cycle of a
normal healthy heart, including typical values of ls and σf ; motivate your answer.
(b) Convert the graphs, given in (a), into a fiber stress-strain loop. What physical
meaning can be attributed to the area enclosed by the loop?
(c) In both numerical and experimental studies, the time course of ls was shown to
change in the paced heart. Sketch the time course of ls in both an early activated
and a late activated region in the heart. Motivate your answer.
Exam Cardiac Modeling (8W160) — February 1, 2007
6. In the ’one-fiber’ model of cardiac mechanics fiber stretch ratio λ is related to ventricular
cavity volume Vc and wall volume Vw according to:
µ
λ=
1 + 3(Vc /Vw )
1 + 3(V0 /Vw )
¶1/3
where V0 represents the cavity volume in the reference state, to which λ is referred to.
This expression was used in a model of left ventricular adaptation. It was assumed
that global function demands, expressed in terms of stroke volume Vstroke and ejection
pressure pe , were matched to optimal mechanics criteria of the myofibers, expressed by
an optimal amount of work wopt per unit of tissue volume per cycle, and an optimal
amount of shortening during the ejection phase, expressed by λopt , by a specific choice
of left ventricular cavity and wall volume.
(a) Derive an expression for wopt .
(b) What are representative values of pe and Vstroke in the healthy human heart?
(c) Assuming that in the healthy human heart adaptation is completed, estimate values
of λopt and wopt .
(d) Use the adaptation model to describe one case of cardiac hypertrophy.