A theoretical evaluation of cardiac output as a function of mean arterial pressure in the human cardiovascular system

A correlation function of cardiac output and mean arterial pressure is presented for the human cardiovasular system. The function is developed using an energy transfer balance for a unit volume of blood which flows in the vascular system between the aorta and the vena cava. The energy transfer balance equates the energy utilized in the vascular system to the algebraic sum of the pulse energy, the kinetic energy and the potential energy in the vascular system. Each of these energies is defined in terms of the physiology of the cardiovascular system. Pulse energy is defined in terms of the work done by the heart on the aorta. Kinetic energy is defined in terms of the cardiac output and the potential energy is defined in terms of the diastolic pressure in the aorta. The utilization energy is equivalent to the energy transfer in the work done by the blood on the viscoelastic blood vessels, and to the frictional energy loss due to drag on the blood mass as it flows through the vascular system.The correlation function of cardiac output with mean arterial pressure demonstrates that the cardiac output is a double-valued function of the mean arterial pressure. The function also varies with the ratio of the fourth power of the Shear Modulus of the blood vessels to the third power of Young's Modulus. The function shows that mean arterial pressure minimizes for a cardiac output of approximately 51 per min when one holds the ratio of the elastic moduli constant. Further discussion indicates how clinicians can use the function, developed in this research, to interpret the experimental data obtained from cardiac output studies.     

Theory of the measurement of arterial compliance in humans

Experimental determinations of the arterial complianceC are discussed, based on a simplified formula involving the systolic and diastolic pressures, the period of diastole and the peripheral resistance. The resistance is expressed in terms of the cardiac output and the mean arterial pressure during diastole. which is derived from an assumed exponential pressure-time curve and leads to a still simpler expression forC. The cardiac output is obtained by a rebreathing method that yields arterial and venous P_{CO 2}, the latter from extrapolation based on a linear difference equation. The experimental standard error of the compliance in terms of those of the various physiological parameters is considered. Numerical examples are given for a healthy man.     

A differentiable,periodic function for pulsatile cardiac output based on heart rate and stroke volume

Many mathematical models of human hemodynamics, particularly those which describe pressure and flow pulses throughout the circulatory system, require as specified input a modeling function which describes cardiac output in terms of volume per unit time. To be realistic, this cardiac output function should capture, to the greatest extent possible, all relevant features observed in measured physical data. For model analysis purposes, it is also highly desirable to have a model function that is continuous, differentiable, and periodic. This paper addresses both classes of needs by developing such a function. Physically, the present function provides an accurate model for flow into the ascending aorta. It is completely specified by a minimal number of standard input parameters associated with left ventricle dynamics, including heart rate, mean cardiac output, and an estimation of the peak-to-mean flow ratio. Analytically, it can be expressed as a product of two continuous, differentiable and periodic factors. Further, the Fourier expansion of this model function is shown to be a finite Fourier series, and explicit closed-form expressions are given for the non-zero coefficients in this series.     

Understanding Guyton's venous return curves

Based on observations that as cardiac output (as determined by an artificial pump) was experimentally increased the right atrial pressure decreased, Arthur Guyton and coworkers proposed an interpretation that right atrial pressure represents a back pressure restricting venous return (equal to cardiac output in steady state). The idea that right atrial pressure is a back pressure limiting cardiac output and the associated idea that "venous recoil" does work to produce flow have confused physiologists and clinicians for decades because Guyton's interpretation interchanges independent and dependent variables. Here Guyton's model and data are reanalyzed to clarify the role of arterial and right atrial pressures and cardiac output and to clearly delineate that cardiac output is the independent (causal) variable in the experiments. Guyton's original mathematical model is used with his data to show that a simultaneous increase in arterial pressure and decrease in right atrial pressure with increasing cardiac output is due to a blood volume shift into the systemic arterial circulation from the systemic venous circulation. This is because Guyton's model assumes a constant blood volume in the systemic circulation. The increase in right atrial pressure observed when cardiac output decreases in a closed circulation with constant resistance and capacitance is due to the redistribution of blood volume and not because right atrial pressure limits venous return. Because Guyton's venous return curves have generated much confusion and little clarity, we suggest that the concept and previous interpretations of venous return be removed from educational materials.     

A method for calculation of human systolic and diastolic blood pressures using an elastic reservoir theory of the systemic arterial system,and some clinical and physiological applications

Two equations have been developed that describe the interrelationship of the clinically measurable variables of the human systemic arterial system. An approximation method is given for their simultaneous solution for systolic and diastolic pressures in terms of heart rate, cardiac output, total peripheral resistance, and aortic distensibility. In this way, blood pressures were calculated for various clinically important and didactically useful situations. The effects on systolic and diastolic pressures due to changing either cardiac output or peripheral resistance or heart rate or aortic distensibility alone are shown. The effects on pulse pressure of varying cardiac output and peripheral resistance while holding mean arterial pressure constant are demonstrated. Compensatory mechanisms in hypertension and exercise are explored. Opinions and conclusions contained in this report are those of the author. They are not to be construed as necessarily reflecting the views or the endorsement of the Navy Department.     

Control of the contractility of the myocardium in a feedback system

Experimental evidence strongly suggests that the contractility of the intact heart in situ, in contrast to that of striated muscle elsewhere in the body, is controlled in a close-cycle system. Thus, the variation of intraventricular pressure during systole follows a complex pattern, whose relative form remains quite constant regardless of the duration of ejection. By use of the single-chambered model of the cardiovascular system, a mathematical representation of a feasible feedback mechanism is developed. The requirement that the feedback system must satisfy mathematical principles eliminates relationships apparently reasonable from a physiological viewpoint. A clinical application which the mathematical development suggests is that early arterial hypertension may arise from an abnormal feedback mechanism with excessively large cardiac output in the initial portion of systole.     

A computer-assisted noninvasive monitoring system with graphic display of online determination of cardiovascular parameters

During anesthesia the cardiovascular system is usually assessed on the basis of heart rate and arterial pressure, although the most important hemodynamic measurement is that of flow. A method for the non-invasive measurement of cardiac output is based on thoracic electrical bio-impedance. The NCCOM3-R7 is a non-invasive cardiac output monitor that makes use of thoracic electrical bioimpedance, which has been shown to provide results comparable with thermodilution in various hemodynamic states both in animals and humans. A new on-line hemodynamic monitoring system has been developed using the non-invasive NCCOM3-R7 (BoMed) cardiac output monitor, a portable microcomputer (NEC Multispeed) in connection with a software package CDDP-1 (BoMed), a Dinamap automatic arterial pressure monitor (Critikon) and an additional 14" display. Every 16 heart beats the cardiac output monitor transfers 11 cardiodynamic parameters in ASCII-format to the microcomputer, where the data are stored. Using the CDDP-1 program the current cardiodynamic status of the patient is displayed numerically and graphically on the monitor screen. Mean arterial pressure is determined by Dinamap and the data must be entered manually in the menu. The program then calculates systemic vascular resistance and left ventricular work index, the CVP being set to 3 torr and PAOP to 6 torr. In the graphic display the current hemodynamic status is shown and the underlying situation is analyzed in terms of systemic vascular resistance and volume-dependent contractility. The reliability of this on-line monitoring system is demonstrated in a high-risk patient.(ABSTRACT TRUNCATED AT 250 WORDS)     

Pulsed nuclear magnetic resonance of potassium (39K) of whole body live and dead newborn mice. Double oscillation frequencies in T1 decay curves.

Pulsed nuclear magnetic resonance relaxation curves (T2 and T1) of potassium (39K) have been measured in detail on whole body newborn mice when alive, and on the same mice after death. The T2 curves are simple exponential with respect to time, but are shorter than for 39K in simple solutions. The T1 curves are not exponential decays, but show large oscillations that may be described approximately as the sum of two separate sine waves of different frequencies. Large T1 oscillations of complex waveform were previously observed by us with 39K in cancer tissues. Gyroscopic motion of adsorbed magnetoelectric dipoles is proposed as a possible physical mechanism accounting for the experimental observations.     

Model studies of blood-glucose regulation

A simplified, linearized model of the system regulating blood-glucose concentrations is reviewed. This model, which predicts a damped sine wave response to an oral glucose load, lumps the large number of kinetic parameters into a much smaller number which can, at least in part, characterize the human glucose regulatory system. The predictions based on the model are compared with measurements of blood-glucose and blood-insulin concentrations during the oral glucose-tolerance test. Various other conditions are simulated and their implications are discussed in terms of the mathematical model used. Presented in part at the meeting of the Federation of American Societies for Experimental Biology in Chicago, April, 1964.     

Cardiovascular mechanics in the early stages of pulmonary hypertension: a computational study

We formulate and study a new mathematical model of pulmonary hypertension. Based on principles of fluid and elastic dynamics, we introduce a model that quantifies the stiffening of pulmonary vasculature (arteries and arterioles) to reproduce the hemodynamics of the pulmonary system, including physiologically consistent dependence between compliance and resistance. This pulmonary model is embedded in a closed-loop network of the major vessels in the body, approximated as one-dimensional elastic tubes, and zero-dimensional models for the heart and other organs. Increasingly severe pulmonary hypertension is modeled in the context of two extreme scenarios: (1) no cardiac compensation and (2) compensation to achieve constant cardiac output. Simulations from the computational model are used to estimate cardiac workload, as well as pressure and flow traces at several locations. We also quantify the sensitivity of several diagnostic indicators to the progression of pulmonary arterial stiffening. Simulation results indicate that pulmonary pulse pressure, pulmonary vascular compliance, pulmonary RC time, luminal distensibility of the pulmonary artery, and pulmonary vascular impedance are much better suited to detect the early stages of pulmonary hypertension than mean pulmonary arterial pressure and pulmonary vascular resistance, which are conventionally employed as diagnostic indicators for this disease.     

A mathematical model of the exercise functional state of the oxygen transport system

Based on the principle of minimum power, a mathematical model of the exercise functional state of the oxygen transport system is presented. Aerobic and anaerobic muscular efficiencies are considered. The energetically optimal arteriovenous oxygen content difference, cardiac output and ventilation during exercise in man are determined depending on mechanical power. Theoretical results are compared with experimental data.     

The leading edge approximation to the nerve axon problem

A negative resistance piece-wise linear model, and one which is the sum of two sine terms are used to solve the nerve axon problem for leading edge waveshape, pulse velocity, maximum rate of rise, and rise time for the “Hodgkin-Huxley axon.” The results are compared analytically and numerically to experiment and the calculated results of Hodgkin and Huxley, a fuse model, and a cubic solution.     

Role of the baroreceptor reflex in the early phases after removing the renal artery constriction in conscious renal hypertensive rats

Hemodynamic studies in unanesthetized rats with chronic one-kidney-Goldblatt hypertension showed a 25% increase in cardiac output and a 42% increase in peripheral resistance. Removal of renal artery constriction under either anesthesia and minor surgical trauma produced an immediate 20% drop in arterial pressure. At the end of the 6 observation period the pressure dropped 30% but still remained at a moderate hypertensive level. The hemodynamic measurement at that time suggested that the pressure drop was the result of a decrease in cardiac output. However, the data obtained 1 hour after removal of the constriction suggested that a vasodilating mechanism may also contribute to pressure normalization in the early phase of reversal of renal hypertension. In the sham-operated hypertensive rats the pressure remained unchanged, while the cardiac output dropped due to compensation by a proportional increase in peripheral resistance. In contrast, in the unclipped animals the same drop in cardiac output produced an equivalent fall in pressure because no change in peripheral resistance occurred. This was not due to an insufficiency of the baroreceptor reflex since bilateral splanchnicectomy performed at that time produced a striking hypotensive response, indicating an overactivity of the sympathetic system possibly due to the baroreceptor still reset to operate at a hypertensive level.     

Cardiac performance of an isolated eel heart: effects of hypoxia and responses to coronary artery perfusion.

The pericardial sac containing the heart was removed from large (2.7-6.3 kg) long-finned eels (Anguilla dieffenbachii). Coronary arteries were cannulated in preparation for perfusion with eel Ringer or red cell suspensions. The hearts were maintained by Ringer perfusion while the performance of the heart was assessed. Responses of the hearts to increases in filling pressure and output pressure were recorded. Maximum cardiac output was 22.3 +/- 1.4 ml/min/kg body mass (mean +/- 1 SEM; N = 9). The highest cardiac power output was measured at maximum cardiac output and was 3.39 +/- 0.32 mW/g ventricle mass (mean +/- 1 SEM; N = 9). Eel hearts could sustain output pressures near 80 cm H2O, but cardiac output was reduced and cardiac power output was 1.89 +/- 0.24 mW/g ventricular mass (mean +/- 1 SEM; N = 9). Maximum cardiac output decreased by 14% when hearts pumped hypoxic Ringer with a PO2 of 11.5 torr. At high input pressures concomitant with high output pressures (80 cm H2O), cardiac power output decreased by 38% upon exposure to hypoxic Ringer. Coronary perfusion of hypoxic hearts with red cell suspensions (mean hematocrit 10.4%) at a rate of 2% of control cardiac output (0.20 ml/min/kg body mass) had no effect on maximum cardiac output. However, coronary perfusion enabled hypoxic hearts to maintain cardiac output when output pressure was raised to 80 cm H2O. Under conditions of high input pressure and high output pressure, power output increased by 20% compared to hypoxic hearts without coronary perfusion.(ABSTRACT TRUNCATED AT 250 WORDS)     

The dead space in a compartmental lung model

A compartmental lung model with any number of synchronously filling and emptying functional chambers and a common dead space or conducting region is considered. It is shown that the model gives rise to an output, in an open circuit washout determination, which is a weighted sum of exponentials. From estimates of these weights and exponential components, estimates of the model parameters can be recovered. Relations giving the unique correspondence between the output parameters and the model parameters are derived and the existence and uniqueness of solutions established.     

Robust, high-resolution, whole cell patch-clamp capacitance measurements using square wave stimulation.

High-resolution, whole cell capacitance measurements are usually performed using sine wave stimulation using a single frequency or a sum of two frequencies. We present here a high-resolution technique for whole-cell capacitance measurements based on square-wave stimulation. The square wave represents a sum of sinusoidal frequencies at odd harmonics of the base frequency, the amplitude of which is highest for the base frequency and decreases as the frequency increases. The resulting currents can be analyzed by fitting the current relaxations with exponentials, or by a phase-sensitive detector technique. This method provides a resolution undistinguishable from that of single-frequency sine wave stimulation, and allows for clear separation of changes in capacitance, membrane conductance, and access resistance. In addition, it allows for the analysis of more complex equivalent circuits as associated with the presence of narrow fusion pores during degranulation, tracking many equivalent circuit parameters simultaneously. The method is insensitive to changes in the reversal potential, pipette capacitance, or widely varying cell circuit parameters. It thus provides important advantages in terms of robustness for measuring cell capacitances, and allows analysis of complicated changes of the equivalent circuits.     

Cardiac output during labour.

Serial measurements of cardiac output and mean arterial pressure were performed in 15 women during the first stage of labour and at one and 24 hours after delivery. Cardiac output was measured by Doppler and cross sectional echocardiography at the pulmonary valve. Basal cardiac output (between uterine contractions) increased from a prelabour mean of 6.99 l/min to 7.88 l/min at greater than or equal to 8 cm of cervical dilatation as a result of an increase in stroke volume. Over the same period basal mean arterial pressure also increased. During uterine contractions there was a further increase in cardiac output as a result of increases in both stroke volume and heart rate. The increment in cardiac output during contractions became progressively greater as labour advanced. At greater than or equal to 8 cm of dilatation cardiac output increased from a basal mean of 7.88 l/min to 10.57 l/min during contractions. There were also further increases in mean blood pressure during contractions. One hour after delivery heart rate and cardiac output had returned to prelabour values, though mean arterial pressure and stroke volume remained raised. By 24 hours after delivery all haemodynamic variables had returned to prelabour values. Haemodynamic changes of the magnitude found in this series are of considerable clinical relevance in managing mothers with complicated cardiovascular function.     

Induction of oscillatory ventilation pattern using dynamic modulation of heart rate through a pacemaker

For disease states characterized by oscillatory ventilation, an ideal dynamic therapy would apply a counteracting oscillation in ventilation. Modulating respiratory gas transport through the circulation might allow this. We explore the ability of repetitive alternations in heart rate, using a cardiac pacemaker, to elicit oscillations in respiratory variables and discuss the potential for therapeutic exploitation. By incorporating acute cardiac output manipulations into an integrated mathematical model, we observed that a rise in cardiac output should yield a gradual rise in end-tidal CO2 and, subsequently, ventilation. An alternating pattern of cardiac output might, therefore, create oscillations in CO2 and ventilation. We studied the effect of repeated alternations in heart rate of 30 beats/min with periodicity of 60 s, on cardiac output, respiratory gases, and ventilation in 22 subjects with implanted cardiac pacemakers and stable breathing patterns. End-tidal CO2 and ventilation developed consistent oscillations with a period of 60 s during the heart rate alternations, with mean peak-to-trough relative excursions of 8.4 +/- 5.0% (P < 0.0001) and 24.4 +/- 18.8% (P < 0.0001), respectively. Furthermore, we verified the mathematical prediction that the amplitude of these oscillations would depend on those in cardiac output (r = 0.59, P = 0.001). Repetitive alternations in heart rate can elicit reproducible oscillations in end-tidal CO2 and ventilation. The size of this effect depends on the magnitude of the cardiac output response. Harnessed and timed appropriately, this cardiorespiratory mechanism might be exploited to create an active dynamic responsive pacing algorithm to counteract spontaneous respiratory oscillations, such as those causing apneic breathing disorders.