Linking cerebral hemodynamics and ocular microgravity-induced alterations through an in silico-in vivo head-down tilt framework

Head-down tilt (HDT) has been widely proposed as a terrestrial analog of microgravity and used also to investigate the occurrence of spaceflight-associated neuro-ocular syndrome (SANS), which is currently considered one of the major health risks for human spaceflight. We propose here an in vivo validated numerical framework to simulate the acute ocular-cerebrovascular response to 6° HDT, to explore the etiology and pathophysiology of SANS. The model links cerebral and ocular posture-induced hemodynamics, simulating the response of the main cerebrovascular mechanisms, as well as the relationship between intracranial and intraocular pressure to HDT. Our results from short-term (10 min) 6° HDT show increased hemodynamic pulsatility in the proximal-to-distal/capillary-venous cerebral direction, a marked decrease (-43%) in ocular translaminar pressure, and an increase (+31%) in ocular perfusion pressure, suggesting a plausible explanation of the underlying mechanisms at the onset of ocular globe deformation and edema formation over longer time scales.


Supplementary Methods
The global cardiovascular (CVS) model is composed of a 1D description of the arterial tree attached to 0D analogues of the systemic peripheral, venous, cardiopulmonary [1][2][3], coronary [4] and ocularcerebrovascular circulations.The global model layout is illustrated in Supplementary Figure 1.
1D Arterial Tree.Blood motion through 1D arteries (arterial tree geometry and features given in Supplementary Table 1) is described by the 1D axisymmetric form of the Navier-Stokes equations for mass and momentum balance: where A(x,t) and Q(x,t) are vessel cross-section area and blood flow rate, respectively, t is time and x the vessel axial coordinate.Blood is modeled as a Newtonian fluid with constant density ρ = 1050 Kg/m 3 and dynamic viscosity µ = 0.004 Pa s.The Coriolis coefficient β and the viscous coefficient N 4 are computed assuming a flat-parabolic velocity profile over each vessel cross-section area.
Gravity is introduced in equations ( 1)-( 2) through the term g sin γ sin α, where g is gravity acceleration magnitude, γ is the vessel orientation with respect to the frontal transverse body axis and α is the vessel inclination with respect to the horizontal reference (i.e., the tilt angle).
The constitutive equation for blood (transmural) pressure p(x,t) added to close the system (1)-( 2) is a function of the local vessel cross-section area A(x,t) Here, coefficients B i (i = 1 . . .5) are function of vessels' geometry and mechanical properties, through the local wave velocity at time t = 0 (c 0 ), and are defined as: where a 3 = 1914 N 2/3 /m 4/3 , a 5 = −45348 N/m 2 , K v is the effective viscosity of the wall, h w is the wall thickness while A 0 and r 0 are the vessels cross-section area and radius at time t = 0, respectively.Local wave velocity is estimated as c 0 = a 2 /(2r 0 ) b 2 [5], where a 2 = 13.3 m 1.3 /s and b 2 = 0.3.An additional partial-collapse hyperbolic model is introduced into eq.( 3) for carotid and vertebral arteries (vessels numbered No. 6,12,13,16,17,20) to cope with very low levels of transmural pressure [2].
Mass and total pressure conservation is imposed at inlet/outlet sections of arterial bifurcations: , where subscripts in, out, 1 and out, 2 indicate the parent vessel and the corresponding two daughter vessels (three for the coronary bifurcation) of a bifurcation, respectively.A 0D model of the aortic valve (see valves model in section 0D Cardiopulmonary Circulation) is coupled with the proximal aorta inlet section, whereas a 0D arteriolar compartment is plugged to each terminal 1D distal artery through a set of lumped characteristic impedances Z c = ρPWV 0 /A 0 .
0D Peripheral Circulation and Venous Return.Equations governing blood motion in the (i, j)-th 0D compartment are where subscript i denotes the considered body region (H, A, UA, LA, L), whereas j indicates the compartment (art, cap, ven, v, svc, ivc and avc).Intraluminal pressure of the (i, j)-th compartment is indicated with p i, j , Q i, j is blood flow rate while R i, j and L i, j are compartmental hydraulic resistance and inertance, respectively.V i, j is the compartmental total blood volume, determined as where C i, j is compartmental compliance, whereas symbol p ext i, j indicates either intrathoracic (IT P) or intracranial pressure (ICP).Where no external pressure is specified p ext i, j = 0 mmHg is assumed.The different constitutive law adopted to mimic non-linear effects of legs veins reads in which ∆V max = 1200 ml is the maximum distending volume of leg veins, C L,v is leg venous compliance.Values of 0D parameters are reported in Supplementary Table 2.
Hydrostatic pressure terms ∆p h i, j in eq. ( 5) are expressed according to Stevino's law: where ρg is blood specific weight, ∆h i, j is the hydrostatic height of the corresponding fluid column (apply only to v, svc, ivc and avc compartments, values reported in Supplementary Table 3), and α is the compartment orientation with respect to the horizontal reference (tilt angle).
0D Cardiopulmonary Circulation.Cardiac chambers are governed by the following constitutive equation where cardiac transmural pressure p ch − IT P (IT P is intrathoracic pressure) is linked to the stressed volume V ch −V un ch (V un ch is the chamber's unstressed volume), whereas the elastance function E ch is given by the relation: in which parameters E ch,A and E ch,B denote chamber's elastance amplitude and baseline value.The normalized shape-elastance function e ch (t) of atria and ventricles are reported in [1].
Cardiac valves are described as non-ideal diodes accounting for several effects onto valve leaflets (i.e. on valve opening angle, θ va ) according to the following relationship: The various terms represent the effects of tissue friction (K f ,va dθ va /dt), pressure (K p,va (p va,u − p va,d )) and inertial (K q,va Q va cos(θ va )) forces, as the influence of downstream vortexes (K v,va Q va sin(2θ va )).
B va is Bernoulli's coefficient of the valve, R va and L va are valve resistance and inertance.Q va is the transvalvular flow, p va,u and p va,d are upstream and downstream pressures governing the valve's leaflets (when p va,u − p va,d ≥ 0 vortexes action is taken into account, otherwise it is neglected).Valve's coefficients K p , K q , K f , K v together with valve's model parameters are reported in [1].
Pulmonary arterial and venous compartments are governed by equations where j − 1 = pu, va (pulmonary valve) and j + 1 = pv (pulmonary veins) when j = pa (pulmonary arteries), whereas j − 1 = pa and j + 1 = lv (left atrium) when j = pv.Values of pulmonary 0D parameters are reported in Supplementary Table 2 Intrathoracic Pressure.Intrathoracic pressure (ITP) applies to all cardiac chambers and pulmonary compartments.It varies with the body position (tilt angle α), according to the following relationship [2] IT P = −4.014+ 1.127 where g is the current gravity acceleration magnitude and g 0 = 9.81 m/s 2 .
Multiscale Coronary Circulation.Specific 0D coronary microvascular districts are linked to each 1D large coronary artery outlet section to describe downstream vascular beds perfusing the myocardium.
The coronary microvascular model is described in detail in [4].
Venous Valves.Venous valves are represented as non-linear hydraulic resistances and inertances within the arms and legs venous compartments, that is R A,v , R L,v , L A,v and L L,v .Eq. ( 5) for the arms/legs venous compartments (subscript A/L, v) modifies as where p A/L,v + ∆p h A/L,v and p A/L,vc are pressures immediately upstream and downstream the venous valve, while the function of the valve opening state ξ A/L,v is expressed as Parameters k vo = k vc = 40 1/(mmHg s) are valve's opening and closing rate, respectively.
with l e f f A/L,v assumed equal to the compartment radius, and the effective area A e f f A/L,v defined as where -ratio of the compartmental total blood volume and the characteristic length of the arms/legs venous compartments.Compartment radius can be determined from A A/L,v (assuming a circular compartment cross-section).Taking as reference the initial state of the system -for which ) -the following expressions can be derived for the non-linear resistance R A/L,v and inertance L A/L,v : where symbols α m , β m , γ m and τ m are saturation and time delay parameters (values in [2]), whereas sympathetic (n s ) and parasympathetic (n p ) activity are with ν = 7, and p corresponding to either pacs or pcp , while ptg to pacs,tg or pcp,tg .
Cerebrovascular Model.The lumped model of the cerebrovascular system is taken from [6] and encompasses 0D descriptions of the large cerebral arteries, of the distal pial circulation and intracerebral arterioles (distal arteries) and of the terminal capillary venous-circulation (see scheme in the Main Manuscript, Figure 5).All cerebrovascular model's details, equations and parameter values can be found in [7].arteries are computed as where p MCA,r , p MCA,l and p BA,w are right and left middle cerebral arteries and basilar (at the circle of Willis) pressures, determined from the pressure constitutive equation combined with the conservation of mass: where C are compartments compliance, Q are the blood flow rates entering and exiting the compartments, and ICP is intracranial pressure.R ICA,r , R ICA,r and R BA are right and left internal carotid and basilar arteries hydraulic resistances.The internal carotid and basilar arteries blood flow rates are used to complete the 1D-0D coupling as outflow condition for the corresponding 1D branches.1D Q ICA,r and Q ICA,l are obtained as where Q eye,r = Q eye,l = 0.5 Q a,eye (half the arterial eye input blood flow, see section Ocular Compartment).Right and left 1D vertebral arteries outflow is determined as Blood flows through large cerebral and distal arteries are computed by Kirchhoff's law at nodes as Distal compartmental blood volumes are obtained through mass conservation. 152 where flow rates Q are determined as Distal blood volumes are used to compute distal compartmental blood pressures according to pressure constitutive law: where p and V are the compartment blood pressure and volume.
The downstream capillary-venous circulation is connected to the superior vena cava of the global CVS model.The dural venous sinus pressure (p dvs ) applied to the outflow branch of the cerebrovascular model is determined as where p svc is superior vena cava pressure, ρg is blood specific weight, α is the tilt angle and L H and L svc are the head and superior vena cava compartment anatomical extensions (Supplementary Table 3), respectively.p dvs is used to compute the venous sinus blood flow (Q vs ) and the cerebrospinal fluid outflow rate (Q o ) through Ohm's law as: where CBF is cerebral blood flow, the overall blood flow drained from the brain.The cerebrospinal fluid rate of formation (Q f ) depends instead on the cerebral capillary pressure p ccap , whereas the cerebral venous pressure is computed through the pressure constitutive law combined with mass conservation: with Intracranial pressure (ICP) is given by the pressure constitutive law combined with mass conservation, taking into account the action of gravity through the cerebrospinal fluid hydrostatic pressure (Stevino's law), so that where with C ic the intracranial non-linear compliance and Q terms the blood flow rates entering and exiting the compartments communicating with the intracranial cavity.
Cerebral (distal) arteriolar resistances and compliances are controlled by cerebral autoregulation and CO 2 reactivity .To each of the six distal regions, the following equations apply: , i=m,a,p; j=l,r, , i=m,a,p; j=l,r, where the subscript n denotes the basal values and , i=m,a,p; j=l,r.
Distal compliances and resistances are expressed as , i=m,a,p; j=l,r, where Parameters and initial values of terms appearing in the cerebrovascular model equations can be found in Tables 4-7 and in [7].
Ocular Compartment.The lumped model of the ocular compartment is adopted from [8,9].The ocular model is multi-compartment and regulates the behavior of the intraocular pressure (IOP) and of the ocular globe volume (V g ) through the following governing equations: where the globe compliance the retrobulbar subarachnoid space-to-globe compliance C rg = 1.1e − 9 l/mmHg, the arterial blood-to- and venous blood-to-globe compliance C vg = 0.7V g (C 1 /IOP + C 2 − 1/(k g IOP)) (with k g = 312 the non-dimensional globe stiffness), the aqueous humor formation rate Q aq,in = 0.048e − 6 l/s, the aqueous outflow facility C tm = 0.0035e − 6 l/(s mmHg) (as long as EV P ≤ IOP, 0 otherwise), the uveoscleral outflow rate Q uv = 0.0067e − 6 l/s (parameters adjusted according to physiological values [10]).p a,eye and p v,eye are arterial and venous pressure at the level of the eye and episcleral venous pressure, respectively, taken as p v,eye = max CV P, EV P , where L f −b = 0.03 m is the perpendicular distance between the globe and the mid-coronal plane.
To integrate the ocular model with the global CV S, the arterial eye input and venous eye output flow rates shall be determined.These are obtained from the mass conservation and the pressure constitutive law applied to the ocular arterial and venous blood compartments as with the eye blood flow rate Q eye = (p a,eye − p v,eye )/R eye , and the eye resistance R eye = 4676 mmHg s/ml such that mean Q eye ≃ 1 ml/min [11].
Numerical Simulation.1D governing equations are discretized and integrated numerically according to a Discontinuous Galerkin Finite Elements approach.The solution is advanced in time employing a 2step Runge-Kutta explicit scheme with constant time step [1].Ordinary differential equations governing 0D compartments are advanced in time via the same 2-step Runge-Kutta explicit scheme.
The cerebrovascular model is attached to the internal carotid and vertebral arteries (No. 6, 12, 16 and 20) of the global CVS model.The lumped characteristic impedance Z c located at the end each 1D terminal artery to achieve the 1D-0D coupling has been virtually removed for arteries No. 6, 12, 16 and 20 of the global model.That is, Z c has been imposed ∼ 0 for internal carotid and vertebral arteries, as the corresponding 0D branching entering the lumped cerebrovascular model should still describe arterial tracts -not arteriolar as done for the rest of the global model.The pressure downstream the right and left internal carotid (p ICA,r , p ICA,l ) and vertebral (p VA,r , p VA,l ) arteries are used as input pressures for the right and left internal carotid and basilar (p BA = 0.5 (p VA,r + p VA,l )) arteries of the cerebrovascular model, respectively.Blood flow rates through the right and left internal carotid (Q ICA,r , Q ICA,l ) and basilar (Q BA )

)
Autonomic Control.The aortic-carotid sinus pressure pacs sensed by the baroreflex control is obtained by averaging the pressure in the aortic arch (p AA (t)) and the right/left carotid sinus pressure (p cs,r (t) and

Table 1 :
1D geometry.Geometric features of the 1D arterial tree (l: vessel length, D in : vessel inlet diameter, D out : vessel outlet diameter, h wall : vessel wall thickness, γ: vessel orientation with respect to the frontal transverse body axis).

Table 2 :
0D parameters values.R, C, L, V un and V are lumped compartmental hydraulic resistance, compliance, inertance, unstressed volume and total blood volume, respectively.H, A, UA, LA and L refer to the head, arms, upper and lower abdomen and legs body regions.svc, ivc and avc are superior, inferior and abdominal veane cavae compartments, while pa and pv denote pulmonary arterial and venous compartments, respectively.

Table 8 :
Tests of significance.P-values resulting from statistical test of significance (two-tailed Wilcoxon paired test, n = 6).BL denotes the baseline seated posture, MAP indicates mean arterial pressure (taken at the finger and corrected at brachial level), HR denotes the heart rate, CO indicates cardiac output, SV indicates stroke volume, and IOP indicates the intraocular pressure.