Parameter estimation tools for cardiovascular ﬂow modeling of fetal circulation Gabriella Bretti1 Roberto Natalini1 Annalisa Pascarella1 Giancarlo

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Parameter estimation tools for cardiovascular ﬂow

modeling of fetal circulation

Gabriella Bretti 1,*, Roberto Natalini1, Annalisa Pascarella1, Giancarlo

Pennati2, Daniele Peri1, Giuseppe Pontrelli1,

1 Istituto per le Applicazioni del Calcolo – CNR, Via dei Taurini 19 – 00185

Rome, Italy

2 Laboratory of Biological Structure Mechanics, Department of Chemistry,

Materials and Chemical Engineering “Giulio Natta”, Politecnico di Milano,

Piazza Leonardo da Vinci, 32 – 20133 Milano, Italy

* gabriella.bretti@cnr.it

Abstract

Usually, clinicians assess the correct hemodynamic behavior and fetal well-

being during the gestational age thanks to their professional expertise, with

the support of some indices deﬁned for Doppler fetal waveforms. Although

this approach has demonstrated to be satisfactory in the most of the cases,

it can be largely improved with the aid of more advanced techniques, i.e.

numerical analysis and simulation. Another key aspect limiting the analysis

is that clinicians rely on a limited number of Doppler waveforms observed

during the clinical examination. Moreover, the use of simple velocimetric

indicators for deriving possible malfunctions of the fetal cardiovascular sys-

tem can be misleading, being the fetal assessment based on a mere statistical

analysis (comparison with physiological ranges), without any deep physio-

pathological interpretations of the observed hemodynamic changes. The use

of a lumped mathematical model, properly describing the entire fetal cardio-

vascular system, would be absolutely helpful in this context: by targeting

physiological model parameters on the clinical reliefs, we could gain deep

insights of the full system. The calibration of model parameters may also

help in formulating patient-speciﬁc early diagnosis of fetal pathologies. In

arXiv:2210.02726v1 [physics.med-ph] 6 Oct 2022

the present work, we develop a robust parameter estimation algorithm based

on two diﬀerent optimization methods using synthetic data. In particular, we

deal with the inverse problem of recognizing the most signiﬁcant parameters

of a lumped fetal circulation model by using time tracings of fetal blood ﬂows

and pressures obtained by the model. This represents a ﬁrst methodological

work for the assessment of the accuracy in the identiﬁcation of model pa-

rameters of an algorithm based on closed-loop mathematical model of fetal

circulation and opens the way to the application of the algorithm to clinical

data.

Keywords: Fetal circulatory system, lumped parameter model, Diﬀerential

algebraic equations, Simulation and numerical modeling, Parameter estima-

tion techniques, Inverse problem.

MSC: 65L80; 81T80; 78M50; 70F17

Introduction

The mathematical study of the adult human blood circulation is a quite

consolidated subject, since the ﬁrst work about it dates back in the late

nineteenth-century [13]. On the contrary, the investigation of fetal (i.e. in

utero) blood circulation is much more recent, with the modeling of sheep fe-

tal cardiovascular system based on animal studies [20] and the ﬁrst modeling

studies of Doppler waveforms in the human umbilical placental circulation,

about a century later [39, 40].

Doppler techniques are a powerful tool to assess fetal blood circulation as

they allow for identiﬁcation of characteristic blood velocity proﬁles in the fe-

tal arterial and venous tree during gestation. Abnormal velocity waveforms

have been associated with adverse perinatal outcome or cardiovascular dis-

eases [19]. A complete understanding of fetal hemodynamics and circulatory

patterns is necessary for the correct application and interpretation of Doppler

ﬁndings [6, 31] and for their best diagnostic use.

It is worth noting that most of the techniques typically adopted for the post-

natal circulation cannot be applied to investigate in utero blood circulation,

and ultrasound based approaches, such as Doppler velocimetry and echo-

graphic imaging, are the only ones applicable in the routine fetal surveillance

and prenatal diagnosis. Indeed, catheter-based measurements are completely

unsuitable because they are too invasive for a fetus (they are applied in few

situations, as the only diagnostic possibility for speciﬁc pathological cases),

thus practically preventing blood pressure measurements. Furthermore, mag-

netic resonance imaging, that is readily used for quantiﬁcation of blood ﬂow

in adult circulation is highly compromised in utero by spontaneous fetal mo-

tions. In turn, these limitations hinder the possibility of evaluating important

fetal parameters like vascular resistances and compliances or ventricular and

atrial elastances. These parameters signiﬁcantly evolve during gestation and

their values can be indicative of a proper or abnormal fetal development and

growth.

In comparison with the information deducible from the individual Doppler

velocimetric tracings that can be only indirectly related to fetal vascular

conditions, computational models have the advantage of providing a more

global view on hemodynamics and, when applied to patient-speciﬁc cases,

could allow the quantiﬁcation of circulatory parameters that are currently

not measurable in the fetus.

A number of existing models have contributed signiﬁcantly to our under-

standing of the fetal circulation, because they allowed the investigation of the

inﬂuence of various parameters on the ﬂow pulse waveforms in fetal districts,

with reference to a generic blood circulation. Diﬀerent mathematical ap-

proaches were adopted to model the whole human fetal cardiovascular system

or an individual portion, and they can be classiﬁed as zero-dimensional open-

loop [6, 27, 32], zero-dimensional closed-loop [26, 30, 45], one-dimensional

open-loop models [16, 17, 36, 43], three-dimensional models [9, 33, 34, 44].

As mentioned above, the study of the fetal blood circulation is quite com-

plicated because of the impossibility in obtaining clinical measurements of

blood pressures and ﬂow rates. The correct hemodynamic behavior at the

diﬀerent stages of the gestation are usually assessed referring to some de-

scriptive indices (e.g. pulsatility indices) deﬁned for each Doppler waveform,

compared with existing normal standard ranges [10].

The use of mathematical models of the human fetal circulation may help to

improve the understanding of the hemodynamic factors determining the in-

dex values and, most importantly, the development of non-invasive mathematical-

based forecasting tools based for the early diagnosis of fetal pathologies.

Indeed, mathematical models predicting Doppler tracing are of interest for

clinicians: unusual shapes of velocimetric waveforms may indicate abnormal

values of vascular parameters, due to a direct inﬂuence of a speciﬁc disease

(e.g. placental disease) or as a compensatory eﬀect of another disease (e.g.

brain-sparing eﬀect during fetal growth retardation). Nevertheless, the diag-

nostic signiﬁcances of the various suggested indices is often limited, as the

anomalies of Doppler indices cannot be uniquely associated to a clear cause,

with the risk that the pathological state in the vascular system is not detected

at very early stages of disease development.

According to the speciﬁc scale of the phenomenon to be studied, various

degrees of simpliﬁcation at some levels have been proposed. One of them

concerns the geometrical dimension of the model. In the current study, we

focus onto zero-dimensional models, or lumped parameters models, where

diﬀerent regions of the vascular system are grouped in blocks and connected

together, in analogy with the electrical circuits. The number of blocks is

related to the desired degree of detail, and, often, the blocks form a closed

loop where both blood circulation and cardiac chambers are modeled with

suitable lumped parameters to describe the whole cardiovascular system.

The main advantage of closed-loop model approach is the possibility to

describe the entire circulatory system and the blood pressure-velocity rela-

tionship in each block with a relatively simple and computationally eﬀective

model. Here, in particular, we refer to a closed-loop lumped model of the

fetal circulation developed and validated by Pennati et al. [30], able to pre-

dict the values for a large number of Doppler indices in a healthy human

fetus. The model, originally set for a fetus at term of the gestation, was then

extended to other gestational months [31].

Although the clinical informations provided by blood ﬂow Doppler measure-

ments are very useful to assess the status of the examined fetus, they are still

limited to few measuring sites and then provide a quite partial description

of the cardiovascular system. The main goal of the methodology proposed in

[30, 31] is to assist the clinicians with a computational tool able to interpret

the collected data (blood ﬂow Doppler velocities), estimate blood pressures

(always not measurable) and ﬂows in the fetal vascular regions not examined

and, more generally, assess the vascular status of fetus across gestational

ages. Namely, the main strength of using a closed-loop lumped model is

the possibility to build what-if scenarios by investigating the impact of vas-

cular modiﬁcations or adaptations due to a disease. For instance, it allows

the investigation of the hemodynamic changes when some model parame-

ters are modiﬁed (e.g. resistances and compliances, as during a peripheral

vasodilation or vasoconstriction occurring in intra-uterine growth retarda-

tion), without directly imposing any ﬂows or pressures. On the contrary,

open-loop circuits, which describe a limited part of blood circulation (e.g.

the arterial tree) see [6], imply to assume ﬁxed boundary conditions to the

model, when instead they should modify due to parameter changes. Hence,

if a lumped parameter model of the fetal cardiovascular system is conceived

to be signiﬁcantly applied in fetal diagnosis, the use of a closed-loop circuit

is mandatory.

An additional important step towards a clinical use of the model is to

obtain patient-speciﬁc parameters able to describe individual fetal circula-

tions. This implies the capability of identifying the speciﬁc values of all the

parameters based on few clinically available information. In fact, a number

of studies suggesting various approaches to estimate patient-speciﬁc lumped

parameters can be found into the literature, applied to either open- [8, 11]

or closed-loop [28, 37] models of the blood circulation. Nevertheless, these

models are generally devoted to the investigation of the postnatal blood

circulation (neonatal or adult, where more and complete clinical data can

be collected), with a single study by Garcia-Canadilla et al. that focuses

on the in utero circulation [7]. Namely, the fetal arterial circulation was

described in an-open loop conﬁguration where two patient-speciﬁc blood-

ﬂow inputs (ventricular outﬂows) and a reference downstream pressure are

imposed as boundary conditions. A constrained nonlinear optimization al-

gorithm, minimizing the mismatch between computed and measured blood

velocity waveforms in three fetal vessels, was employed to estimate 13 model

parameters. The methodology was applied to 37 real cases (22 healthy and

15 growth-retarded fetuses), and a good the matching of the target func-

tions was reported. Nevertheless, when the number of unknown parameters

is high, and the available patient-speciﬁc data are limited and aﬀected by

uncertainties (e.g., variable heart rates can be observed across the various

time tracings), a preliminary analysis is suggested to verify the identiﬁability

of the parameters, before applying the method to real patients [37].

For this reason, as a ﬁrst step, a virtual patient should be considered,

where the clinical targets consisting in time tracings of blood ﬂows and pres-

sures are generated through a forward model solution (for the sake of simplic-

ity, in the following we will indicate these targets as synthetic data). A deep

check of the ability of the identiﬁcation methodology to properly estimate

the vascular and biomechanical model parameters is crucial before the model

can be used as a descriptive tool of the fetal circulatory system and as a pre-

dictive tool for an early detection of fetal pathologies. The use of synthetic

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ParameterestimationtoolsforcardiovascularowmodelingoffetalcirculationGabriellaBretti1,*,RobertoNatalini1,AnnalisaPascarella1,GiancarloPennati2,DanielePeri1,GiuseppePontrelli1,1IstitutoperleApplicazionidelCalcoloCNR,ViadeiTaurini1900185Rome,Italy2LaboratoryofBiologicalStructureMechanics,Department...

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Parameter estimation tools for cardiovascular ﬂow modeling of fetal circulation Gabriella Bretti1 Roberto Natalini1 Annalisa Pascarella1 Giancarlo

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