A review of multiscale 0D-1D computational modeling of coronary circulation with applications to cardiac arrhythmias

Computational hemodynamics is becoming an increasingly important tool in clinical applications and surgical procedures involving the cardiovascular system. Aim of this review is to provide a compact summary of state of the art 0D-1D multiscale models of the arterial coronary system, with particular attention to applications related to cardiac arrhythmias, whose effects on the coronary circulation remain so far poorly understood. The focus on 0D-1D models only is motivated by the competitive computational cost, the reliability of the outcomes for the whole cardiovascular system, and the ability to directly account for cardiac arrhythmias. The analyzed studies show that cardiac arrhythmias by their own are able to promote significant alterations of the coronary hemodynamics, with a worse scenario as the mean heart rate (HR) increases. The present review can stimulate future investigation, both in computational and clinical research, devoted to the hemodynamic effects induced by cardiac arrhythmias on the coronary circulation.

Under these hypotheses, the 1D model is composed by continuity and momentum equations in one-dimensional form, obtained as the integro-differential form of the balance equations of mass and momentum, and assuming: (i) flow properties as function of a single spatial variable (i.e., the vessel axial coordinate x), and (ii) uniform flow properties over the vessel crosssectional area A, normal to the axial coordinate x. In so doing, x and t (time) are the independent variables, while area (A), pressure (P), and flow rate (Q) are the dependent variables. A constitutive equation -being linear or non-linear to account for the vessel elasticity or viscoelasticity respectively -links P and A, along the x direction and completes the differential system. The general form of the system of hyperbolic partial differential equations reads where τ is the wall shear stress, ρ is the blood density, r is the vessel internal radius, and is a linear or non-linear function of A.
If instead the coronary circulation is modelled through a 0D lumped parameter approach [2,13,14], the hemodynamics is modeled through a suitable combination of electrical counterparts: resistances (R) accounting for the viscous-dissipative effects, compliances/elastances (C/E) accounting for distensibility/elasticity effects, and inertances (L) accounting for inertial effects, giving rise to the most general 3-elements (or RLC) Windkessel model. Through the electric analogue, the only independent variable is time (t) and each vascular compartment is described by three time-dependent hemodynamic variables: pressure (P), flow rate (Q), volume (V). With reference to the RLC circuit reported in Fig. 1, each i-th compartment is characterized by three ordinary differential equations: an equation for the conservation of mass (expressed in terms of volume variation), a momentum equation (accounting for the flow variation), and a linear state equation between pressure and volume.
Namely: Equations (1) e (2) represent the general and common form for the 1D and 0D hemodynamic modeling, respectively. For the sake of simplicity and space, we neglect details of the different modeling choices adopted in literature, which however are formulations obtained starting from the above set of equations (1) and (2);  Upstream coupling with systemic circulation is usually handled by 1D or 0D models [2,13,14] as described in Eq. (1) and (2). Typical flow rate and pressure signals at the aortic root are sometimes directly imposed as inlet boundary conditions for the coronary circulation;  Downstream coupling with distal coronary circulation is in most of the cases dealt with 3elements Windkessel models [2,13,14] as described in Eq. (2) and sometimes with a single impedance element;  Mechanical coupling between the myocardial tissue and coronary walls can be included to account for the vessel squeezing during the heart contraction phase [10,12,15,16];  Different mechanisms of myocardial blood flow regulation, such as myogenic, flow, and metabolic control, can be added to the coronary modeling [12,17,18].
Both mechanical heart contraction coupling and autoregulation mechanisms are modeled through algebraic or ordinary differential equations, which are coupled to the arterial coronary model.

Results
We here report a chronological overview of 0D-1D coronary blood flow modeling appeared in the last two decades, by mainly focusing on the methodological solutions adopted, results, and aim of each study. Then, recent applications of 1D coronary modeling in presence of cardiac arrhythmias are described more in detail.

Overview of 0D-1D coronary blood flow modeling
We start with the paper by Smith et al. (2002) [19], which is a seminal and numerically-oriented work in this area . The authors implemented a very detailed and artificially-built coronary network through   a classical 1D model, with parabolic velocity profile, elastic walls and no time dependent resistances to account squeezing due to heart contraction. Distal coronary circulation was modeled through appropriate 0D models. The aim of the paper was to describe a mathematical model and discuss its numerical properties, through idealized configurations, such as propagation along the network of an impulsive pressure input and wash-out of the network. The results showed physiologically realistic flow rates, washout curves, and pressure distributions. The 0D model was divided into three transmural layers (subendocardium, midwall, and subepcardium), while autoregulation mechanisms related to the intramyocardial pressure and myocardial filling were as well accounted for. In a later paper of 2015 by Mynard et al. [22], the coronary modeling considered only the main coronary arteries and branches, was extended to the right coronary circulation and coronary veins, and was eventually included into a 0D-1D model of the entire adult circulation. The purpose was to present and in vivo validate a coronary model able to describe microvascular properties that differ regionally and transmurally, as well as wave propagation effects in the conduit arteries. The authors were able to quantify forward and backward wave intensity as well as the interaction between cardiac function/mechanics and wave dynamics. No greater prominence of wave propagation effects was found in adults compared with newborns. The 0D-1D coronary is upstream coupled with the left heart and systemic arterial tree, which is modeled as the large coronary conduits (i.e., by 1D models with non-linear viscoelastic arterial wall).
The aim was to study age-induced effects on the wave propagation and reflection, as well as on the ventricular function. The authors found that the subendocardial viability ratio decreases with age, the total coronary flow slightly reduces, and the left ventricular work increases, resulting in a possible oxygen supply-demand unbalance due to physiological aging.
The works by Ge et al. in 2018 [25,26] are as well based on the models by Mynard et al. [21,22], contraction-relaxation mechanism, by considering the interplay between coronary perfusion and the corresponding regional contractility. The most recent paper [29] also included three mechanisms of flow regulation: shear response (inducing relaxation of the vessel radius due to changes in the vascular wall shear stress), myogenic responses (i.e., the ability of vascular smooth muscle to constrict a vessel in response to an increase of the transvascular pressure) and flow regulation controlled by a vasodilator signal related to physiological necessity. Aim of the two works was to investigate the mechanical dyssynchrony, namely the contractile dyssynchrony between the septum and left ventricular free wall due to asynchronous activation of the heart. The outcomes revealed that regional intramyocardial pressures play a significant role in affecting regional coronary flows in mechanical We conclude this overview by briefly mentioning few recent works, where coronary blood flow was modeled through a fully lumped parameter (0D) approach, by means of electrical counterparts adequately tuned on the basis on measured data. The aim of these studies was diversified but in all cases quite specific: to obtain a personalized framework for the coronary perfusion [31,32], to have a faster computational tool for planning coronary artery bypass surgery [33], and to focus on myocardial mechanical function coupled with coronary perfusion [16]. bradycardia, long QT syndrome and premature ventricular contraction. The assigned discharge at aorta root was always modelled as a sinusoidal function. The investigated cases were obtained by deterministically changing the frequency and the amplitude of such function, while no stochastic term was introduced. The authors found that coronary blood flow (CBF), which is the net blood flow through coronary arteries (LCA and RCA) per beat, was significantly affected by arrhythmias. In particular, with respect to the baseline pacing case (which is here reported in Fig. 2a for the overall CBF, accounting for LCA+RCA), for long QT syndrome CBF decreased at rest (60 bpm) by 26% in LCA and 22% in RCA (see Fig. 2b). During bigeminy, trigeminy, and quadrigeminy, overall CBF decreased by 28%, 19%, and 14% with respect to the baseline pacing at rest (60 bpm), respectively (see Fig. 2c). RCA, and their main branches) was implemented similarly to [22], through 1D models with explicit non-linear viscoelastic wall properties. A 0D model, divided into three transmural layers (subendocardium, midwall, and subepcardium), was adopted as distal condition for each penetrating vasculature and microcirculatory district downstream of each 1D coronary artery. The coronary model was included into a 0D-1D model of the entire adult circulation, accounting for 1D modeling of the systemic arterial tree (which served as the upstream boundary condition for the coronary circulation), and a 0D lumped parameter representation of the venous return, contractile heart, and pulmonary circulation. The model also included a short-term baroreceptor mechanism accounting for the control of the systemic vasculature, chronotropic and inotropic effects. AF was simulated by stochastic beating extraction as in [35], with both atrial elastances constant to mimic the absence of atrial kick. Three mean HR were simulated (75, 100, 125 bpm) in both sinus rhythm (SR) and AF.

Multiscale coronary modeling: application to cardiac arrhythmias
An inter-layer and inter-frequency analyses were conducted downstream of the three main coronary arteries (LAD, Cx, RCA), by focusing on the ratio between mean beat-to-beat blood flow in AF compared to SR, / . As reported in Fig. 4, AF caused a direct reduction in microvascular coronary flow particularly at higher HR, with the most important decrease seen in the subendocardial layers perfused by LAD and Cx.

Discussion
The 0D-1D computational modeling of coronary circulation here reviewed highlights interesting This is probably due to the fact that LCA is considered a conduit artery being much more close to the aortic root and less representative of the intralayer perfusion and the microvasculature.
In the specific case of cardiac arrhythmias, the analyzed studies agree in identifying a maximum of CBF (either computed on the LAD or overall on the LCA + RCA) around an average HR of 90-110 bpm. However, the reduced coronary perfusion beyond this HR threshold is not matched by a reduction of oxygen consumption, resulting in an unbalance of the coronary supply-demand ratio. QT syndrome appears to affect right and left coronary perfusion similarly, but to a greater extent the lower HRs. Instead, left coronary microvasculature is much more affected by AF than the right counterpart and this can be explained by the reduced driving pressure (aortic pressure) and the increased extravascular force (left ventricular end-diastolic pressure), both exacerbated by increasing HR. These latter mechanisms can also explain why the subendocardial layer is the most prone to AFinduced alterations. Both in AF and in quadrigeminy, trigeminy and bigeminy cases, coronary perfusion worsens for increasing HR.
Even though still small in number, computational studies already outline important clinical implications. First, the role of irregular RR beating in AF on the coronary perfusion is poorly understood. Several mechanisms -such as reduced aortic pressure related to short RR intervals, coronary vasoconstriction, and reduced coronary blood flow -have been proposed to explain how irregular ventricular rhythm can exert negative effects on the coronary hemodynamics. However, the precise mechanisms through which AF impacts the coronary circulation still remain to be explored [7]. Another clinical implication is related to the impact of ventricular rate on the coronary perfusion.
Although lenient (resting HR < 110 bpm) and strict (resting HR < 80 bpm) rate control strategies were found not to differ in terms of mid-term cardiovascular outcomes [37], definitive data are still missing [38] and specific findings related to coronary flow impairment are overall scarce. For both rhythm and rate alterations, computational modeling can suggest useful evidences to clinical practice and better address further necessary in vivo investigation.
Given the frequent coexistence of AF and other cardiac diseases -such as coronary artery disease and heart failure [39,40] -and the substantial absence of computational studies considering the concomitant presence of different cardiac pathologies, future modeling efforts should be devoted to investigate to what extent the interaction of AF with other cardiac diseases affects the coronary circulation.

Limitations
3D computational approaches were excluded from this review, as the focus was on 0D-1D models only. While the first ones are much more detailed but computationally expensive, being not feasible to describe the entire circulation but only the local hemodynamics, the latter represent a good settlement between the computational cost, the level of the hemodynamic details, and the reliability of the outcomes for the whole cardiovascular system. 0D-1D models are able to capture wave transmission and reflection phenomena and are well suited to study the effects of cardiac arrhythmias on the coronary circulation. In addition, the reported 0D-1D studies offer a diversified scenario of the modeling approaches adopted. If 1D coronary modeling is quite analogous in literature data (only arterial network details usually change), other physiological mechanisms, such as coronary flow regulation, viscolelastic arterial walls, heart contraction feedbacks and proper distal boundary conditions, are accounted for or not depending on the aim pursued. Thus, to date, an overall concordant framework of optimal modeling solutions cannot be easily identified. Last, the focus on cardiac arrhythmias led to exclude investigations mostly aimed at different goals, such as surgery procedures, coronary flow regulation, and other cardiovascular diseases.

Conclusions
The literature reviewed in the current work highlights a lively research activity related to 0D-1D models focused on coronary hemodynamics. In particular, cardiac arrhythmias by their own have been shown to promote relevant alterations of the coronary hemodynamics. The computational approaches here discussed are not merely able to describe the direct hemodynamic effects of cardiac arrhythmias, but also provide unique hints regarding the possible mechanisms behind these phenomena. The current small number of coronary models applied to cardiac arrhythmias stresses the need for future studies focused on this topic. The present review can boost future research devoted to the hemodynamic effects exerted by AF and other arrhythmias on the coronary circulation, exploiting the great potential of multiscale mathematical modeling in cardiovascular fluid dynamics.

Author contributions
SS and LR conceived and designed the review study, analyzed the literature data, wrote, revised and approved the final version of the manuscript.

Ethics approval and consent to participate
Not applicable.    Table 1.  Table 1.   Table 1.