Models of Right Ventricular Shape and Function
Edward Marcus , Damien Craig*, Ira Cheifetz *
Marcus Laboratories USA Newton, Ma.
and *Duke Children's Hospital and Medical Center
Durham, North Carolina
1. Abstract
The right ventricle (RV) is an irregularly shaped organ with asynchronous motions during the systolic phase of its contraction. We have developed advanced modeling methods to assess RV shapes averaged through a population and applied our shape results to assess RV function in the presence of high pressures induced by pulmonary hypertension (ref 1). In our model ‘s result of the RV shape, we observe a late systolic exponential decay of RV volume with time and hypothesize this decay as reflecting RV structural interaction with outflow resistance in a discharge modality - where the ventricle’s capacitance releases blood with a characteristic time constant proportional to flow resistance. To validate this capacitive and resistive model of RV contraction, we measure the pressure, outflow, and volume of piglets and calculate the pulmonary artery resistance R, volumetric capacitance C, and volumetric time constant T.
Results: With seven piglets, regression fits of volume V(t) to an exponent yield a regression range from .940 <= r2<=.996 . The measured time constants T are correlated to the product of pulmonary resistance and ventricular capacitance as T = 1.05*R x C + .03 , r2=.933.
Conclusion: Volume is released from the right ventricle with a systolic time constant equal to the product of ventricular capacitance and pulmonary load resistance.
2. Introduction
In this paper we investigate the response of the Right Ventricle’s volume to changes in pressure and experimentally investigate the role of a capacitance ratio (capacitance = volume/pressure) in the timing of ventricular contraction. In our model, as the ventricle decreases its size (and stored capacitive energy) during systole, we propose the energy change experienced by the ventricle also powers the outflow energy of ejected blood which overcomes resistance to pulmonary flow. Equating energy transfer between the ventricle’s capacitance and the pulmonary artery’s resistance, we mathematically derive a characteristic systolic time constant (T) as a predicted exponential rate of the stroke volume’s release by the ventricle during systole. Time constant T is derived from a simple differential equation of capacitive discharge by the ventricle into the pulmonary artery, and our model predicts T = R x C where R=pulmonary resistance and C= ventricular capacitance. To experimentally investigate this rate prediction, we measure the pressure, volume, and outflow of in-vivo right ventricles in young piglets.
3. Methods
3.1 Animal Measurement
Seven young swine (9–12 kg, aged 6-8 weeks) are premedicated with intramuscular acepromazine (1.0 mg/kg) and ketamine (20 mg/kg) with each animal then receiving an i.v. sodium thiopental bolus (25 mg/kg). The trachea was intubated, and the animal was placed on an SV300 ventilator (Siemens Inc.; Solno, Sweden) in the volume control mode (ref 2).
One-millimeter epicardial ultrasonic dimension transducers were sutured to the epicardium measuring the base to apex major axis diameter, the anterior to posterior minor axis diameter of the left ventricle, the left ventricular free wall to septal distance, and the RV free wall to septal distance. A septal crystal was placed into the interventricular septum and guided close to the right side of the septum. An ultrasonic transit time flow probe (Transonic Systems Inc.; Ithaca, NY, U.S.A.) was placed around the pulmonary artery. Tapes were placed around both cavae. A micromanometer pressure transducer (MPC-500; Millar Instruments Inc.; Houston, TX, U.S.A.) was placed into the right cavity. Pacing wires were sutured onto the right atrium. Heart rate, cardiac rhythm, PA flow, and ventricular pressure were continuously monitored and all data was recorded at 500 hz sampling rate and stored digitally to computer.
3.2 The Model
Capacitance (C) of the right ventricle is defined by a ratio of volume to pressure.
C = V/P (1)
V= average systolic RV volume and P = average systolic RV pressure
Resistance (R) of the pulmonary artery is defined as a ratio of arterial pressure to flow.
R=P/Q (2) Q=average systolic pulmonary flow cc/sec
Equating the pressure of the ventricle in eqn1 to the arterial pressure in eqn2 , and specifying a conservation between ventricular outflow (Q) and the ventricle’s stroke volume rate (Q=-dV/dt) , the following differential equation of the ventricle volume V is derived
V(t)/C= - R dV(t)/dt (3)
The solution to this equation is
V(t) = Vmax e – t/ T where T = R x C (4)
This exponential formula is familiar in the theory of capacitive electronic discharge (ref 3). In the context of an analogous fluid circuit consisting of the heart and arterial circulation, this formula also models the ventricle’s capacitive release of its pre-ejection blood reservoir into the pulmonary artery during systole. The release of blood by the ventricle is modeled to occur with a characteristic systolic time constant T = R x C.
3.3 Measuring T and Validation
Right ventricular volume V(t) through the systolic interval is measured by subtraction of the integrated outflow Q(t) from the ventricle’s initial volume state at a systolic interval start time t1.
V(t) = V(t1) – t1 ∫ t Q(t) dt (5)
Initial volume V(t1) is estimated from ultrasonic transducer dimensions reconstructed into the RV volume at time t1 (method of Fenely ( ref 4) ).
Linear regression fitting of ln(V(t)) obtains the time constant T of each animal. The capacitance and resistance measures of each animal are then multiplied and the resulting product of R x C is correlated to this time constant.
4. Results
Correlation between T and R x C is illustrated in Figure 1 with a resulting near unity slope and regression illustrating the agreement of the systolic stroke volume rate with an R x C time constant model.
Figure 1: The model’s estimate of the volumetric time constant is plotted with the measured arterial resistance R and ventricular capacitance C products from seven animals.
Based on time constant correlation to the model, the T=RC relation is then applied to model the ventricle’s Ejection Fraction EF defined as
EF= 100*(Vmax)-Vmin)/Vmax (6)
where Vmax=maximum volume and Vmin =minimum volume
The ejection fraction, a common index of ventricular function from clinical cardiology, is modeled by first applying the exponential volume V(t)=Vmax e-t/RC. Eqn 6 then becomes
EF = 100x (1-e-(Tejection)/RC)) (7)
where Tejection=time of Vmin – time of Vmax
This EF model, which is based on resistance and capacitance, is correlated to the actual ejection fraction as measured from the Vmax and Vmin from each of the seven study animals in Figure 2.
Figure 2: The Ejection Fraction is a standard clinical measure of the ventricle’s function. When V(t)=Vmax e-t/RC, EF= (Vmax-Vmin)/Vmax = (1-e-(Tejection)/RC)) .
5. Discussion
Our results show the right ventricle’s systolic function is primarily influence by two factors- ventricular capacitance and pulmonary resistance. Based on the measured capacitive and resistive elements of a discharge circuit, the pressure and outflow data we observe in piglets are very consistent with a capacitive time constant model of systolic ejection by the right ventricle. Our experimental data is from the right side of the heart, however the left side of the heart has similar outflow timing (ref 5). We then hypothesize a capacitance and resistance based timing model also describes stroke volume delivery to the systemic circulation by the left ventricle.
Practical applications of our model follow from current imaging capabilities available to non-invasively measure a volumetric time constant T and apply these measures to quantify independent changes in arterial resistance or ventricular elasticity without the need of invasive catheterization. Such non-invasive time constant assessments may provide a simple method to assess circulatory changes resulting from arterial stenosis, interventions by arterial dilating stents, or quantification of pulmonary hypertension following application of Nitric Oxide. Future studies of time constant methods on larger patient groups may provide a clearer indication of these applications to easily assess the function of the ventricle and the circulatory tree.
6. References
1. Marcus E, Munoz R, Margossian R, Colan S, & Wessel D. Echocardiographic Assessment of Right Ventricular Response to Hypertension in Neonates on the Basis of Average-shaped Contraction Models. Journal of American Society of Echocardiography 15:2, 1145-1153,2002.
2. McGovern JJ, Cheifetz IM, Craig DM, Bengur AR, Quick G, Ungerleider RM, & Meliones JN , Right Ventricular Injury in Young Swine: Effects of Catecholamines and Right Ventricular Function and Pulmonary Vascular Mechanics, Pediatr Res. 2000 Dec;48(6):763-9.
3. S.D. Senturia & B.D.Wedlock , Electronic Circuits and Applications, John Wiley and Sons, 1975.
4. MP Feneley, JR Elbeery, JW Gaynor, SA Gall Jr, JW Davis & JS Rankin, Ellipsoidal shell subtraction model of right ventricular volume. Comparison with regional free wall dimensions as indexes of right ventricular function Circ Res. 1990 Dec; 67(6): 1427-1436.
5. W.W. Nichols & M.F. O'Rourke. McDonald's Blood Flow in Arteries, theoretical, experimental and clinical principles, Lea & Febiger, 1990.