Sigmoid isostiffness-lines: An in-vitro model for the assessment of aortic stenosis severity

1. Introduction

Patients with symptomatic severe aortic stenosis, one of the two most common valvular heart diseases, benefit from aortic valve replacement (1). This underlines the importance of a correct diagnosis. The aortic valve opening area (AVA) is the main parameter used to assess the severity of aortic stenosis (1). However, AVA depends on the transvalvular flow rate (Q) and the nature of this relation is unclear. Around a third of patients with severe aortic stenosis have reduced Q due to reduced left ventricular function (2, 3). This situation is ambiguous because the reduced AVA can be due to reduced Q alone, with or without increased stiffness of the valve. However, aortic valve replacement is indicated only for stiffened valves (e.g., due to calcification). Dobutamine stress echocardiography is used as an additional test to increase Q and observe the corresponding change in AVA (46). The main assumption, currently used for clinical decisions, is that the relation between Q and AVA is linear. Previous study showed that the relation between Q and transvalvular pressure loss (ΔP) under stress is non-linear and difficult to predict (7) and that severe aortic stenosis does not seem to behave like an orifice with a fixed area (8) To account for the large interindividual variability of Q-increases during dobutamine stress, the AVA has been projected to a standardized Q-value (set arbitrarily to 250 ml/s) using linear interpolation (3). The corollary of the assumption of linearity, however, is that AVA would always continue to increase without boundaries with increasing Q. In this in vitro experiment with varying stiffness grades of porcine aortic valves, we compared the accuracy of a linear and a sigmoid, saturating model for the prediction of valve stiffness and AVA. We constructed “isostiffness-lines” over a large spectrum of Q that also include values encountered during low-flow situations and stress tests.

2. Methods

We harvested aortic valves from 4 months old pigs (≈ 120 kg) which were slaughtered within 24 h and kept thereafter at 4°C before the preparation of the valves. A valve identifier scheme was defined as follows: AXXX, where A stands for aortic valve and XXX is the ID number of the valve starting from 001 defined as the harvested valve number. We cut the valve with human surgical instruments as follows: on the side of the left ventricle (LV), we preserved 1 cm of the left ventricular outflow tract (LVOT) below the lower plane defined by the cusps of the aortic valve and cut the ascending aorta 0.5 cm above the plane defined by the 3 commissures of the aortic valve. We then sutured the LVOT on a wedge of neoprene sheet with a central hole. We then secured the neoprene sheet with the sutured valve between two POM (Polyoxymethylene) flanges clamped together with screws (Figures 1A,B). We sutured the aortic side of the valve to a loosely tied indented ring so that the valve cusp would not collapse during diastole, thus allowing the proximal ascending aorta to dilate during systole. We measured the area of the LVOT by counting the number of pixels within the LVOT in an image of the mounted valve taken with the camera in the axial direction from the ventricle side. We calibrated the pixel size by measuring the number of pixels of the inner portion of a circular hole of the known area of the flange on the same image (Figures 1C,D). We placed the valve inside a distal ascending aorta phantom made of silicone (ELASTOSIL®RT 601 A/B Wacker Chemie AG, München, Germany). The aortic valves were tested in a flow loop simulating the left heart as described previously (9). The cardiac output was measured by a transit-time flow probe (TS410/ME-11PXL, Transonic Systems, Inc., Ithaca, NY, USA) which was positioned directly upstream of the mechanical mitral valve between the left atrium and the LV (Figure 2A). The blood mimicking fluid, composed of 40/60% (by weight) glycerine and deionized water at room temperature was used to mimic the viscosity of the blood (9). We recorded the pressure with pressure transducers in the LV (XtransVR, CODAN pvb Critical Care GmbH, Forstinning, Germany) and in the compliance chamber (PBMN flush, Baumer Electric AG, Switzerland) of the flow loop. The two pressure sensors were calibrated with a water column. The distance between the two pressure sensors was 23.2 cm and the distance between the valve and the pressure sensor was 20.5 cm (with a length of the ascending aorta phantom of 15.5 cm, the pressure sensor residing 5 cm inside the compliance chamber. The signals of the pump position, flow-meter, pressure in the LV and the compliance chamber, and the trigger were acquired via a data acquisition system (DAQ USB-6221, National Instruments, Austin, Texas, USA) at a sampling frequency of 20,000Hz.

FIGURE 1

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Figure 1. (A,B) Valve mounting: A harvested valve loosely attached to a ring with a sewing thread on the side of the ascending aorta and sewn to a neoprene sheet entrapped between two POM (Polyoxymethylene) flanges. (C,D) LVOT area measurement.

FIGURE 2

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Figure 2. Sketch of the setup (A). Video frames of valve opening (B).

2.1. Aortic valve opening area (AVA)

Aortic valve opening area of the mounted aortic valves in a pulsatile flow loop was filmed during the ejection time with a high speed camera with a frame rate of 2,000 Hz (Photron FASTCAM Mini AX 100, Reutlingen, Germany).

A light source was placed behind the valve and the image contrast was optimized before acquisition. The image was binarized for every pixel during post-processing to dichotomize valve tissue and AVA (Figure 2B). The pixels were counted and the pixel size was measured by optically measuring a calibration checkerboard with squares of known size while keeping the same camera focus and focal length. The AVA divided by the LVOT area was reported for each valve in order to account for different valve sizes.

2.2. Transvalvular flow rate (Q)

The instantaneous Qinst was calculated from the piston velocity of the pump multiplied by the area of the piston. The retrograde flow (Qretro) measured by the flow sensor positioned proximal to the mechanical mitral valve was subtracted. This resulted in a notch in the flow signal (Figure 3A). In order to impose the same vascular afterload in all experiments, the resistor and the water level of the compliance chamber of the flow loop were adjusted to obtain a constant systolic pressure of 110 mmHg and a diastolic pressure of 70 mmHg (Figures 4A–C). The mean systolic transvalvular flow (Qsyst¯) was computed by taking the average of all the Qinst values over the ejection time. Both Qsyst¯ in [ml/s] [as commonly used in the clinical literature (3, 4)] and Qsyst¯ indexed to the LVOT area in [m/s] were reported in order to account for different valve sizes. For each time point, both AVA and Qinst were measured at 10 different cardiac output values ranging from 0.5 to 5.0 liters/min (Figure 3).

FIGURE 3

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Figure 3. Selection of Qpeak¯ and corresponding AVApeak¯ points (for one valve: A019). (A) Qinst with synchronous corresponding AVAinst. Ten lines corresponding to the 10 different cardiac output values for one stiffness grade from 0.5l/min (in light green) to 5.0l/min (in dark blue). Stiffness grade 1 is depicted (cardiac output of 0.5l/min was missing at stiffness grade 0). (B): Instantaneous retrograde Q (Q retro) measured with flow probe placed proximal to the mitral valve for one stiffness grade (1). (C): All Qinst and corresponding AVAinst for one valve at one stiffness grade 1 and 10 cardiac output values. (D): Selection of the Qinst higher than 97% of max and corresponding AVAinst for one valve at one stiffness grade (1) and 10 cardiac output values. (E): All Qinst and corresponding AVAinst for one valve at the three stiffness grades and 10 cardiac output values. (F): Selection of the Qinst higher than 97% of max and corresponding AVAinst for one valve at three stiffness grades 1 and 10 cardiac output values and their mean (Qpeak¯ and AVApeak¯) for each stiffness grade and each cardiac cycle: (G).

FIGURE 4

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Figure 4. Valve A019: (A–C): Instantaneous aortic pressure. (D–F): Instantaneous transvalvular pressure. (G–I): Instantaneous LV work. In each plot, there are 10 lines corresponding to the 10 different cardiac output values from 0.5l/min (in light green) to 5.0l/min (in dark blue) for one stiffness grade.

2.3. Transvalvular gradient

The instantaneous transvalvular pressure gradient (ΔP) was computed by subtracting the pressure in the compliance chamber from the pressure in the LV (Figures 4D–F) which were recorded with the two transducers in LV and compliance chamber as described in Section 2. The mean transvalvular gradient was computed by averaging all the positive values during valve patency.

2.4. Cumulative LV work

The cumulative work performed by the LV was calculated for each time point of the cardiac cycle as previously described (10) (Figures 4G–I):

WLV(T)=∫0TPLVdVLVdt·dt    (1)

where WLV(T) is the cumulative work performed by the pump from the start of the cycle to the time point T, PLV is the pressure in the LV and dVLV is the instantaneous change in volume in the LV.

2.5. Valve stiffening and relative stiffness computation

The valves were stiffened by treating them with formaldehyde, a protein cross-linking agent, to obtain a total of three stiffness grades (stiffness grades a, b, and c). The relative stiffness s of the native stiffness grade a was defined as sa = 1 and the relative stiffness of grades b and c was computed as ratio (k) between the LV work at grades b and c and the LV work at grade a at the four highest cardiac output values as follows:

[Wmax 5.01/mina·k5.0b,cWmax 4.51/mina·k4.5b,cWmax 4.01/mina·k4.0b,cWmax 3.51/mina·k3.5b,c]=[Wmax 5.01/minb,cWmax 4.51/minb,cWmax 4.01/minb,cWmax 3.51/minb,c]    (2) sb,c=k5.0b,c+k4.5b,c+k4.0b,c+k3.5b,c4    (3)

where Wmax =max0≤T≤Tcycle{WLV(T)} is the LV work or the work performed by the ventricle over the whole cycle at one particular stenosis grade and one particular cardiac output value. The average of the ratios of the four highest cardiac output values was calculated, corresponding to the range of physiological cardiac output values.

2.6. Post-processing

The delay between the camera and the pump position sensor was measured as well as the delay between the camera and the flow probe to synchronize the three signals using a circular cross-correlation [Figure 5; (11)]. The AVA signal was smoothened by performing a centered moving average over 40 frames (0.02 s) for each time point, thus keeping the signal at 2,000 Hz. Qinst signal was smoothened by centered moving average over 800 samples (0.04 s) and down-sampled by a factor of 10 to a sampling rate of 2,000 Hz. Only the Q values (together with the corresponding AVA) which where higher than 97% of max were selected. This corresponds to the phase of the cycle were the flow is the least pulsatile. From this subset, the averages (Qpeak¯, AVApeak¯) were computed for each cardiac output value of each stiffness grade (Figure 3) for further analysis.

FIGURE 5

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Figure 5. Measurement of delay between the camera and pump position sensor via the DAQ (A,B): Detection of the position of the pump with the camera. (C,D): Circular cross-correlation results of the position of pump: DAQ vs. camera signal. We obtained the signal of the position of the pump through the DAQ and filmed its displacement with the camera. We then made a binary copy of the region of interest and searched the leading edge of the pump by setting a point on a horizontal line in front of the pump on its displacement axis. For each frame, we searched the first white pixel, marked it as black for quality control, and thus recorded the pump position through the entire circle (A,B). The entire series included 9 cycles with the camera and 9 with the position sensor via the DAQ. We then calculated the lag between the camera and the position of the pump via the DAQ using circular cross-correlation. We found that the signal of the camera was in advance of 14 ms with respect to the DAQ (C,D). The circular cross-correlation of two signals x, y, ∈ size of N can be defined by [§8.8.1, Smith (11)]: r^x,y(l)=1N(x⋆y)(l)=1N∑n=0N-1x(n)¯y(n+l)l = Where ⋆ is the Discrete Fourier Transform correlation operator. The delay between the camera and the flow probe measured in another experiment was of 11 ms.

2.7. Statistics of baseline characteristics

From the experiments with the different valves, the mean and standard deviation of the relative stiffness, mean transvalvular gradient [mmHg], LV work [J], Qsyst¯ [ml/s] and Qsyst¯ indexed to LVOT [m/s], and maximum AVA [% of LVOT] were reported for each stenosis grade and the 3 following cardiac output values: 0.5, 2.5, and 5.0 l/min in Table 1. In a linear mixed effect model, we tested the fixed effect of the cardiac output and the relative stiffness on each of those 5 variables, setting the valve identifiers as the random effect.

TABLE 1

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Table 1. Baseline characteristics.

2.8. Prediction of relative stiffness and AVA in a modified K-fold cross-validation algorithm

The relative stiffness of each grade was predicted in K-fold cross-validation, a machine learning algorithm (12).

2.8.1. Linear and sigmoid models

First, a linear (with respect to Q) model was used as follows:

AVA^1=F1(Q,s1,θ1,θ2)=θ1·Q·θ2s1    (4)

By analyzing a scatter plot of points, we postulated a saturating sigmoid behavior (with respect to Q) and modeled it mathematically as follows:

AVA^2=F2(Q,s2,θ3,θ4,θ5)=θ3s2θ4(1e(−Q·θ5)+1−0.5)    (5)

where AVA^1 and AVA^2 are the respective predicted AVA for each model, s1 and s2 are the two relative stiffnesses of each model, θ1,…,5 are the hyperparameters to be fitted on the training set and to be kept constant for all the valves of the test set and the final clinical decision tool, F1 and F2 are the two functions describing the relation between those variables.

2.8.2. Fitting of hyperparameters and relative stiffness

We sequentially trained the hyperparameters θ1,…,5 and relative stiffness s1,2 in a modified K-fold cross-validation algorithm (Figure 6). As previously described (12), we first split the entire dataset comprising Qpeak¯ and corresponding AVA and s data points of all the analyzed valves into a training dataset which included all the valves except one and a test dataset which included the valve set aside in the training dataset. The entire dataset was composed of K=11 valves, with 3 different stiffness grades at 10 different cardiac output values making a total of 330 data points. During the training step (Equation 6), we fitted the parameter θ1 − 5 on the training dataset. We repeated the procedure sequentially setting each valve in the test set such that:

θ^j=argminθ∑i=1m1[AVAi−F(Qi,si,θ)]2    (6)

where

• j ∈ is the index of the split. For each split, there is a training set (noted as xtrainj) and a test set (noted as xtestj).

• each data point i ∈ corresponds to one Qi with one AVAi at one particular stiffness sitrainj of the training set trainj of size m1=(K−1)·ns·nf. In our case: (K−1) = 10 valves, ns = 3 number stiffness grades and nf = 10 different cardiac output values. Therefore m1 = 300.

• θ^j is the optimal vector θ (θ=[θ1,θ2]⊤ for F1 and θ=[θ3,θ4,θ5]⊤ for F2) obtained on training set j.

• The least square optimization uses the Levenberg-Marquardt algorithm.

FIGURE 6

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Figure 6. Schematic depiction of the K-fold cross-validation algorithm.

Once θ1-5^j for F1 and F2 were obtained, we could use them as a fixed variable during the test step (Equation 7) to fit the predicted relative stiffness ŝj on the set composed of the remaining test valve:

s^j=argmins∑i=1m2[AVAitestj−F(Qitestj,s,θ^j)]2    (7)

m2=ntrainj··ns·nf. In our case: 1 valve, ns=1 stiffness grades nf=10 different cardiac output values, m2=10. For each test set, the test step was repeated three times for the three stiffness grades.

2.8.3. Accuracy assessment of the linear and sigmoid models

We assessed the accuracy of the models by evaluating the agreement between measured and predicted relative stiffnesses (s and ŝ), we computed the Pearson correlation coefficient, the bias, and its respective 95% CIs, the coefficient of variation, and the mean squared error (MSE) by performing a linear regression and Bland-Altman analysis. In order to assess the goodness of fit of the model while taking into account its complexity, we computed the Akaike information criteria (AIC) which we defined as the endpoint (13), with lower values indicating a superior model. The MSE and AIC were computed as follows:

MSE=1n∑i=1n(AVA^1,2-AVAi)2    (8) AIC=n·log(MSE)+2k    (9)

where n is the total number of data points (one for each cardiac output of each stenosis grade of each valve) and k is the number of hyperparameters plus one (corresponding to the variance estimate) (14). We reported the mean values of θ1,…,5 and their SD obtained during the K-fold cross-validation. Finally, using the same test set, we predicted the AVApeak¯ points corresponding to the Qpeak¯ of the 3 and 5 highest cardiac output values, respectively, for each stiffness grade of each valve, by predicting the ŝ1,2 using the data points with nf = 10-3 = 7 respectively nf = 10-5 = 5 Qpeak¯ of the lowest cardiac output values using Equation (7). We then used ŝ1,2 and θ1,…,5 and the Qpeak¯ of the 3, respectively, 5 highest cardiac output values to predict AVApeak¯ using the linear [Equation (4)] and sigmoid model [Equation (5)]. This prediction scheme takes into account that, in the clinical routine, low-flow low-gradient aortic stenoses are common and require projecting the AVA at normal Q from low Q-values. The number of AVApeak¯ to be predicted (3 and 5) were chosen arbitrarily.

2.9. Software used

Data processing and analysis were written in Python and Julia programming languages (15, 16). Image processing was performed in Python. Mixed models were computed using the lme4 packages (17) of R programming language (18).

3. Results

Three valves were excluded from the data analysis because their neoprene sheet was accidentally torn during the valve preparation process. The baseline characteristics of the 11 valves included in the final data analysis are presented in the Figure 7 and Table 1. There were 4 data points missing making a total of 330-4=326 effective data points . The obtained Qsyst¯ largely encompassed the reported mean physiological Q encountered in the clinic (134 ± 8 to 508 ± 28 ml/s at 0.5 and 5.0 l/min read at the flow probe). At normal physiological flow (cardiac output of 5.0 l/min) and native stiffness grade, there was a Qsyst¯ of 508 ± 27 ml/s, Qsyst¯ indexed to LVOT of 0.96 ± 0.15 m/s, a maximum AVA of 37.2 ± 6.5 % of LVOT, a transvalvular mean gradient of 23.3 ± 7.5 mmHg and a Wmax of 1.01 ± 1.01 J. There was a significant positive effect of the cardiac output on those five 5 variables: (p < 0.001, Table 2). On the other hand, both the Qsyst¯ (p = 0.277) and the Qsyst¯ indexed to LVOT (p = 0.378) were not influenced by the relative stiffness, confirming that Qsyst¯ was, as expected, very similar between different stiffness grades. Moreover, the relative stiffness had a significant negative effect on the maximum AVA and a significant positive effect on both the LV work and the mean transvalvular gradient (p<0.001 for the three values, Table 2). The linear model F1 could predict the stiffness with good accuracy (ŝ1=0.860 · s1 + 0.095, r = 0.794, p < 0.001, θ1=3.69 ± 0.66, θ2 = 0.066 ± 0.009) with a higher bias and equally high coefficient of variation compared to the sigmoid model (bias: 0.07, 95% CI = [–0.15; 0.29], CV: 57%, Figures 8C,D). The sigmoid model F2 could predict the relative stiffness with good accuracy (ŝ2= 0.822 · s2 + 0.196, r = 0.758, p < 0.001, θ3=0.72 ± 0.01, θ4=3.14 ± 0.11, θ5=2.80 ± 0.23) with a relatively low bias but a high coefficient of variation (bias: 0.01, 95% CI = [–0.23; 0.25], CV = 57%, Figures 8A,B). Overall, the sigmoid model better predicted the relative stiffness than the linear model (AIC: –242 vs. –239). The sigmoid models also better predicted the AVApeak¯ corresponding to the Qpeak¯ of the 3 (AIC = –1,743 vs. AIC = –1,048) and 5 highest cardiac output values (AIC = –1,471 vs. AIC = -878) than the linear model for each stiffness grade of each valve (Figure 9). The MSE was more than five times higher in the linear model than in the sigmoid model (MSE = 12.69e−5 vs. 2.24e−5 and MSE = 12.63e−5 vs. 2.40e−5 for the prediction of AVApeak¯ with Qpeak¯ of the 3 and 5 highest cardiac output values, respectively). The linear model systematically overestimated the predicted AVApeak¯ as can be observed with the slope value (slope = 1.40 respectively slope = 1.58 for 3 respectively, 5 AVApeak¯ predictions) whereas there was no such bias in the sigmoid model in which the slope was much closer to 1 (slope = 0.98 respectively slope = 1.07 for 3 respectively, 5 AVAs predictions (Figure 10). Interestingly, even after having carefully subtracted the delay between the signal allowing to synchronize the Qinst and the AVAinst signal, we observed that a subset of points had positive computed Qinst with a closed valve. This could be attributed to a bulging effect of the valve where the cusps move during the isovolumetric contraction time without opening (Figure 3C). Finally, we plotted all the “isostiffness

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