Riassunti Advanced numerical methods for coupled problems and living systems

Caratteristiche del flusso sanguigno nelle arterie e nelle vene

Il flusso sanguigno nelle arterie è caratterizzato da una pressione più elevata rispetto alle vene. La viscosità del sangue è un fattore importante che influisce sul flusso sanguigno, con valori più alti nelle arterie rispetto alle vene. La velocità del flusso sanguigno è più elevata nelle arterie rispetto alle vene.

Le arterie presentano un flusso sanguigno pulsatile, con un'onda di flusso più concentrata. La pressione arteriosa assume valori compresi tra 70 e 130 mmHg.

Le vene, invece, presentano un flusso sanguigno più costante e meno pulsatile. La pressione venosa è generalmente più bassa rispetto a quella arteriosa, con valori compresi tra 7 e 12 mmHg.

Le vene possono essere dotate di valvole che impediscono il reflusso del sangue. Il flusso sanguigno nelle vene dipende in gran parte dalla contrazione muscolare e dagli effetti respiratori.

La compliance dei vasi sanguigni è un fattore importante che influisce sul flusso sanguigno. La dissipazione di pressione è trascurabile prima di raggiungere la microcircolazione, ma la pressione non è costante nello spazio in un dato istante.

È presente un gradiente di pressione spaziale, con una grande caduta di pressione nella microcircolazione, dove le forze viscose dominano completamente quelle inerziali.

between proximal and distal regions facilitating the blood movement. Propagating nature of the pressure, which is in fact a wave traveling along the arterial network. Wave speed ranges from 500 cm/s in the aorta to 1200 cm/s in coronaries. The presence of bifurcations or high resistance regions (such as the microvasculature) produces wave reflections propagating backward to the heart. The vessel compliance (cont'd). This is due to the vessel's compliance, i.e. the ability of the vessel to distend under the forces exerted by the blood. Vessel wall displacements are quite large (10% of the lumen diameter). The arteries more proximal to the heart, in particular the aorta, are very extendible and store during systole about 50% of the entering blood. This blood reserve is then discharged during diastole owing to the vessel wall elastic response (windkessel effect). This is responsible for the smoothing of the blood flow waveform going downstream along the arterial network. This guarantees an almost continuous flow.exchange of oxygen with the tissuesFurther details in Nichols and ORourke, eds (2005), McDonalds Blood Flow in Arteries, Hodder Arnold;Quarteroni, Tuveri and Veneziani (2000), Computational vascular fluid dynamics: Problems, models andmethods, CVS, 2LANDSCAPE ON AVAILABLE DATAMandatory to consider data that are patient-specificGeometric dataUsually to obtain the computational domain, the following steps are perfor-med (Antiga, Piccinelli, Botti, Ene-Iordache, Remuzzi and Steinman (2008), An image-based modelingframework for patient-specific computational hemodynamics, Med Biol Eng Comput 46)- Acquisition of clinical images:An angiography exploits the property that a liquid inside the vessel appearsbrighter than the vessel wallMost common techniques are X-ray imaging, e.g. computed tomography(CT), and Magnetic Resonance (MR)- Image enhancement:Usually performed to improve images quality which is often affected by noiseand artifacts- Image segmentation:Construction of the shape of

A vascular district detecting those points which are supposed to belong to the boundary of the vessel lumen

Geometric data (cont'd)

- Building the computational mesh:

Usually it is made by unstructured tetrahedra, because of their flexibility to fill volumes of complex shape

In arteries, the mesh should be fine enough to capture Wall Shear Stresses (WSS) expressing the magnitude qP 22t t(j)rv · of tangential viscous forces on the lumen boundary: on where is W SS µ n) ⌧ , v⌃ = ⌃j=1(j) the fluid velocity, the outward unit vector, and the tangential unit vectors, To this aim, then ⌧ j = 1, 2. construction of a boundary layer mesh is essential, even at low Reynolds numbers

Images detecting the vessel wall are not routinely acquired in the clinical practice. In this case, a reasonable approach to obtain the vessel wall geometry is to extrude the reconstructed boundary lumen

Boundary data

Fluid problem: ⇣ ⌘D,t N,t Trvon onv g , pn µ n h ,= + + (rv) = ff f fStructure

problem: D,t N,ton ond g , T n h ,= =s ss ss

Physical boundary: Blood velocity is known, either in the case of rigid walls or for fluid-structure interaction (FSI)v ḋ=

For the structure, at the internal surface it is common to prescribe T n =s where is the fluid pressure, either measured or computed in FSI P n, P

Boundary data (cont’d)

Physical boundary (cont’d): On the outer wall boundary , the interaction with the surrounding tissue is considered: on↵ d T n P n ,+ =s ext extST which assimilates the surrounding tissue to a sequence of elastic springs with rigidity and where is the external pressure↵ P extST

Artificial boundary: It is introduced by the trunction of the domain.!We cannot use physical principles to prescribe bc We need measurements

For the fluid, often Doppler Ultrasound technique is used due to its non-invasiveness. This provides at each time a flow rate condition Z ·⇢ v n d Q=f t

Phase Contrast (PC)-MR technique acquires velocity measures in

white blood cells and platelets 3The diameter of blood cells is approximately whereas that ofcm,101the smallest arteries/veins is about cm10Thus, blood in medium and large vessels is considered as Newtonian⇣ ⌘µ TrvT p) pn µD(v) pn (rv)(v, = + = + +f 2In the smallest arteries, such as coronaries, or in presence of a vessel narro-wing, a non-Newtonian rheology is more appropriately assumedpThe viscosity depends on the shear rate ,D D˙ = 2 =II IIP 31 D Dij iji,j=12Physical assumptions (cont’d)In particular, is higher than the Newtonian one at small shear rateµThis is due to the aggregation of red cells at low shear rates forming rouleauxFor example, the Carreau-Yasuda model is given by⇣ ⌘ 0.781.251.25µ(x, t) µ µ t) ,˙= + (µ ) 1 + ( (x,1 10Further details can be found in Robertson, Sequeira and Owens (2009), Rheological models forblood, in Cardiovascular mathematics, edited by L. Formaggia, A. Quarteroni, A. Veneziani,

Springer

Laminar or turbulent flow?

In healthy conditions, blood flow is mainly laminar

Transitional flow may develop in some pathological instances

This is the case e.g of stenotic carotids and abdominal aortic aneurysms

A possible strategy is to refine the mesh performing a direct numerical simulation (Lee, Lee, Fischer, Bassiouny, Loth, J Biomech 41, 2008)

Alternatively, a turbulence model (like large eddy simulation) could be used on a coarser mesh (Rayz, Berger, Saloner, Computer Methods in Applied Mechanics and Engineering, 196, 2007).

Examples of numerical results

MODELING VASCULAR WALL DYNAMICS

Anatomy of vacular walls

The total thickness is about 10% of the lumen diameter (1% for the pulmunary circulation)

Three layers: intima, media, adventitia

The inner part of the intima is the endothelium

The media and the adventitia mainly have a major role in characteri-zing the mechanical response of the vessel wall

Their main structural components are elastin and collagen

The media is also formed by

smooth muscle cells which provide the tone to the vessel wall

Elastin provides the elasticity of the vessel wall at small strain

Collagen forms sti↵ fibers that provide tensile strenght at large strain! highly non-linear elastic properties

The quantity of elastin and collagen decreases going downstream along the arterial network, whereas the quantity of smooth muscle cells increases.

The equation of elasto-dynamics

The problem is written in a reference domain 3R⇢ using a Lagrangian framework ⌦ = ⌦ (0)s s L

We use the notation to denote in any function g g= ⌦ sts defined in the current solid configuration g ⌦

Find, at each time the structure displacement such that t, d2@ d r ·⇢ T 0 ,(d) = in ⌦s s s2@t

The first Piola-Kirchho↵ tensor is related to the Cauchy tensor T T(d) (d)s s T rx thanks to the relation , where is the deformation T JT F F= =s s tensor and represents the change of volume between the reference and the current configurations

Piola-Kirchhoff stress tensor is obtained by differentiating a suitable Strain Energy Density Function. The arterial constitutive laws separate the isotropic and elastic behavior due to the elastin from the anisotropic and stiff one due to the collagen. A common choice for the isotropic part is the Neo-Hookean law: Giso T, C F F(tr(C) - 3) / 2 whereas for the anisotropic part, an exponential law is often considered: aniso ⇥ (C) = 1 / 2k^2 (a · (Ca) - 1)^2 e^(2k) Here, a is the unit vector of the direction of collagen fibers, G characterizes the stiffness of the material for large displacements, and k is the shear modulus. Reference: - G. Holzapfel and R. Ogden (2010), Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 466(2118) - Holzapfel, Gasser, Ogden (2000), J Elast 61 When not available from medical images, the fibers direction can be determined using other methods.