Physics Based and Data Driven Algorithms for the Simulation of the Heart Function - Alfio Quarteroni Politecnico di Milano, Italy EPFL Lausanne ...
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Physics Based and Data Driven Algorithms for the Simulation of the Heart Functio Alfio Quarteron Politecnico di Milano, Ital EPFL Lausanne, Switzerlan i y d n
Physics-based vs Data-driven modeling
Model
M(u,d) = 0
Data First principles Solution Output
d E(w,d) = 0 u y(u)
+
Constitutive relations
C(u,w,d) = 0 Physics-based modeling
(white box)
a separation line ?
training Data-driven modeling
Training data (black box)
{di , yi }
Artificial Neural Network
Data Output
d y
Alfio Quarteroni Politecnico di Milano - EPFL
The Pumping Function - The Heart and the Circulation https://www.stanfordchildrens.org Al o Quarteroni Politecnico di Milano - EPFL fi
Cardiac Electrical Activity Purkinje fibers https://www.stanfordchildrens.org Al o Quarteroni Politecnico di Milano - EPFL fi
The Mathematical Heart - Conceptual Design
The Mathematical Heart - The Equations Al o Quarteroni EPFL Lausanne fi
The core cardiac models 1. Electrophysiolog 2. Active mechanic 3. Passive mechanic 4. Fluid dynamic 5. Valves motio 6. Myocardium perfusio … and their integratio Al o Quarteroni Politecnico di Milano - EPFL fi n s s y s n n …
Whole Heart electrophysiology simulation (R.Piersanti) Time evolution of the ventricular and atrial transmembrane potential Al o Quarteroni Politecnico di Milano - EPFL fi
Whole Heart electrophysiology simulation (R.Piersanti) Activation Map Wave front propagation Al o Quarteroni Politecnico di Milano - EPFL fi
Identifying possible targets of ablation for ventricle arrhythmias (S.Pagani) Entry: lines of block at slow-conduction isthmus boundaries Exit: slow-conduction late activation possible targets of ablation A. Frontera, S. Pagani et al (2020) Heart Rhythm Al o Quarteroni MOX - Politecnico di Milano - EPFL fi
The coupled electromechanical model Al o Quarteroni Politecnico di Milano - EPFL fi
Electro-mechanical simulation (R.Piersanti, F.Regazzoni, M.Salvador) 12 Al o Quarteroni Politecnico di Milano - EPFL fi
Fluid Dynamics Al o Quarteroni Politecnico di Milano - EPFL fi
Fluid dynamics of the left heart (A.Zingaro) Blood velocity Blood pressure Vortex formation Q-criterion and velocity magnitude: • when the mitral valve opens, high speed jets coming from the LA ll the LV. • This produces the formation of a O-ring shaped vortex, a coherent structure rolling through the lea ets of the mitral valve. • This big vortex breaks into smaller coherent structures lling the LV and moving towards the apex • As the systole begins, marked by the opening of the aortic valve, the structures are ushed out in the aorta. • At the same time, new jets are entering in the LA, but weaker with respect to the ones observed in diastole • Al o Quarteroni Politecnico di Milano - EPFL fi fi fi fl fl .
3D-0D multiscale model 0D 3D←0 0D D 3D←0 D 3D 3D 3D 3D 3D 3D 3D 3D [F. Regazzoni, M. Salvador, P. Africa, M. Fedele, L. Dede’, A. Quarteroni 2021 (submitted)] iHeart Project
Fluid dynamics of the whole heart (A.Zingaro) - 1.34 M DOFs, havg = 2 mm - Δt = 2 ⋅ 10−4s - Computational cost: more than 5 days on 48 cores for a single heartbeat Al o Quarteroni Politecnico di Milano - EPFL fi ,
Simulation of a pathological scenario: mitral regurgitation Healthy Mitra regurgitation regurgitan flow A. Zingaro et al. MOX Report, 2021 Al o Quarteroni Politecnico di Milano - EPFL fi l t
Modeling the cardiac response to COVID-19 Goal: Investigate the effects of COVID-19 on the cardiovascular system. Methods: In silico study of the interplay between hemodynamic alterations associated with COVID-19 (increased pulmonary resistance, decreased oxygen saturation, altered heart rate) and the cardiac DIGITAL function. TWIN Clinical Partners Dede’, Regazzoni, Vergara, …, Cogliati, Pontone, Quarteroni; Math. Bio. Eng. (2021) Alfio Quarteroni Politecnico di Milano - EPFL :
A broader picture: from myocardial perfusion to cell contractility Coronary blood flow Fluid dynamics stress strain stress strain Mechanics strain Blood perfusion Electro Active force physiology u calcium Blood Oxygen velocity/flux transport [Ca2+]i ΦO2 SO2 Blood O2 flux Metabolism saturation [ATP] Gas exchange (pulmonary ATP Cardiomyocytes respiration) (adenosine contraction triphosphate) concentration Al o Quarteroni Politecnico di Milano - EPFL fi , , -
On Problem’s Data Al o Quarteroni Politecnico di Milano - EPFL fi
Where data come from : Clinical Images… and how we use them Al o Quarteroni Politecnico di Milano - EPFL fi
Mesh generation tools for cardiac geometries join disconnected tag different regions generate the generate the surfaces through a in order to treat surface mesh volumetric mesh connecting them independently triangulation M. Fedele, A. Quarteroni (2021), Int J Numer Method Biomed Eng Al o Quarteroni Politecnico di Milano - EPFL fi
Preprocessing: from medical images to shapes and grids the computational mesh of the computational mesh of the myocardium the internal cavities Zygote Media Group (2014), Technical repor M. Fedele, A. Quarteroni (2021), Int J Numer Method Biomed Eng Al o Quarteroni Politecnico di Milano - EPFL fi t
Anatomy of cardiac Muscle Fibers Illustrative representation of multiscale cardiac musculature anatomy (b) (d) (c) Myofibers orientation the results of clustered cardiomyocytes R. Chabiniok, M. Sermesant, E. Kuhl, P. Moireau, D. Chapelle, D. Nordsletten (2016), Interface Focus Al o Quarteroni Politecnico di Milano - EPFL fi :
The role of Cardiac Muscle Fibers (Myofibers) EP M Myofibers (Cardiac Muscle fibers determine the electric wave propagation (EP and the muscle contraction (M) Cardiac Muscle Fiber Conductivity Tenso Piola-Kirchhoff Tenso Fiber field can be obtained by: • Imaging techniques (DT-MRI) usually too nois • Mapped from atla complex registration algo • Rule-Based Methods (RBM) Surrogate of myofibers Example: Laplace-Dirichlet-Rule-Based-Methods (LDRBMs) Al o Quarteroni Politecnico di Milano - EPFL fi s y r s r . ) )
LDRBM whole heart fibers Anterior view Posterior view R. Piersanti, P.C. Africa, M. Fedele, C. Vergara, L. Dede', A.F. Corno, A. Quarteroni (2021), CMAME Al o Quarteroni Politecnico di Milano - EPFL fi
From Math to War A Few Clinical “Postcards” Alfio Quarteroni Politecnico di Milano - EPFL d
Electromechanical Modeling of Ischemic Cardiomyopathy Modeling of post-infarction hearts with reduced ejection fraction (EF) taken from magnetic resonance imaging. Application of the virtual-heart arrhythmia risk predictor (VARP) Goal: computational simulations of patient-specific ventricular tachycardia by combining DIGITAL electrophysiology, mechanics and hemodynamics TWIN Salvador, Fedele, Africa, …, Trayanova, Quarteroni; Computers in Biology and Medicine (2021) Alfio Quarteroni Politecnico di Milano - EPFL
Outer Loop and Isthmus in Ventricular Tachycardia Circuits Conduction velocity Goal: characterize the vector approximation electrophysiological substrate of ventricular tachycardia (VT) reentrant circuits. Methods: numerical approximation of conduction velocities during sinus rhythm and ventricular tachycardia in six postmyocardial outer-loop isthmus infarction patients. POPULATION Clinical Partners A. Frontera, S. Pagani,…, A. Quarteroni, P. Della Bella, Heart-Rhythm (2020) Alfio Quarteroni Politecnico di Milano - EPFL :
Systolic Obstruction in Hypertrophic Cardiomyopathy Goal Clinical Partner - computational assessment of haemodynamical effects of Hypertrophic Cardiomyopathy (HCM DIGITAL TWIN - indication on surgical treatment (septal myectomy) non-obstructive obstructive HCM HCM + SAM of the MV suggested region for surgical treatment Fumagalli, Fedele, Vergara, Dede’ et al.; Computers in Biology and Medicine (2020) Fumagalli, Vitullo, Vergara, Fedele et al.; Frontiers in Physiology (2021, under review) Alfio Quarteroni Politecnico di Milano - EPFL s ) s
Cardiac Resynchronisation Therapy Clinical Partner: Cardiology Division, Ospedale S. Maria del Carmine, Rovereto Goal 1: predict the Latest Electrically Activated Segment (LEAS) on the epicardial veins with a reduced, less impactful mapping (limited to the coronary sinus) Goal 2: optimize CRT, evaluating mechanical indices for selecting: - best position for left electrode - best delay between stimuli Left pacing: endocardial veins Right pacing: At the apex/septum Alfio Quarteroni Politecnico di Milano - EPFL
Main numerical challenges ✦ Capturing Multiple Scale ✦ Lack of Data (IC,BC, Coef cients) ✦ Geometric Patient-specific Nonconformit femoropopliteal bypass ✦ Simulation Splitting of aAlgorithm patient-specific femoropopliteal bypass [Marchandise et al, 2011]. Fluid Structure Coupling Geometry Total ✦ Mesh Scalable #1 Preconditioner 9’029’128 2’376’720 338’295 8’674’950 20’419’093 Mesh # 2 71’480’725 9’481’350 1’352’020 68’711’934 151’026’029 ✦ Model Order Reductio 100 Mesh # 1 90 Mesh # 2 ✦ Accounting for Variability and Uncertaint Iterations GMRES [-] 80 70 60 ✦Sensitivity Analysi 50 40 30 20 128 256 512 1024 2048 4096 8192 16384 Number of cores 1000 Vh Vh Vh Mesh # 1 Vh Time step computation [s] Mesh # 2 Mh Mh Mh Mh A. Manzoni, A. Quarteroni, An Overview on Optimal uh(µ1) Control of PDEs 2 uh(µ1) u (µ1) h 1 100 N uh(µ ) uh(µ ) uh(µN ) uh(µN ) µ1 µ2 uNu(µ) 2 h (µ ) uN (µ) uh(µN ) uN (µ) ⇣1 1 ⇣1 Definition of optimal control problems uN (µ) 10 ⇣N 128 256 512 1024 2048 4096 8192 16384 ⇣N ⇣1 In abstract Numberterms, ⇣1 1. of coresa control problem can be expressed by the paradigm illustrated in Fig. There is a system expressed by a state problem that can be⇣Neither an algebraic problem, an Al o Quarteroni initial-value problem for ordinary differential equations, Politecnico di Milano or a boundary-value ⇣-N for problem EPFL Left: post-processing of the solution at times t = 1.8 s, 1.9 s and 2.0 s, respectively. partial differential equations. Its solution, that will generically be denoted by y, depends fi s s n s s y fi y
Physics Base Meet Data-Driven Modeling Alfio Quarteroni Politecnico di Milano - EPFL s d
Artificial Neural Networks Number of hidden layers (nL): • nL = 1 : shallow network ANN • nL ≫ 1 : deep network input layer output layer hidden layers Definition Definition An Artificial Neural Network (ANN) is a function where Tj are affine functions ( ( )= − ) and is a nonlinear activation function acting componentwise. Examples of activation functions Alfio Quarteroni Politecnico di Milano - EPFL
ANNs as functions approximators (see also Joan Bruna’s lecture)
Theorem (Cybenko 1989) – Density of ANNs
Consider a sigmoidal continuous activation function, that is
{1,
0, →−∞
( )→
→+∞
0
Then, given a continuous function ∈ ( ), (where is the hypercube in ℝ ), for any ε > 0, there exists a
single-layer (i.e. nL = 1) ANN such that
( ) − ( ) < ε for any ∈
Theorem (Mhaksar 1996) – Approximation rate
∞ ,
Consider a sigmoidal activation function ∈ (ℝ). Then, given a function ∈ ( ), for any ε > 0 there
exists a single-layer (i.e. nL = 1) ANN such that
( ) − ( ) < ε for any ∈
with space dimension
= (ε− ) regularity
G. Cybenko. Math. Control Signals Systems (1989)
H. N. Mhaskar, Neural Comput. (1996)
Similar results are available: Pinkus, Acta Numer. (1999)
• For in (ℝ ), in Lipschitz spaces, in Sobolev spaces, … S. Liang, R. Srikant arXiv (2016)
T. Poggio et al. Internat. J. Automation Comput. (2017)
• For other types of activation function (e.g. ReLU)
D. Yarotsky, Neural Netw. (2017)
I. Gühring, M. Raslan, G. Kutyniok, arXiv (2020)
Alfio Quarteroni Politecnico di Milano - EPFL
Two Kinds of ANNs Fully connected ANNs Usage: • Regression input output • Classification • Approximation of functions Autoencoders Usage: «code» • Dimensionality reduction • Anomaly detection The ANN is trained to reproduce outputs as close as possible to the inputs (i.e. approximating the identity function) input output Input/output: high-dimensional representation of the datum Code: low-dimensional representation of the datum Encoder: from high dimension to low dimension encoder decoder Decoder: from low dimension to high dimension Alfio Quarteroni Politecnico di Milano - EPFL
Physics Based & Data Driven: a cooperative game Knowledge incorporated into a mathematical model is essential to - make predictions outside the range of the training data se - account for variations of the system (what happens if the rules of the game change? - account for data variability, missing data and errors “Models based on first principles are essential components of systems that extract valuable insights from massive data, insights that tend to go far beyond what can be recovered by black-box statistical modeling alone” (Rüde et al., 2016, arXiv:1610.02608, SIAM Review) Alfio Quarteroni Politecnico di Milano - EPFL ) t :
Dictionary Partial/ordinary differential equations describing problems in Engineering, PDE Life sciences, Natural Sciences, and Finance. High-fidelity numerical solvers (e.g. based on Finite Elements, Finite HF Volumes) that approximate the solution of partial differential equations. Artificial neural networks are architectures made of interconnected nodes (neurons) which can learn arbitrary complex input-output functions by ANN processing paired input and output samples (training data). Reduced-order models based on Machine Learning dimensionality reduction techniques: ROM ● Proper Orthogonal Decomposition; ● Randomized SVD; ● (Discrete) Empirical Interpolation Method (DEIM). Alfio Quarteroni Politecnico di Milano - EPFL
Physics-informed neural networks (PINN): general framework Objective: identify the solution and the unknown parameters given: ANN where: with noisy observations: PINN approach: PDE enforced as a penalization term ROM Find the weights and biases of an ANN and the unknown parameters s.t. where: HF data model (PDE) PDE model (BC) Alfio Quarteroni Politecnico di Milano - EPFL
Physics-informed neural networks: general framework Find the weights and biases of an ANN and the unknown parameters s.t. ANN ROM HF automatic differentiation PDE M. Raissi, P. Perdikaris, G. E. Karniadakis, J. of Comp. Physics (2019) Alfio Quarteroni Politecnico di Milano - EPFL
Multifidelity PINN, I Objective: solve parameter estimation problems in presence of either ● low data ● high noise ANN Method: PDE solution can be expressed as a combination of a low fidelity guess and a high-fidelity correction Find the the weights and biases of an ANN and the unknown parameters s.t. ROM where: data HF model (PDE) model (BC) surrogate or correction PDE reduced-order model F. Regazzoni, S. Pagani, A. Cosenza, A. Lombardi, A. Quarteroni, Rend. Lincei Mat. Appl. (2021) Alfio Quarteroni Politecnico di Milano - EPFL
Multifidelity PINN, II Method: ANN - PDE solution can be expressed as a combination of a low fidelity guess and a high-fidelity correction; ROM - the low-fidelity solution acts as an a priori information; HF - the correction is enforced by regularizing the mathematical equation in the loss function. PDE F. Regazzoni, S. Pagani, A. Cosenza, A. Lombardi, A. Quarteroni, Rend. Lincei Mat. Appl. (2021) Alfio Quarteroni Politecnico di Milano - EPFL
ANN-based emulators Consider a PDE whose solution depends on a set of parameters where ANN Goal: build a computationally inexpensive surrogate of the high-fidelity solvers (e.g. a FEM-based solver), able to approximate the solution given the parameters Through a high-fidelity solver, generate a database of solutions = ( ; ) ROM Train an ANN surrogating the parameter-to-solution map (Reduced Order Model, ROM): ( ; ) HF (+ possibly suitable regularization terms) where − ( ; ) PDE F. Regazzoni, S. Pagani, A. Cosenza, A. Lombardi, A. Quarteroni, Rend. Lincei Mat. Appl. (2021) Alfio Quarteroni Politecnico di Milano - EPFL
ANN-based emulators: issues Issue 1: large number of high-fidelity simulations (that are typically computationally demanding might be required) regularize through the PDE residual (in similar manner to PINNs) Issue 2: how to deal with time-dependent problems? Possible solution: treat time as a further space variable ( ; ) We cannot consider time-dependent inputs We loose physical soundness (no “arrow of time”) Model Learning: learn a time-dependent differential equation rather than a static map Alfio Quarteroni Politecnico di Milano - EPFL
Application 1: ANN-enhanced multiscale simulations Objective: reducing the computational cost associated with the numerical approximation of microscale models (HF model) in multiscale simulations ANN Method: - Through the HF microscale model, we generate a collection of simulations (input-output pairs); - We construct a ROM that surrogates the dynamics of the HF model, built in a data-driven manner from the HF model results; - The ROM consists of a system of ODEs, whose right-hand side is represented ROM by an ANN; - Once trained, the ANN-based ROM is used to surrogate the dynamics of the HF model at the microscale level, within multiscale simulations. Set of input-output pairs Data-driven MOR HF PDE F. Regazzoni, L. Dede’, A. Quarteroni Jour. Comp. Phys. (2019) F. Regazzoni, L. Dede’, A. Quarteroni CMAME (2020) Alfio Quarteroni Politecnico di Milano - EPFL
Application 1: ANN-enhanced multiscale simulations
validation
ANN
High-fidelity model Reduced-order
of force generation model
actin filament
upscaling
downscaling
myosin filament
ROM
a priori
knowledge
+ Model
Learning
HF
Collection of 3D FEM
simulations electromechanic
{ui(t), yi(t)} al model
PDE
F. Regazzoni, L. Dede’, A. Quarteroni CMAME (2020)
Alfio Quarteroni Politecnico di Milano - EPFLApplication 1: ANN-enhanced multiscale simulations Results Accuracy Calcium wave Tissue deformation Indicator HF-EM ANN-EM Relative error propagation ANN Stroke volume (mL) 58.45 58.42 5.64 · 10-4 Ejection fraction (%) 43.03 43.01 5.65 · 10-4 Max pressure (mmHg) 112.5 112.3 2.18 · 10-3 Work (mJ) 739.2 737.2 1.71 ∙ 10-3 ROM Computational time (20 cores) Force Mechanic Ionic Potential Total gen. s Cardiac fibers Pressure-volume HF-EM 3.13 % 0.47 % 83.07 % 13.33 % 20h 18’ loop HF model ANN model ANN-EM 41.21 % 4.80 % 2.54 % 51.45 % 2h’ 03’ HF 400 x speedup (force generation model) 10 x speedup (overall) Memory usage from 2198 (HF-EM) to 24 (ANN-EM) variables per nodal point PDE 100 x memory saving F. Regazzoni, L. Dede’, A. Quarteroni CMAME (2020) Alfio Quarteroni Politecnico di Milano - EPFL
A look ahead - How will AI shape computational medicine MACHINE learning from data LEARNING statistical improved shapes and learning imaging diagnosis morphologies optimal catheter-based surgery or POPULATION measurements biomarkers DIGITAL device PATIENT and clinical TWIN indicators clinical history and therapy comorbidities planning Databases learning from physics MATHEMATICAL MODELS VIRTUAL in-silico clinical trial POPULATION Alfio Quarteroni Politecnico di Milano - EPFL
THANK YOU The iHEART Team @ PoliMi
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