# Lumped Parameter Model for Heart Failure with Novel Regulating Mechanisms of Peripheral Resistance and Vascular Compliance

To research the change in the physiological mechanism between the heart failure (HF) patient and healthy persons, the physiology parameters, which include the myocardial contractility, systemic vascular resistance (SVR), and vascular compliance, were studied. Through clinical data of HF patients, the computing method of the myocardial contractility was proposed; SVR as a function with respect to mean arterial pressure (MAP) was represented; and the vascular compliance was defined as a function with respect to MAP and cardiac output. Based on these parameters, a lumped parameter model of the cardiovascular system was established to reproduce the hemodynamic status of HF patients and verify the validity of the physiological mechanism. The simulation results demonstrate that the proportional error of mean flow, arterial pressure, and systolic blood pressure is 11.9%, 2.3%, and 4.7% compared with the clinical data, respectively. The proportional error of end-diastolic volume, end-systolic volume , and ejection fraction is 13.4%, 3%, and 3.9%, respectively.

^{*}School of Life Science and BioEngineering, Beijing University of Technology, Beijing, People’s Republic of China

^{†}Peking University Third Hospital, Beijing, People’s Republic of China

Disclosures: This work was partly sponsored by the National Natural Science Foundation of China (Grant No 11072012) and also sponsored by the National Natural Science Foundation of China (Grant No 10872013, 31070754) and the Furtherance Program of 11 Beijing University of Technology on Strengthening Talents Education (015000543111503).

Reprint Requests: Dr. Yu Chang, School of Life Science and BioEngineering, Beijing University of Technology, Beijing 100124, People’s Republic of China. Email: changyu@bjut.edu.cn; Dr. Zhe Zhang, Peking University Third Hospital, Beijing 100191,

People’s Republic of China. Email: zhangzhe@bjmu.edu.cn.

Affecting around 2% of the general population, the prevalence of heart failure (HF) increases to 10–20% between the ages of 70 and 80 years and is probably higher in subjects aged 80–100 years.^{1} More than 5 million people in the United States and more than 500,000 cases are diagnosed each year^{2}; more than one million in the Europe are around 0.4–2.0% of the overall population.^{3}

Heart failure^{4} is defined as a dysfunction of the ventricular function, where the heart is no longer able to pump enough blood to satisfy the body’s metabolic needs either during effort or at rest. The decrease in cardiac output (CO) and the decline in mean arterial pressure (MAP) are recognizable symptoms for HF in clinical.

Lumped parameter mathematical models as a research method of the cardiovascular system have been developed for many years for a variety of purposes.^{5} Shi and Korakianitis^{6} proposed a numerical model of the human cardiovascular system and physiologically pulsatile type ventricular assist devices to simulate the health, left ventricle (LV) failure and LV failure assisted by a left ventricular assist device. In this model, some regulating mechanisms were not considered. Harvard-MIT^{7} established the model of the Research Cardiovascular SIMulator. The model includes the cardiovascular system and the feedback system. For a nonlinear lumped parameter model of the cardiovascular system coupled with a rotary blood pump was presented by University of Pittsburgh to research the control of the pump. In addition, there are numerous recent literatures in modeling of the cardiovascular system with impaired heart function in the field of blood pump research. Lim et al.^{9} proposed that a failed heart was modeled by decreasing the end-systolic elastance value to 40% of normal. Moscato et al.^{10} proposed that systolic LV failure was simulated by using a depressed slope of end- systolic pressure–volume relationship (ESPVR), which is equal to the 30% of that of a healthy LV. Heart failure is a complex syndrome, it is not only related to the cardiac function but also peripheral circulation system. As to the model mentioned above, because some parameters were derived from a normal person instead of *congestive heart failure* (CHF) patients, the systemic vascular resistance (SVR) and vascular compliance remained constant, which is essentially different from what is observed in the clinical data. Those models only consider the cardiac function to simulate the HF. However, the peripheral circulation system includes the vascular resistance and compliance should be considered. Hence these models can reproduce the hemodynamic status of the cardiovascular system but cannot obviously reproduce variability when the change in hemodynamic parameters of CHF patients that clearly limit the scope of application. In addition, the mechanism of the cardiovascular system has changed in CHF. To solve this problem, a model of the cardiovascular system based on HF patients was proposed. Through the clinical data of HF patients, the model of individual patients was obtained and the variation of the parameter that included SVR and vascular compliance with respect to time.

In this article, the physiology parameters, which include the myocardial contractility, SVR, and vascular compliance, are studied. According to clinical data of HF patients, the ESPVR and the end-diastolic pressure–volume relationship (EDPVR) were calculated. By the slope of ESPVR and EDPVR, the myocardial contractility will be obtained. We found that SVR as a function of MAP was represented; and vascular compliance was defined as a function of MAP and CO. To verify the validity of the physiological mechanism, a lumped parameter model of the cardiovascular system was proposed based on these parameters.

## Materials and Methods

### Myocardial Contractility

The myocardial contractility as an essential for the evaluation of cardiac function has high clinical value. Analysis of pressure–volume (PV) relationship is still regarded as the most reliable method to obtain a load of independent parameters of contractility and diastolic compliance.^{11},^{12} The LV PV relationship has been extensively used for examination of ventricular function in a variety of research and clinical settings.^{13}

Figure 1 illustrates the LV PV relationship of one cardiac cycle. The PV loop moving toward the upper right in HF patients is named HFPV loop because of the increase in preload and after load. Meanwhile, the slope of ESPVR is reduced in HF patients, which reflected the decline of myocardial contractility^{14} and the augment of after load. The EDPVR showed practical value for the assessment of patients with diastolic dysfunction. Through ESPVR and EDPVR, the features of myocardial contractility will be represented. However, many parameters, such as left ventricular pressure (LVP), the pressure of end-systolic and volume of the ventricle, are difficult to real-time measure of the clinical.

In this article, the myocardial contractility has been estimated by clinical data. Suga^{15} found that the LV could be considered as a time-varying elastance (*E*(*t*)). The *E*(*t*) functions and “double hill” function^{11} have been used to calculate the contractility. Various parameters for the heart, including the internal diameter of the ventricle end-diastolic and end-systolic, ejection fraction (EF), left atria pressure (LAP), and so forth, have been measured by routine examination. According to the internal diameter of the ventricle end-diastolic and end- systolic, volume is calculated by Teichholz formula.^{16}

where *D* is internal diameter of ventricle, and *V* is the volume of ventricle. Based on Figure 1, the EDPVR was defined as an exponential function, denoted as:

where *V* _{ed} is the volume of end-diastolic, *P* _{ed} is the pressure of end-diastolic, *V* _{0} is a volume when pressure is 0 mmHg, and *a* _{0} and *a* _{1} are constant coefficients. In this work, we assumed the LAP is equal to the pressure of end-diastolic. The approximate calculation of function obtained the constant coefficient and EDPVR function by clinical data, and then calculated the slope d*p* _{ed}/d*V* that is *E* _{min}, which reflects the diastolic function. The ESPVR function can be expressed by a simple equation:

where *V* _{es} is the volume of end-systolic, *P* _{es} is the pressure of end-systolic, *V* _{0} is a volume at 0 mmHg, and *b* is the constant coefficient. The *P* _{es} is hard to measure in routine examination. Therefore, *P* _{es} is defined as a function of systolic blood pressure (SBP) and can be expressed by a simple equation:

where *k* is the constant coefficient. The ESPVR function will be obtained by clinical data, and the coefficient *b* is the *E* _{max′} which reflects the systolic function.

### Systemic Vascular Resistance

Systemic vascular resistanceis an important parameter, which influences the arterial pressure and blood flow.^{17} In addition, it is a crucial parameter to research the cardiovascular system in a lump parameter model. The decrease in CO and the insufficient blood perfusion are obvious symptoms for HF.^{18} When those symptoms are caused, the nervous system and renin-angiotensin-aldosterone system (RAAS)^{4} will be activated to adjust the capability of the blood supply, named compensatory adjustment. In short-term, the myocardial contractility and the CO were perceptibly increased to satisfy the demands of the body. However, the nervous system and RAAS overactivity will lead to an increase in the concentration of norepinephrine and aldosterone in plasma,^{19} which impact myocardium and SVR. For decompensated HF patients, the myocardial contractility has been impaired, characterized by the low CO and arterial pressure. At the same time, the vasoconstriction leads to the increase in SVR that caused the increase in the heart rate after the load, which results in more reduction of the CO and arterial pressure that leaded to worsening HF.

Fu^{20} proposed that the baroreceptor activities on peripheral resistance (R9) are a nonlinear function of MAP (which Fu refers to PAS) with a negative slope, which corresponds to negative feedback in the closed loop system (Figure 2A). The regulating mechanisms were coincided with the healthy physiology. When arterial pressure declined, the compensatory mechanism operated to maintain the pressure by an increase in the SVR. However, the regulating mechanisms were unclear for decompensated HF patients. In this article, the relationship of MAP and SVR will be determined by clinical data. The SVR was defined as an exponential function (SVR-MAP), denoted as:

where MAP is the mean arterial pressure, *m* _{0} and *m* _{1} are the constant coefficients, and MAP and SVR were the clinical data.

To verify the validity, the clinical data of five HF patients have been collected and analyzed by using Statistic Package for Social Science (SPSS) software. Baseline characteristics are shown in Table 1, which are clinical data of five HF patients. The clinical data come from Peking University Third Hospital. The Spearman correlations between MAP and SVR of five HF patients are significantly correlated (*p* < 0.01), and the correlation coefficients are 0.435, 0.582, 0.731, 0.808, and 0.741, respectively. The SVR of HF patients and the calculated value of SVR by SVR-MAP were analyzed with a paired sample correlation. Results of paired samples of correlations show a significant linear correlation between the SVR of HF patients and the calculated value of SVR (*p* < 0.01). The paired samples *t-*test shows that SVR of HF patients and that calculated according to **Equation 5** are not significantly different (*p* > 0.1). According to the paired sample correlations and the paired samples *t-*test, the SVR-MAP function can reflect the SVR and then reveal the relationship between the SVR and MAP, which are positively correlated.

Figure 2B illustrates the relationship between SVR and MAP of five patients. Although the relationship of the different patients is atypical, it is common that the SVR increased along with the increase in MAP. The relation between Figure 2A and 2B was quite distinct because the research conditions, which are the compensatory mechanism (Figure 2A) and decompressed HF (Figure 2B), were changed.

### Vascular Compliance

Vascular compliance represents the vascular elastic property, and it is also an important factor of afterload. Grey et al.^{21} reported that artery elasticity index of arteriole is an independent risk indicator of cardiovascular diseases. Other research found that compliance of the artery of HF patients was reduced compared with that of a healthy human.^{22} Therefore, this article researched the relationship of compliance. We found that the compliance is in contact with CO and MAP according to clinical data. The vascular compliance was defined as a function of CO and MAP(C-CO and MAP), denoted as:

where *C* is vascular compliance; CO is cardiac output; MAP is mean arterial pressure; and *n* _{0}, *n* _{1}, *n* _{2}, and *n* _{3} are the constant coefficients. The paired sample correlations were analyzed by SPSS, which analyzes the correlation of compliance.

The results show that the vascular compliance derived from clinical data and that calculated according to **Equation 6** are significant linear correlation (*p* < 0.01), except the fourth patient (*p* = 0.084). The paired samples *t-*test shows that vascular compliance of HF patients and that calculated according to **Equation 6** are not significantly different (*p* > 0.80). According to the results of paired samples of correlations and the paired samples *t-*test, the vascular compliance is calculated accurately by **Equation 6**. A group of coefficients (*n* _{0}, *n* _{1}, *n* _{2}, and *n* _{3}) have been calculated by clinical data of Table 1, which is 1.0868, −0.0995, 0.0147, and −0.0509, respectively.

### Model of Cardiovascular System at HF Stage

According to the above researches, we established a lumped parameter model of the cardiovascular system for HF. It includes the LV, left atrium, right ventricle, right atrium, systemic circulation, and pulmonary circulation. *R* represented the resistance of the vessel, *L* represented the inertia of blood, *C* represented the compliance of the vessel, and diode represented the valve. The cardiovascular model can therefore be represented as an electric circuit as shown in Figure 2.

Table 2 lists the state variables of the model, and Table 3 provides the system parameters and their associated values.^{8} Suga^{15} found that the LV could be considered as a time-varying elastance (*E*(*t*)). In this model, LV is described as a *E*(*t*) function, proposed by University of Pittsburgh.^{8} Myocardial contractility has been calculated according to **Equations 1–4** and substituted into *E*(*t*). In this article, data of one patient were used for numerical simulations and to verify the relationships of SVR-MAP and C-CO and MAP. The vascular resistance and vascular compliance of systemic circulation were computed in response to clinical data of left ventricular end-diastolic dimension,left ventricular end-systolic dimension, LAP, and SBP in Table 4. Meanwhile, the data of MAP, EF, and HR were used to set the initial value of simulation.

The state equations are written according to the valve state and divided into three periods in a whole cardiac cycle, which is the isovolumic phase, ejection phase, and the filling phase. The state equations for the model in Figure 3 can be written in the form:

where *A* _{i} is coefficient matrix of X state vector and *i* = 1, 2, 3.

1. Isovolumic phase*: A* _{1} *,B* _{1}

In this period, the volume of heart has unchanged with the change in pressure. All over the valves have been closed, which means AV, TV, and MV are open circuit.

2. Ejection phase: *A* _{2} *,B* _{2}

In this period, the heart ejected the blood from ventricular to aorta. The volume and the flow rate have acute change. Aortic valve and pulmonary valve are open, and tricuspid valve and mitral valve are close.

3. Filling phase: *A* _{3} *,B* _{3}

In this period, the ventricular volume expands and the pressure and flow rate reduce. Tricuspid valve and mitral valve are open, and aortic valve and pulmonary valve are close.

## Result

To verify the validity of the model, a numerical simulation was conducted. The values of parameters used in this model were listed in Table 3. To mimic the status of HF, the initial value of the model is set according to the clinical data of patients in Table 4. The simulation is continued for 4 seconds, and heart rate (HR) is 77 bpm.The results of simulation are shown in Figures 4 and 5. Figure 4 shows the curve of flow rate and pressure. The first panel illustrates the flow rate waveform. In the ejection phase, the aortic valve is open and the flow rate exceeded 0 ml/second. Besides, the flow rate is 0 ml/second. The second curve is pressure, which includes LVP, right atria pressure (RAP), and aortic pressure (AOP). The SBP is about 117.2 mmHg from the cure of AOP, and the proportional error is 4.7% compared with the clinical data. The diastolic blood pressure (DBP) is about 60 mmHg, and the RAP is about 6 mmHg. It is found that the mean flow is 4.7 L/minute, and the MAP is about 84 mmHg in third and fourth curves. The mean flow is close to 5 L/minute, which is physiological demand of human, and the MAP is lower than 90 mmHg. The proportional errors are 11.9% and 2.3%, respectively. The pulsatility index, which defines as a ratio of pulse pressure and MAP, is about 0.65.

Figure 5 shows the waveform of LV volume and pressure. The end-diastolic volume is about 86 ml, the end-systolic volume (ESV) is about 149 ml, and the EF is approximately 42.3%.The proportionate error is 13.4%, 3%, and 3.9%, respectively. The ESV is obviously greater than the routine of 30 ml,^{23} which reflects the systolic dysfunction.

## Discussion

Myocardial contractility is essential for the evaluation of cardiac function and has high clinical value. At present, cardiac magnetic resonance (CMR) and ultrasonic cardiogram are used for the diagnosis of the clinical. Cardiac magnetic resonance is established to measure global pump function.^{24} It can estimate ESPVR by combining ventricular pressure. However, the CMR technique has been limited to a certain extent in clinical. The primary routine examination includes ultrasonic cardiogram in clinical. Some parameters, which are the internal diameter of ventricle end-diastolic and end-systolic, EF, LAP, SBP, DBP, MAP, HR, and so on, can be obtaining by the ultrasonic cardiogram and daily monitoring. In this article, the clinical data are obtained by ultrasonic cardiogram and the routine examination (*i.e.*, electrocardiogram and and blood pressure). Because of otherness of daily monitoring, there is error for the calculation of myocardial contractility. In the future, how to more accurately estimate the contractility is still a problem.

Systemic vascular resistance is an important index to reveal the information of arterial stiffness and provide the reference of cardiovascular disease that was researched in this article. Systemic vascular resistance is impacted by nervous regulation and then reflects the physiological mechanism. The regulatory mechanism mentioned above that can reflect the physiology for a healthy human. However, the mechanism of the HF patient is in decompensated condition and the relation of SVR and MAP will be changed. For instance, Malpas^{25} demonstrates that the sympathetic nervous system is overactive, when the ventricular function is impaired. Owing to the overactivity of the sympathetic nervous system, the balance of sympathetic and parasympathetic system has been changed.^{26} According to the study of Boilson et al.^{27}, the peripheral resistance of HF patients increases, when the LV assist device is implanted to improve perfusion. It indicates that the mechanism of baroreflex regulation has been impaired, which is consistent with the results of our work. From the clinical data, it is found that SVR are a nonlinear function of MAP with a positive slop shown in Figure 2B, which reflects the developing tendency of HF. It is common that the SVR increased along with the increase in MAP. Meanwhile, a problem would be that it is the vasoconstriction and the decline of MAP. The condition mentioned above may significantly affect the function of vital organs. For instance, significant blood pressure reduction is strongly associated with worsening renal function.^{28}

In **Equation 5**, the relationship of MAP and SVR was determined by clinical data. The Spearman correlations *p* < 0.01 illustrates that SVR and MAP are a correlation. If *p* > 0.01, this research is insignificant. Systemic vascular resistance was impacted by multiple factors, such as the nervous system and RAAS. In this article, MAP as an influencing factor of SVR was considered. In the clinical data, the otherness of daily monitoring can lead to difference. So that the Spearman correlation coefficient of patients 1 and 2 is a moderate correlation rather than solid correlation. In the future, the relationship of SVR and the multiple parameters will be researched to improve the veracity.

The change in vascular compliance is a complex process,^{29} which is related to nervous and hormonal regulation. The vascular compliance as an indicator of clinical care was estimated solely to report the vascular elastic property.

From the curve of flow rate and pressure (Figure 4), we found that the flow rate increased and the pressure declined in one cardiac cycle. In medication, the aim was to increase the blood perfusion to satisfy the demands of the human so that the SVR was regulated to obtain more CO by pharmacotherapy. Meanwhile, the MAP was decreased because of hemangiectasis.

In the lumped parameter model, the SRV and system compliance as a function have to be considered, which produces higher reassembling than constant value. Adding to the clinical data of HF patient, there is a change in the general model to the individuality model. In the future, the resistance and compliance of pulmonary circulation will be considered, and the right ventricle function will be added to the model. Moreover, neurohumoral regulation is also added to the model to more accurately reflect the physiology.

## Conclusion

The myocardial contractility, SVR, and vascular compliance were studied and calculated by clinical data of HF patients. Based on these parameters, a lumped parameter model of cardiovascular system was proposed to reflect the hemodynamic parameters of HF. The simulation results demonstrate that the proportional error of mean flow, arterial pressure, and SBP is 11.9%, 2.3%, and 4.7% in comparison with the clinical data, respectively. Hence, the computing method of myocardial contractility, SVR function, and vascular compliance function can reflect the physiological mechanism of HF patient.

## References

*in vitro*model. II. The influence of peripheral resistance, perfusion pressure and blood flow. Scand J Clin Lab Invest. 1989;49:459–464