reflected waves return to this point. Rp is obtained similarly, but with waveforms measured at the pe- ripheral sites. This is equivalent to our definition of characteristic resistance of an artery, eq. (2.27), i.e., the impedance of an infinite artery, where all points are free of influence from the reflections generated at the artery’s terminal. The authors then proceed to integrate the RCR WK’s equation (2.28) over the time period(t1< t < t2), obtaining an integral formula for the periphery compliance:
Cp =
Rt2
t1 pavg(t)dt−(Rp+Rc)Rt2
t1 qavg(t)dt
Rp(pavg(t1)−pavg(t2))−RpRc(qavg(t1)−qavg(t2))) (2.33) Equation (2.33) provides us with the means of overcoming the difficulties associated with parameter choice for WK models. It is especially useful as it is presented completely in time domain variables, and assumes a compact, convenient shape, easily computed.
2.4 Proximal boundary conditions - Description of the left ventri-
vary slightly in portions of their formulations, but are all driven by the notion of left ventricle time-varying elastance. In this work, we shall formulate a reduced formulation of left ventricle behaviour, as a means to supplying adequate input into the 1D simulation domain.
The LV is one of the four chambers of the heart, composed mostly of muscular tissue that allows it, as the rest of the heart, to contract under orders from the nervous system. Upstream of the LV, through the mitral valve, lies the left atrium, which discharges blood into the LV whenever the mitral valve is open.
Downstream from the LV, through the aortic valve, is the systemic arterial system, towards which the heart’s pumping action, formed by a combination of contraction and valve dynamics, drives the fluid. We define, following classic works such as Suga et al. (1973) and Suga and Sagawa (1974), the LV’s elas- tance (symbolised byE¯in this work, to avoid confusion with the Young’s modulus) as the instantaneous ratio between intraventricular pressure and volume ((2.34)), and we accept the findings of these works (see Section 1.2), which are generalised for human LVs (Senzaki et al. (1996)). These works state that the elastance values through the cycle are only weakly influenced by ventricular after load, this means that the elastance, as defined in (2.34) behaves as an intrinsic property of the heart chamber, and has a time distribution that is similar for most mammals (Segers et al. (2003)). Also, Senzaki et al. (1996) managed to prove that elastance distributions are fairly constant for the human LV, even for diseased individuals (Segers et al. (2003)), when normalised by tpeak(time to peak),HR(heart rate) andE¯peak (peak elastance). Changes in these parameters, depending on HR variations can be estimated through the relations existent in Ottensen and Steele (2003). Even though this notion of an independent elas- tance is known to be inaccurate (Danielsen and Ottensen (2001)), we assume the classic definition of the term, as it is still able to describe the cardiac organ’s influence on flow into the aorta, providing credible ejection patterns and pressure intervals at the root node of an arterial network.
E(t) =¯ pLV(t) VLV(t)−V0
(2.34) Physically, elastance (E(t)) ((2.34)) can be regarded as the inverse function of the compliance quantity¯ introduced in Section 2.3. Its time-varying nature mimics the ventricle’s periodic contraction, and drives the LV’s internal pressure pLV. As E¯ rises, so does pLV, and this escalation eventually results in a pressure difference across the aortic valve sufficient to cause its opening, provoking ejection of blood unto the aorta and the systemic arterial system, draining the ventricle’s volumeVLV. In (2.34),V0 is a asymptotic value of volume, and does not correspond to the LV’s volume when discharged, assuming sometimes even negative values. We ignore the LA’s behaviour, and choose to simply instantly refill the LV at convenient times, to ensure the constant driving of blood flow into the 1D domains. This simplifica- tion is also used in Alastruey et al. (2008), and is valid, as we mainly wish to model the LV’s behaviour during ejection, to provide adequate BC’s to the 1D model.
Unlike the WK 0D model, the LV description is based upon 2 main states, which we, inspired by For- maggia et al. (2006), shall denominate OV and CV (open ventricle, closed ventricle). In the CV state, the ventricular pressure-volume relation is strictly described by (2.34) alone, and we consider the LV and the 1D domains to be uncoupled. For OV status, the coupling between the two systems is engaged. A
schematisation of this interaction can be put in analogue circuit form, as in fig. 2.3, for which the electric line analogy, with (2.34) yields equation (2.35), which describes the LV’s behaviour during ejection.
Figure 2.3: A schematic of the reservoir - resistance - 1D model coupling used to model the LV. During discharge into the aorta, we model the LV as closed. It is only during diastole, when the LV and 1D arterial system are uncoupled thatVLV is reset to its maximum operating valueVLVmax.
−pLV(t) + ¯E(t)(VLV(t)−V0)−RLVqLV = 0 (2.35) Shapes for the elastance function are derived in several works as double Hill functions (Lankhaar et al.
(2009), Mar ˚ak (2013)), because the shape of these functions greatly resembles the original measured elastance distribution’s form, as derived by Suga for the canine heart, and in Senzaki et al. (1996) for the human case. Indeed the findings of Senzaki and co-workers extend the usefulness of the works of Suga, by uncovering the similarities between both elastance functions (the human and the canine), upon normalisation of time and peak elastance. Presently, we choose to use a function definition borrowed from Mar ˚ak (2013), originally introduced by Stergiopulos, that is written in (2.36). This function’s expres- sion outputs normalised values of elastance (normalised by maximum elastance), and offers control, via tpeak, of the positioning of the peak value in time. ξ,n1 andn2are shape parameters, given in Mar ˚ak (2013).
E(t) = ¯¯ E0+ ∆ ¯E( (ξTt )n1
1 + (ξTt )n1)( 1
1 + (ξTt )n2) (2.36) As mentioned, we define two primary states, as in Formaggia et al. (2006) to characterise the proximal BC:
• CV – Closed ventricle – When the valve is fully closed the 1D domain is uncoupled from system (2.35). qv=0, and VLV = constant, isovolumic compression takes place, and pLV varies solely driven by the elastance functionE(t). On the arterial side, the proximal node of the first 1D seg-¯ ment acts as a high resistance, reflecting most of, if not all of the incoming backwards wavesW2 back into the forward direction in the 1D domain.
• OV – Open Ventricle – As the aortic valve becomes fully open the coupling between the LV and
Figure 2.4: Between OV and CV phases, a third, small fraction of the cardiac cycle is dedicated to enforcing the descent of the proximaluvalue to zero, thus achieving the CV status. uthris the negative value of velocity, that triggers valve closure.p1Drefers to proximal arterial pressure.
the 1D domain is enforced by (2.35), noting thatqrootis directioned opposite toqLV. This relation, together with the pressure correspondence betweenpLV andproot, and with the aid of the charac- teristic variables, and their expressions in the(p, q)system, will drive the 1D(A, u)system. RLV
is the resistance to the flow provoked by the aortic valve.
Switching fromOV toCV and vice-versa is achieved by two conditional statements, that depend on the states at the lumped LV and proximal 1D ascending aorta (fig 2.4). The intermediate phase, as well as a non-nulluthris enabled to allow for a certain amount of reverse flow to escape through the valve, out from the 1D domain, as this is a common feature observable in ”in vivo” aortic waveforms, named the dicrotic notch. We achieve this, by defining an initial sub phase in theCV period where we varyux=0in a forced way, fromuthr down to zero at a constant time rate, and impose this upon the 1D solution by means ofW1 = 2u−W2 ((2.21)). This models the gradual closure of the valve, and it does permit for the desired period of inverse flow in early diastole, and to obtain root conditions of more physiological substance.