Fast and robust parameter estimation with uncertainty quantication for the cardiac function Matteo Salvador1 Francesco Regazzoni1 Luca Dede1 Alo Quarteroni12

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Fast and robust parameter estimation with uncertainty

quantiﬁcation for the cardiac function

Matteo Salvador1,∗, Francesco Regazzoni1, Luca Dede’1, Alﬁo Quarteroni1,2

1MOX-Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

2´

Ecole Polytechnique F´ed´erale de Lausanne, Lausanne, Switzerland (Professor Emeritus)

∗Corresponding author (matteo1.salvador@polimi.it)

Abstract

Parameter estimation and uncertainty quantiﬁcation are crucial in computational cardiol-

ogy, as they enable the construction of digital twins that faithfully replicate the behavior of

physical patients. Robust and eﬃcient mathematical methods must be designed to ﬁt many

model parameters starting from a few, possibly non-invasive, noisy observations. Moreover, the

eﬀective clinical translation requires short execution times and a small amount of computational

resources. In the framework of Bayesian statistics, we combine Maximum a Posteriori estima-

tion and Hamiltonian Monte Carlo to ﬁnd an approximation of model parameters and their

posterior distributions. To reduce the computational eﬀort, we employ an accurate Artiﬁcial

Neural Network surrogate of 3D cardiac electromechanics model coupled with a 0D cardiocir-

culatory model. Fast simulations and minimal memory requirements are achieved by using

matrix–free methods, automatic diﬀerentiation and automatic vectorization. Furthermore, we

account for the surrogate modeling error and measurement error. We perform three diﬀerent in

silico test cases, ranging from the ventricular function to the entire cardiovascular system, in-

volving whole-heart mechanics, arterial and venous circulation. The proposed method is robust

when high levels of signal-to-noise ratio are present in the quantities of interest in combination

with a random initialization of the model parameters in suitable intervals. As a matter of fact,

by employing a single central processing unit on a standard laptop and a few hours of com-

putations, we attain small relative errors for all model parameters and we estimate posterior

distributions that contain the true values inside the 90% credibility regions. With these beneﬁts,

our approach meets the requirements for clinical exploitation, while being compliant with Green

Computing practices.

Keywords: Cardiac electromechanics, Machine Learning, Surrogate modeling, Parameter estima-

tion, Uncertainty quantiﬁcation

1 Introduction

Personalization of computational heart models is necessary to better addressing patient–speciﬁc

pathophysiology and for assisting the clinicians in the decision–making process for medical treatment

[26, 51]. In this ﬁeld, sophisticated cell-to-organ level mathematical models comprising systems of

nonlinear diﬀerential equations, along with eﬃcient and accurate numerical methods, have been

developed to properly describe the physical phenomena underlying the cardiac function [1, 9, 11,

29, 33, 39, 49, 50].

arXiv:2210.03012v1 [math.NA] 6 Oct 2022

Numerical simulations using these biophysically detailed and anatomically accurate mathematical

models call for high computational costs and for a signiﬁcant amount of computational resources

[40]. Imaging techniques are combined with numerical simulations to perform robust parameter

estimation in patient–speciﬁc cases [24, 25, 43, 47]. On the other hand, multi-ﬁdelity models of

cardiac electromechanics, deep learning-based models of cardiac mechanics or simpliﬁed lumped

circulation models are also employed for the same purpose [6, 17, 38, 46]. All these mathematical

tools mainly focus on the ventricular activity of the human heart.

In this paper, we present a numerical strategy to perform parameter calibration with uncertainty

quantiﬁcation (UQ) by means of a reduced–order model (ROM) of 3D cardiac electromechanics

coupled with closed–loop blood circulation [40]. The ROM, which is based on Artiﬁcial Neural Net-

works (ANNs), encodes the dynamics of the pressure-volume relationship obtained from an accurate

full–order model (FOM) of the cardiac function [9, 33, 39]. Moreover, it allows for real-time numer-

ical simulations on a personal computer while embedding electromechanical parameters of the 3D

mathematical model [40].

Parameter estimation is carried out by solving a constrained optimization problem with an

eﬃcient adjoint-based method that exploits matrix–free methods, automatic diﬀerentiation and au-

tomatic vectorization [5, 23]. Then, we account for the uncertainty coming from possible model

and measurement errors. Speciﬁcally, we employ the Hamiltonian Monte Carlo (HMC) algorithm

to perform inverse UQ [2, 14].

We verify our approach against in silico data with diﬀerent levels of signal–to–noise (SNR)

ratio. We consider several non-invasive time-dependent quantities of interest (QoIs), such as arterial

systemic pressure, atrial and ventricular volumes, in order to estimate many model parameters,

ranging from cardiac mechanics to cardiovascular hemodynamics. This can be done in a few hours

of total execution time while simply employing one Central Processing Unit (CPU) of a standard

laptop. Our method can therefore be applied to clinical data, where accuracy, robustness and

timeliness are certainly essential.

2 Mathematical models

We display in Figure 1 several mathematical models for the cardiac function featuring a diﬀer-

ent degree of physical accuracy and computational complexity. These mathematical models can

be regarded as a FOM for cardiac electromechanics EM3D and three diﬀerent ROMs. Although

model EM3D presents a high level of biophysical accuracy, it is also associated to high performance

computing and signiﬁcant computational costs. This motivates the use of ROMs, which are compu-

tationally cheaper than the corresponding FOM in terms of execution time and computer resources.

Moreover, they do not signiﬁcantly compromise the accuracy of the FOM. These ROMs simulate

the pressure-volume relationship of one or multiple cardiac chambers and can be all employed for

fast and robust parameter estimation. We will use the following taxonomy:

•EMANN: ANN based surrogate models of cardiac electromechanics, built as black-box from a

collection of pre-computed numerical simulations through a data-driven approach [40];

•EMEMULATOR: parametric emulators of cardiac electromechanics built with a grey-box ap-

proach, that is by ﬁtting a priori deﬁned physics-inspired curves from data obtained by means

of numerical simulations [38];

•EM0D: fully 0D electromechanical models, i.e. time-varying elastance models assuming a

linear relationship between pressure and volume [13, 39].

The aforementioned mathematical models are coupled with a generic circulation model Cof the

remaining part of the cardiovascular system by exchanging pressures and volumes. For the sake of

simplicity, in this paper we consider a ROM built with the EMANN model for the left ventricle (LV)

only.

2.1 3D electromechanical model

We model the electromechanical activity of the LV by means of a set of diﬀerential equations. In

compact form we can write the model as [38, 40]:







∂y(t)

∂t =L(y(t), pLV(t), t;θEM) for t∈(0, T ],

y(0) = y0.

(1)

The nonlinear diﬀerential operator Lencodes the diﬀerential equations and boundary conditions,

while the state vector y(t) contains several variables associated with the cardiac function, such as the

action potential, intracellular calcium concentration, sarcomere length and mechanical deformation.

y0represents the initial condition. pLV(t) indicates the endocardial pressure of the LV. We introduce

ΘEM ⊆RNEM , that is the parameter space, being NEM the number of parameters, and we denote

as θEM the model parameters for cardiac electromechanics such that θEM ∈ΘEM. Examples of

electromechanical parameters are electric conductivities, ﬁbers direction, contractility and passive

stiﬀness of the myocardium.

From now on, we label Equation (1) as EM3D, which requires a closure relationship to determine

the pressure pLV(t). Indeed, we couple EM3D with a generic circulation model of the cardiovascular

system, henceforth denoted as C[1, 29, 33, 39]. The coupled problem reads [38, 40]:











∂y(t)

∂t =L(y(t), pLV(t), t;θEM) for t∈(0, T ],

dc(t)

dt =f(c(t), pLV(t), t;θC) for t∈(0, T ],

V3D

LV (y(t)) = V0D

LV (c(t)) for t∈(0, T ],

y(0) = y0,

c(0) = c0.

(2)

The state variables c(t) of the cardiocirculatory model contain pressures, volumes and ﬂuxes in

diﬀerent compartments of the vascular network, while θC∈ΘC⊆RNCis a vector of NCparame-

ters that contains, for instance, resistances, conductances or elastances. The volumetric constraint

V3D

LV (y(t)) = V0D

LV (c(t)) between EM3D and Cmodels is enforced by means of pLV(t) [39], which acts

numerically as a Lagrange multiplier [39]. An alternative approach would be to adapt the closure

relationships to the diﬀerent phases of the heartbeat [21, 42], e.g. by using preload and afterload

windkessel models [54].

The surrogate models that will be introduced in the next sections are built from the 3D-0D

closed-loop mathematical model EM3D − C presented in [33, 39]. We consider a LV obtained from

the Zygote 3D human heart model [55], endowed with a ﬁber architecture generated by means of

suitable Laplace-Dirichlet-Rule-Based methods [32]. Nevertheless, our mathematical and numerical

models directly generalize to patient-speciﬁc geometries [43]. For cardiac electrophysiology, we

employ the monodomain equation [7] coupled with the ten Tusscher-Panﬁlov ionic model [52]. We

model mechanical activation in the active stress formulation by means of the biophysically detailed

θCθEM

pressure

volume

Circulation model Electromechanical model

displacement [mm]

0 21

p Q

E(t)

pressure

volume

pressure

volume

pressure

volume

Complexity

Computational cost

L R C

Figure 1: Representation of diﬀerent EM −C models. A generic circulation model Cmay be coupled

with biophysically detailed and anatomically accurate EM3D, ANN-based EMANN, emulator-based

EMEMULATOR or 0D EM0D electromechanical models according to the requirements in terms of

accuracy, computational eﬃciency and memory storage.

and anatomically accurate RDQ20-MF model [35]. We consider the Guccione constitutive law in a

quasi-incompressible regime for passive mechanics [53]. We adopt spring-damper Robin boundary

conditions at the epicardium to account for the presence of the pericardial sac [12, 30]. We prescribe

energy-consistent boundary conditions to model the interaction with the part of the myocardium

beyond the artiﬁcial ventricular base [37]. Blood circulation over the whole cardiovascular system

is modeled by using a closed-loop mathematical model Cproposed in [13, 39]. For the sake of

completeness, we report the equations of the 0D cardiocirculatory model in Appendix A.

2.2 Artiﬁcial neural network based reduced-order model

Following the ANN-based ROM introduced in [40], we build a set of ordinary diﬀerential equations

(ODEs), whose right hand side is represented by an ANN, that learns the pressure-volume dynamics

of the 3D cardiac electromechanical model EM3D reported in Equation (1). In this framework, the

ANN-based ROM EMANN for the LV reads:







dz(t)

dt =N N z(t), pLV(t),cos( 2πt

THB ),sin( 2πt

THB ),θEM;b

wfor t∈(0, T ],

z(0) = z0.

(3)

The fully connected feedforward ANN is deﬁned by N N :RNz+NE M +3 →RNzand z(t)∈RNz

represents the reduced state vector. The ANN receives Nzstate variables, NEM scalar parame-

ters, pressure pLV, and two periodic conditions in time as input. Indeed, the cos(2πt/THB) and

sin(2πt/THB) terms account for the periodicity THB of the heartbeat, thus allowing for arbitrarily

long real-time numerical simulations of cardiac electromechanics [40]. Finally, weights and biases of

the ANN are encoded in b

w∈RNw.

We train the ANN to enable the predicted LV volume to coincide with the ﬁrst state variable, i.e.

z(t)=[VANN

LV (z(t)),e

z(t)]Twith e

z(t)∈RNz−1. In this manner, by following the notation introduced

in [36], we consider an output-inside-the-state approach. The remaining components of vector z(t),

that is e

z(t), are latent variables without immediate physical interpretation. For all the details

regarding the model training strategy of this ANN-based ROM we refer to [40].

The coupled EMANN − C model gives rise to a diﬀerential-algebraic system of equations (DAEs)

that reads:











dz(t)

dt =N N z(t), pLV(t),cos( 2πt

THB ),sin( 2πt

THB ),θEM;b

wfor t∈(0, T ],

dc(t)

dt =f(c(t), pLV(t), t;θC) for t∈(0, T ],

V0D

LV (c(t)) = VANN

LV (z(t)) for t∈(0, T ],

z(0) = z0,

c(0) = c0.

(4)

In order to perform parameter estimation by employing an adjoint sensitivity method [5], we need

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FastandrobustparameterestimationwithuncertaintyquanticationforthecardiacfunctionMatteoSalvador1;,FrancescoRegazzoni1,LucaDede'1,AloQuarteroni1;21MOX-DipartimentodiMatematica,PolitecnicodiMilano,Milan,Italy2EcolePolytechniqueFederaledeLausanne,Lausanne,Switzerland(ProfessorEmeritus)Correspondi...

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Fast and robust parameter estimation with uncertainty quantication for the cardiac function Matteo Salvador1 Francesco Regazzoni1 Luca Dede1 Alo Quarteroni12

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