Universality of Vortex Ring Decay in the Left Ventricle

Journal Pre-proofs Universality of Vortex Ring Decay in the Left Ventricle Melissa C. Brindise, Brett Meyers, Pavlos P. Vlachos PII: DOI: Reference:

S0021-9290(20)30111-1 https://doi.org/10.1016/j.jbiomech.2020.109695 BM 109695

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Journal of Biomechanics

Received Date: Revised Date: Accepted Date:

17 September 2019 8 February 2020 21 February 2020

Please cite this article as: M.C. Brindise, B. Meyers, P.P. Vlachos, Universality of Vortex Ring Decay in the Left Ventricle, Journal of Biomechanics (2020), doi: https://doi.org/10.1016/j.jbiomech.2020.109695

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UNIVERSALITY OF VORTEX RING DECAY IN THE LEFT VENTRICLE Short Title: Vortex Ring Decay in the LV

Melissa C. Brindise1*, Brett Meyers1*, Pavlos P. Vlachos1 Author Affiliation: 1

Purdue University, Department of Mechanical Engineering *Authors contributed equally to this work

Corresponding author: Pavlos P. Vlachos Keywords: Left Ventricular Diastolic Dysfunction, Vortex Ring, Circulation Decay, Confined Vortex Ring

A BSTRACT We present clinical measurements and a theoretical model for the decay of the left ventricular (LV) vortex ring. Previous works have postulated that the formation of the vortex ring downstream of the mitral annulus is affected by LV diastolic impairment. However, no previous works have considered how the strength of the vortex ring will decay inside the ventricle, after its formation. Although the vortex ring formation relates to the very initial stage of the filling, the decay process is governed by a large portion of the diastolic time and will be affected by the interaction of the ventricle walls and the vortex ring. Here we performed in-vivo measurements and presented a mechanistic model to calculate the evolution of the vortex ring strength and predict the rate of vortex ring decay within the left ventricle. The results demonstrated the actual circulation decay rate was universal, remaining nearly unchanged across all subjects of varying LV geometry or diastolic function. Furthermore, using the model-predicted circulation decay rate, differentiation between normal and abnormal filling was observed.

I NTRODUCTION During left ventricular (LV) diastolic filling, a vortex ring is formed downstream of the mitral valve (Pasipoularides, 2009). Many previous studies have concluded that vortex ring formation is important in left ventricular filling and may be associated with diastolic function (Charonko et al., 2013; Gharib et al., 2006; Ghosh et al., 2010; Poh et al., 2012). Although the dynamics of the left ventricular vortex ring have been studied previously (Baccani et al., 2002; Domenichini et al., 2007; Ishizu et al., 2006; Kheradvar et al., 2007; Kilner et al., 2000), the majority of the aforementioned studies assume that the vortex ring is generated in a large volume of fluid, as opposed to the confined volume of the left ventricle. Recently a few studies have investigated the formation and propagation of vortex rings within confined domains motived by their relevance to biological flows, such as left ventricular filling or flow within a stenosed artery (Blackburn and Sherwin, 2007; Pasipoularides, 2009; Peterson and Plesniak, 2008; K. C. Stewart et al., 2012).

2

During early diastolic filling blood flows into the left ventricle, and the inflow jet produces a vortex ring. The vortex ring strength, quantified by its hydrodynamic circulation†, continues to increase until it is pinched-off, and that instant is defined as the vortex ring formation time. It has been shown that the left ventricular vortex ring stores kinetic energy and redirects the blood flow towards the aortic valve for ejection (Faludi et al., 2010; Kilner et al., 2000). Previous work on radially confined vortex rings has demonstrated that the vortex ring formation process is unaffected by the confinement domain. The vortex ring formation number and peak circulation remain unaltered for varied radial confinement volumes when the stroke ratio (piston length-to-diameter ratio, L/D0) is less than or equal to 4. When L/D0 is greater than 4, a slight delay in the formation number was observed (Stewart and Vlachos, 2012). However, after vortex ring pinch-off, the normalized vortex ring circulation decayed at a rate proportional to the confinement. In (K. C. Stewart et al., 2012), a model for a thin-core vortex ring traveling within a rigid cylindrical domain was developed and it was shown that the decay coefficient exponentially grew with increasing vortex ring confinement, resulting in an increased circulation decay rate. This model exhibited strong agreement with experimental data over the duration of circulation decay and supports the notion that the radial confinement of the vortex ring— although it does not affect the formation time—governs the vortex circulation decay. The LV geometry creates the confinement of the mitral inflow jet and resulting vortex ring during filling. For patients with diastolic dysfunction the confinement is altered by factors including decreased myocardial relaxation, reduced compliance, and mild ventricle wall thickening. Based on the previous vortex ring studies, we hypothesize that this change can lead to an increased decay rate of the vortex-ring circulation strength during early-filling, which will be dependent on the mitral valve and LV chamber diameter, as well as the vortex ring circulation and diameter at peak diastole. Hence, the rate of decay of the diastolic vortex ring strength for

Circulation (Γ) is a measure of the rotation within a volume of fluid and is defined as: Γ = ∮𝑈𝑑𝑙, the line integral of the velocity (𝑈) , or equivalently, Γ = ∬𝑠𝜔 ∙ 𝑛𝑑𝑠, the area integral of vorticity (), where 𝑛 is the unit normal vector. †

3

normal and abnormal filling ventricles will be governed by the radial confinement imposed by the left ventricle walls. Moreover, in this work, we test the ability of a theoretical model for a thin-core vortex ring traveling in a cylindrical confinement domain to predict the circulation decay rate for the LV diastolic vortex ring.

M ETHODS We used phase contrast MRI (pcMRI) to measure the velocity field and the LV geometry, invivo, for a cohort of normal and diseased human subjects. These measurements are subsequently used as inputs for the vortex ring circulation decay rate model and prediction comparisons. P HASE C ONTRAST M AGNETIC R ESONANCE I MAGING AND V ELOCIMETRY

pcMRI velocimetry measurements were performed on an Avanto 1.5T Scanner from Siemens Medical Solutions, located at the Wake Forest University Baptist Medical Center in WinstonSalem, NC. 2D pcMRI scans were acquired for twelve patients classified within various stages of left ventricular diastolic dysfunction (as determined by physicians) in accordance with the institutional review board (IRB) guidelines pre-established for the study. Velocity encoding (VENC) for each scan was 100-130 cm/s, with a repetition time (TR) of about 20 ms with 40, 45, or 50 reconstructed phases (depending on patient heart rate). Echo time (TE) was 3.3, and there was 1 view per segment. Flip angle was 20 degrees, and the resolution was 320x256 at 1.25 mm/pixel in-plane with a 5 mm slice thickness. Three signal averages were used for each scan. In addition, a separate high signal to noise ratio (SNR) imaging scan was acquired immediately following each pcMRI over the same field of view and used to perform image segmentation on the velocity portraits, as noise in the real part of the pcMRI images often made boundary detection difficult. These images were registered to the pcMRI scans via common anatomical landmarks, and the boundaries of the left ventricle (LV), left atrium (LA), aortic outflow tract, right ventricle, and descending aorta were mapped and transferred to the pcMRI images.

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P ATIENT P OPULATION

The twelve subjects analyzed within this study were selected from patients undergoing clinically indicated cardiac MRI procedures. Information about the group is shown in Table 1. These patients were previously included in another study by Charonko et al. (Charonko et al., 2013). V ELOCITY F IELD P OST -P ROCESSING

Proper Orthogonal Decomposition (POD) was used to post-process the results and reconstruct a reduced order model of the system. POD is a data denoising technique that decomposes a flow field into its fundamental components, or eigenmodes (Doligalski, 1994; Sirovich, 1987). The eigenmodes used to reconstruct the flow fields were determined using the automated entropy line fit (ELF) thresholding criterion which better preserves small-scale modes, reducing the effects of random error and improving the accuracy of post processing metrics (Brindise and Vlachos, 2017). Subsequently, one pass of universal outlier detection (UOD) was used to remove any remaining spurious vectors (Westerweel and Scarano, 2005). V ORTEX I DENTIFICATION S CHEME

A local vortex identification scheme was used to determine the LV vortex ring locations using the λci criterion, where the vortex cores are defined using the imaginary part of the complex eigenvalue of the velocity gradient tensor (Zhou et al., 1999). A threshold was performed to remove λci2 values less than 7% of the maximum value in order to eliminate any falsely identified vortices. This λci2 threshold criteria had negligible effects on the subsequent analysis because the maximum strength vortices are of interest. Vorticity was calculated using a compactRichardson finite difference scheme, minimizing noise amplification and bias error (Etebari and Vlachos, 2005). The circulation strength Γ of each vortex was calculated by computing the line integral of the velocity 𝑈 according to Equation 1. Γ = ∮𝐶𝑈 ∙ 𝑑𝑙

5

1

C ALCULATING THE V ORTEX R ING D ECAY

We analytically modelled the evolution and decay of the confined vortex rings in the left ventricle subject to the effect of the confinement, as presented in (K. C. Stewart et al., 2012). Here, for completeness, we summarize the basic steps of this model. The self-propagation velocity of a thin vortex ring 𝑉 is calculated as (Saffman, 1985): Γ𝐶1

𝑉 = (2𝜋𝐷𝑉𝑅) ,

2

with C1 is defined as

(

𝐶1 = 𝑙𝑛

4𝐷𝑉𝑅 𝑎

),

1

―4

3

where 𝐷𝑉𝑅 is the vortex ring diameter and a is the vortex ring core radius, which is assumed to be much smaller than the vortex ring radius (DVR/2). For a given vortex ring core radius 𝑎 and ring diameter 𝐷𝑉𝑅 this equation provides a relationship between propagation velocity 𝑉 and circulation Γ. Furthermore, this equation captures the evolution of circulation, assuming a constant vortex ring diameter 𝐷𝑉𝑅 and a constant drag coefficient 𝐶𝑑. Equation 4 displays the time varying estimate of the circulation based on the peak circulation, Γ0, at time t0 and a constant parameter β known as circulation decay rate for all values of t greater than t0, Γ0

Γ(𝑡) = Γ0𝛽(𝑡 ― 𝑡0) + 1,

4

where β is defined as 𝑎𝐶𝑑𝐶1

𝛽 = 𝜋2𝐷3 . 𝑉𝑅

5

The parameter β is dependent on the drag coefficient 𝐶𝑑, the vortex ring diameter 𝐷𝑉𝑅, and the vortex core radius a, and was shown to be a function of the radial confinement ratio 𝐷𝑉𝑅 𝐷 (D: 6

confinement diameter). Additionally, β is associated with the rate of vorticity production along the wall (K. C. Stewart et al., 2012) as the vortex ring cores interact with the confinement wall (K. C. Stewart et al., 2012). The confinement ratio 𝐷𝑉𝑅 𝐷 corresponds to the distance between the vortex ring cores and the confinement wall. A decreased separation distance between the vortex ring cores and the radial confinement walls leads to increased circulation decay due to interaction and annihilation of opposite signed vorticity (K. C. Stewart et al., 2012). A relationship for the drag coefficient was determined in (K. C. Stewart et al., 2012), and found to be:

(

𝐶𝑑 = 𝐶0𝑑𝑒𝑥𝑝 9.50

𝐷𝑉𝑅 𝐷

).

6

where 𝐶0𝑑 is a calculated constant, 𝐶0𝑑 = 0.0052. C ALCULATING LV V ORTEX R ING D ECAY

The rate of circulation decay of the vortex ring formed within the LV just beyond the mitral annulus can be modeled using the above equations, assuming the LV walls do not drastically move during early filling. In order to estimate the circulation decay rate, four parameters must be calculated: (1) peak positive and negative circulation values, (2) vortex ring diameter at the time of peak mitral inflow velocity, (3) vortex core radius at the time of peak mitral inflow velocity, (4) LV diameter during early filling at the time of peak mitral inflow velocity. The peak positive and negative circulation values are calculated using the vortex identification scheme discussed in the previous section. The vortex ring diameter 𝐷𝑉𝑅 and LV diameter 𝐷𝐿𝑉 were calculated at the time instant of peak velocity inflow. The 𝐷𝐿𝑉 was calculated at the widest point of the ventricle. The mitral valve diameter 𝐷𝑀𝑉 was considered to be the “piston diameter” and the confinement ratio was assumed as 𝐷𝐿𝑉 𝐷𝑀𝑉. The average confinement ratio for all cases was approximately 1.5. According to (K. C. Stewart et al., 2012), this confinement ratio corresponds to an expected ratio of the vortex ring diameter to the piston diameter 𝐷𝑀𝑉 of 1.25. Thus, in this work, the vortex ring diameter was assumed to be 1.25 times the mitral valve diameter 𝐷𝑀𝑉. The vortex ring diameter 𝐷𝑉𝑅 was assumed to equivalently equal the mitral valve diameter plus 2 times the 7

vortex ring core radius (DVR = DMV + 2a = 1.25DMV). Thus, the vortex ring core radius was assumed to be 0.125 times DMV. Using the pcMRI data from the twelve subject and the vortex identification scheme described above, the circulation decay rate parameter β, or model predicted rate of circulation decay, was calculated using Equation 5 for each patient. This predicts the rate of circulation decay of a vortex ring, starting from the measured peak circulation within the LV. An increase in β corresponds to faster circulation decay. For simplicity we will refer to β as predicted circulation decay rate. In addition, a circulation decay rate β* was also calculated for each patient directly from the pcMRI velocity data. The β* was determined by calculating the best fit on the data by iteratively estimating the β* value and comparing the calculated circulation curve with the circulation values measured during early diastole. The curves were compared by calculating the Chi-squared merit function, χ2, between them. The Chi-squared merit function is a goodness of fit parameter for non-linear curves. The β* value which provided the highest χ2 value was chosen as the measured and fitted circulation decay value. Hereafter, β* will be referred to as data-fit circulation decay rate. Equation 7 was used to compute the difference between the predicted and data-fit circulation decay trends, 1

𝑡

|Γ(t)𝑑𝑎𝑡𝑎 𝑓𝑖𝑡 ― Γ(t)𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑|

∆(Γ𝑑𝑎𝑡𝑎 𝑓𝑖𝑡, Γ𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑) = 𝑁∑𝑡𝑓= 𝑡 1

0 (Γ(t)𝑑𝑎𝑡𝑎 𝑓𝑖𝑡 2

+ Γ(t)𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑)

7

where Γ(𝑡) is the circulation strength at time t, t0 is the time of peak circulation, tf is the time at the end of the circulation decay process, and N is the number of data points contained within the range 𝑡0 < 𝑡 < 𝑡𝑓. The positive and negative circulation difference sums computed in Equation 7 were averaged.

8

R ESULTS Figures 1 and 2, display representative normal and abnormal filling cases respectively. In Figures 1a and 2a, the vorticity is shown with the vortex core contours and vortex ring centers denoted. The circulation versus time (Γ(t)) trends are shown in Figures 1b and 2b. The circulation is displayed as positive and negative circulation approximately mirrored at zero showing the circulation of the positive and negative vortex core. For each patient, the measured circulation at each time is shown (blue-dots) and the two lines represent the predicted circulation (solid line), and the data-fit circulation from the measured data (dashed line). Shown in Figure 1 are Patient 1 (top) and Patient 2 (bottom) from Table 1. In Figure 2, Patients 6 (top) and 12 (bottom) from Table 1 are provided. Figure 1b (top), shows for Patient 1 the circulation increased quickly and reached a peak value at a dimensionless time of approximately 0.5 and then quickly decayed until a small secondary peak is observed. This is typical for a healthy patient where the first peak, which corresponds to the early filling (E-wave: diastolic relaxation), is larger compared to the second peak, corresponding to late filling (A-wave: atrial contraction). The time of peak circulation during the early filling corresponds to the instant of the mitral vortex ring pinch off. Beyond this time, the vortex ring does not receive any additional influx of kinetic energy and the decay process ensues. Qualitatively, very good agreement between the predicted and data-fit circulation is observed suggesting that the circulation decay process within normal filling closely follows that of the model predicted. For Patient 2 (Figure 1 - bottom), the circulation increased quickly and reached a peak value at a non-dimensional time of 0.6. Patient 2 decayed slower than Patient 1, with no secondary peak in circulation from the late filling observed. The agreement between the predicted and data-fit circulation was lower than for Patient 1 but still qualitatively similar. For both abnormal patients, shown in Figure 2, the LV shape was altered. Patient 6 (Figure 2 – top) had a highly enlarged atrium. The vorticity distributions (Figure 2a) for both patients indicate a significantly weaker vortex ring formation process. Similarly, in Figure 2b, the circulation evolution of both patients was characterized by an initial increase reaching a peak during the E-wave, and a second peak at the A-wave. However, in this case the initial rise was 9

slower (delayed) and to a smaller peak value, suggesting impaired LV relaxation. Further, the second peak was much stronger which is representative of a very strong atrial contraction. These features are consistent with diastolic impairment and cardiac remodeling. Moreover, as shown in The top row shows Patient 1 and the bottom row shows Patient 2 from Table 1. Spatial scale was not preserved between vorticity maps; however, velocity scale was preserved. Figure 2b, the predicted and data-fit circulation comparison for both abnormal patients qualitatively displays poor agreement. Although the true circulation decay appears to be exponential, the model overestimates the rate of decay as shown by the sharp decrease of the (solid) predicted line just after the peak circulation strength. Figure 3a quantitatively considers the difference between the predicted and data-fit circulation curves using Equation 7. The average difference between the predicted and data-fit circulation trends was 36 ± 35% for the normal filling cases and 109 ± 33% for the abnormal filling cases. This confirms from the analysis above that the abnormal LV filling causes the circulation decay process to deviate from the predicted model. Figure 3b shows the circulation decay rates β and β* for all patients, grouped based on normal and abnormal filling. On average, patients with normal filling maintained a β of 0.10 ± 0.066 1/cm2 and similar β* values of 0.055 ± 0.065 1/cm2. Patients with abnormal filling had increased β values of 1.15 ± 0.950 1/cm2. In contrast, for abnormal patients, β* values were 0.07 ± 0.029 1/cm2. For normal filling, all patients showed exceptional agreement between β and β* decay rate values. However, the prediction discrepancy is apparent for abnormal filling where β overpredicts compared to β*. Most striking is the agreement of β* for normal and abnormal filling. For both patient groups β* was within the same range, suggesting that the actual mitral annulus vortex ring decay rate, after formation, is universal for both normal and abnormal filling conditions. In Figure 4, other commonly considered cardiac function parameters are compared across the normal and abnormal filling patients. Included here are LV ejection fraction (EF) (Figure 4a), early peak filling velocity (E) (Figure 4b), late peak filling velocity (A) (Figure 4c), and maximum pressure difference between the mitral valve and apex during early diastole (ΔP) (Figure 4d). Due to the small sample sizes for the normal (N=5) and abnormal (N=6) 10

populations, statistical significance testing was not conducted. Across all parameters, no significant trends are observed, with abnormal filling patients generally spanning a large range of values. In all cases, substantial overlap between the normal and abnormal ranges are observed. Principally, this indicates that changes within these variables cannot individually explain the observed differences in circulation trends. Moreover, in the previous analysis it was demonstrated that β* cannot differentiate between normal and abnormal filling. Here, similar to β*, it is observed that the E, A, ΔP, and EF variables cannot differentiate between normal and abnormal filling. Figure 5 provides the relationship of both β and β* against DLV/DMV and demonstrates the analysis of peak circulation Γ0 and 𝐷𝐿𝑉 𝐷𝑀𝑉 across normal and abnormal filling. In Figure 5a, abnormal filling patients had higher confinement, resulting in lower 𝐷𝐿𝑉 𝐷𝑀𝑉 (𝐷𝐿𝑉 𝐷𝑀𝑉 ≤ 1.4), while normal filling patients had elevated DLV/DMV (𝐷𝐿𝑉 𝐷𝑀𝑉 > 1.4). The 𝐷𝐿𝑉 𝐷𝑀𝑉 was the only variable considered that maintained mean and standard deviation ranges which did not overlap between normal and abnormal cases. For peak circulation Γ0 (Figure 5b), a trend is observed where, on average, patients with normal filling maintained high Γ0 during early filling (Γ0 = 136.4 ± 61.6 cm2/s), while abnormal filling subjects typically maintained lower Γ0 values (Γ0 = 82.5 ± 51.3 cm2/s). However, there are some outliers in this trend, leading to mean and standard deviation ranges which overlap across normal and abnormal filling. Nonetheless peak circulation exhibited improved differentiation between normal and abnormal filling as compared to cardiac function parameters shown in Figure 4. Figure 5c demonstrates that across the 𝐷𝐿𝑉 𝐷 range β* remained approximately unchanged. Conversely, an exponential relationship 𝑀𝑉 was observed between β and 𝐷𝐿𝑉 𝐷𝑀𝑉. This is not a particularly unexpected result, as 𝐷𝐿𝑉 𝐷𝑀𝑉 is a principle variable which indirectly governs β. However, it does provide a potential means to differentiate normal and abnormal filling.

D ISCUSSION In this work, a model which was previously developed to predict the circulation decay rate for radially confined vortex rings was applied to clinical phase contrast MRI data of patients with 11

normal filling and clinically determined abnormal filling. The model, utilizing the peak measured circulation value for each patient, accurately predicted the circulation decay for the normal patients and drastically over predicted the circulation decay in abnormal filling patients. Normal filling patients displayed a high peak vortex ring circulation (136.4 ± 61.6 cm2/s) and 𝐷𝐿𝑉 𝐷 above 1.4. The vortex ring circulation was found to quickly increase to a large peak 𝑀𝑉 and then decay throughout the remainder of early diastolic filling for normal filling patients. Conversely, patients with abnormal filling maintained lower peak circulation (82.5 ± 51.3 cm2/s) and 𝐷𝐿𝑉 𝐷𝑀𝑉 at or below 1.4. For abnormal filling, the vortex ring circulation maintained a slower increase to the early filling peak circulation and was primarily characterized by a larger peak circulation caused by atrial contraction. In accordance with these differences, the predicted circulation decay parameter β was found to grow with increasing 𝐷𝐿𝑉 𝐷𝑀𝑉, such that it was expected that 𝐷𝐿𝑉 𝐷𝑀𝑉 values would cause faster decay of the vortex ring. However, as demonstrated by the measured circulation decay values β*, the LV vortex ring circulation did not decay as predicted by the circulation decay model. The true circulation decay for the abnormal filling patients was much slower, resulting in the LV vortex ring maintaining strength and most often enduring until the start of the late diastolic filling wave. In combination, these two results punctuate the universality of the actual decay rate of the mitral annulus vortex ring, β*. Specifically, in (K. C. Stewart et al., 2012) it was demonstrated that in an ideal experimental environment, the rate of vortex ring decay is exponentially proportional to the confinement ratio. Hence, the model predicted circulation rate β computed here maintained an exponential ratio with the confinement ratio 𝐷𝐿𝑉 𝐷𝑀𝑉, as expected from the previous experiments. However, the actual circulation decay rate β* did not adhere to this exponential model, and instead remained unchanged across varying confinement ratios, violating the confined vortex ring decay model (Equations 2-6). This suggests that in order to maintain the universal circulation decay rate, even in the presence of high confinement, the left heart adapts. For example, preservation of the left ventricular vortex ring circulation throughout diastole, even in the case of extremely confined volumes, may assist in maintaining efficient systolic ejection. Overall, this finding generally supports the idea that left ventricular filling velocities and 12

pressures adapt in order to ensure a sufficient volume of oxygenated blood enters the LV per beat, even as diastolic function changes. Moreover, the exact reason why the model fails for abnormal filling cases cannot be specifically pinpointed from these results. As demonstrated in Figure 4, abnormal filling conditions can yield large ranges of specific flow properties and changes such that no realizable relationships between the changes in flow properties (E, A, E/A, ejection fraction, etc.) and changes to vortex ring peak circulation Γ0 could be discerned. However, one possible explanation for the model failing stems from transmitral waveform alteration due to LV remodeling, essentially the velocity ramp function, which affects the vortex ring formation. This is supported by the fact that the abnormal filling cases were principally denoted by a large late filling wave. Furthermore, the model is based on experiments using a piston pump with a sharp velocity ramp function, which produces a waveform qualitatively similar to normal filling. Hence, when this ramp function changes for the normal filling cases, they no longer match the model conditions, and the model again fails. While the model does not provide an accurate estimation of the circulation decay rate for abnormal filling, it nonetheless shows merit in differentiating normal and abnormal filling, unlike the measured decay rate β*. According to Figure 5, a confinement ratio 𝐷𝐿𝑉 𝐷𝑀𝑉 of 1.4 appears to be the critical value below which adaptation must occur. This roughly coincides with the critical juncture where the exponential fit model flattens out. The exponential fit model provided in Figure 5 can be figuratively thought of as the projection of the experimental confined vortex ring model within the LV domain. Furthermore, using only the morphologically defined confinement ratio, 𝐷𝐿𝑉 𝐷𝑀𝑉, it is possible to discern between normal and abnormal filling according to the model. This is a notion that should be explored in future work and across more patients. It should be noted that in this work the abnormal filling primarily consisted of hypertrophic LVs and no true dilated LVs were considered. Overall, in this work, we demonstrated the universality of the rate of decay of the mitral annulus vortex ring, regardless of diastolic function. This finding suggests that one inherent outcome of LV remodeling is maintaining the homeostatic circulation decay rate. This also expands on previous work (Kelley C Stewart et al., 2012) which demonstrated a universality in the vortex 13

ring formation time of patients with normal and specific abnormal filling cases despite the variation in early diastolic filling time and transmitral inflow velocity magnitudes. Further, we applied a theoretical model based on the decay rate of confined vortex rings to the LV vortex ring. We demonstrated that normal filling abides by the theoretical model while abnormal filling violates the model. A clear separation between normal and abnormal filling was observed at a 𝐷𝐿𝑉 𝐷 of 1.4, yielding a possible means to determine diastolic function. 𝑀𝑉 C ONFLICT

OF INTEREST AND AUTHORSHIP

PPV has performed the conception and design, and initial analysis and interpretation of the data; drafting the article, revising it critically for important intellectual content, and approval of the final version. MCB and BM have participated in the final analysis and interpretation of the data, revising it critically for important intellectual content, and approval of the final version. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.

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17

F IGURE L EGENDS Table 1: Patient Specific Data Figure 1: Representative normal filling cases. (a) Vorticity map during early filling. (b) Circulation strength versus time throughout diastole. The top row shows Patient 1 and the bottom row shows Patient 2 from Table 1. Spatial scale was not preserved between vorticity maps; however, velocity scale was preserved. Figure 2: Representative abnormal filling cases. (a) Vorticity map during early filling. (b) Circulation strength versus time throughout diastole. The top row shows Patient 6 and the bottom row shows Patient 12 from Table 1. Spatial scale was not preserved between vorticity maps; however, velocity scale was preserved. Figure 3: (a) Average difference between the model predicted circulation decay rate and the datafit circulation decay rate curves for normal and abnormal filling patterns. (b) Comparison of the model predicted β and data-fit β* for normal and abnormal filling patterns. Open symbols display the mean values with the bars representing plus or minus one standard deviation. Filled symbols represent individual patient values. Figure 4: Common cardiac function parameters including (a) ejection fraction, (b) E-filling velocity, (c) A-filling velocity, and (d) P, the maximal pressure difference between the mitral valve to the apex in the early filling wave compared across normal and abnormal filling patterns. Open symbols display the mean values with the bars representing plus or minus one standard deviation. Filled symbols represent individual patient values. Figure 5: Comparison of (a) peak early-filling circulation and (b) confinement ratio, DLV/DMV, across normal and abnormal filling patterns. Open symbols display the mean values with the bars representing plus or minus one standard deviation. Filled symbols represent individual patient values. (c) Relationship of β and β* versus DLV/DMV. An exponential fit of the β values is provided.

18

Table 1 Pat #

Age

Sex

E

A

Lat E'

Sep E'

(cm/s)

(cm/s)

(cm/s)

(cm/s)

1

2

3

4

echo E/A

E/E'

EF

5

(%EDV)

MRI E

MRI A

(cm/s)

(cm/s)

7

8

MRI E/A

MRI EF (%EDV) 9

Additional Comments

6

Normal Filling 1

19

M

*

*

*

*

*

*

*

90

43

2.1

61

--

2

18

M

*

*

*

*

*

*

*

71

47

1.5

51

--

3

24

M

102

78

12.3

12.4

1.3

8.3

45

70

45

1.6

44

Chemo-therapy

4

38

F

*

*

*

*

*

*

48

47

48

1.0

62

LV Hypertrophy

5

54

F

58

76

7

3.2

0.8

11.4

60

55

57

1.0

67

--

Left Ventricle Diastolic Dysfunction History of 6

65

F

*

*

*

*

*

*

*

52

79

0.7

47

coronary artery disease and bypass surgery History of

7

73

F

*

*

*

*

*

*

46

65

41

1.6

53

8

61

M

*

*

*

*

*

*

*

43

68

0.6

48

Hypertensive

9

61

M

*

*

*

*

*

*

*

59

78

0.7

43

--

10

60

F

68

80

10.6

7.7

0.9

7.4

69

57

52

1.1

53

--

11

52

F

90

84

7.9

5.6

1.1

13.3

67

59

67

0.9

70

Chemo-therapy

12

52

M

39

43

4.3

5.5

0.9

8

30

27

33

0.8

38

*No contemporaneous measurement was performed;1 peak early filling velocity;

2

bypass surgery

Acute anterior MI

peak late

filling velocity; 3 motion of the lateral side of the mitral annulus using tissue Doppler in an apical 4-chamber view;

4

motion of the septal side of the mitral annulus using tissue Doppler in an

apical 4-chamber view; 5 calculated from lateral E'; 6 ejection fraction as percent of end diastolic 19

volume from apical 4-chamber echocardiography; 7 ejection fraction as percent of end diastolic volume from MRI. Figure 1

20

Figure 2

21

Figure 3

22

Figure 4

23

Figure 5

24