*2.4. Computational Setup*

The Tonstad power plant sand trap is 184 m-long, and the main tunnel has a crosssectional area of around 9.9 m × 11 m (around 119 m2). A picture of the prototype sand trap is shown in Figure 3. A 3D model of the sand trap is developed using engineering CAD software, see Figure 4. It was created from engineering drawings provided by the Sira-Kvina power company—the owner of the plant—and consists of a rectangular inlet section, a diffuser, a long tunnel section with a gentle slope, and a weir in the invert where the tunnel transitions into the penstock. This model is used as a base to add upgrades to in two other models. The additional models were created to test the effects of the upgrades, where one model includes only ribs and another includes both ribs and v-shaped rakes.

**Figure 3.** Picture of a prototype sand trap tunnel at hydropower plant; courtesy of Sira-Kvina kraftselskap.

**Figure 4.** (**a**) Sand trap without upgrades. The distance from the end of the diffuser to the weir is 184 m. The diffuser is 16 m-long. The sand trap tunnel has a cross-sectional area of 119 m2. (**b**) Ribs. The ribs are placed just upstream of the penstock, and are 1 m-wide and spaced 1 m-apart. The ramp has an 8% inclination. (**c**) Top view of the v-shaped rakes in the diffuser. The v-shaped rakes measure 6 m in height. Distances tip-to-tip between rakes are 1 m and 0.8 m for the upstream and downstream row, respectively. Zoomed part shows rake dimensions.

The ribs are placed just upstream of the penstock, in combination with the weir. This setup was tested in experiments [4,5]. The purpose of the ribs is to allow bed load sediments to fall between the ribs, while water is minimally affected. There are five ribs in total. The length of the ribs and the gap between ribs both measure 1 m. A ramp is placed upstream of the ribs to raise them above the bed and create space for sediments to accumulate. Further, in this way, excavating into the tunnel floor is not necessary. In

the work by Richter et al., the ramp was also found to protect sediments that had fallen between the ribs from being resuspended and flushed into the penstock.

The rake structure is made up of two rows of v-shaped rakes, with the tip of the rakes facing downstream. The rakes measure 6 m in height. Distances between rakes are 1 m and 0.8 m for the upstream and downstream row, respectively. The purpose of the rakes is to even the flow of the jet from the inlet and enhance the diffuser effect of slowing down the flow. The distance from the inlet to the rakes is 25 m, which is approximately 3 times the diameter. To ensure that stable and developed flow reaches the rakes, a distance of 5–10 times the inlet diameter is necessary. The fact that the flow reaching the rakes may not be fully developed in the simulations should be taken into consideration when analysing the results.

The three models of the sand trap are meshed similarly. In all three models, the diffuser and the invert are both given tetrahedral mesh structures, while the simpler geometries of the inlet section, tunnel, and penstock have structured hexahedral meshes. The number of cells in each mesh is >23 × 106. Inflation layers are added along the tunnel walls so that global *y*+ < 30. This ensures accurate representation of flow conditions in the boundary layers. The mesh surrounding the ribs and the rakes are refined further to a maximum cell size of 0.01 m. Average element sizes in the diffuser, tunnel, and rib section are 1.05 mm, 1.39 mm, and 0.43 mm, respectively.

The chosen advection scheme is the "High Resolution" scheme available in CFX. This scheme is second-order accurate in smooth regions and reduces its order of accuracy in regions of high gradients, where unboundedness is a factor. A first-order-accurate scheme is generally more robust and reaches convergence criteria faster than a second-orderaccurate scheme. However, this comes at the cost of higher numerical diffusion, resulting in a less-accurate solution. The transient scheme used is the Second-Order Backward Euler scheme.

Steady state multiphase simulations are run to create initial conditions for the transient multiphase simulations. The total simulated time for the transient simulations is set to allow sediments enough time to either reach the bed or exit through the outlet. The simulations use a discharge of 80 m3/s, which is the discharge when the power plant is operating at design conditions. The inlet velocity boundary condition represents this mass flow rate. The wall roughness is 10−3 m in the numerical model. An overview of solution parameters and boundary conditions is presented in Table 1.

**Table 1.** Selected parameters and boundary conditions for unsteady simulations.

Multiphase flow was implemented by enabling the particle transport solid model in CFX, also known as the discrete phase model (DPM). DPM uses the Eulerian–Lagrangian multiphase model to track the paths of individual particles as they travel through the domain. It is well-suited for situations such as in the present work, where the volume fraction of the solid phase is low. The one-way coupled fluid pair model was chosen to solve the fluid–particle interactions. This model is computationally cheaper than the two-way coupled model. As the sediment phase's effect on the fluid phase is negligible in this case, the one-way coupled model was found to produce satisfactory results. In the transient multiphase simulations, sediments were injected with uniform distribution over the inlet during the first 100 s. The mass flow of sediments was set to 1000 kg/s, leading to a total mass of sediments injected into the sand trap of 105 kg. The sediments were tracked as they travelled through the model, and the sand trap efficiency was given by the time-integrated mass flow report at the outlet by the end of the simulation. The sediment diameters have a normal distribution with a mean of 0.75 mm and a standard deviation of 0.25 mm.

By also ensuring that the solution is mesh independent, the accuracy of the results are further improved. Mesh independence is achieved when a defining value of the simulation no longer changes significantly. The purpose of a mesh-independence study is to minimise the discretisation error resulted from approximating the geometry during the meshing stage [18]. The model with both rakes and ribs was used for the meshindependence study. Three different mesh qualities were used. Following the procedure described by Celik I. B. et al. [19], an estimation of the discretisation error was obtained. In the calculations, *N* is the number of cells; *h* is the representative cell size; *r* is the refinement factor; *φ* is the pressure drop from inlet to outlet; *p* is the apparent order of the method; *φext* are the extrapolated values of *φ*; *ea* and *eext* are the approximate and extrapolated relative errors, respectively; and GCIfine is the fine-grid convergence index. The mesh discretisation error was found to be 1.4%. The results of the calculations are presented in Table 2. The fine mesh with 26.9 million cells is considered for further analysis and will be used to conduct the final numerical studies.

**Table 2.** Parameters of the mesh-independence study.

The simulation results are verified by ensuring that residuals reach a satisfactory convergence criteria and that there is a stable mass flow through the domain. This signifies that the solution is computationally correct. The hydraulic representation of the numerical model will be validated by comparison with PIV measurements on a physical scale model of the sand trap and to ADCP measurements from the prototype sand trap [4,20]. As the flow state is highly stochastic, the velocity distributions will never be identical. However, it is useful to make sure that the jet in the diffuser observed in the PIV measurements also exists in the simulated flow. It is also important to ensure that the simulated velocities are of reasonable magnitudes.


*Communication* **Longitudinal Surveillance of Fetal Heart Failure Using Speckle Tracking Analysis**

**Julia Murlewska 1,\*, Oskar Sylwestrzak 1,2, Maria Respondek-Liberska 1,2, Mark Sklansky 3,4 and Greggory Devore 4,5**

**Abstract:** Long-term monitoring of a fetus with heart failure is an undeniable challenge for prenatal cardiology. Echocardiography is constrained by many fetal and maternal factors, and it is difficult to maintain the reproducibility of the measured and analyzed parameters. In our study, we presented the possibilities of using modern speckle tracking technology in combination with standard echocardiography parameters that may be insufficient or less sensitive in the context of monitoring life-threatening fetal conditions. Our analysis shows the superiority of the parameters used to assess fetal cardiac architecture, such as the GSI Global sphericity Index, and fetal cardiac function, such as the FAC fractional area change and the EF ejection fraction, which temporal change may indicate a worsening condition of the fetus with heart failure. The significant increase in the parameters of fetal heart size in speckle tracking allows for an improved echocardiographic diagnosis and monitoring of the fetus with heart failure and the prognostic conclusions about the clinical condition after birth. Significant decreases in FAC for the left and right ventricles and EF for the left ventricle may indicate an unfavourable prognosis for the monitored fetus due to heart failure.

**Keywords:** fetal echocardiography; fetal strain; fetal speckle tracking; fetal heart failure

### **1. Introduction**

For longitudinal echocardiographic monitoring during fetal life, there is no consensus on the best approach to evaluate fetal cardiovascular function. Compared to postnatal echocardiographic evaluation, fetal evaluations may be complicated by various fetal positions and orientations, making serial assessment with a single measurement more challenging. Therefore, we compared conventional echocardiographic measurements with speckle-tracking derived assessments [1] (Figure 1) to determine which approach, in a single fetus with progressive heart failure, may be most useful to predict fetal well-being.

**Citation:** Murlewska, J.; Sylwestrzak, O.; Respondek-Liberska, M.; Sklansky, M.; Devore, G. Longitudinal Surveillance of Fetal Heart Failure Using Speckle Tracking Analysis. *J. Clin. Med.* **2022**, *11*, 7102. https://doi.org/10.3390/ jcm11237102

Academic Editor: Yariv Yogev

Received: 9 October 2022 Accepted: 23 November 2022 Published: 30 November 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** (**A**,**B**) Fetal strain at 30 weeks 4 days. (**C**,**D**) Fetal strain at 32 weeks 4 days. End-Systolic/ End-Diastolic Borders of LV-left ventricle and RV-right ventricle displacement.

### **2. Materials and Methods**

During the first pregnancy of a 26-year-old woman, the first trimester evaluation demonstrated active herpes labialis and shingles, which were treated with Aciclovir. New onset of ascites, hydrops testis, and ductus venosus reversal flow was seen at 30 weeks of gestation. With the subsequent development of fetal cardiomegaly, cardiac hypertrophy, pericardial effusion, mitral, tricuspid, and pulmonary valve regurgitation, and fetal heart failure, transplacental digoxin treatment was initiated, along with dexamethasone and cefuroxime. The standard echo-sonographic findings are presented in Table 1. Speckle tracking analysis (Table 2, video S1) has been demonstrated to provide both qualitative and quantitative assessments of fetal size, shape, and function, as recently described by DeVore [1–4]. Speckle tracking forms a characteristic pattern of spots and kernels on the irregular surfaces of fetal myocardium to measure longitudinal, radial, and circumferential

regional strain [5]. Pregnancy was maintained for over two more weeks, but was then complicated by polyhydramnios, maternal hypertension, and premature rapture of membrane. Due to abnormal fetal tracings, an emergency CS was performed at 32 weeks of gestation. The newborn baby boy had a birth weight of 2500 g, Apgars 5/6/6, and was admitted to the ICU. Following aggressive postnatal medical management, as well as PDA-patent ductus arteriosus ligation at 1 month of age, he was discharged home after 79 days with a diagnosis of congestive herpetic cardiomyopathy.

**Table 1.** Standard selected echocardiographic parameters analyzed in the present case indicating the progression of fetal heart failure.

GA: gestational age; HA/CA: heart area/chest area; RV/Tei LV: MPI-myocardial performance index for right and left ventricle; RV/LV SF: shortening fraction for right and left ventricle; CVPS: cardiovascular profile score.

Left Ventricular Ejection Fraction (%) 41.61 −2.93 24.61 −5.69 ↓

**Table 2.** Advanced echocardiographic parameters of the fetal heart using speckle tracking technology.

### **3. Results**

Our analysis and interpretation of the standard and advanced echocardiographic parameters using speckle tracking technology allowed us to evaluate the efficacy of prenatal treatment with digoxin. Digoxin was administered from week 30 to week 32. Standard size of the HA/CA (heart area/chest area) was constant, and a detailed speckle tracking analysis revealed a significant prolongation of the TL transverse length dimension and a shortening of the BAL-basal–apical length dimension, thereby reducing the GSI-global Sphericity Index GSI = BAL/TL) [2]. We also observed an increase in the following parameters: the area and circumference of the heart and the area for the left and right ventricles. Standard cardiovascular parameters, such as the Tei index or MPI, for the left and right ventricles increased without significant difference over a two-week period (values between 0.7 and 0.9), and the CVPS cardiovascular profile score remained unchanged at level 5. Fetal global cardiac contractility was also analyzed using the parameters FAC-fractional area change (%) and EF-ejection fraction (%) [3,4], which significantly decreased over time (Tables 1 and 2 show the analyzed echocardiographic parameters). The tables contain selected parameters that allow the analysis of fetal heart failure.

### **4. Discussion**

Fetal circulatory failure may have various causes and, therefore, it is very difficult to assess the effectiveness of any treatment [6]. So far, cardiovascular profile score has been considered a very useful diagnostic tool to quantify the severity of heart failure, which is a composite score based on five different echocardiographic parameters [7]. Our long-term analysis presents a modern approach of monitoring a fetus with circulatory failure. We tried to show the superiority of speckle tracking analysis, which is a more detailed method; although it is more involved and time-consuming, it can have a very important place precisely for the studies of fetuses with the most severe cardiovascular and non-cardiovascular conditions that lead to circulatory failure and fetal death. Speckle tracking may indicate some changes in fetal cardiac architectonics not seen by standard echocardiography, and may also have superiority over standard fetal cardiac function exponents, which may be less sensitive than FAC and EF; we have highlighted these findings which seem very promising in the context of monitoring a fetus with circulatory failure. The pattern of speckles within a defined region, known as a kernel, can be conceptualized as an 'acoustic fingerprint'. The spatial movement of multiple kernels can be used to measure longitudinal, radial, and circumferential regional strain. Changes in myocardial strain precede the changes in ejection fraction in patients with cardiomyopathy. STE is less dependent on the angle of insonation than conventional echocardiography, allowing more flexibility in image acquisition than Doppler-based techniques. STE is less susceptible to fetal and maternal movements. The accuracy of STE has been confirmed in postnatal life in animal models [5].

FAC should generally take values ≥35% [3], while EF should be >50% [4]. In our case, these values corresponded to <−2.93 Z-score and significantly decreased over a two-weeks below the mean, the CVPS was constant, and its analysis made it difficult to predict an impending premature birth. M-mode analysis of the myocardial shortening fraction can also be made more accurate using speckle tracking and 24-segment examination [8] Finally, a HA/CA analysis associated with fetal cardiomegaly may not be adequate for serious cases of fetuses with heart failure, in which more demanding and more sensitive parameters to the deterioration of circulatory performance, such as GSI, LV Area, or RV Area, could be used.

#### **5. Conclusions**

The speckle tracking analysis performed in our case allows a detailed monitoring of a fetus with heart failure.

**Author Contributions:** Conceptualization: J.M.; methodology: J.M. and O.S.; software: G.D.; validation,: O.S., M.R.-L. and M.S., formal analysis: J.M. and O.S.; investigation: J.M.; resources: J.M. and M.R.-L.; data curation: J.M.; writing—original draft preparation: J.M.; writing—review and editing: J.M., G.D. and M.S.; visualization: J.M.; supervision: M.S., M.R.-L. and G.D.; project administration: J.M. and G.D.; funding acquisition: none. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Ethical review and approval were waived for this study, due to the retrospective analysis.

**Informed Consent Statement:** Written informed consent was obtained from the patient to publish this paper.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**