A systematic review of small for size syndrome after major hepatectomy and liver transplantation. This section discusses the results presented in Results section and their implications for future developments. This is proving the clinical relevance of these results to define a virtual population. However (i) measured pre-hpx \(P_{\text {pv}}\) has higher values than simulated one (top left panel of Fig. Variance-based sensitivity analysis: theory and estimation algorithms. The Sensitivity study step can be enabled from Show More Options. 21 for a more detailed recent review of SA methods applied in this context. 2.2 Sensitivity Analysis. The business use this method to measure their profitability position in the market. (4) is 0.03 vs. 0.02, respectively. \end{aligned}$$, $$\begin{aligned}&\forall i \in \left\{ {\text {RA}}, {\text {LA}} \right\} \quad e_{i}(t) = \left\{ \begin{array}{ll} \frac{1}{2} \left[ 1 + \cos \left( \pi \frac{t+T_{\text {cc}}-t_{\text {ar}}}{T_{\text {ar}}}\right) \right] &{} 0 \le t \le t_{\text {ar}} + T_{\text {ar}} - T_{\text {cc}}, \\ 0 &{} t_{\text {ar}} + T_{\text {ar}} - T_{\text {cc}} \le t \le t_{\text {ac}}, \\ \frac{1}{2} \left[ 1- \cos \left( \pi \frac{t-t_{\text {ac}}}{T_{\text {ac}}}\right) \right] &{} t_{\text {ac}} \le t \le t_{\text {ac}} + T_{\text {ac}}, \\ \frac{1}{2} \left[ 1 + \cos \left( \pi \frac{t-t_{\text {ar}}}{T_{\text {ar}}}\right) \right] &{} t_{\text {ac}} + T_{\text {ac}} \le t \le T_{\text {cc}}, \end{array}\right. As displayed by Fig. Model parameters are tuned based on each patient data. Sensitivity Analysis. \end{aligned}$$, $$\begin{aligned} R_{\text {pv}}&= \dfrac{P_{\text {pv}} - P_{\text {liver}}}{Q_{\text {pv}}}, \end{aligned}$$, $$\begin{aligned} R_{\text {ha}}&= \dfrac{{\text {MAP}} - P_{\text {liver}}}{Q_{\text {ha}}}, \end{aligned}$$, $$\begin{aligned} R_{\text {hv}}&= \dfrac{P_{\text {liver}}-P_{\text {vc}}}{Q_{\text {pv}} + Q_{\text {ha}}}, \end{aligned}$$, $$\begin{aligned} R_{\text {DO}}&= \dfrac{{\text {MAP}} - P_{\text {pv}}}{Q_{\text {pv}}}, \end{aligned}$$, $$\begin{aligned} R_{\text {OO}}&= \dfrac{{\text {MAP}}-P_{\text {vc}}}{{\text {CO}} - Q_{\text {pv}} - Q_{\text {ha}}}, \end{aligned}$$, \(P_{\text {vc}} = P_{\text {pv}} -{\text {PCG}}\), \(P_{\text {liver}} = P_{\text {pv}} - \alpha _{\text {liver}} \, {\text {PCG}}\), https://doi.org/10.1007/s10439-022-03098-6, Human Cardiovascular Lumped-Parameter Model, Impact on the Performances of the Calibration Step, Sensitivity Analysis Results Using the Full Model, Sensitivity Analysis Results Using the Novel PCE-Based Approach, http://creativecommons.org/licenses/by/4.0/, S.I. \ldots N_{{{\text{outputs}}}} ], $$, $$ S_{{ij}}^{{{\text{tot}}}} = 1 - \frac{{\text{var} [{\mathbb{E}}(Y_{i} |X_{{ - j}} )]}}{{\text{var} [Y_{i} ]}} = 1 - S_{{( - ij)}} \quad \forall i \in [1. Ask & get answers from experts & other users. 7), and (iv) simulated pre-hpx CO has higher values than the data distribution from Ref. Which equipment is used to level the ground and spread the loose material? The model consisted of a very detailed closed-loop system with a refined description of the liver structure. Note that for every input and output couple the first index is close to the associated total index, which means that higher order interactions are negligible. Google Scholar. B. In particular, the value of the weights are the following: \(w_{i} = 1\) for PCG, MAP and CO, \(w_{i} = \frac{2}{3}\) for \(P_{\text {pv}}\) and \(w_{i} = \frac{1}{3}\) for \(Q_{\text {pv}}\) and \(Q_{\text {ha}}\). $$, $$\begin{aligned}&\forall i \in \left\{ {\text {RV}}, {\text {LV}} \right\} \quad e_{i}(t) = \left\{ \begin{array}{ll} \frac{1}{2} \left[ 1- \cos \left( \pi \frac{t}{T_{\text {vc}}}\right) \right] &{} 0 \le t \le T_{\text {vc}}, \\ \frac{1}{2} \left[ 1 + \cos \left( \pi \frac{t-T_{\text {vc}}}{T_{\text {vr}}}\right) \right] &{} T_{\text {vc}} \le t \le T_{\text {vr}} + T_{\text {vc}}, \\ 0 &{} T_{\text {vc}} + T_{\text {vr}} \le t \le T_{\text {cc}}, \end{array}\right. 12, in particular from their patient cohort data (more details in Appendix 2). \(P_{\text {pv}}\) is mainly influenced by \(E_{{\text {b}},{\text {LV}}}\), Hpx, \(R_{\text {DO}}\), \(R_{\text {pv}}\), and \(R_{\text {hv}}\); PCG is mainly influenced by Hpx and mildly by \(E_{{\text {b}},{\text {LV}}}\), \(R_{\text {DO}}\), \(R_{\text {pv}}\), \(R_{\text {hv}}\) and \(R_{\text {OO}}\); MAP and CO are mainly influenced by \(E_{{\text {a}},{\text {LV}}}\), \(E_{{\text {b}},{\text {LV}}}\) and \(R_{\text {OO}}\); \(Q_{\text {ha}}\) is mainly influenced by Hpx, \(R_{\text {ha}}\), \(E_{{\text {b}},{\text {LV}}}\) and \(R_{\text {OO}}\); \(Q_{\text {pv}}\) is mainly influenced by \(R_{\text {DO}}\), \(E_{{\text {b}},{\text {LV}}}\) and \(R_{\text {OO}}\). 15,600 to RS. High order interactions can also be evaluated with high order Sobol indices, see Ref. The post-hpx discussion (right panels of Figs. \ldots N_{{{\text{outputs}}}} ] $$, \(X_{(-j)} = \left( X_{1}, \dots , X_{j-1}, X_{j+1}, \dots , X_{d} \right) \), \({{\,{\text{var}}\,}}[{\mathbb {E}}[Y_{i}|X_{j}]]\), \({{\,{\text{var}}\,}}[{\mathbb {E}}(Y_{i}|X_{j})]\), $$\begin{aligned}&Y = {\mathcal {M}}(X) = \sum _{k=0}^{\infty } \beta _{k} \varPsi _{k} (X), \\&\Rightarrow Y \approx {\mathcal {M}}^{\text {PCE}} (X) = \sum _{k=0}^{P} \beta _{k} \varPsi _{k} (X), \end{aligned}$$, $$\begin{aligned}&Y = {\mathcal {M}} (X) = \sum _{k=0}^{P} \beta _{k} \varPsi _{k}(X) + \varepsilon _P \Rightarrow \beta ^{\text {T}} \varPsi (X) \approx {\mathcal {M}}(X) \\&\Rightarrow \beta ^{*} = {\text {argmin}}_\beta {\mathbb {E}} \left[ \left( \beta ^{\text {T}} \varPsi (X^{(N_{\text {s}})}) - {\mathcal {M}}(X^{(N_{\text {s}})}) \right) ^{2} \right] , \end{aligned}$$, $$ Q^{2} = 1 - \dfrac{\sum _{l=1}^{N_{\text {test}}} \left( Y^{(l)} - {\mathcal {M}}^{\text {PCE}}(X^{(l)} \right) ^{2}}{N_{\text {test}} \, {{\,{\text{var}}\,}}(Y)}, $$, \(Y^{(l)} = {\mathcal {M}}(X^{(l)})\, \forall l = 1, \dots , N_{\text {test}}\), \(\left\{ X^{(l)} \right\} _{l = 1, \dots , N_{\text {test}}} \cap \left\{ X^{(k)} \right\} _{k = 1, \dots ,N_{\text {s}}} = \emptyset \), \(N_{\text {test}}^{*} = 4 \times 10^{4}\), \(N = \left[ 5 \times 10^{3}, \,10^{4}, \,2 \times 10^{4},\, 4 \times 10^{4} \right] \), $$\begin{aligned}&{\text {err}}^{S_{1}, Y}_{N_{1},N_{2}} = \max _{j \in \left[ 1, \dots , d \right] }\left|S^{N_{1}}_{1,X_{j}} - S^{N_{2}}_{1,X_{j}}\right|,\\&{\text {err}}^{S_{\text {tot}}, Y}_{N_{1},N_{2}} = \max _{j \in \left[ 1, \dots , d \right] } \left|S^{N_{1}}_{{\text {tot}},X_{j}} - S^{N_{2}}_{{\text {tot}},X_{j}} \right|, \end{aligned}$$, $$ {\text {Err}}_{L^{2}} = \sqrt{\sum _{i=1}^{6} w_{i} \left( \dfrac{Y_{i}^{\text {target}} - Y_{i}^{\text {sim}}}{Y_{i}^{\text {target}}} \right) ^{2}}, $$, $$ f \left( \dfrac{{\text {d}}y}{{\text {d}}t}, y, t, x \right) = 0, $$, $$ \left\{ \begin{array}{ll} \frac{{\text {d}}V_{i}}{{\text {d}}t} = Q_{{\text {in}},{\text {i}}} - Q_{{\text {out}},{\text {i}}} &{} \forall i \in \left\{ {\text {RA}}, {\text {RV}}, {\text {LA}}, {\text {LV}} \right\} \\ P_{i} = E_{i}(t) (V_{i} - V_{0,i}) &{} \forall i \in \left\{ {\text {RA}}, {\text {RV}}, {\text {LA}}, {\text {LV}} \right\} \\ Q_{{\text {out}},{\text {i}}} = G_{i}(\Delta P) \; \Delta P &{} \forall i \in \left\{ {\text {RA}}, {\text {RV}}, {\text {LA}}, {\text {LV}} \right\} , \end{array}\right. 6b). A. Wiley Online Library, 2004. In the context of using Simulink Design Optimization software, sensitivity analysis refers to understanding how the parameters and states (optimization design variables) of a . Using as baseline value the median of the clinical measurements from Ref. The y-axis displays the relative frequency, which is the ratio of the frequency of a particular event to the total frequency of that event to happen. 6b indicate that the pre-hpx value of MAP and CO can be exploited to have a good estimation of \(E_{{\text {a}},{\text {LV}}}\), \(E_{{\text {b}},{\text {LV}}}\)and \(R_{\text {OO}}\). The remaining heart elastances (right ventricle and left atrium) are changed proportionally to \(E_{{\text {a}},{\text {RA}}}\), \(E_{{\text {b}},{\text {RA}}}\), \(E_{{\text {a}},{\text {LV}}}\) and \(E_{{\text {b}},{\text {LV}}}\)12. From a theoretical viewpoint, the computational cost required by the number of model evaluations in this approach can still be very high, depending on the computational cost of a single model evaluation. Refs. SR authors can conduct sensitivity analyses to explore whether their results are sensitive to exclusion of low quality studies or a high RoB. In particular, the couple of heart elastances in the left ventricle combined with the other organ resistance \(R_{\text {OO}}\) have the largest impact on the driving force of the cardiovascular system (MAP and CO, pre-hpx and post-hpx). PubMed Central Figure 4 displays the predicted probability density functions for the major hemodynamics outputs Y and compares them with the associated clinical measurement distributions from Ref. Each heart chamberright atrium (RA), right ventricle (RV), left atrium (LA), left ventricle (LV)is described by the following system of equations: where \(V_{i}\) and \(V_{0,i}\) are the volume and unloaded volume of the heart chamber i, respectively; \(Q_{{\text {in}},{\text {i}}}\) and \(Q_{{\text {in}},{\text {i}}}\) are the incoming and outgoing flows of the heart chamber i, respectively; \(P_{i}\) is the heart chamber pressure; \(\Delta P\) is the pressure drop across the valve; \(G_{i}(\Delta P)\) is the valve conductance of heart chamber i dependent on \(\Delta P\); \(E_{i}\) is the elastance function, defined by. A sensitivity analysis is a repeat of the primary analysis or meta-analysis, substituting alternative decisions or ranges of values for decisions that were arbitrary or unclear. Central panels of Fig. 138, 2016. Username or email * Password * Sorry, you do not have permission to ask a question, You must login to ask a question. In this study \(P=\dfrac{(d + q)!}{d!\,q! Comparison with clinical measurements. SA results using the full model \({\mathcal {M}}\) before (pre-hpx) and after (post-hpx) the virtual hepatectomy (\(N=10^{4}.\)). 1 for the SA study are: portocaval gradient PCG, which is the pressure difference between the PV and the inferior vena cava; systemic arterial pressure, called MAP in the clinics; blood flow in the HA (\(Q_{\text {ha}}\)) and in the PV (\(Q_{\text {pv}}\)). In particular, the orthonormal basis of the PCE is built using only such couples employing the adaptive Stieltjes algorithm,23 a more stable alternative to the well-known GramSchmidt algorithm. D. Economics of cost and benefits of the project. The Sobol indices analysis using the Saltelli algorithm (see Sobol Indices section) is performed applying the previous empirical distributions shown in Fig. Sensitivity analysis is a study of_____? Sensitivity analysis is a major approach to re-examining an already concluded viability study in order to determine what the investment appraisal outcome would be, if same or all the factor elements were to vary. Sensitivity analysis frequently uses in both business and economics in order to study the impact on variable to the others. Let me illustrate this on a simple example: Imagine that you research a problem with three variables: Income ($/yr. In particular for MAP the pre-hpx and post-hpx filtered medians have an increased accuracy of 74% and 68%, respectively. Sensitivity analysis and model assessment: mathematical models for arterial blood flow and blood pressure. Associate Editor Joel Stitzel oversaw the review of this article. Moreover, considering only the virtual patient cases in which the original algorithm had reached the maximum number of iterations allowed in the calibration step, the speed up of the new algorithm is on average 41% faster and with comparable precision. In particular for pre-hpx and post-hpx \(P_{\text {pv}}\) medians the difference is below 0.4 mmHg (\(3\%\)), while for pre-hpx and post-hpx PCG medians the difference is 0.82 mmHg (\(17\%\)) and 1.22 mmHg (\(20\%\)), respectively. As example Fig. We then propose an efficient approach based on PCE to perform GSA based on the already computed simulations. The use of a SA methodology to investigate the influence of inputs (Input Parameters section) to clinically relevant quantities of interest (Quantities of Interest section) is fundamental due to the presence of several organ compartments and nonlinear elements, which makes the interactions among parameters and outputs non trivial. Google Scholar. \(T_{\text {vc}}\), \(T_{\text {ac}}\), \(T_{\text {vr}}\), and \(T_{\text {ar}}\) are the ventricular and atrial contractions and relaxations duration, respectively; \(t_{\text {ac}}\) and \(t_{\text {ar}}\) are the starting times of contraction and relaxation, respectively. 6a) is consistent with the difficulty for surgeons to foresee postoperative portal hypertension due to hepatectomy. Mesh Sensitivity Study (also known as Grid Independence Analysis or Mesh Sensitivity Analysis) is a crucial part of any CFD analysis.When performing an analysis it is important to remember that your solution is the numerical solution to the problem that you posed by defining your mesh and boundary conditions.The more accurate your mesh and boundary conditions, the more . This article is intended as a tutorial on sensitivity analyses, in which we discuss three methods to conduct sensitivity analysis. 8 shows the temporal evolution during the simulation of the simulated portal vein pressure (blue line). The cause is that ranges of the measurements and the ones specified in the physiological filter (Table 2) do not always match. Biol. Company financials. 4 the authors conducted a synthetic-data-based parametric study for a closed-loop cardiovascular system in order to investigate sensitive and insensitive model parameters trying to decrease the complexity of the model. A sensitivity analysis can also be referred to as . Article In particular the ambition of the current research is twofold: (i) perform a sensitivity analysis (SA) study to identify the most significant model parameters (inputs) with respect to the main postoperative clinical outputs of interest, and (ii) create a virtual population representative of a real patient cohort available for future studies. This work focuses on Sobol indices, a variance decomposition-based method, which expresses the share of variance of an output that is due to a given input or input combination. A summary of these results, denoting the sensitivewhen \(S_{ij} \gg 0.1\)and insensitive parameterswhen \(S^{\text {tot}}_{ij} \approx 0\)for each clinical output is displayed in Table 3. Reproduced for the elastances in the literature ), and ( iv ) pre-hpx! The fact that SA is employed with various goals: e.g not always match you evaluate potential outcomes to better! Twitter ; Share on WhatsApp ; you must login to add an answer A. D., S.! 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