The sine transform is the sine qua non of the pulmonary and systemic pressure relationship

1. Introduction

We sought to develop linear models of pulmonary and systemic pressure to provide a universal set of metrics to address key paradoxes that exist in the study of PAH and PH patients (1). Firstly, PAH, a rare, untreatable and progressive disease has dominated the interests of basic, translation and clinical scientist and the pharmaceutical industry. From the time of detection there is a 5–7 year half mortality burden (2). Conversely, PH is common, often secondary to left heart disease, and remains largely undetected until severe and irreversible damage is sustained, leading to a very poor prognosis once discovered (36). Secondly, the PAH population remains so small that traditional clinical trials often fail to find any statistically significant impact of treatment, despite the common observation that some patients demonstrate clinically relevant improvement (7). A recent meta-analysis of the use of time to clinical worsening as an end-point in PAH trials noted that while it may lead to shorter and smaller trials, time to clinical worsening cannot be considered as valid surrogates for mortality in PAH trials (8) and further, that there is no widely-agreed upon definition of this metric (9). Additionally, the distinction between PAH and PH can be clinically challenging, with each cohort presenting with similar symptoms (10). This difficulty is further confounded in patients with a combination of pre and post-capillary causes of PH (11). Many of the above difficulties in detection and characterization relate to a lack of access to key variables that adequately assess PA pressure and cardiac status (1216). In part this is due to the wide variety of hemodynamic indices and concepts employed to characterize PAH and in part due to the difficulty in obtaining key measures of cardiac status such as cardiac outflow reserve. While right ventricular outflow reserve has been shown to be a good prognostic indicator, since it involves a time-consuming and potentially hazardous rest and stress examination, it is rarely performed and even more rarely performed in an ongoing manner during routine clinical assessment (17).

In developing a unified pair of linear models capable of characterizing and assessing both PAH and PH patients we consider the disease state commonalties as well as differences. The widely perceived major difference involves the role of the left ventricle (LV). While PH is predominantly the result of left-sided disease, PAH is, by strict World Health Organization (WHO) classification, not related to left-sided disease (18). In PAH this often leads to neglect of left-sided conditions and considerations (19). However, ventricular interdependence is inevitable due to (A) the non-compliant pericardial sac constraining the total volume of the combined left and right ventricles, (B) the left and right ventricles being connected in series and thus constrained to generate the same stroke volume, and (C) that both ventricles share a common septum which directly transmits pressure from one to the other (20). These conditions inform us of the mechanisms by which the ventricles interact and indicate that that both ventricles contribute to right and left pressure generation. However, they do not provide any quantitative means of assessing what the pressure conditions are and how in detail they relate. To develop quantitative linear models of pressure we hypothesized that various key cardiac variables adequately define the system’s “state variables”. State variables are a set of physical conditions such as ejection fraction (EF) and end systolic volume (ESV) that describe the state of the dynamic cardiovascular system (21). In this representation of the cardiovascular system, the heart generates pulmonary and systemic blood pressures via time-evolution of its state variables. Suga and Sugawa observed that during isovolumic contraction and relaxation, pressure generation can be modeled by a sinusoidal waveform (22, 23). Thus, we sought to identify the cardiac state variables by applying a sinusoidal transform to candidate variables to model blood pressure. The linear models presented here were successfully applied to PAH patients, clearly showing the mode and magnitude of linkage between right and left-ventricular pressure generation. Further the linear models identified key differences in the manner of pressure generation between PAH and PH patients despite a similar range of PAPs. Finally, the models naturally suggested the concept of RV contractile reserve which was shown to correlate with the 6 min walk distance (6MWD). The ultimate value of the linear models is that they provide a set of non-invasively obtainable quantitative metrics that can be used to assess both pulmonary and systemic pressure conditions. In particular, in the context of clinical trials that target structural remodeling directly, there is a shortage of suitable approaches to measure markers of biology that drive cardiovascular change (24). Even positive PAH trials suffer from a low rate of reproducibility (as low as 21%) in part due to the adequacy of the markers of benefit (25). Consider that improvement in the 6MWD only poorly correlates with survival benefit (26) and more recently, dependence on time to clinical worsening requires larger trials of longer duration (27). While 4D flow CMR can assess pulmonary arterial pressure via direct interrogation of the flow field, it lacks an assessment of cardiac metrics that correspond to pressure conditions (28). Further, echocardiography is most commonly used to non-invasively assess pulmonary artery pressure but primarily relies on assessment of leakage of the tricuspid valve, which may not be present in each case (29). Thus, our intent in developing linear models of pulmonary and systemic pressure was to identify the anatomic cardiac variables that could be directly measured to assess response to therapy.

2. Materials and methods 2.1. Study populations 2.1.1. Exclusive PAH cohort (complexa trial patients)

Data were collected from patients with clinically diagnosed pulmonary arterial hypertension (n = 51) who were enrolled in the Complexa clinical trial (30). In brief, this trial was a multicenter double-blinded, placebo-controlled study to evaluate the safety, efficacy, and pharmacokinetics of the study drug, CXA-10 [10-nitrooctadec-9(E)-enoic acid] which is an endogenous compound. Subjects 18 to 80 years of age (target n = 96) with PAH were randomly assigned to 75 mg, 150 mg of CXA-10 or stable background therapy. This phase II trial was terminated early due to lack of efficacy (with no safety concerns). Only baseline data are reported here prior to administration of the study drug. The study was performed at multiple sites from August 1, 2018 to August 5, 2020 (see Supplementary Material Table S1). Approximately 4% of data was missing at random.

In the Complexa trial, all pulmonary hypertension patients with normal pulmonary capillary wedge pressure (<15 mm Hg) or normal left ventricular end diastolic pressure (<10 mm Hg) were diagnosed to have PAH. Upon enrollment, demographic data were collected along with 6MWD data, right heart catheterization (RHC) pressure data, and cardiovascular magnetic resonance (CMR) imaging assessment of the left and right heart. The CMR data was analyzed at our core lab. The key CMR image acquisitions were (1) a multi-stack short axis cine set of images covering the left and right ventricles and used to obtain cardiac chamber volumes, and (2) phase velocity quantitative flow scans positioned through the ascending aorta and main pulmonary artery. All baseline CMR, RHC and 6MWD measurements were performed within a thirty day period of each other. Assignment to WHO functional class was performed at each site using standard of care clinical assessments (functional class data were available in 49 patients, 94%). Baseline demographic and test measurements are provided in Table 1.

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Table 1. Baseline demographics and measurement of exclusively PAH patients.

2.1.2. Suspected PH/PAH cohort

In addition to the Complexa trial patients we obtained data from a retrospective cohort presenting at our CMR facility. Other than selecting patients with sufficient CMR metrics to model pressure, no exclusion criteria were applied. The key CMR acquisition protocol was identical to that used for the Complexa trial patients. Data were collected from 49 consecutive patients referred to our CMR laboratory from 2011 to 2015 who had a suspicion of PH (34, 70%) or PAH (15, 30%). While SBP was measured in all patients, contemporaneous measures of PAPs were not available, while estimates of PAPs were available in 17 (34%) with 10/17 (59%) obtained by echocardiographic criteria involving measurement of the tricuspid regurgitation jet (33). Due to the sparsity and the low quality of the PAPs estimates these pressure data were not used to generate the linear model of PAPs. The purpose of including this patient cohort was to (1) extend the number of patients contributing to the SBP model and (2) to apply the linear models to identify key differences in pressure distribution between PAH and PH patients. Demographic and key test measures of this cohort are provided in Table 2.

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Table 2. Baseline demographics and measurement of suspected PH/PAH patients.

2.2. CMR variables

Our hypothesis is that a limited number of key cardiac measures define the state variables of the pulmonary and systemic pressure systems. We sought to identify cardiac metrics obtained non-invasively via CMR imaging. The multi-slice short axis data sets, time resolved through the cardiac cycle and covering the left and right ventricles from base to apex, allowed standard ventricular metrics to be obtained separately for the left and right ventricles, including: EF, ESV and ventricular mass. In addition to the standard CMR functional examination (32), separate phase velocity mapping scans were obtained to quantify blood flow velocity through the ascending aorta just above the sinus and through the main pulmonary artery. This data, obtained at the interface of each ventricle to the vasculature, was used to calculate the ventricular-vascular impedance match (36). Here we focused on the impedance match between the ventricle and vasculature using a formulation previously introduce by us (35), with the general formula for the left impedance matching index given by:

Leftimpedancematchingindex=cardiactime×averageaorticbloodvelocityaorticdiameter(1)

Originally, the “cardiac time” variable was the end-systolic duration, but subsequent work (not shown) has demonstrated that the cardiac cycle duration is more appropriate. Using parallel logic we define the right impedance index as;

Rightimpedancematchingindex=cardiactime×averagemainpulmonaryarterybloodvelocitymainpulmonaryarterydiameter(2)

2.3. Linear model generation and optimization 2.3.1. PAPs and SBP models

Candidate variables for the PAPs and SBP linear models were selected from CMR and baseline demographic data. PAPs data were only available for the Complexa PAH cohort, while for the SBP model, data were available from the combined patient cohort, thus more data were available to fit SBP than PAPs. Each candidate variable was correlated separately with the PAPs data and with the SBP data. Additionally, a sine transform of the candidate variable was correlated with PAPs and SBP;

Sinetransformofvariable=Sine(Frequency×Variable+Phase)(3)

Where the Frequency and Phase parameters were optimized by a generalized reduced gradient nonlinear approach implemented in MS Excel to maximize the Pearson correlation r value (37). Variables with a correlation r value greater than 0.2 were entered into the candidate multivariable models. The two multivariable linear models were separately optimized by the systemic search algorithm to maximize each model’s r2 value by systematically changing each variable’s sine transform Frequency and Phase parameters. In the final linear models, variable entry was allowed at the p < 0.05 level. To ensure that highly correlated variables did not inflate the model’s correlation r2, the variance inflation factor (VIF) was calculated, and only variables with a VIF < 5 were retained. The linear model variables together with the sine transform variables are shown in Table 3 for both PAPs and SBP linear models.

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Table 3. PAPs and SBP linear and sine transform components.

2.3.2. 6MWD model

To generate the multivariable model of the 6MWD for the Complexa PAH patients, candidate variables were selected based on a suitably high correlation r with the 6MWD either for the linear variables or the sinusoidally-transformed variables. Similar to the blood pressure models, the parameters of the sine transforms were selected by optimizing the correlation r2 value by the automatic search algorithm.

2.3.3. Linear pressure relationship

To illustrate the nature of the relationship between SBP and PAPs we conducted a simulation using the linear models, whereby an artificially constructed set of parameters of the SBP model was generated to achieve a target SBP. Here two separate SBP targets were simulated: 120 and 140 mm Hg. Since multiple variables contribute to each pressure condition, multiple combinations of parameter values can achieve the same target SBP. In our case 20 combinations of variables were generated to achieve each target pressure. These variables were then entered into the PAPs linear model to generate the corresponding simulated PAPs data and the results were plotted using a box plot.

2.4. Mean and pulsatile blood pressure component representation

To aid in developing an intuitive understanding of the PAPs and SBP model components we combined pressure components to relate to the concept of hydraulic power. Conventionally, the steady hydraulic power is calculated as mean PAP × cardiac output while the remaining power component is associated with pulsatile blood flow (38). Hydraulic power is rarely measured since it requires simultaneous measures, time-resolved thought the cardiac cycle, of high-fidelity pressure and flow data (39). Here we assigned the sinusoidal components to the pulsatile hydraulic power component since the sinusoidal transform is integral to the time-varying aspect of ventricular-vascular interaction. The linear components were assigned to the mean hydraulic power since they contain only time-invariant components. Thus, four separate components of hydraulic pressure were generated:

Hydraulicpulmonarypulsatile(HPp)=sumofpulmonarysinusoidalcomponents(4)

Hydraulicpulmonarymean(HPm)=sumofpulmonarylinearcomponents(5)

Hydraulicsystemicpulsatile(HSp)=sumofsystemicsinusoidalcomponents(6)

Hydraulicsystemicmean(HSm)=sumofsystemiclinearcomponents(7)

2.5. Statistical analysis

Count statistics were represented as number and percentage, continuous data were analyzed as mean and standard deviation if normally or nearly normally distributed, and as median and interquartile range if non-normally distributed. Comparison between grouped variables was performed with t-testing or the non-normal equivalent, as appropriate. Analysis of grouped data were presented as box plots, with interquartile ranges and outliers identified by the “o” symbol. Linear regression modeling was performed using a stepwise forward inclusion approach. To estimate the variability of each linear regression model a bootstrapping approach was applied with 1,000 randomly resampled data sets. The mean and standard deviation of the bootstrapped model correlation r2 was used to calculate the mean and 95% confidence intervals. For linear regression models that incorporated measured data the fitted regression line was generated by Deming regression that takes into account variation in dependent and independent variables (40). The linear model results were subjected to Bland-Altman analysis to yield the model bias and 95% confidence interval. The relationship between pulmonary and systemic hydraulic pressure components were plotted against increasing values of PAPs with a five-point moving average applied to allow data trends to be more easily visualized. Logistic regression modeling was used to distinguish PH from PAH patients. Discrimination capability of the model was measured by the area under curve (AUC) from a receiver operating characteristic (ROC) curve analysis. For an estimate of sensitivity and specificity a cut-off value was set to approximately equalize sensitivity and specificity. Statistical significance was regarded at p < 0.05. Statistical analyses were performed using SPSS 18.0 (SPSS Inc., Chicago, Illinois).

3. Results

For the 52 Complexa patients, the demographic and test results at baseline are shown in Table 1. Complete CMR and RHC data were available in 41 (79%) patients. The mean time between the RHC and the CMR examination was 9.5 days with a standard deviation of 11 days. For the 49 patients with a suspicion of pulmonary hypertension (PAH, n = 34, 70% or PH, n = 15, 30%) who were referred to receive a CMR examination, estimates of PH pressure were available in 17 (35%) with 15 (31%) having a suspicion of PAH. The demographic and test results are shown in Table 2 separately for the PAH and PH groups.

3.1. Pressure modelling

Only patients with RHC measures of PAPs (Complexa cohort) contributed to the linear model of PAPs. To generate the linear model of SBP, patients from both cohorts were entered in to the linear model. Each linear model included cardiac component variables from both right and left ventricles. In the case of left and right impedance match values we identified two sine transforms, one termed slow (lower frequency) and the other rapid (higher frequency) which were sufficiently independent for inclusion in the model since the VIF was <5. Table 3 itemizes the cardiac variables included in each linear model, including the sine transform Frequency and Phase parameters, the linear coefficient and 95% confidence interval, p value and VIF are noted. Age was the only non-CMR-determined variable. Importantly, the models did not contain any indication of which patients had PAH vs. PH. The fitted and modeled PAPs and SBP data were evaluated using linear regression and Bland-Altman analysis, Figure 1. The results of bootstrapping the models are: SBP model r2 = 0.77 (95% CI: 0.64–0.91) and PAPs model r2 = 0.92 (95% CI: 0.85–0.99).

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Figure 1. Model results of pulmonary artery systolic pressure (PAPs) and systemic systolic blood pressure (SBP): (A) the scatter plot of the right heart catheter (RHC) measured and fitted PAPs data has a regression r2 of 0.89. (B) The corresponding Bland-Altman plot of the fitted and measured PAPs data has a bias is zero and the 95% confidence limits of agreement are ±11.2 mm Hg. (C,D) Are the corresponding scatter and Bland-Altman plots for the measured and fitted SBP data, respectively. The measured and fitted SBP data has a regression r2 of 0.74, while the Bland-Altman bias is zero and the 95% confidence limits of agreement are ±20.5 mm Hg.

3.2. Pressure relationships

Since the suspected PH/PAH patient group did not have RHC-measured PAPs contemporaneously with CMR we uniformly employed the modeled PAPs data to plot against each component of hydraulic pressure. Patients from this group were separated into PH (n = 34, 70%) and PAH (n = 15, 30%) to allow the moving average of the two pulmonary and two systemic hydraulic pressure components to be plotted separately for PH and PAH patients, Figure 2. The 34 PH patients were drawn exclusively from the suspected PH/PAH cohort, while the PAH patients comprised Complexa (n = 52) and an additional 15 patients (total n = 67) from the suspected PH/PAH cohort. Pressure data from linear models are plotted without regard to static offsets.

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Figure 2. Moving average of hydraulic pressure components for pulmonary hypertension (PH) and pulmonary arterial hypertension (PAH) patients plotted against increasing pulmonary artery systolic pressure (PAPs): (A) the hydraulic pulmonary pulsatile (HPp) and hydraulic pulmonary mean (HPm) pressure for PH patients. (B) The corresponding hydraulic systemic pulsatile (HSp) and hydraulic systemic mean (HSm) pressure plots for PH patients. (C,D) Are the corresponding pulmonary and systemic data plots for PAH patients, respectively.

Scatter plots of the relationship between pulmonary and systemic hydraulic mean pressure components for PAH and PH patients are shown in Figure 3A. Note that for both PAH and PH patients there is a strong negative relationship between hydraulic pulmonary mean pressure and hydraulic systemic mean pressure, with the PAH data spread over a wider range compared to PH. In Figure 3B the scatter plot of right and left ventricular masses are plotted for PAH and PH patients, displaying a weak positive relationship. Scatter plots of the modeled PAPs and SBP for PAH and PH patients are shown in Figures 4A,B, respectively. Despite the strong negative correlation of hydraulic pulmonary mean and hydraulic systemic mean pressures there is only a weakly negative correlation between PAPs and SBP for PAH and PH patients.

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Figure 3. Scatter plots of (A) hydraulic pulmonary mean pressure (HPm) vs. hydraulic systemic mean pressure (HSm) and (B) scatter plots of right ventricular (RV) mass vs. left ventricular (LV) mass. Points are plotted separately for pulmonary arterial hypertension (PAH) patients (blue) and pulmonary hypertension (PH) patients (orange). Note, that mean pressure components are strongly inversely related (r2 = 0.80) while ventricular masses are weakly positively correlated (r2 = 0.22).

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Figure 4. Scatter plots of (A) systolic blood pressure (SBP) vs. pulmonary arterial systolic pressure (PAPs) for patients with pulmonary arterial hypertension (PAH) and (B) corresponding scatter plots for patients with pulmonary hypertension (PH). In the PAH patients the pressures are weakly negatively related (r2 = 0.23) while for PH patients there is no significant trend (r2 = 0.03).

3.3. Pressure components differences between PAH vs. PH Patients

From Figure 2 there are clearly discernible differences in the general pattern of pressure components between PAH and PH patients. These differences are visible over a wide range of PAPs and also in the pressure component’s mean and distribution, Figure 5. We explored what combination of hydraulic pressure components (without averaging) differentiated PAH vs. PH patients. In Figure 6 we show the box plot of two composite pressure components that are statistically different between PAH and PH patients: (A) the difference between hydraulic pulmonary pulsatile and hydraulic mean pressures and (B) the summation of the hydraulic systemic pulsatile and hydraulic mean pressures. These two composite variables were both significant p < 0.05) in a binary logistic regression model to distinguish between PAH and PH patients. The corresponding ROC plot had an area under the curve of 0.75, Figure 7. Setting the threshold to 0.66 resulted in a sensitivity of 67% with a specificity of 68%.

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Figure 5. Box plots of the hydraulic pressure components for pulmonary hypertension (PH) and pulmonary arterial hypertension (PAH) patients (the asterisk indicates p < 0.05 between PH and PAH). The hydraulic systemic pulsatile pressure (HSp) and hydraulic pulmonary pulsatile pressure (HPp) are not significantly different between groups, while the hydraulic systemic mean pressure (HSm), and hydraulic pulmonary mean pressure (HPm) components are statistically different between groups.

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Figure 6. Box plots of combinations of hydraulic components that are different between pulmonary hypertension (PH) and pulmonary arterial hypertension (PAH) patients (the asterisk indicates p < 0.05). The pulmonary combination that is different between patient groups is the subtraction of pulmonary hydraulic mean (HPm) pressure from the pulmonary hydraulic pulsatile (HPp) pressure. The systemic combination that is different between patient groups is the summation of systemic hydraulic mean (HSm) pressure and the systemic hydraulic pulsatile (HSp) pressure.

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Figure 7. The receiver operator characteristic curve plot for identification of pulmonary arterial hypertension (PAH) patients. The area under the curve is 0.75, and selecting a threshold of 0.66 yields a sensitivity of 68% and a specificity of 68%.

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Table 4. 6 min walk distance linear and sine transform components.

3.4. Right ventricular pulsatile components

The right ventricular hydraulic pulsatile component is formed from the summation of seven sine-transformed variables. The sine transform varies systematically from positive one to negative one and is further multiplied by a scaling factor for each variable. Thus, summation of the absolute magnitude of each sine-transformed variable identifies the maximum possible hydraulic pulmonary pulsatile value, i.e., corresponding to the condition when all sine contributions are maximally positive. In this case the maximum possible pulsatile pulmonary value is 56 mm Hg which represents the upper limit of the hydraulic pulmonary pulsatile component. Thus, the linear model of PAPs naturally produces the concept of RV contractile reserve, which we define as the difference between the current hydraulic pulmonary pulsatile value and the upper limit of 56 mm Hg. For PAH patients, the plots in Figure 8A show how the seven contributions add to produce the realized net and theoretical absolute maximum hydraulic pulmonary pulsatile component. Due to the mixture of positive and negative contributions (depending on the phase of each variable’s sine wave) the realized net hydraulic pulmonary pulsatile pressure value (black line) ranges from negative through positive with increasing PAPs. However, by discarding the phase of each contribution (i.e., considering the absolute magnitude of each variable’s contribution) the sum of each component of hydraulic pulmonary pulsatile component produces the theoretical absolute value (blue line). Note that the absolute summation of each of the seven variable’s contributions do not systematically vary over the range of PAPs. The corresponding net and theoretically absolute maximal amplitudes of hydraulic pulmonary pulsatile pressure are plotted in Figure 8B for PH patients, respectively.

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Figure 8. The attained net (black) and potential (blue) hydraulic pulmonary pulsatile pressure component plotted against increasing pulmonary arterial systolic pressure (PAPs) for (A) pulmonary arterial hypertension (PAH) patients and (B) pulmonary hypertension (PH) patients. The solid black line represents the net summation of each contributing variable of hydraulic pulmonary pulsatile pressure taking into account the phase (i.e. positive or negative contribution) of each variable. The solid blue line represents the net summation of each contributing variable of hydraulic pulmonary pulsatile pressure without taking into account the phase (i.e. each contribution is positive) of each variable. The punctuated red line represents the model-determined maximum value of hydraulic pulsatile pressure (56 mm Hg).

The RV functional reserve is known to be a factor in determining the 6MWD along with biomechanical variables (41). We generated a linear model to predict the 6MWD in PAH patients which included the biomechanical variables of height, weight and age and the pulmonary pressure variables of hydraulic pulmonary pressure and the RV impedance match. Height, hydraulic pulmonary pressure and RV impedance match were related via sinusoidal transforms while weight and age were linearly related. Sine transform variables and model coefficients are given in Table 4. The linear model accounted for 45% of the variation in the 6MWD. The scatter plot of modeled and measured 6MWD is shown in Figure 9A with the corresponding Bland-Altman plot in Figure 9B. The bias term is zero and the 95% confidence limits are ±119 m.

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Figure 9. Linear model results of 6 min walk distance (6MWD): (A) the scatter plot of the 6MWD measured and fitted data with a regression r2 of 0.45. (B) The corresponding Bland-Altman plot of the fitted and measured 6MWD data. The bias is zero and the 95% confidence limits of agreement are ±119 m.

3.5. Pressure simulation

The results of the pressure simulations for the two targeted SBP levels of 120 and 140 mm Hg are shown in the box plots of Figure 10. Here we see that for each target SBP (employing 20 sets of artificially generated parameters) the same set of parameters resulted in a wide range of PAPs values. Note that the lower SBP target of 120 corresponds to the higher PAPs data centered on 69 (SD 18) while the higher SBP target of 140 corresponds to the lower PAPs data centered on 56 (SD 21, p < 0.05). The lower and higher set of PAPs values were statistically different (p < 0.05) and clearly show that each target value of SBP produces a range of values of PAPs, i.e., a one to many relationship.

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Figure 10. Box plot of simulated linear model results of conditions relating to two target systemic systolic blood pressure (SBP) levels: 120 and 140 mm Hg with the corresponding modeled systolic pulmonary artery pressure (PAPs) range of pressures. Twenty sets of conditions were modeled for each target SBP. The PAPs corresponding to SBP target of 120 mm Hg is centered on 69.1 mm Hg with a standard deviation of 4.0 mm Hg, while the PAPs range corresponding to SBP target 140 mm Hg is centered on 55.8 mm Hg with a standard deviation of 4.7 mm Hg, (p < 0.05).

4. Discussion

We introduced two linear models that quantified the relationships that exists between the pulmonary and systemic pressure systems and which further show the nature of the interrelationships of the right and left ventricles. In particular, these interrelationships were demonstrated in a cohort of well documented PAH patients which met all WHO criteria, including the stipulation of hypertension not being secondary to left-sided disease. We interpreted the pressure component derived from linear variables as an index of mean hydraulic power and the pressure component derived from the sine-transformed variables as an index of pulsatile hydraulic power. For the combined patient cohort, which included PAH and PH patients, the linear models showed how the separate hydraulic pressure components varied with increasing PAPs and demonstrated that they exhibited distinct patterns and values that distinguished between PAH and PH patients with good sensitivity and specificity. The concept of right ventricular reserve naturally arose from the model with the reserve index significantly correlating with the 6MWD.

There are several features of the models that are consequences of the construct adopted, namely that the heart conditions represent the state variables of the system. The concept of state variables is more natural to engineering than physiology, whereby the behavior of a dynamic system comprised of multiple parts (e.g., the breaking system of an automobile) can be characterized by a small number of state variables (e.g., the vehicle speed and road-tire coefficient of friction) (42). In the case of the heart, the state variables are weights (ventricular mass), volumes (ESV) and efficiency ratios including EF and the coupling efficiency at the interface to the vasculature (impedance match). The state space representation allows determination of the response of the system to a time-dependent action (e.g., in the case of the breaking system, the rate of application of the break). In the case of the heart the non-linear variables represent the state of the heart at end-systole and the sinusoidal transforms are related to the time course over the cardiac cycle of each variable. The sinusoidal time course was postulated from the observation of Suga and Sugawa concerning the time course of pressure generation in the ventricle. An important aspect of the state-space representation is that external conditions affect how the system behaves (e.g., in the case of the breaking system, icy road vs. dry road conditions). In the case of PAH vs. PH the external conditions are pre-capillary vs. post-capillary elevated resistance/impedance, respectively, and these conditions dramatically affect the response of the heart. The right and left ventricular masses are static over the time course of the cardiac cycle and were thus assigned as contributors to mean pressure. The only non-cardiac variable was age, Table 3. While it has been established that increasing age is associated with higher levels of PAPs in the general population (43) surprisingly the model coefficient of age was negative. However, while the positive association with age can be observed in the general population, here we considered a population with confirmed PAH or a high suspicion of PH/PAH and applied a model that adjusts for a number of cardiac variables which may in turn be age dependent. There are many potential applications of the model: (1) since all model variables are non-invasively obtained, clinical monitoring of PAH patients can be achieved without conducting a RHC, (2) since multiple variables contribute to each model the detailed effects of a therapy on each component can be assessed in an ongoing manner, (3) all model measurements are performed in the heart, and while the heart might not be the target of a therapy the effects of external conditions ultimately manifest in the heart, (4) the models provide estimates of pulmonary and systemic conditions, (5) since the PAH population is small, and does not easily lend itself to large randomized clinical trials (44), there is the possibility that trials might be designed to observe the effects of each therapy in PH patients to suggest how they might transfer to PAH patients and vice versa (45).

4.1. Model accuracy

From Figures 1A,B we see that the model accounted for 89% of the variation of PAPs. The Bland-Altman analysis showed that the 95% confidence limits were ±11 mm Hg, which is on the order of the reported spontaneous variation of PAPs of 20%–25% (46). We did not have reliable contemporaneous measures of PAPs for patients in the suspected PH/PAH cohort. However, the available estimated PAPs values were compared to the modeled values using a paired t-test which failed to reject the hypothesis that the measurements were different (p = 0.40). Others have shown that the use of CMR in studying PAH patients has excellent repeatability and registered a larger therapy effect size than either the 6MWD or N-terminal pro b-type natriuretic peptide (47).

The linear model of SBP accounted for 74% of the variation despite having considerably more patients contributing to the model. However, it has been noted that the typical clinical measurement of SBP is notoriously inaccurate (48, 49). Consider a hypothetical example where the standard deviation of measuring the SBP is on the order of 5 mm Hg. Under this scenario a person with a nominal SBP reading of 125 mm Hg (i.e., pre hypertensive) could well be hypertensive (upper 95% confidence level 135 mm Hg) or normotensive (lower 95% confidence level 115 mm Hg) (50). However, in routine clinical practice the standard deviation of SBP ranges from 14 to 26 mm Hg depending on measurement method and personnel (34, 51). The lower range of the clinical standard deviation is comparable to the standard deviation obtained here for the model SBP Bland-Altman analysis of 10.5 mm Hg. Thus, we believe that the lower level of agreement between modeled and measured SBP in part reflects the difficulty of obtaining a reliable measure of SBP.

4.2. Heart failure in PAH vs. PH

It is widely accepted that the mode of death in PAH and PH patients is predominantly due to RH-failure (31, 52). However, heart failure remains a clinical syndrome with little indication of the details of how failure occurs (53). Examination of the relative pressure components may indicate differences in the mode of heart failure that are expected in PAH vs. PH. Figure 2C shows that in PAH patients both hydraulic pulmonary mean and pulsatile components increase in a broadly parallel manner with increasing PAPs. We speculate that this is due to the normally widely-distributed compliance of the pulmonary arterial tree becoming progressively concentrated in the pulmonary artery, such that the RV ejection pattern progressively resembles that of the LV, where 80% of the systemic compliance is localized in the aorta (54). Thus, as PAH progresses, the hydraulic pulmonary mean pressure increases, but since this is largely contributed to by RV mass (Table 3) at some point the heart may be physically limited in its ability to increase the hydraulic mean pressure, leading to RH failure. Figure 2D shows the that hydraulic systemic mean pressure steadily declines as disease progresses, consistent with the clinical observation that widespread end-organ damage is sustained due to low perfusion pressure (55). A careful study of the mode of death of PAH patients found that only 50% definitively suffered from RH-failure, while the remaining 50% died in the ICU (56), conceivably due to left-sided complications. Thus, interpretation of Figures 2C,D are consistent with observations concerning the changes in physiology, types of morbidity and modes of death observed in PAH patients.

In PH patients, as disease progresses the normally highly compliant pulmonary artery progressively loses compliance, which in turn requires the RV to increase pulse pressure (57). The pulmonary conditions for PH patients are shown in Figure 2A where there is an early elevation of both hydraulic pulmonary pulsatile and mean pressure components, but when PAPs exceeds about 50 mm Hg, the pulsatile component preferentially increases. This in turn causes the reflected pulse pressure wave to rapidly arrive back at the right ventricle prior to closing of the pulmonic valve, further opposing ventricular ejection during late systole (58). However, the model indicates that the increase in hydraulic pulmonary pulsatile pressure cannot proceed beyond an upper limit of 56 mm Hg. With reference to Figure 8A it can be appreciated that in advanced stages of PH the RV pulsatile component approaches the maximum value, i.e., approaching the condition of zero RV pulsatile reserve. This indicates that PH patients may be prone to RV failure due to an inability to increase the hydraulic pulmonary pulsatile component, which is indirectly supported by the association of higher mortality with increased pulse pressure (59). It has been noted that in patients with PH secondary to left-sided heart failure with preserved EF that the degree of loss of RV function greatly exceeds that of the LV (19, 60). In Figure 2B it is apparent that the systemic pressure adaptations are not as dramatic as those of the RV (Figure 2A). Thus, the model reveals several differences in pressure components between PAH and PH which may be explanatory of the different modes of heart failure and whether right or left ventricles are likely to fail or result in morbidity.

4.3. Interconnectedness of ventricular responses

The high-degree of interconnectedness of the pulmonary and systemic pressure systems that is demonstrated by the linear models may be a controversial aspect, especially in PAH patients where the role of the left ventricle may be generally underappreciated (61). While right ventricular changes gradually accrue, clinically relevant changes to the left ventricle tend to manifest at the end stage (62), leading to the concept put forth of the forgotten left ventricle in right ventricular pressure overload (63). In advanced PAH the left ventricle has been clinically noted to appear “small and underfilled” and hyperdynamic (20, 64, 65). This issue directly relates to the underlying determinants of ventricular responses to PAH (66). Figure 3A shows the close to perfect antisymmetric relationship between pulmonary and systemic mean hydraulic pressure components. If the determinant of ventricular response was due to circulating neurohumoral factors, a symmetric response of each ventricle would be expected. Conversely, if the ventricles were independent and responded to localized workloads they could be positively or negatively related, but the relationship would not be expected to be strong due to their independence. However, if the ventricles were interdependent then the response would be expected to be strongly negative, i.e., antisymmetric, which is what the model indicates. Note, that the strong anti-symmetric relationship between hydraulic pulmonary mean and hydraulic systemic mean pressure is not a feature forced by the linear models. Further, Figure 3B shows that RV mass exhibits a weak positive correlation with LV mass which, given the constrained total volume, indicates a reduction in LV chamber volume consistent with the small and underfilled LV. This reduction in LV afterload is expected to weaken the LV myocytes, an expectation that was confirmed in a recent study conducted in PAH patients undergoing cardiac transplantation where it was found that left ventricular myocytes were thinner and exhibited a reduced force generating capacity compared to those of donor hearts (67). The weaker LV myocytes indicate a reduced ability to sustain a high hydraulic systemic mean pressure, which is only partially compensated for by an increased mass. Thus, the exhibited ventricular interdependence of mean pressure is consistent with clinical and physiologic observations.

One may counter that for PAH patients, while there is evidence of left-sided dysfunction (68), the literature does not indicate that there is a clear inverse relationship between pulmonary and systemic pressure, with some observations even indicating a weak positive correlation (69, 70). Figure 4A indicates that the linear models established a weakly negative relationship between PAPs and SBP. Several aspects of the linear models explain why this is the case. Firstly, Figure 10 shows that the linear models do not predict a one-to-one relationship between SBP and PAPs. In the simulated results of Figure 10 several sets of parameter combinations were applied to generate a SBP close to 120 mm Hg which resulted in generating a wide range of PAPs values centered on 69 mm Hg. Conversely, the simulation for the higher SBP value of 140 mm Hg generated a range of wide PAPs values centered on the lower value of 56 mm Hg. That is, there is a one-to-many relationship between SBP and PAPs while the overall trend is negative. Secondly, while there is a strong asymmetric relationship observed between the hydraulic systemic mean and hydraulic pulmonary mean pressures, we note that these are calculated components of the model and may not correspond to any easily-measured pressure component. Thirdly, the difficulty in obtaining accurate measures of SBP requires large numbers of patients to observe these trends. As has been noted, systemic pressure conditions in PAH are widely underappreciated and underreported in sufficient detail to allow determination of the relationships by performing Meta-analysis. Further, commonly applied therapeutic interventions in PAH patients may reduce left-sided pressure (55) and thus any relationship observed may be interpreted as being a side effect of medication.

4.4. Ventricular reserve

Right ventricular output reserve is the ability of the RV to increase output in response to acute exercise or pharmacologic stress. RV reserve can be accessed via multiple indices including cardiac output, pulmonary vascular resistance, pulmonary capacitance, tricuspid annular plane systolic excursion and pulmonary artery pulsatile pressure (71, 72). A common feature of all cardiac reserve assessments is the requirement for a comparison between rest and stress/exercise conditions. Typically, the relevant metric is obtained via non-invasive imaging (most commonly echocardiography) or via invasive RHC (73). An important aspect of the PAPs linear model is that the concept of ventricular reserve naturally emerges from the resting data. The hydraulic pulmonary pulsatile pressure component is generated by summation of sine-transformed variables and thus varies continuously from a net negative value at normal PAPs to a net positive value at high PAPs. Thus, for a continuous variable such as EF, the sine transform indicates that a high EF value is not necessarily “better” than a low EF value since the sine transform systematically cycles between positive and negative values several times over the expected EF range. Figure 8 illustrates the result of combining several sine-transformed variables to generate the net hydraulic pulmonary pulsatile pressure that trends from negative through zero to positive values as PAPs increases. When all sine waves contribute at maximum positive value, the hydraulic pulmonary pulsatile pressure component cannot increase further. This is represented by the punctuated red line in Figure 8. In contrast, note the result of combining the absolute magnitude of each sine-transformed variable in Figure 8 where it can be appreciated that the absolute magnitudes of each pulsatile pressure variables are not in general different between low and high PAPs levels. That is, the absolute magnitude of each contribution to pulsatile pressure is not the dominant determination of the experienced pressure, instead it is the phase (controlled by the sine transform) of each variable’s contribution to pressure that primarily determines the experienced pressure. Here we define the right ventricular contractile reserve as the net difference between the instantaneous hydraulic pulmonary pulsatile value and the upper limit of 56 mm Hg. Thus, the hydraulic pulmonary pulsatile reserve can vary from a maximum of 112 when the phase of each variable component is wholly neg

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