Elasticity of the aorta

Nico A.M. Schellart, Dept. of Med. Physics, AMC

Principle

When the heart contracts, blood is pumped into the aorta. Then, pressure P_a, flow and volume V of the aorta increase. We consider now firstly how under assumed validity of the law at of Hooke and Laplace the P-V relation will be. Firstly, we calculate using the law of Laplace how the flow of the aorta relates to P_a.

Fig. 1 Pressure increase causes dilatation of the cylinder

Since the force in the aorta wall is F_w = P_ar_il, where r_i the aorta inner diameter and l its length. Without overpressure, the width of the upper halve of the aorta cylinder is w = πr_i,0 with r_i,0, the inner radius before the blood ejection of the left ventricle (Fig. 1, right upper part). Then the material cross section is A = ld₀. After ejection the inner radius is increased to r_i,t. and the change of w, the strain of the elastic aorta wall, is Δw = π(r_i,t.- r_i,0). The wall stress goes from (assumed) zero (equilibrium before ejection) to a value given by Hooke’s law:

σ = Eε = EΔL/L =F/A, (1)

(see Elasticity and Hooke's law), and after ejection, we obtain:

σ = F_w /A = EΔw/w = E(r_i,t.- r_i,0)/r_i,0. (2)

Fig. 2 The relation between the radius r of a cylinder and the pressure in the cylinder

This gives to how the wall tension increases at a cylinder, if by pressure increase the radius of the cylinder increases (Fig. 4-4). Substitution from σ = P_ar_i in (2) provides (after conversion):

r_i,t = r_i,0(1- P_ar_i,0/Ed)⁻¹. (3)

This function is illustrated in Fig. 2. In literature one finds except wall stress (in N/m²) also wall tension T= Pr/d, generally given in N/m.

When the pressure in the aorta increases, also the volume V increases. Using V = πr_i,0²l it follows that:

V(P_a)= V₀(1- P_ar_i,0/Ed)⁻². (4)

All values must be expressed in preferably the SI system (otherwise convert, e.g. 100 mmHg ~ 13600 N/m2 = 13.6 kPa). At the age of 50 years, realistic values for the aorta are R_i,0 = 5.6 mm, E = 5,105 N/m2, d = 2 mm, V_o=40 mL and l = 40 cm.

Fig. 3 depicts equation (4). Above 70 mmHg, this P-V relation ceases to be quadratic. V increases much less and finally the relation becomes sigmoid. The model, derived from the linear approach of elasticity (law of Hooke), only holds well at low values of P_a. At higher values of P_a, collagen in the vessel wall will dominate the properties of the elastine.

Fig. 3 Aorta volume V as a function of the pressure of the aorta

More Info

As a measure for the ductility of the aorta (compliance) C is used, either in its dynamic (C_dyn) or static (C_stat) form. C_dyn is the pressure derivative of the volume/pressure curve (drawn line in Fig. 3): C_dyn = dV/dP_a. Fig. 4 shows the impact of the compliance of the aorta. The drawn line represents the pressure in the left ventricle and the dotted line that in the aorta. An approach of the dynamic compliance between 80 and 120 mmHg (the regular values) is obtained by taking dV/dP_a at 100 mmHg, which can be approximated by C_dyn,100 = (185-160)/(120- 80) = 0.625 mL/mmHg). The static compliance C_stat is V/P_a. For P_a = 100 mmHg it follows that C_stat,100 ≈ 170/100 = 1.7 mL/(mmHg). When the volume-pressure relation of the aorta is approached by a straight line between the origin and the point with pressure P_a (the ' work point '), the slope of this line represents static compliance. This definition of C_stat holds for the aorta as a whole, here with an assumed length of 40 cm. In the literature, C_stat is sometimes expressed per unit length of the vessel. The compliance of the aorta results in a rather regular flow, in spite of the pulse wise time course due to the contractions of the heart.

Fig. 4 Effect of the compliance on the aorta pressure

In Fig. 4 shows that after closing the aorta valve P_a (and volume) decreases about exponentially. The total pulsation of P_a is much smaller than that of the ventricle due to the high aorta compliance. If the aorta would be a rigid tube, the blood flow would be more strongly pulsating. This will result in a larger mechanical load of the rest of the circulatory system as follows from the Windkessel model of the circulatory system. Aging diminish the 'empty' volume V₀ of the aorta. This hardly affects C_stat but it seriously decreases C_dyn: a decrease of approximately 0.6 mL/mmHg at the age of 50 to approximately 0.2 mL/mmHg at 80 years.

Literature

Van Oosterom, A and Oostendorp, T.F. Medische Fysica, 2^nd edition, Elsevier gezondheidszorg, Maarssen, 2001.

N. Westerhof, Noble M.I.M and Stergiopulos N. Snapshots of hemodynamics: an aid for clinical research and graduate education, 2004, Springer Verlag.