Higher ventricular rate during atrial fibrillation relates to increased cerebral hypoperfusions and hypertensive events

Atrial fibrillation (AF) is associated with cognitive impairment/dementia, independently of clinical cerebrovascular events (stroke/TIA). One of the plausible mechanisms is the occurrence of AF-induced transient critical hemodynamic events; however, it is presently unknown, if ventricular response rate during AF may impact on cerebral hemodynamics. AF was simulated at different ventricular rates (50, 70, 90, 110, 130 bpm) by two coupled lumped parameter validated models (systemic and cerebral circulation), and compared to corresponding control normal sinus rhythm simulations (NSR). Hemodynamic outcomes and occurrence of critical events (hypoperfusions and hypertensive events) were assessed along the internal carotid artery-middle cerebral artery pathway up to the capillary-venous bed. At the distal cerebral circle level (downstream middle cerebral artery), increasing ventricular rates lead to a reduced heart rate-related dampening of hemodynamic signals compared to NSR (p = 0.003 and 0.002 for flow rate and pressure, respectively). This response causes a significant progressive increase in critical events in the distal cerebral circle (p < 0.001) as ventricular rate increases during AF. On the other side, at the lowest ventricular response rates (HR 50 bpm), at the systemic-proximal cerebral circle level (up to middle cerebral artery) hypoperfusions (p < 0.001) occur more commonly, compared to faster AF simulations. This computational study suggests that higher ventricular rates relate to a progressive increase in critical cerebral hemodynamic events (hypoperfusions and hypertensive events) at the distal cerebral circle. Thus, a rate control strategy aiming to around 60 bpm could be beneficial in terms on cognitive outcomes in patients with permanent AF.

for the distributions of the mean flow rate per beat (Q) and mean pressure per beat (P ) during AF. The reference distribution is taken at 70 bpm, and the test is performed for each average hemodynamic variable comparing the following AF simulations: 70-50 bpm, 70-90 bpm, 70-110 bpm, 70-130 bpm. When p-values are below 10 −20 , the notation < 10 −20 is adopted.

Cardiovascular model
The cardiovascular model proposed by [2,6,7] is coupled to a baroreceptor model [4] to describe the whole (arterial and venous) circulation system in presence of short-term autoregulation.

Equations
For the sake of simplicity, the equations are grouped into cardiac (with the four chambers), circulatory (with systemic and pulmonary loops), and baroreceptor mechanism sections.

Left atrium
dV la dt = Q pvn − Q mi , P la = P la,un + E la (V la − V la,un ), (S1) where the subscript un denotes the unstressed pressure and volume levels of each cardiovascular section.
The time-varying elastance is and the atrium activation function is where E la,min and E la,max are the minimum and maximum elastance values, respectively, while T ac is the beginning of atrial contraction.
The valve opening is decided by the angular position of the leaflets: and the valve motion is governed by Left ventricle The time-varying elastance is and the ventricle activation function is where E lv,min and E lv,max are the minimum and maximum elastance values, respectively. T me and T ce are the instants where the elastance reaches its maximum and constant values, respectively.
The valve opening is decided by the angular position of the leaflets: and the valve motion is governed by

Right atrium
The time-varying elastance is where the activation function is given by Eq. (S3), while E ra,min and E ra,max are the minimum and maximum elastance values, respectively.
The valve opening is decided by the angular position of the leaflets: (S13) and the valve motion is governed by

Right ventricle
The time-varying elastance is where the ventricle activation function is given by Eq. (S8), while E rv,min and E rv,max are the minimum and maximum elastance values, respectively.
The valve opening is decided by the angular position of the leaflets: and the valve motion is governed by Pulmonary Circuit The sympathetic, n s , and parasympathetic, n p , activities [Hz] are expressed as where P cs is the carotid sinus pressure averaged over one cardiac cycle, which is here taken equal to systemic arterial pressure averaged per beat, P a [mmHg s]. µ [mmHg] is the mean systemic arterial pressure in physiological conditions at steady state for HR=70 bpm, while ν is a dimensionless parameter characterizing the steepness of the curves. subtracting from the total volume, V tot , all the volume contributes at the generic vascular section i,

Equations
For the sake of simplicity, the equations are grouped following the three main partitions of the model.
The autoregulation and CO 2 activity equations for the distal district are separately reported.

Large arteries
where the flow rates, Q, are: Q P CA2,lef t = P P CA,lef t − P dp,lef t R P CA2,lef t + R dp,lef t /2 , Q P CA2,right = P P CA,right − P dp,right R P CA2,right + R dp,right /2 ,

Distal arterial circulation
where the flow rates, Q, are: Q dp,lef t = P dp,lef t − P c R dp,lef t /2 , Q dp,right = P dp,right − P c R dp,right /2 , Q cpm,right = P dp,right − P dm,right R cpm,right , while the constitutive relations are: P dp,lef t = V dp,lef t C dp,lef t + P ic , P dp,right = V dp,right C dp,right

Capillary-venous circulation
where the flow rates, Q, the compliances C ic , C vi , and the resistance R vs are defined as follows:

Autoregulation and CO 2 reactivity equations
For each of the six distal regions, the following equations hold: , i=m,a,p; j=left,right, where the subscript n denotes the basal values and , i=m,a,p; j=left,right.
Distal compliances and resistances are ruled by the following relations: i=m,a,p; j=left,right i=m,a,p; j=left,right (S44)

Initial Conditions and Cerebral Parameters
Initial conditions are given in terms of pressures, volumes, autoregulation and CO 2 reactivity state variables, and are equal in both NSR and AF conditions. The initial setting is chosen according to flow rate repartition given by [8] with the following total basal values: basal cerebral blood flow Q n = 12.5 ml/s, basal distal resistance R dn = 5.4 mmHg s/ml, basal distal compliance C dn = 0.2 ml/mmHg. State variables for autoregulation, x aut , and CO 2 reactivity, x CO 2 , are initially considered as no activity is present. Initial pressures in the cerebral regions are chosen in the physiological range reached during a normal cardiac cycle. Volumes in the distal region j are obtained at t=0 using the constitutive relation V dj,t=0 = C dj,t=0 (P dj,t=0 − P ic,t=0 ), where P dj,t=0 and P ic,t=0 are imposed as previously said.
The adopted initial conditions are listed in

Numerical Scheme
The differential equations of both models are solved by means of a multistep adaptive solver, implemented by the ode15s Matlab function. This variable order solver is based on the numerical differentiation formulas (NDFs) and is chosen because is one of the most efficient and suitable routines for stiff problems. In fact, the differential system of both models shows some stiffness features, that is the equations include some terms that can lead to rapid variation in the solutions. This aspect is particularly relevant during end-diastolic and end-systolic phases, when the valves opening and closure cause sudden variations of the leaflet angular position.      po 7 rad/(s ml) θ max 5/12 π rad   0.78 ml P dp,lef t,0 58.5 mmHg V dp,lef t,0 1.18 ml P dp,right,0 58.5 mmHg V dp,right,0 1.18 ml P ICA,lef t,0 100 mmHg P ICA,right,0 100 mmHg P BA,willis,0 100 mmHg x aut,i,j,0 0 i=m,a,p, j=left,right