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144 8. TOWARDS THE REAL-TIME MODELING OF THE HEART

The active tension force varies throughout the heart, as the actual sarcomere length is computed as a function of the
fiber strain, E ff ¼ E :V 1
V 1 :

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l sarcomere ¼ l sarcomere p 2E ff +1, (8.12)
R
sarcomere
where l R corresponds to the resting sarcomere subjected to zero stress. The active stress model has been suc-
cessfully applied to hearts belonging to different species such as rats [2], dogs [54], sheeps [10, 55, 56], and humans
[57, 58].

8.2.3 Windkessel Model
To model the ejection phase, an additional model needs to take into account the behavior and characteristics of the
ventricles when connected to the systemic blood circulatory system in order to control the change in pressure, volume,
and flow rate. This model also needs to reproduce the duration of the ejection period while incorporating details such
as rapid ejection and slow ejection phases. This is facilitated by a so-called Windkessel (WK) model. In the cardiac-
related literature, its occurrence is very common. For example, it was used by Creigen et al. [59] to model an artificial
heart pump, and by Molino et al. [60] to understand the arterial mechanical characteristics.
In the context of cardiac modeling, many applications can also be found such as in Ottesen and Danielsen [61],
where the authors made use of the WK model to simulate an LV with arbitrary heart rate or in the work of Bovendeerd
et al. [62], Usyk and McCulloch [49], Sainte-Marie et al. [63], Kerckoffs et al. [64], Sermesant et al. [65], and Kerckhoffs
et al. [66] where single ventricular and biventricular heart models successfully reproduced the ejection phase. The WK
model is formulated as an ordinary differential equation where three different forms can be distinguished: the two-
element WK model, the three-element WK model, and the four-element WK model. As outlined in the works of Par-
ragh et al. [67] and Westerhof et al. [68], the predictive accuracy increases from two- to four-element WK models but
less from the three-element WK to the four-element one.
Even though the four-element WK model includes most of the physiological mechanisms of the ejection phase and is
very accurate, it has, however, two major drawbacks. First, its formulation consists of a second-order flow rate deriv-
ative. Such an order term adds more complexity and instability in a solving scheme. And second, the inductance con-
stant is very difficult to quantify from experimental data [68]. Due to these reasons, the three-element WK model is
utilized here. The equation of the three-element WK model is given as follows:

R a dIðtÞ PðtÞ dPðtÞ
1+ IðtÞ + CR a ¼ + C , (8.13)
R p dt R p dt
where R p is equivalent to the peripheral resistance, R a is the flow resistance, and C is the elastic arterial compliance. P is
the cavity pressure of the ventricle, which is approximately equal to the aortic pressure [69], while I is the negative rate
of change of the ventricular cavity volume.

8.3 REDUCED ORDER METHOD

With the cardiac mechanics models relevant for this research introduced, the concept of ROM is now outlined. ROM
is a technique commonly used to decrease the complexity of a large system of equations. This is achieved by compres-
sing the whole system to such a point that accuracy is not, in an excessive way, negatively impacted and that the gen-
eral behavior of the problem, for example, the mechanics, is preserved. As mentioned before, one widely used method
classified as an ROM is POD; this will be utilized in the form of the PODI method to achieve real-time modeling of
the heart.

8.3.1 Proper Orthogonal Decomposition
The POD is a method that can be used to extract features from any dataset consisting of either linear or nonlinear
data. In the literature, it is usually found in the form of Kharhunen-Loève decomposition (KLD) [70, 71], singular
value decomposition (SVD), or principal component analysis. Even though each of these methods has different der-
ivations, Wu et al. [72] showed the equivalence between those methods and how they can all produce the same
solution.

I. BIOMECHANICS