Leaflet Remodeling Reduces Tricuspid Valve Function in a Computational Model

Abstract

Tricuspid valve leaflets have historically been considered “passive flaps”. However, we have recently shown that tricuspid leaflets actively remodel in sheep with functional tricuspid regurgitation. We hypothesize that these remodeling-induced changes reduce leaflet coaptation and, therefore, contribute to valvular dysfunction. To test this, we simulated the impact of remodeling-induced changes on valve mechanics in a reverse-engineered computer model of the human tricuspid valve. To this end, we combined right-heart pressures and tricuspid annular dynamics recorded in an ex vivo beating heart, with subject-matched in vitro measurements of valve geometry and material properties, to build a subject-specific finite element model. In this model, we then increased leaflet thickness and stiffness and reduced the stretch at which leaflets stiffen, which we call “transition-λ.” Subsequently, we quantified mean leaflet stresses, leaflet systolic angles, and coaptation area as measures of valve function. We found that leaflet stresses, leaflet systolic angle, and coaptation area are sensitive to independent changes in stiffness, thickness, and transition-λ. When combining thickening, stiffening, and changes in transition-λ, we found that anterior and posterior leaflet stresses decreased by 26% and 28%, respectively. Furthermore, systolic angles increased by 43%, and coaptation area decreased by 66%; thereby impeding valve function. While only a computational study, we provide the first evidence that remodeling-induced leaflet thickening and stiffening may contribute to valvular dysfunction. Targeted suppression of such changes in diseased valves could restore normal valve mechanics and promote leaflet coaptation.

Graphical Abstract

1. Introduction

Heart valves have historically been thought of as inert tissues, or “passive flaps” [1]. However, we now know that they are highly sensitive to their mechano-chemical environment and are “active tissues” [2, 3]. In fact, their extra-cellular matrices are populated by valve interstitial cells (VIC) that maintain tissue homeostasis [4]. To this end, VICs continually secrete matrix proteins – namely collagen and elastin – and matrix remodeling enzymes to repair tissue and ensure matrix durability[5, 6]. However, VICs may also acquire a hyper-contractile and myofibroblast-like phenotype in response to injury or disease[7]. These “activated” VICs may drive excess collagen deposition in the extra-cellular matrix. This fibrotic tissue remodeling leads to tissue thickening and stiffening and alters the nonlinearity of their constitutive response. In turn, these tissue-level changes alter the mechanics of valve leaflets[8, 9]. Notably, such changes were reported in the mitral valves of patients with functional mitral regurgitation[10], where remodeling-induced leaflet thickening and stiffening increased leaflet stresses and reduced coaptation[11].

Inspired by these findings, we sought to determine if the tricuspid valve remodels similarly in response to disease – namely, functional tricuspid regurgitation. Moderate to severe tricuspid regurgitation affects nearly 1.6 million patients in the US and is an independent predictor of morbidity and mortality[12, 13]. In most cases, the origin of regurgitation is extrinsic to the valve and occurs as a consequence of other pathologies, e.g. pulmonary hypertension, that induce right-ventricular remodeling[14]. In turn, right-ventricular remodeling leads to annular dilation and papillary muscle displacement that prevent leaflet coaptation. As such, the valve leaflets are thought to remain unaffected and the regurgitation is considered “functional” in nature[15]. Interestingly, we found evidence of leaflet remodeling in sheep with functional tricuspid regurgitation[16, 17]. As in the mitral valve, we found leaflet thickening and stiffening and alterations in the material nonlinearity of the leaflets’ constitutive response. Additionally, we and others found that tricuspid leaflets grow in area[18]. We hypothesize that these remodeling-induced changes fundamentally alter the leaflets’ ability to bend and stretch, thereby affecting valve mechanics. This, in turn, may prevent the enlarged leaflets from sealing the dilated tricuspid orifice; thus impeding valve function. Furthermore, we posit that suppressing these changes in tricuspid leaflets will, in fact, restore valve mechanics and promote leaflet coaptation. The objective of our current study is to test this hypothesis.

2. Methods

2.1. High-fidelity finite element model

To test our hypothesis we use Texas TriValve 1.0, a reverse-engineered and openly accessible computer model of a healthy human tricuspid valve, see Figure 1. Briefly, we first loaded the right side of a healthy, beating human donor heart in an organ preservation system (OCS, TransMedics, Andover, MA)[20]. In the beating heart, we then recorded transvalvular hemodynamics via miniaturized pressure sensors (PA4.5- X6; Konigsberg Instruments Inc., Pasadena, CA), recorded tricuspid annulus dynamics via sonomicrometry crystals (Sonometrics Inc., London, Ontario, Canada)[21], and imaged tricuspid valve coaptation via epicardial echocardiography (Vivid S6, GE Healthcare, Chicago, IL). Upon arresting the heart, we excised the tricuspid valve complex, digitized the leaflet geometry, and characterized the tissue material properties via in vitro experiments. To rebuild the valve in silico, we non-rigidly warped the digitized leaflet geometry onto a sonomicrometry-based 3D reconstruction of the tricuspid annulus. Next, we created chordal insertion sites based on in vitro images of the leaflets. We then assigned subject-specific hyperelastic material properties to the leaflets and chordae tendineae; informed by planar biaxial and uniaxial tensile tests, respectively. In our model, we then imposed annular displacement and transvalvular pressure gradient boundary conditions based on sonomicrometry crystal and pressure sensor measurements, respectively. Finally, we simulated leaflet coaptation in Abaqus/Explicit (6.20-1, Dassault Systemes, Vélizy-Villacoublay, France) and validated our model deformations against echocardiography images of the tricuspid valve in the same beating heart. Detailed descriptions of model creation and validation are provided in our previous work[19].

Figure 1:

Texas TriValve 1.0 modeling pipeline. We reverse-engineered a subject-specific computational model of a healthy, human tricuspid valve. To this end, we recorded annular dynamics, transvalvular pressures, and echo images of the tricuspid valve in an ex-vivo beating heart. We then arrested the heart, digitized the valve morphology, and characterized tissue material properties in-vitro. Next, we combined the subject-specific geometry, material properties, and boundary conditions in a finite element model of the valve. Finally, we simulated leaflet coaptation through structural simulations in Abaqus/Explicit and validated our results against echo images of the tricuspid valve in the same beating heart. Reproduced with permission from Mathur et al[19]

In this validated model, we modify annular geometry and boundary conditions to reflect changes seen in patients with pulmonary hypertension, which we refer to as our baseline model[18]. Specifically, we non-uniformly dilate the tricuspid annulus to increase annular area by 62%[22], see Figure 2. Moreover, we tether the papillary muscles and apply a pathological pressure gradient to the ventricular surface of the valve leaflets[23, 24].

Figure 2:

Non uniform annular dilation. We dilate the healthy tricuspid annulus along the Anterior-Posterior (A-P) and Septal-Lateral (S-L) axes as observed in patients with pulmonary hypertension[22]. Reproduced with permission from Mathur et al[19]

2.2. Remodeling-induced changes to the baseline valve model

To investigate the impact of remodeling-induced leaflet changes on valve mechanics, we alter three quantities as observed in our studies of remodeling tricuspid valves: i) leaflet stiffness, ii) leaflet thickness, and iii) the stretch at which leaflets show strain-stiffening, which we subsequently call transition-λ. Previously, we have demonstrated that the tricuspid valve remodels in sheep with biventricular heart failure that develop tricuspid valve leakage. In those animals, we found that the anterior leaflet thickness increased by 40%, leaflet stiffness increased by 30%, and transition-λ decreased by 3%. Interested readers may find further details of our findings in our prior publication[16]. Please note that we did not observe any remodeling in the tricuspid chordae tendineae of sheep with functional tricuspid regurgitation. Thus, we have excluded such effects in our computational model.

In this study, we first examine the sensitivity of our model to independent changes in leaflet thickness, stiffness, and transition-λ. Specifically, we increase leaflet thickness by 20-100%, increase leaflet stiffness by 20-100%, and finally decrease transition-λ by 1-5%, as shown in Figure 3. To quantify transition-λ, we first determine the slope of the leaflets’ tension-stretch curve at low and high stretch values, i.e. the toe- and calf-stiffness, respectively. We then identify the closest data point on the tension-stretch curve to the intersection of these slopes as the transition-λ[25]. See Tables A1 and A2 for altered stiffness and transition-λ values, respectively. We then examine the combined effects of remodeling-induced changes as observed in our previous sheep study. That is, we increase leaflet thickness by 40%, increase leaflet stiffness by 30%, and decrease transition-λ by 3%. To obtain the aforementioned values of leaflet thickness, leaflet stiffness, and transition-λ we modify the material parameters used in our computational model, as detailed in the online supplement to this article.

Figure 3:

Remodeling-induced macro- and micro-structural changes to the tricuspid valve. (A) Here, we uniformly increase leaflet thickness by 20-100% in our simulations. (B) Furthermore, to alter the highly non-linear constitutive properties of the leaflets we first determine the slope of the tension-stretch curve at low and high stretch values, i.e. the toe- and calf-stiffness, respectively. Next, we identify the closest data point on the tension-stretch curve to the intersection of these slopes as the transition-λ. (C) We then simultaneously increase leaflet toe- and calf-stiffnesses by 20-100% in our models. (D) Furthermore, we decrease leaflet transition-λ by 1-5%

Furthermore, to quantify the impact of the remodeling-induced changes on valve function, we compare three metrics between all cases: i) the average leaflet stress in leaflet centers, ii) the leaflet systolic angle at end-systole, and iii) the coaptation area. To average leaflet stresses, we first create a planar projection of the tricuspid leaflets, see Figure 4. On this planar projection, we then create square regions of size 7 mm X 7 mm at the belly region of each tricuspid leaflet. Finally, for each tricuspid leaflet, we identify all element centroids within each square and compute the average maximum principal Cauchy stress[26].

Figure 4:

Square regions where maximum principal Cauchy stresses are averaged for each tricuspid leaflet

3. Results

3.1. Isolated changes to leaflet thickness, stiffness, and transition-λ impact tricuspid valve function

To first study the “sensitivity” of the tricuspid valve to remodeling-induced leaflet changes, we separately study the leaflets’ response to increased leaflet thickness and leaflet stiffness, and decreased transition-λ. Specifically, we first increased leaflet thickness by 20-100%, before increasing leaflet stiffness by 20-100%, and finally decreasing transition-λ by 1-5%.

3.1.1. Impact on leaflet stresses

First, we investigated the impact of isolated remodeling-induced leaflet changes on leaflet stresses. To this end, we averaged leaflet stresses in a square region located in the center of each leaflet, see Figure 5. We found that increasing leaflet thickness reduced average end-systolic stresses in each leaflet, see Figure 5A. Specifically, increasing the leaflet thickness by 20-100% led to a reduction in average stress of 23-59% in the anterior, 18-59% in the posterior, and 26-61% in the septal leaflet. In contrast, we found that increasing leaflet stiffness increased average end-systolic stresses in each leaflet, see Figure 5B. Here, we found that increasing leaflet stiffness by 20-100% led to an increase in average stress of 9-34% in the anterior, 5-7% in the posterior leaflet, and 20-154% in the septal leaflets. Finally, we found that decreasing transition-λ altered average end-systolic stresses in each leaflet, see Figure 5C. In detail, we found that decreasing transition-λ by 1-5% led to a decrease in average stress of 1-4% in the anterior leaflet and 3-5% in the posterior leaflet. In contrast, average end-systolic stresses increased by 4-47% in the septal leaflet. Leaflet stresses for the above cases are provided in Table 1. In summary, average end-systolic leaflet stresses are highly sensitive to changes in leaflet thickness, leaflet stiffness, and transition-λ.

Figure 5:

Isolated remodeling-induced changes alter leaflet stresses in the regurgitant tricuspid valve. (A) Contours of stress overlaid on the end-systolic configuration of the tricuspid valve show that an isolated increase in leaflet thickness reduces leaflet stresses. In contrast, an isolated increase in stiffness or decrease in transition-λ increases leaflet stresses, as shown in (B) and (C), respectively. We compute average stresses for each leaflet in the regions indicated by a dashed square in (A)

Table 1:

Leaflet stresses in the centers of tricuspid valve leaflets for the baseline valve with isolated changes to leaflet thickness, stiffness, and transition-λ. All values are presented as mean ± 1 S.D

Condition	Change	Anterior Leaflet		Posterior Leaflet		Septal Leaflet
		Stress (kPa)	Change	Stress (kPa)	Change (%)	Stress (kPa)	Change (%)
Control	–	90.95 ± 25.25	–	65.88 ± 14.90	–	87.53 ± 32.74	–
Thickness Increase	20%	70.19 ± 19.74	−22.82	53.92 ± 12.60	−18.17	64.36 ± 24.65	−26.47
	40%	60.15 ± 19.50	−33.87	44.14 ± 11.50	−33.01	57.71 ± 18.60	−34.07
	60%	49.49 ± 16.81	−45.59	36.26 ± 11.52	−44.97	41.15 ± 17.29	−52.99
	80%	44.95 ± 16.17	−50.57	32.48 ± 11.01	−50.70	41.67 ± 14.98	−52.40
	100%	37.64 ± 14.76	−58.61	27.05 ± 9.90	−58.95	33.90 ± 10.95	−61.28
Stiffness Increase	20%	99.46 ± 30.92	9.37	69.29 ± 17.57	5.17	104.87 ± 43.09	19.81
	40%	107.35 ± 37.09	18.03	69.68 ± 20.45	5.76	132.46 ± 56.47	51.32
	60%	112.93 ± 43.35	24.17	69.19 ± 24.14	5.01	101.57 ± 45.74	16.04
	80%	105.47 ± 43.97	15.97	71.55 ± 26.68	8.59	187.31 ± 92.57	113.98
	100%	121.90 ± 54.81	34.04	70.39 ± 29.12	6.85	222.44 ± 115.69	154.12
Transition-λ Decrease	1%	90.45 ± 24.53	−0.54	63.99 ± 14.03	−2.87	90.79 ± 34.22	3.72
	2%	89.11 ± 23.36	−2.02	63.58 ± 13.44	−3.50	97.81 ± 36.05	11.74
	3%	87.81 ± 22.83	−3.45	63.12 ± 12.84	−4.20	106.42 ± 38.37	21.57
	4%	87.86 ± 22.45	−3.39	62.56 ± 12.33	−5.05	115.06 ± 41.34	31.44
	5%	87.03 ± 22.29	−4.31	62.30 ± 11.94	−5.43	128.83 ± 46.79	47.18

3.1.2. Impact on leaflet systolic angle

Next, we investigated the impact of isolated remodeling-induced leaflet changes on the systolic angle of the anterior leaflet, see Figure 6A. Additionally, we investigated the impact of isolated remodeling-induced leaflet changes on the leaflet contact area, i.e., coaptation area, see Figure 6B.

Figure 6:

Isolated remodeling-induced changes alter leaflet motion in the regurgitant tricuspid valve. (A) Here, we define the systolic angle as the angle between a point on the anterior leaflet and the annular plane during systole. (B) Furthermore, we quantify leaflet coaptation by considering the area of all finite-element faces in contact at end-systole, depicted here in red and projected on a 2D representation of the leaflet surface. (C) Increasing leaflet thickness non-uniformly decreases systolic angles. (D) In contrast, increasing leaflet stiffness substantially increases systolic angles. (E) Similarly, decreasing transition-λ increases systolic angles. (F) Here, we also see that an increase in leaflet thickness increases leaflet coaptation area. (G) In contrast, an increase in leaflet stiffness reduces leaflet coaptation area. (H) Finally, a decrease in transition-λ also decreases coaptation area

Firstly, we found that increasing leaflet thickness reduced the anterior leaflet systolic angle, see Figure 6C. Specifically, increasing the leaflet thickness by 20-100% first led to a 19-47% decrease in systolic angle at end-systole. In contrast, we found that increasing leaflet stiffness increased the anterior leaflet systolic angle, see Figure 6D. In particular, we found that increasing stiffness by 20-100% led to a 31-133% increase in systolic angle at end-systole. Furthermore, we found that decreasing transition-λ increased leaflet systolic angles as well, as seen in Figure 6E. That is, a 1-5% decrease in transition-λ led to a 4-22% increase in systolic angle at end-systole. Anterior leaflet systolic angles for the above cases are provided in Table 2. In summary, anterior leaflet systolic angles are highly sensitive to changes in leaflet stiffness and transition-λ but not leaflet thickness.

Table 2:

Anterior leaflet systolic angle and valve contact area for the baseline tricuspid valve as well as those vales with isolated changes to leaflet thickness, stiffness, and transition-λ

Condition	Change	Systolic angle (°)	Change (%)	Contact Area (mm2)	Change (%)
Control	–	10.77	–	251.64	–
Thickness Increase	20%	8.73	−18.94	257.77	2.44
	40%	9.75	−9.52	271.30	7.82
	60%	8.13	−24.50	301.03	19.63
	80%	8.15	−24.37	287.71	14.34
	100%	5.72	−46.91	324.12	28.81
Stiffness Increase	20%	14.13	31.11	132.51	−47.34
	40%	17.05	58.27	76.44	−69.62
	60%	18.63	72.91	70.28	−72.07
	80%	25.08	132.76	36.85	−85.36
	100%	21.44	99.03	22.53	−91.05
Transition-λ Decrease	1%	11.20	3.94	229.96	−8.61
	2%	11.47	6.46	211.16	−16.08
	3%	11.94	10.86	176.33	−29.93
	4%	12.52	16.23	161.70	−35.74
	5%	13.16	22.20	134.84	−46.41

3.1.3. Impact on coaptation area

Secondly, we found that increasing leaflet thickness increased end-systolic coaptation area, see Figure 6F. Specifically, increasing the leaflet thickness by 20-100% led to a 2-29% increase in coaptation area. In contrast, we found that increasing leaflet stiffness reduced end-systolic coaptation area, see Figure 6G. In detail, increasing leaflet stiffness by 20-100% led to a 47-91% decrease in coaptation area. Finally, we found that decreasing transition-λ also reduced end-systolic coaptation area, see Figure 6H. That is, a 1-5% decrease in transition-λ led to a 9-46% decrease in coaptation area. Leaflet contact areas for the above cases are provided in Table 2. In summary, leaflet contact areas are highly sensitive to changes in leaflet stiffness, leaflet thickness, and transition-λ.

3.2. Combined changes to leaflet thickness, stiffness, and transition-λ impact tricuspid valve function

In addition to studying the isolated effect of increasing leaflet thickness, increasing leaflet stiffness, and decreasing transition-λ, we also studied their combined effect. To this end, we chose values to match those observed in our previous sheep study[16]. Specifically, we combined a 40% increase in thickness with a 30% increase in stiffness and a 3% decrease in transition-λ.

3.2.1. Impact on leaflet stresses

Here, we introduced eight study cases to contrast the combined effect of remodeling-induced leaflet changes on valve function with their isolated impacts. A control case (I), cases investigating isolated changes only (II a/b/c), cases combining two of the leaflet changes (III a/b/c), and a final case in which we combine all changes (IV). In case IV we found that simultaneously increasing leaflet thickness and stiffness, and decreasing transition-λ by previously measured magnitudes led to a decrease in average stresses of 26% in the anterior and 28% in the posterior leaflets, see Figure 7A. Additionally, this led to an increase in average stresses of 17% in the septal leaflet. As demonstrated by case IIIa, this response was driven by simultaneous changes in thickness and stiffness. Together, an increase in thickness and stiffness led to a decrease in average stresses of 24% in the anterior , 28% in the posterior leaflets, and 9% in the septal leaflet. Leaflet stresses for the above cases are provided in Table 3. In summary, the changes in average end-systolic leaflet stresses are driven by a simultaneous increase in leaflet thickness and stiffness.

Figure 7:

Remodeling-induced changes alter leaflet stresses and motion in the regurgitant tricuspid valve. (A) Leaflet stresses in the tricuspid valve are sensitive to the combination of remodeling-induced changes in functional tricuspid regurgitation, as shown in (B). Similarly, changes to anterior leaflet systolic angles and valve coaptation area depend on the combination of remodeling-induced changes applied, as shown in (C) and (D), respectively

Table 3:

Maximum principal Cauchy stresses in the central region of the baseline tricuspid valve as well as those valves with isolated and combined remodeling-induced changes. Specifically, a 40% increase in leaflet thickness, a 30% increase in stiffness, and a 3% decrease in transition-λ. All values are presented as mean ± 1 S.D

Case	Condition	Anterior Leaflet		Posterior Leaflet		Septal Leaflet
		Stress (kPa)	Change (%)	Stress (kPa)	Change (%)	Stress (kPa)	Change (%)
I	Control	90.95 ± 25.25	–	65.88 ± 14.90	–	87.53 ± 32.74	–
IIa	Δ Thickness	60.15 ± 19.50	−33.87	44.14 ± 11.50	−33.01	57.71 ± 18.60	−34.07
IIb	Δ Stiffness	103.27 ± 34.83	13.55	72.65 ± 19.45	10.26	120.52 ± 49.50	37.68
IIc	Δ Transition-λ	87.81 ± 22.83	−3.45	63.12 ± 12.84	−4.20	106.42 ± 38.37	21.57
IIIa	Δ Thickness & Δ Stiffness	69.48 ± 28.64	−23.61	47.20 ± 15.88	−28.36	79.41 ± 27.80	−9.28
IIIb	Δ Stiffness & Δ Transition-λ	96.80 ± 28.45	6.44	65.73 ± 18.14	−0.23	129.25 ± 60.14	47.66
IIIc	Δ Transition-λ & Δ Thickness	57.96 ± 18.03	−36.27	43.02 ± 10.29	−34.71	71.72 ± 20.83	−18.07
IV	FTR Maladaptation	67.41 ± 25.30	−25.88	47.74 ± 14.03	−27.55	102.71 ± 35.69	17.34

3.2.2. Impact on systolic angle

Next, in case IV we found that remodeling-induced leaflet changes increase anterior leaflet systolic angle, see Figure 7C. Specifically, a simultaneous increase in thickness and stiffness, and decrease in transition-λ led to a 43% increase in systolic angle. This change was primarily driven by an increase in stiffness, as exemplified by case IIb. Specifically, an increase in stiffness led to a 52% increase in systolic angle. Systolic angles for the above cases are provided in Table 4. In summary, the increase in systolic angles is primarily driven by an increase in leaflet stiffness.

Table 4:

Anterior leaflet systolic angle and valve contact area for the baseline tricuspid valve as well as those vales with isolated and combined remodeling-induced changes. Specifically, a 40% increase in leaflet thickness, a 30% increase in stiffness, and a 3% decrease in transition-λ

Case	Condition	Systolic angle (°)	Change (%)	Contact Area (mm2)	Change (%)
I	Control	10.77	–	251.64	–
IIa	Δ Thickness	9.75	−9.52	271.30	7.82
IIb	Δ Stiffness	16.32	51.51	103.70	−58.79
IIc	Δ Transition-λ	11.94	10.86	176.33	−29.93
IIIa	Δ Thickness & Δ Stiffness	14.66	36.03	142.90	−43.21
IIIb	Δ Stiffness & Δ Transition-λ	15.41	43.07	82.16	−67.35
IIIc	Δ Transition-λ & Δ Thickness	10.88	1.00	214.83	−14.63
IV	FTR Maladaptation	15.44	43.28	85.51	−66.02

3.2.3. Impact on coaptation area

Additionally, in case IV we found that remodeling-induced leaflet changes decrease leaflet coaptation area, see Figure 7D. Specifically, a simultaneous increase in thickness and stiffness, and decrease in transition-λ led to a 66% decrease in coaptation area. This change was primarily driven by an increase in stiffness as seen in case IIb. To that end, an increase in stiffness by previously measured magnitudes led to a 59% decrease in coaptation area. Coaptation areas for the above cases are provided in Table 4. In summary, the decrease in leaflet contact areas is primarily driven by an increase in leaflet stiffness.

4. Discussion

Tricuspid valve leaflets have long been considered inert or “passive” structures. However, we recently demonstrated in sheep with functional tricuspid regurgitation that tricuspid valve leaflets may thicken, stiffen, and alter their material non-linearity. We hypothesize that these changes impede tricuspid valve coaptation and that suppressing them may restore valve function. The goal of our current work was to test this hypothesis, see Figure 8.

Figure 8:

We use a high-fidelity computational model of the human tricuspid valve to determine the impact of remodeling-induced changes to leaflet thickness, stiffness, and material nonlinearity. We found that these changes, both in isolation and when combined, impact leaflet stresses and motion. Thus, suppressing these changes may restore tricuspid valve function

In our first study, we examined the isolated effects of a remodeling-induced increase in thickness and stiffness as well as a decrease in transition-λ. We found that an isolated increase in leaflet thickness reduced stresses in the tricuspid valve. These computational findings agree with those of Kong and colleagues, who found a 40% decrease in leaflet stresses for a 62% increase in leaflet thickness[27]. Additionally, we found that an isolated increase in leaflet stiffness increased stresses in the tricuspid valve. Kong et al. report a similar increase in stresses for stiffer tricuspid leaflets. Furthermore, we are the first to report that an isolated decrease in transition-λ increased leaflet stresses. While the sensitivity of leaflet stresses to transition-λ has never been directly investigated by others, in a computational model, Wu et al. observed that tricuspid valve stresses are highly sensitive to changes in leaflet constitutive properties [28]. Next, we found that an isolated increase in stiffness and a decrease in transition-λ substantially increased anterior leaflet systolic angles. Interestingly, similar trends for leaflet stiffness were observed by others in a computer model of the regurgitant mitral valve[29]. Finally, we found that an isolated increase in stiffness and decrease in transition-λ led to a decrease in coaptation area; thereby impacting valve function. While there are no similar studies for the tricuspid valve, others have previously observed such findings in computational models of the mitral valve[11, 29, 30].

In our second study, we examined the combined effect of a remodeling-induced increase in thickness and stiffness as well as a decrease in transition-λ. We found reduced stresses in valves with thicker leaflets (case IIa) – even when accompanied by an increase in stiffness (case IIIa), decrease in transition-λ (IIIc), or both (case IV). This is likely due to the leaflets’ folding resistance dominating their constitutive stiffness. That is, the leaflets increasingly resist folding and limit coaptation. As a result, the transvalvular pressure load is supported by a larger area. This, in turn, reduces inflation of the leaflets which decreases in-plane forces and reduces stresses. Next, we found that remodeling-induced changes, together (case IV), increase systolic angle. Notably, we previously observed a similar increase in experimentally measured anterior leaflet systolic angles in sheep with regurgitant valves[31]. Finally, we found that remodeling-induced changes, together (case IV), significantly decrease coaptation area. This implies that tricuspid valve function is negatively impacted. Furthermore, the coaptation area lost in a valve with a simultaneous increase in thickness and decrease in transition-λ (case IIIc) is less than a quarter of the coaptation area lost in the fully remodeled valve.

The significance of our findings is two-fold. Firstly, our findings may inspire novel, pharmacological strategies to treat functional tricuspid regurgitation[32]. For example, pharmaceutical treatments may be used to inhibit the thickening and stiffening of valve leaflets while permitting beneficial area growth. This strategy was successfully used by Bartko and colleagues to contain the fibrotic response of leaflets in the mitral valve in sheep. Specifically, they used Losartan, an angiotensin-II receptor blocker, to arrest an increase in leaflet thickness[33]. A similar strategy was used by Marsit et al. to reduce fibrotic thickening in ovine mitral leaflets using Cyproheptadine, a serotonin receptor 2B antagonist[34]. Secondly, our findings may be used to help understand the sub-optimal outcomes of surgical tricuspid valve repair. Specifically, we may better understand why some centers report failure rates as high as 25% within one year of patients being treated with the surgical gold standard – ring annuloplasty[35]. In the porcine mitral valve, Sielicka and colleagues experimentally demonstrated that ring annuloplasty led to leaflet thickening and stiffening which, subsequently, resulted in valvular dysfunction[36]. Suppressing similar changes due to tricuspid annuloplasty may, thus, improve valve function and repair outcomes.

Naturally, our study is subject to limitations. Firstly, we emulate leaflet remodeling as observed in a chronic model of functional tricuspid regurgitation in sheep. As such, inter-species differences as well as the gradual onset of disease may alter the extent of leaflet remodeling and its effects in patients. Secondly, we apply remodeling-induced changes observed in the centers of the ovine anterior leaflet to the posterior and septal leaflets. Furthermore, we homogenize these changes across the entire leaflet area. As such, we do not consider inter- and intra-leaflet variations in leaflet remodeling[37]. We hope future studies will elucidate these properties so that we may use them in subsequent models. Moreover, we represent the constitutive behavior of the tricuspid valve leaflets using a homogenized, isotropic strain energy function in our current finite element model. As such, we do not consider remodeling-induced changes in leaflet microstructure, as observed in our previous animal study[16]. To overcome this limitation, we hope to use a microstructurally-informed, anisotropic strain energy function to model the hyperelastic response of tricuspid valve leaflets in future releases of Texas TriValve[38]. In addition to the above limitations, we may not generalize our results to all tricuspid valves due to the large variation in their leaflet morphology[39]. Last but not least, this is a computational study and caution is warranted.

5. Conclusions

In a virtual case study, we found that tricuspid valve function is sensitive to remodeling-induced leaflet changes. Specifically, leaflet stresses, anterior leaflet systolic angles, and valve contact area were all impacted by changes in leaflet thickness, leaflet stiffness, and constitutive response. Thus, these findings suggest that suppressing leaflet thickening and stiffening may, in fact, restore tricuspid valve function. In turn, our results may inspire novel surgical and pharmacological treatments for tricuspid regurgitation. Future experimental studies will be needed to support these findings.

Highlights.

• We introduce the effects of leaflet remodeling in a computer model of the human tricuspid valve

• We demonstrate that remodeling-induced changes alter leaflet stress, leaflet motion, and valve coaptation area

• We show that suppressing leaflet thickening and stiffening may improve valve function

Appendix A. Stiffness and Transition-λ Data

Table A1:

Toe- and calf-stiffness values identified for tricuspid valve leaflets under normal and stiffened material responses

Stiffness Increase	Anterior Stiffness (N/m)		Posterior Stiffness (N/m)		Septal Stiffness (N/m)
	Toe	Calf	Toe	Calf	Toe	Calf
0%	6.89	3046.50	9.02	3400.45	3.98	1881.91
20%	8.27	3655.72	10.82	4080.40	4.77	2258.17
40%	9.64	4264.97	12.63	4760.50	5.57	2634.53
60%	11.02	4874.28	14.43	5440.60	6.36	3010.95
80%	12.40	5483.59	16.24	6120.72	7.16	3387.39
100%	13.78	6093.00	18.04	6800.88	7.96	3763.89

Table A2:

Transition-λ values identified for triscuspid valve leaflets under normal and stiffened material responses

Transition-λ Decrease	Anterior Transition-λ (−)	Posterior Transition-λ (−)	Septal Transition-λ (−)
0%	1.1524	1.1031	1.2613
1%	1.1409	1.0921	1.2486
2%	1.1294	1.0811	1.2362
3%	1.1179	1.0701	1.2234
4%	1.1064	1.0591	1.2110
5%	1.0948	1.0480	1.1982