Fractional Flow Reserve Evaluated as Metric of Coronary Stenosis - A Mathematical Model Study.

Introduction: Coronary arterial stenosis may impair myocardial perfusion with myocardial ischemia and associated morbidity and mortality as result. The myocardial fractional flow reserve (FFR) is clinically used as a stenosis-specific index. Aim: This study aims to identify the relation between the FFR and the degree of coronary arterial stenosis using a simple mathematical model of the coronary circulation. Methods: A mathematical model of the coronary circulation, including an arterial stenosis of variable degree, was developed. The relation between the FFR and the degree of stenosis (defined as the fractional cross sectional area narrowing) was investigated, including the influence of the aortic and venous pressures and the capillary resistance. An additional study concerning 22 patients with coronary artery disease permits comparison of clinical data and in silico findings. Results: The FFR shows an S-shaped relationship with the stenosis index. We found a marked influence of venous and aortic pressure and capillary resistance. The FFR is accompanied by a clinically relevant co-metric (FFR C ), defined by the Pythagorean sum of the two pressures in the definition formula for FFR. In the patient group the FFR C is strongly related to the post-stenotic pressure (R = 0.91). The FFR C requires establishment of a validated cut-off point using future trials. Conclusion: The S-shaped dependence of FFR on the severity of the stenosis makes the FFR a measure of the ordinal scale. The marked influences of the aortic and venous pressures and the capillary resistance on the FFR will be interpreted as significant variations in intra- and inter-individual clinical findings. These fluctuations are partly connected to the neglect of considering the FFR C . At otherwise identical conditions the FFR as measured at baseline differs from the value obtained during hyperemic conditions. This expected observation requires further investigation, as the current hyperemia based evaluation fails to take advantage of available baseline data.

Conductivity Rise During Irreversible Electroporation: True Permeabilization or Heat?

PURPOSE: Irreversible electroporation (IRE) induces apoptosis with high-voltage electric pulses. Although the working mechanism is non-thermal, development of secondary Joule heating occurs. This study investigated whether the observed conductivity rise during IRE is caused by increased cellular permeabilization or heat development. METHODS: IRE was performed in a gelatin tissue phantom, in potato tubers, and in 30 patients with unresectable colorectal liver metastases (CRLM). Continuous versus sequential pulsing protocols (10-90 vs. 10-30-30-30) were assessed. Temperature was measured using fiber-optic probes. After temperature had returned to baseline, 100 additional pulses were delivered. The primary technique efficacy of the treated CRLM was compared to the periprocedural current rise. Seven patients received ten additional pulses after a 10-min cool-down period. RESULTS: Temperature and current rise was higher for the continuous pulsing protocol (medians, gel: 13.05 vs. 9.55 °C and 9 amperes (A) vs. 7A; potato: 12.70 vs. 10.53 °C and 6.0A vs. 6.5A). After cooling-down, current returned to baseline in the gel phantom and near baseline values (Δ2A with continuous- and Δ5A with sequential pulsing) in the potato tubers. The current declined after cooling-down in all seven patients with CRLM, although baseline values were not reached. There was a positive correlation between current rise and primary technique efficacy (p = 0.02); however, the previously reported current increase threshold of 12-15A was reached in 13%. CONCLUSION: The observed conductivity rise during IRE is caused by both cellular permeabilization and heat development. Although a correlation between current rise and efficacy exists, the current increase threshold seems unfeasible for CRLM.

Vascular narrowing in pulmonary arterial hypertension is heterogeneous: rethinking resistance.

In idiopathic pulmonary arterial hypertension (PAH), increased pulmonary vascular resistance is associated with structural narrowing of small (resistance) vessels and increased vascular tone. Current information on pulmonary vascular remodeling is mostly limited to averaged increases in wall thickness, but information on number of vessels affected and internal diameter decreases for vessels of different sizes is limited. Our aim was to quantify numbers of affected vessels and their internal diameter decrease for differently sized vessels in PAH in comparison with non-PAH patients. Internal and external diameters of transversally cut vessels were measured in five control subjects and six PAH patients. Resistance vessels were classified in Strahler orders, internal diameters 13 µm (order 1) to 500 µm (order 8). The number fraction, that is, percentage of affected vessels, and the internal diameter fraction, that is, percentage diameter of normal diameter, were calculated. In PAH, not all resistance vessels are affected. The number fraction is about 30%, that is, 70% of vessels have diameters not different from vessels of control subjects. Within each order, the decrease in diameter of affected vessels is variable with an averaged diameter fraction of 50-70%. Narrowing of resistance vessels is heterogeneous: not all vessels are narrowed, and the decrease in internal diameters, even within a single order, vary largely. This heterogeneous narrowing alone cannot explain the large resistance increase in PAH We suggest that rarefaction could be an important contributor to the hemodynamic changes.

Proportional Relations Between Systolic, Diastolic and Mean Pulmonary Artery Pressure are Explained by Vascular Properties.

Recently, it was shown that proportional relationships exist between systolic, diastolic and mean pulmonary artery pressure (P(sys), P(dia) and P(mean)) and that they are maintained under various conditions in both health and disease. An arterial-ventricular interaction model was used to study the contribution of model parameters to the ratios P(sys)/P(mean), and P(dia)/P(mean). The heart was modeled by a time-varying elastance function, and the arterial system by a three-element windkessel model consisting of peripheral resistance, R(p), arterial compliance C(a), and pulmonary artery characteristic impedance Z(0). Baseline model parameters were estimated in control subjects and compared to values estimated in patients with pulmonary hypertension. Results indicate that experimentally derived ratios P(sys)/P(mean) and P(dia)/P(mean) could be accurately reproduced using our model (1.59 and 0.61 vs. 1.55 and 0.64, respectively). Sensitivity analysis showed that the (empirical) constancy of P(sys)/P(mean) and P(dia)/P(mean) was primarily based on the inverse hyperbolic relation between total vascular resistance (R(T); calculated as R(p) + Z(0)) and C(a), (i.e. constant R(T)C(a) product). Of the cardiac parameters, only heart rate affected the pressure ratios, but the contribution was small. Therefore, we conclude that proportional relations between systolic, diastolic and mean pulmonary artery pressure result from the constancy of R(T)C(a) thus from pulmonary arterial properties, with only little influence of heart rate.

Evaluation of model-independent deconvolution techniques to estimate blood perfusion.

This report evaluates several methods to estimate blood perfusion and residue functions in dynamic contrast enhanced (DCE) MRI. Among these are model-dependent and model-independent techniques. All methods were applied to series of Monte Carlo simulations to evaluate the accuracy in order to reproduce different underlying vascular residue functions and blood perfusions. Of the model-independent approaches the use of B-splines with Tikhonov regularization was shown to have a reasonable accuracy in blood perfusion estimations and was less biased than all model-dependent approaches. This technique seems most promising for application to experimental data.

Estimation of three- and four-element windkessel parameters using subspace model identification.

A windkessel model is widely used to operationalize vascular characteristics. In this paper, we employ a noniterative subspace model identification (SMI) algorithm to estimate parameters in a three- and four-element windkessel model by application of physical foreknowledge. Simulation data of the systemic circulation were used to investigate systematic and random errors in the parameter estimations. Results were compared with different methods as proposed in the literature: one closed-loop and two iterative methods for the three-element model, and one iterative method for the four-element model. For the three-element model, no significant systematic errors were observed using SMI. Concerning random errors, SMI appeared more robust in parameter estimations compared with the other methods (P < 0.05 for a signal-to-noise ratio of 18 dB). For the four-element model, a significant systematic error in the estimate of the arterial inertance L was observed (P = 0.011). However, for all methods, an increasing number of outliers in parameter estimates were observed at increased noise levels. These outliers were almost exclusive due to errors in estimates of L. In conclusion, with SMI physical parameters can mathematically be derived by application of physiological foreknowledge. For a three-element windkessel model, SMI appeared a very robust method to estimate parameters. However, application to a four-element windkessel model was less accurate because of low identifiability of L. Therefore, based on the simulation results, the use of the four-element windkessel model is questionable.

Assessing fluid responses after coronary surgery: role of mathematical coupling of global end-diastolic volume to cardiac output measured by transpulmonary thermodilution.

BACKGROUND: Mathematical coupling may explain in part why cardiac filling volumes obtained by transpulmonary thermodilution may better predict and monitor responses of cardiac output to fluid loading than pressures obtained by pulmonary artery catheters (PACs). METHODS: Eleven consecutive patients with hypovolaemia after coronary surgery and a PAC, allowing central venous pressure (CVP) and continuous cardiac index (CCIp) measurements, received a femoral artery catheter for transpulmonary thermodilution measurements of global end-diastolic blood volume index (GEDVI) and cardiac index (CItp). One to five colloid fluid-loading steps of 250 ml were done in each patient (n = 48 total). RESULTS: Fluid responses were predicted and monitored similarly by CItp and CCIp, whereas CItp and CCIp correlated at r = 0.70 (P < 0.001) with a bias of 0.40 l min(-1) m(-2). Changes in volumes (and not in CVP) related to changes in CItp and not in CCIp. Changes in CVP and GEDVI similarly related to changes in CItp, after exclusion of two patients with greatest CItp outliers (as compared to CCIp). Changes in GEDVI correlated better to changes in CItp when derived from the same thermodilution curve than to changes in CItp of unrelated curves and changes in CCIp. CONCLUSIONS: After coronary surgery, fluid responses can be similarly assessed by intermittent transpulmonary and continuous pulmonary thermodilution methods, in spite of overestimation of CCIp by CItp. Filling pressures are poor monitors of fluid responses and superiority of GEDVI can be caused, at least in part, by mathematical coupling when cardiac volume and output are derived from the same thermodilution curve.

Modeling the instantaneous pressure-volume relation of the left ventricle: a comparison of six models.

Simulations are useful to study the heart's ability to generate flow and the interaction between contractility and loading conditions. The left ventricular pressure-volume (PV) relation has been shown to be nonlinear, but it is unknown whether a linear model is accurate enough for simulations. Six models were fitted to the PV-data measured in five sheep and the estimated parameters were used to simulate PV-loops. Simulated and measured PV-loops were compared with the Akaike information criterion (AIC) and the Hamming distance, a measure for geometric shape similarity. The compared models were: a time-varying elastance model with fixed volume intercept (LinFix); a time-varying elastance model with varying volume intercept (LinFree); a Langewouter's pressure-dependent elasticity model (Langew); a sigmoidal model (Sigm); a time-varying elastance model with a systolic flow-dependent resistance (Shroff) and a model with a linear systolic and an exponential diastolic relation (Burkh). Overall, the best model is LinFree (lowest AIC), closely followed by Langew. The remaining models rank: Sigm, Shroff, LinFix and Burkh. If only the shape of the PV-loops is important, all models perform nearly identically (Hamming distance between 20 and 23%). For realistic simulation of the instantaneous PV-relation a linear model suffices.

GREIT: a unified approach to 2D linear EIT reconstruction of lung images.

Electrical impedance tomography (EIT) is an attractive method for clinically monitoring patients during mechanical ventilation, because it can provide a non-invasive continuous image of pulmonary impedance which indicates the distribution of ventilation. However, most clinical and physiological research in lung EIT is done using older and proprietary algorithms; this is an obstacle to interpretation of EIT images because the reconstructed images are not well characterized. To address this issue, we develop a consensus linear reconstruction algorithm for lung EIT, called GREIT (Graz consensus Reconstruction algorithm for EIT). This paper describes the unified approach to linear image reconstruction developed for GREIT. The framework for the linear reconstruction algorithm consists of (1) detailed finite element models of a representative adult and neonatal thorax, (2) consensus on the performance figures of merit for EIT image reconstruction and (3) a systematic approach to optimize a linear reconstruction matrix to desired performance measures. Consensus figures of merit, in order of importance, are (a) uniform amplitude response, (b) small and uniform position error, (c) small ringing artefacts, (d) uniform resolution, (e) limited shape deformation and (f) high resolution. Such figures of merit must be attained while maintaining small noise amplification and small sensitivity to electrode and boundary movement. This approach represents the consensus of a large and representative group of experts in EIT algorithm design and clinical applications for pulmonary monitoring. All software and data to implement and test the algorithm have been made available under an open source license which allows free research and commercial use.

Cardiac phase-dependent time normalization reduces load dependence of time-varying elastance.

The time-varying elastance concept provides a comprehensive description of the intrinsic mechanical properties of the left ventricle that are assumed to be load independent. Based on pressure-volume measurements obtained with combined pressure conductance catheterization in six open-chest anesthetized sheep, we show that the time to reach end systole (defined as maximal elastance) is progressively prolonged for increasing ventricle pressures, which challenges the original (load-independent) time-varying elastance concept. Therefore, we developed a method that takes into account load dependency by normalization of time course of the four cardiac phases (isovolumic contraction, ejection, isovolumic relaxation, filling) individually. With this normalization, isophase lines are obtained that connect points in pressure-volume loops of different beats at the same relative time in each of the four cardiac phases, instead of isochrones that share points at the same time in a cardiac cycle. The results demonstrate that pressure curves can be predicted with higher accuracy, if elastance curves are estimated using isophase lines instead of using isochrones [root-mean-square error (RMSE): 3.8 +/- 1.0 vs. 14.0 +/- 7.4 mmHg (P < 0.001), and variance accounted for (VAF): 94.8 +/- 1.3 vs. 78.6 +/- 14.8% (P < 0.001)]. Similar results were found when the intercept volume was assumed to be time varying [RMSE: 1.7 +/- 0.3 vs. 13.4 +/- 7.4 mmHg (P < 0.001), and VAF: 97.4 +/- 0.5 vs. 81.8 +/- 15.5% (P < 0.001)]. In conclusion, phase-dependent time normalization reduces cardiac load dependency of timing and increases accuracy in estimating time-varying elastance.

Pulmonary vascular resistance and compliance stay inversely related during treatment of pulmonary hypertension.

AIMS: Pulmonary arterial compliance (C) is increasingly being recognized as an important contributor to right ventricular afterload, but for monitoring of treatment of pulmonary hypertension (PH) most often still only pulmonary vascular resistance (R) is used. We aimed at testing the hypothesis that R and C are coupled during treatment of PH and that substantial changes in both R and C would result in more haemodynamic improvement than changes in R alone. METHODS AND RESULTS: Data were analysed of two right-heart catheterizations of 52 patients with pulmonary arterial hypertension and 10 with chronic-thromboembolic PH. The product of R and C (= stroke volume over pulse pressure) did not change during therapy (P = 0.320), implying an inverse relationship. Changes in cardiac index correlated significantly (P < 0.001) with changes in R (R(2) = 0.37), better with changes in C (R(2) = 0.66), and best with changes in both (R(2) = 0.74). CONCLUSION: During therapy for PH, R and C remain inversely related. Therefore, changes in both R and C better explain changes in cardiac index than either of them alone. Not only resistance but also compliance plays a prominent role in PH especially in an early stage of the disease.

Quantification of right ventricular afterload in patients with and without pulmonary hypertension.

Right ventricular (RV) afterload is commonly defined as pulmonary vascular resistance, but this does not reflect the afterload to pulsatile flow. The purpose of this study was to quantify RV afterload more completely in patients with and without pulmonary hypertension (PH) using a three-element windkessel model. The model consists of peripheral resistance (R), pulmonary arterial compliance (C), and characteristic impedance (Z). Using pulmonary artery pressure from right-heart catheterization and pulmonary artery flow from MRI velocity quantification, we estimated the windkessel parameters in patients with chronic thromboembolic PH (CTEPH; n = 10) and idiopathic pulmonary arterial hypertension (IPAH; n = 9). Patients suspected of PH but in whom PH was not found served as controls (NONPH; n = 10). R and Z were significantly lower and C significantly higher in the NONPH group than in both the CTEPH and IPAH groups (P < 0.001). R and Z were significantly lower in the CTEPH group than in the IPAH group (P < 0.05). The parameters R and C of all patients obeyed the relationship C = 0.75/R (R(2) = 0.77), equivalent to a similar RC time in all patients. Mean pulmonary artery pressure P and C fitted well to C = 69.7/P (i.e., similar pressure dependence in all patients). Our results show that differences in RV afterload among groups with different forms of PH can be quantified with a windkessel model. Furthermore, the data suggest that the RC time and the elastic properties of the large pulmonary arteries remain unchanged in PH.

Correction of phase offset errors in main pulmonary artery flow quantification.

PURPOSE: To investigate whether an existing method for correction of phase offset errors in phase-contrast velocity quantification is applicable for assessment of main pulmonary artery flow with an MR scanner equipped with a high-power gradient system. MATERIALS AND METHODS: The correction method consists of fitting a surface through the time average of stationary pixels of velocity-encoded phase images, and subtracting this surface from the velocity images. Pixels are regarded as stationary if their time standard deviation falls into the lowest percentile. Flow was measured in the main pulmonary artery of 15 subjects. Each measurement was repeated on a stationary phantom. The phase offset error in the phantom was used as a reference. Correction was applied with varying polynomial surface orders (0-5) and stationarity percentiles (5-50%). The optimal surface order and stationarity percentile were determined by comparing the fitted surface with the phantom. RESULTS: Using a first-order surface and a (noncritical) 25% percentile, the correction method significantly reduced the phase offset error from 1.1 to 0.35 cm/second (RMS), which is equivalent to a reduction from 11% to 3.3% of mean volume flow. Phase error correction strongly affected stroke volume (range -11 to 26%). CONCLUSION: The method significantly reduces phase offset errors in pulmonary artery flow.

Interventricular septal configuration at mr imaging and pulmonary arterial pressure in pulmonary hypertension.

PURPOSE: To investigate whether a relationship exists between septum shape and systolic pulmonary arterial pressure (PAP) in patients with pulmonary hypertension. MATERIALS AND METHODS: Study protocol was approved by institutional ethics review committee; all patients gave informed consent. Right-sided heart catheterization with vasodilator testing was performed in 39 adult subjects suspected of having pulmonary hypertension. There were 11 men and 28 women, aged 21-75 years (mean, 46 years). Only two patients showed favorable response to vasodilators, defined by a decrease in PAP of more than 20%. Synchronous right- and left-ventricular pressure measurements and four-chamber magnetic resonance (MR) imaging were used to identify timing of maximal leftward ventricular septal bowing within cardiac cycle. Septal bowing was evaluated with MR, measured on short-axis cine heart images, and expressed as curvature (reciprocal of radius). Curvature was quantified on one image (the one that showed the most severe distortion of normal septal shape). The relationship between systolic PAP and septal curvature was tested with linear regression analysis. P <.05 was considered to indicate a statistically significant difference. RESULTS: Of 39 subjects, 37 had pulmonary hypertension. Maximal distortion of normal septal shape was found during right ventricular relaxation phase. Systolic PAP was proportional to septal curvature: r=0.77 (P < .001), slope=-114.7, and intercept=67.2. In the two vasodilator responsive subjects, a significant reduction of leftward ventricular septal bowing was observed in response to reduction of right ventricular pressure. CONCLUSION: In 37 patients with pulmonary hypertension, systolic PAP higher than 67 mm Hg may be expected when leftward curvature is observed.