# Hemodynamic Effects of Alpha-Tropomyosin Mutations Associated with Inherited Cardiomyopathies: Multiscale Simulation

^{1}

^{2}

^{3}

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## Abstract

**:**

^{2+}regulation of cardiac muscle caused by these mutations were introduced into the myocardial model of the left ventricle (LV) while the LV shape remained the same as in the model of the normal heart, the cardiac output and arterial blood pressure reduced. Simulations of LV hypertrophy in the case of the Ile284Val mutation and LV dilatation in the case of the Asp230Asn mutation demonstrated that the LV remodeling partially recovered the stroke volume and arterial blood pressure, confirming that both hypertrophy and dilatation help to preserve the LV function. The possible effects of changes in passive myocardial stiffness in the model according to data reported for HCM and DCM hearts were also simulated. The results of the simulations showed that the end-systolic pressure–volume relation that is often used to characterize heart contractility strongly depends on heart geometry and cannot be used as a characteristic of myocardial contractility.

## 1. Introduction

^{2+}regulation of the actin–myosin interaction in sarcomeres of striated muscles. The Tpm molecules bind to each other in a head-to-tail manner and form a continuous strand located in a helical groove on the surface of an actin filament. The strand controls the availability of actin sites for the binding of motor domains of myosin molecules—myosin heads. Another regulatory protein, troponin (Tn), binds Tpm in a 1:1 stoichiometry and forms, together with the fibrillary actin and Tpm, a regulated thin filament. Tn controls Tpm movement with respect to the axis of an actin filament in a Ca

^{2+}-dependent manner [8,9]. The regulation of muscle contraction is highly cooperative: relatively small changes in intracellular Ca

^{2+}concentration cause large changes in force and actin–myosin ATPase rate. Modeling [10,11,12] suggests that the local movements of a stiff Tpm strand (caused by Ca

^{2+}binding to Tn or myosin binding to actin) are transmitted to neighbor parts of the strand, providing high cooperativity.

^{2+}regulation of the myosin–actin interaction measured in the in vitro motility assay or experiments with single cardiomyocytes [13,14,15]. These molecular and cellular level changes are believed to underlie the impairment of the heart function. Two questions remain unanswered: (1) how the changes in the actin–myosin interaction at the molecular and cellular levels caused by the Tpm mutations affect the pumping function of the heart and the LV particularly; and (2) how does the remodeling of the LV wall associated with the cardiomyopathies change the heart function? To address these questions, we performed a computer simulation of the heart mechanics using a recently developed multiscale LV model [16] incorporated into a simple lumped parameter model of circulation [17]. A model of myocardial mechanics used in the multiscale model was described previously [18]. It describes all major mechanical features of cardiac muscle including the force–velocity and stiffness–velocity relations, tension responses to step-like and ramp changes in muscle length, high cooperativity of Ca

^{2+}-force relation, and its length-dependence, etc. Here, we used available experimental data to modify model parameters to account for the changes in the Ca

^{2+}regulation of myocardial mechanics at the molecular and cellular levels caused by the cardiomyopathy-associated Tpm mutations. Then we simulated the effect of these changes on the movement of the LV wall during heartbeats. We also simulated the effects of remodeling of the left ventricle—hypertrophy and stenosis of the outflow tract for the Ile284Val mutation and the dilatation for the Asp230Asn one—on the calculated cardiac output. The effects of changes in the passive myocardial stiffness associated with HCM and DCM [5,6,7] were also estimated.

## 2. Materials and Methods

#### 2.1. Cardiac Muscle Mechanics and Regulation

_{1}and A

_{2}, which represent the fractions of available myosin binding sites on actin in the overlap zone and outside this zone, respectively. The kinetic equations for the variables accounted for the Ca

^{2+}regulation of the thin filaments. The balance equation for Ca

^{2+}concentration in cytoplasm included the terms of Ca

^{2+}influx (set as a given function of time), Ca

^{2+}binding to troponin and to other cytoplasmic proteins, and Ca

^{2+}uptake from cytoplasm. Variation of the average micro-distortion depended on the sliding velocity of the myosin and actin filaments, thus being defined by the macroscopic strain rate tensor. The full set of equations describing the passive and active stress components in terms of continuum mechanics and the kinetic equations for the interaction of contractile and regulatory proteins and Ca

^{2+}dynamics are given in [18] and Appendix A. For the values of the model parameters, see Supplementary Materials Table S1.

#### 2.2. Geometry of the Left Ventricle

#### 2.3. Model of Circulation

#### 2.4. Numerical Simulation and the Model Validation

#### 2.5. Modeling Cell-Level Effects of Two Cardiomyopathy-Associated Mutations

^{2+}release and uptake in myocardial cells was also assumed to be the same as in normal myocardium. Only the parameters of the equations that describe Ca

^{2+}binding to Tn, strain-dependence of the binding, and the number of myosin filaments per cross-section area in the myocardial model were changed to simulate the effects of the mutations.

^{2+}–force relationship in single permeabilized myocardial cells from the LV of a patient with HCM-associated Ile284Val Tpm mutation and healthy donors. They had found that maximal tension at saturating Ca

^{2+}concentration decreased by approximately 55% upon the Ile284Val Tpm mutation, while the Ca

^{2+}concentration required for half-maximal activation decreased by a third. Besides, the length-dependent activation [19] in the cells from the HCM patient was significantly reduced as compared to that in healthy donors [15]. The maximal sliding velocity of reconstructed thin filaments containing Ile284Val Tpm in vitro was the same as of those with wild-type (WT) Tpm [20]. To simulate these experimentally observed changes, we decreased the density of myosin filaments per cross-section area (leading to a decrease in the maximal active tension) and the parameter of length-dependency of the activation while increasing the parameter of the Tn affinity for Ca

^{2+}(thus increasing Ca

^{2+}sensitivity). The model parameters and changes introduced to simulate the effects of the Tpm mutations listed above are described in Supplementary Materials Table S4. The resulting dependencies of isometric tension and the activation level in the overlap zone A

_{1}on dimensionless Ca

^{2+}concentration are shown in Figure 1 (red).

^{2+}regulation and mechanical properties of human cardiac muscle caused by the Asp230Asn Tpm mutation. Several individuals from two families carrying the mutation have shown a mild or severe heart failure with the ejection fraction reduced down to 20% and significantly increased end-diastolic diameter of the left ventricle [13]. In vitro studies of the effects of the Asp230Asn Tpm mutation on Ca

^{2+}regulation of myosin ATPase in the presence of regulated thin filaments and Ca

^{2+}binding to thin filaments have shown a reduced Ca

^{2+}sensitivity and decreased Ca

^{2+}affinity for the thin filaments [13]. LV mechanics in vivo and in situ at the cellular and molecular levels were studied in transgenic mice carrying the Asp230Asn mutation [14]. A significantly reduced Ca

^{2+}sensitivity and the cooperativity of Ca

^{2+}regulation were found in vitro in the presence of the Asp230Asn Tpm mutation compared to WT Tpm [14,21]. To simulate the experimentally observed changes, we varied two model parameters in the regulation block of our model: the cooperativity parameter and the Tpm affinity for Ca

^{2+}. The details are described in Supplementary Materials Table S4. The effects of these changes on the calculated dependencies of isometric tension and the activation level on Ca

^{2+}in the overlap zone A

_{1}are shown in Figure 1 (green).

#### 2.6. Modeling LV remodeling for HCM and DCM-Associated Tpm Mutations

#### 2.7. Modeling Changes in Passive Myocardial Stiffness Accompanying HCM and DCM

## 3. Results

#### 3.1. Simulation of Hemodynamic Changes Caused by the Asp230Asn and Ile284Val Tpm Mutations Without the LV Remodeling

^{2+}regulation were changed as described in Section 2.5 (see also Supplementary Materials, Table S4) and shown in Figure 1. All other parameters of the model were the same as those for the normal heart model. The simulations performed with the model parameters corresponding to the normal right ventricle showed blood redistribution from the pulmonary circulation to the systemic one and a significant LV overfill. To overcome this problem in the absence of available data regarding changes in the right ventricle geometry and function caused by the mutations, we decreased the parameter of maximal isovolumetric pressure for the right ventricle to obtain the same end-diastolic LV volume as that in the normal heart model. The parameter was decreased from 85 to 70 mm Hg for the simulation of the Ile284Val Tpm mutation and to 60 mm Hg for the case of the Asp230Asn one. The duration of the right ventricle systole remained unchanged for simplicity. The hemodynamic variables obtained during the heartbeat simulations are shown in Figure 2.

#### 3.2. Changes in Ca^{2+} Transients Caused by the Tpm Mutations

_{1}and n increased compared to their normal values, while the Ca

^{2+}peak decreased (Figure 3, HCM). These changes are caused by the shift of the force-Ca

^{2+}toward lower Ca

^{2+}concentration (Figure 1) and by a reduction of the free intracellular Ca

^{2+}concentration due to its binding to Tn. In contrast, in the DCM model, the peaks of A

_{1}and n were reduced, while the peak of the free Ca

^{2+}concentration was enhanced (Figure 3, DCM).

#### 3.3. Simulation of Hemodynamic Changes Caused by the Tpm Mutations and the LV Remodeling

^{2+}sensitivity (Figure 5b). The stroke volume and the aortic blood pressure reduced, although remaining close to those in the normal heart model (70 mL, 107/71 mm Hg versus 82 mL and 121/81 mm Hg, respectively) while the ejection fraction was reduced significantly to 39% (compared to 65%).

_{3RV}, Table S3). Such conditions resulted in an increased LV preload of 8.6 mm Hg and the end-diastolic volume of 105 mL. Simulation of the effect of LV hypertrophy for the HCM-associated Ile284Val Tpm mutation also resulted in a more complete recovery of the hemodynamic parameters (Figure 5c). The aortic pressure of 120/80 mm Hg and the stroke volume of 79 mL were close to those calculated for the normal heart model (121/81 mm Hg, 82 mL), while the ejection fraction increased to 75%. When the stenosis of the LV output tract (maximal orifice area 1 cm

^{2}at aortic area of 3 cm

^{2}) was taken into account in the model of HCM caused by the Tpm mutation, the peak LV systolic pressure increased up to 184 mm Hg, while the aortic pressure (112/79 mm Hg) and the stroke volume (78 mL) reduced slightly compared to the HCM model without the stenosis (Figure 5d). The peak pressure gradient (72 mm Hg) was similar to that in a patient with this mutation [15].

#### 3.4. Simulation of the Effects of the Tpm Mutations and the LV Remodeling on the Pressure-Volume Loops

^{2+}sensitivity of active tension, while the HCM-associated mutation decreases the maximal active tension and length-dependent activation (Figure 1).

## 4. Discussion

#### 4.1. Cell-Level Changes in Ca^{2+} Regulation and Cardiac Function

^{2+}transient increased as was observed in transgenic mice [14], while the activation level (A

_{1}) and the fractions of actin-bound myosin heads (n) decreased compared to the model of normal heart (Figure 3, DCM). When changes in Ca

^{2+}regulation found in myocardial cells with the HCM-associated Ile284Val Tpm mutation [15] were simulated, the amplitude of Ca

^{2+}transient decreased, while the fraction of actin bound myosin heads increased (Figure 3, HCM). These effects result from a combination of several factors: change in Ca

^{2+}sensitivity of the thin filaments caused by the mutations (Figure 1) and Ca

^{2+}binding to Tn affecting the free Ca

^{2+}concentration. Despite the difference, changes in the cell-level model parameters corresponding to those caused by both the HCM and DCM Tpm mutations decreased arterial blood pressure, stroke volume, and ejection fraction compared to the model of normal LV (Figure 2).

#### 4.2. Effects of LV Remodeling

^{2+}activation caused by the Ile284Val Tpm mutation. However, an increase in the isotropic or anisotropic (titin) component of passive myocardial stiffness along with the wall thickening led to an even more severe impairment of LV diastolic and systolic function (Table 1). The hypertrophy also led to heterogeneity in sarcomere length distribution across the LV wall (Figure 4) and to an increase in the LV twist.

#### 4.3. PV Loops Depend on Not Only on Myocardial Contractility but Also LV Geometry

^{2+}sensitivity for the DCM mutation and the decreased maximal force with a simultaneous reduction in the length-dependence of Ca

^{2+}activation for the HCM mutation.

^{2+}sensitivity of cardiac muscle. From our results and published clinical data, we can suggest that the ESPVR slope is strongly affected by changes both in the contractile properties of the myocardium and the LV geometry and cannot be used as a characteristic of myocardial contractility for hypertrophic LVs.

#### 4.4. Relation to Previous Works

^{2+}sensitivity of myofilaments. A more sophisticated 3D electromechanical model of a DCM heart was used to estimate the efficiency of an LV-assist device [42]. Although these authors used more realistic anatomy of the ventricles than the one in our model and included electrophysiological processes, the description of the active tension was too simplified to reproduce the changes in active tension and its regulation caused by the mutations. Several models were suggested to describe heart remodeling (see reviews [43,44]). Some models of this kind describe electromechanics of failing heart including concentric and eccentric hypertrophy. For example, model [45] described electrophysiology, ventricle anatomy, and passive myocardial mechanics in some detail, while the specification of active tension did not allow one to simulate the effects of the Tpm mutations. A 3D simulation of the diastolic function of DCM heart that considered its detailed remodeling including the changes in a number of sarcomeres in cardiomyocytes was presented [46]. Concentric hypertrophy was simulated by a detailed 3D electromechanical model [47], where cardiac muscle electrophysiology was described by a bidomain model and its mechanics were specified by a detailed cardiac cell model [48]. The authors examined the effects of heart remodeling (concentric hypertrophy) caused by aortic stenosis on heart performance and did not investigate any effects of mutations of sarcomere proteins.

#### 4.5. Limitations

^{2+}regulation seems to be a reasonable simplification.

## Supplementary Materials

## Author Contributions

## Funding

**.**

## Conflicts of Interest

## Appendix A

#### A.1. Cell Model of Myocardium [18]

**T**was specified by the following equation:

**F**is the Finger deformation tensor, I

_{1}, I

_{2}are its first and second invariants;

**E**is the unit tensor, p is the Lagrange factor caused by incompressibility;

**B**is the anisotropy tensor equal to the tensor square of the unit vector $\overrightarrow{{c}_{f}}$ aligned with the direction of muscle fibers in deformed muscle $B=\overrightarrow{{c}_{f}}\otimes \overrightarrow{{c}_{f}}$.

_{0}, a

_{1}are constant parameters. The difference from the work [53] is the absence of anisotropic part, which, in our model, was included only in the last term of the Equation (A1).

_{tit}is the anisotropic passive tension of the intra-sarcomere cytoskeleton mainly caused by titin filaments; T

_{A}is the active tension produced by the actin–myosin interaction. Titin tension was specified by the equation based on the worm-like chain model [26].

_{s}and L

_{s}

_{0}are the deformed and reference sarcomere lengths, N

_{M}is the number of the myosin filaments per unit cross-section area of muscle in its initial reference state; L

_{c}is the total, or ‘contour’, length of a titin molecule, L

_{p}is so-called persistence length; k

_{B}and T are the Boltzmann constant and absolute temperature. The deformed sarcomere length is described by the equation ${L}_{s}={L}_{s0}\sqrt{\overrightarrow{{{c}_{f}}_{0}}G\overrightarrow{{{c}_{f}}_{0}}}$, where

**G**is the right Cauchy–Green deformation tensor, $\overrightarrow{{{c}_{f}}_{0}}$ is the unit vector aligned with fibers in unstrained muscle.

_{cb}is the total number of myosin heads per one-half of a myosin filament; W(L

_{s}) is the length of the overlap zone of the thick and thin filaments in a half-sarcomere normalized for its maximal value; E

_{cb}is the constant cross-bridge stiffness, n is the probabilities of a myosin head being attached to the actin filament, θ is the fraction of strongly bound cross-bridges among n, and h is a cross-bridge distortion during transition from the weakly bound state to the strongly bound one. The kinetic equations for n and θ are as follows

_{+}, f

_{−}, H

_{+}, H

_{−}are kinetic rates that depend on ensemble-averaged cross-bridge distortion δ. The equation for the normalized cross-bridge distortion ${\delta}^{\prime}=\delta /h$ was

_{1}is the probability that a binding site of actin in the overlap region of the actin and myosin filaments is in the open state for the myosin head. The similar probability for the region outside the overlap zone was denoted by A

_{2}. The kinetics of these variables depended on c (Ca

^{2+}concentration in cytoplasm normalized by ‘normal’ half-activation concentration c

^{0}

_{50}), cooperativity parameter m (the Hill coefficient), sarcomere length via parameter k

_{s}, and the number of strongly bound cross-bridges via parameter k

_{n}. W

_{a}is the length of the overlap zone normalized by the actin length per half sarcomere.

_{+}is a characteristic rate constant of the Ca-troponin; ${I}_{Ca}\left(t\right)={I}_{Ca}^{0}\left(exp\left(-{k}_{1Ca}t\right)-exp\left(-{k}_{2Ca}t\right)\right)$ is a given inflow of Ca

^{2+}ions to the cell normalized by с

^{0}

_{50}. The parameters of this block of the model and their values are presented in Table S1.

#### A.2. Left Ventricle Approximation

_{in}, r

_{out}, h

_{in}, and h

_{out}[16,22].

#### A.3. Hemodynamics Model [17]

_{**_Pas}are passive elastic parts of chamber pressures, V

_{**}are their volumes, and V

_{0**}, E

_{1}, E

_{2}, and μ

_{a}

_{1}are constant parameters. Subindices LA, RA, and RV stand for the left atrium, right atrium, and right ventricle, respectively. Active pressures of the right ventricle and the atria were found from ordinary differential equations. Due to the introduction of time delay with relaxation time τ and the analogue of force–velocity equation, those described the pressure time-course more accurately than the pressure–volume dependencies with time-dependent stiffness coefficients commonly used in other lumped parameter models.

_{**_Act}are active parts of chamber pressures, and E3, μ

_{v}, and μ

_{a2}are constant parameters. F

_{**_Act}depended on activation time-functions e(t) and on the volumes V providing the Starling’s law for the atria and the right ventricle; k

_{V}, V

_{Min}, and V

_{Max}are the parameters for the pressure–volume relation.

_{a}and t

_{v}are the times of contraction initiations for the atria and ventricles, respectively; and T

_{a}, T

_{v}are the systole durations.

_{iRV}, Q

_{oRV}, Q

_{iLV}, Q

_{oLV}, Q

_{iRA}, and Q

_{iLA}are blood flows through the tricuspid valve, pulmonary valve, mitral valve, aortic valve, and the flows through systemic and pulmonary veins, respectively; Q

_{A}is the arterial flow. R and L are hydraulic and inertial resistances of the vessels and chamber entrances (subindex i) and exits (subindex o); R

_{per}and R

_{perPulm}are the peripheral vascular resistances of systemic and pulmonary circulation systems; C represents the compliances of the vascular reservoirs presented in the model; R

_{C}and C

_{C}characterize the viscoelastic properties of the systemic arteries. Indexes A1, A2, V, APulm, and VPulm corresponded to the aorta, large arteries, systemic veins, pulmonary arteries, and veins, respectively.

#### A.4. Local Hydraulic Valve Resistances in the Cases of the Valve Pathologies

_{0}is an area of a completely open valve in healthy conditions; Re is the Reynolds number, C

_{ζ}and b

_{i}are parameters.

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**Figure 1.**The dependence of the normalized isometric tension (

**a**) and thin filament activation A

_{1}(

**b**) on the dimensionless concentration of Ca

^{2+}ions for the simulation of normal cardiac muscle (blue) and of those with the Ile284Val (red) and Asp230Asn (green) tropomyosin (Tpm) mutations. Continuous, dashed, and dotted lines correspond to sarcomere lengths of 2.2 μm, 2.0 μm, and 1.8 μm, respectively.

**Figure 2.**The results of the simulations of heartbeats for model of the normal (Norm) left ventricle (LV) cardiac muscle and those with the Ile284Val (hypertrophic cardiomyopathy, HCM) and Asp230Asn (dilated cardiomyopathy, DCM) Tpm mutations. V

_{RV}, V

_{LV}are volumes of the right and left ventricles, respectively; P

_{A}, P

_{LV}, P

_{RV}, and P

_{PA}are pressures in aorta, left and right ventricles and pulmonary artery, respectively. The color code is shown on top of the plots.

**Figure 3.**The time course of the normalized Ca

^{2+}concentration in the cytosol (c), the level of activation of the Tpm–Tn (troponin) system in the overlap zone of sarcomeres (A

_{1}), and the fraction of myosin heads bound to actin (n) in a mid-wall finite element located near the LV equator. The results obtained in the normal LV model (Norm) and those with normal LV geometry and cell-level parameters characteristic for the HCM and for DCM Tpm mutations are shown.

**Figure 4.**The end-diastolic (top) and end-systolic (bottom) LV geometry for the models of the normal (Norm), dilated (DCM), and hypertrophic (HCM) LV obtained during the simulations. The color code shows sarcomere length.

**Figure 5.**The results of simulations of hemodynamic variables during a heartbeat for the model of normal LV myocardium and normal LV geometry (

**a**) and those with the DCM- and HCM-associated Tpm mutations and LV remodeling (

**b**,

**c**,

**d**). (

**b**) The cell level effects of the Asp230Asn TPM mutation were combined with LV dilatation as described in Methods; (

**c**) the cell-level effects of the Ile284Val Tpm mutation were combined with LV hypertrophy as described in Methods; (

**d**) the same as (

**c**) plus the stenosis of the LV outflow tract. The color codes for the pressures and volumes are shown.

**Figure 6.**The LV pressure–volume loops obtained from the simulation of the LV with normal cardiac muscle and normal geometry (blue) and the simulations of DCM (green) and HCM (red). Different loops were obtained at different preloads. (

**a**) PV loops obtained with normal LV geometry; (

**b**) PV loops obtained for the remodeled LVs with DCM and HCM at default passive myocardial stiffness. The dashed straight lines are the ESPVR lines plotted for each simulation case.

LV Hemodynamic Characteristics | HCM + Normal Passive Stiffness | HCM + Stiff Titin Component | HCM + Stiff Isotropic Component | DCM + Normal Passive Stiffness | DCM + Soft Titin Component |
---|---|---|---|---|---|

Low Preload | |||||

End-diastolic pressure, mm Hg | 6.4 | 4.3 | |||

Peak pressure, mm Hg | 115.2 | 112.6 | 92.5 | 105.4 | 108.5 |

End-diastolic volume, mL | 92.2 | 88.3 | 73.2 | 157.5 | 165.1 |

Stroke volume, mL | 67.7 | 64.1 | 48.3 | 64 | 66.1 |

Ejection fraction, % | 73 | 73 | 66 | 41 | 40 |

Average Preload | |||||

End-diastolic pressure, mm Hg | 8.7 | 5.1 | |||

Peak pressure, mm Hg | 128.4 | 123.1 | 99.3 | 110.7 | 113.2 |

End-diastolic volume, mL | 104.9 | 99.2 | 80 | 169.8 | 176.6 |

Stroke volume, mL | 79 | 74 | 54.8 | 67.3 | 68.9 |

Ejection fraction, % | 75 | 75 | 69 | 40 | 40 |

High Preload | |||||

End-diastolic pressure, mm Hg | 11.5 | 5.8 | |||

Peak pressure, mm Hg | 141.1 | 133 | 108.6 | 115.2 | 118.3 |

End-diastolic volume, mL | 117.9 | 109.5 | 88.8 | 181.5 | 191 |

Stroke volume, mL | 89.9 | 83.1 | 63 | 70.2 | 72.1 |

Ejection fraction, % | 76 | 76 | 71 | 39 | 38 |

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**MDPI and ACS Style**

Syomin, F.; Khabibullina, A.; Osepyan, A.; Tsaturyan, A.
Hemodynamic Effects of Alpha-Tropomyosin Mutations Associated with Inherited Cardiomyopathies: Multiscale Simulation. *Mathematics* **2020**, *8*, 1169.
https://doi.org/10.3390/math8071169

**AMA Style**

Syomin F, Khabibullina A, Osepyan A, Tsaturyan A.
Hemodynamic Effects of Alpha-Tropomyosin Mutations Associated with Inherited Cardiomyopathies: Multiscale Simulation. *Mathematics*. 2020; 8(7):1169.
https://doi.org/10.3390/math8071169

**Chicago/Turabian Style**

Syomin, Fyodor, Albina Khabibullina, Anna Osepyan, and Andrey Tsaturyan.
2020. "Hemodynamic Effects of Alpha-Tropomyosin Mutations Associated with Inherited Cardiomyopathies: Multiscale Simulation" *Mathematics* 8, no. 7: 1169.
https://doi.org/10.3390/math8071169