Fig 1.
Glagov remodeling in atherosclerotic arteries.
(a) Glagov et al. [4] analyzed the left main coronary artery in 136 hearts to deduce changes in morphology with respect to plaque burden. Remodeling behavior depends on whether stenosis is less than or greater than 30%. With the same data, the authors were also able to fit a piecewise linear curve with a jump in derivative at about 40% (not shown). In this case, the curve gently decreased when the stenosis was <40% and rapidly decreased when the stenosis was >40%. (b) In Glagov remodeling, initially, the lumen area increases slightly while the internal elastic lamina (IEL) increases in area. After the plaque reaches about 40% of the IEL area, the luminal area starts to decrease. Adapted from [4].
Fig 2.
Multi-layer axisymmetric model for arterial remodeling.
Geometry in the (a) reference, unstressed configuration when t < 0, (b) pressurized configuration at t = 0 and (c) grown, pressurized configuration for t > 0.
Table 1.
Parameter values for a three-layer model of vessel remodeling.
Fig 3.
Comparison of model results to Glagov’s data.
(a) Data from Glagov’s paper of lumen area L versus stenosis fraction S. Superimposed are solid curves L = f(S), generated by our model initialized with reference lumen areas πa(0)2 = πA2 = 5 (dash-dotted), 8 (dotted), 12 (dashed) and 15 (solid) mm2. (b) Data and model prediction of the internal elastic lamina area πb2 as a function of plaque area π(b2 − a2).
Fig 4.
Growth of the intima in-vivo and ex-vivo.
Ex-vivo configurations are quickly obtained in our model either by taking P = 0 at each step of the in-vivo growth or taking P = 0 from the start of the simulation.
Table 2.
Qualitative differences in vessel geometry for ex-vivo and in-vivo vessels. Stenosis fraction is defined in Eq (2).
Fig 5.
Comparison of ex-vivo and in-vivo vessel dimensions from (a) Glagov’s paper [4] and (b) the PROSPECT trial [8].
Four model vessel cross-sections are simulated with unloaded initial lumen areas of 5, 8, 12 and 15 mm2 in (a). The growth of corresponding vessels loaded with pressure P = 50, 80, 110 mmHg is simulated in (b) (for a fixed stenosis, lumen area increases as P increases). The ex-vivo vessels in (a) are predicted to exhibit a local maximum with respect to stenosis fraction, while the equivalent, loaded lumen areas of in-vivo vessels in (b) are monotonically decreasing with respect to stenosis.
Fig 6.
Remodeling of ex-vivo and in-vivo vessels.
(a) and (b): Time evolution of ex-vivo and in-vivo vessels as they undergo identical uniform growth in the intima. The ex-vivo vessel is unpressurized with P = 0 while the in-vivo vessel is subject to a luminal blood pressure of P = 110 mmHg. (c) and (d) are the respective plots of lumen area vs. plaque area which are qualitatively different. Glagov’s ex-vivo data suggested that during the early stages of atherosclerosis, the lumen area increased by 2.3 mm2 for every mm2 growth of plaque (dashed line in (c)).
Fig 7.
Best-fit smoothing cubic-splines through Glagov’s ex-vivo data (top row) and in-vivo data from the PROSPECT trial (bottom row).
In each case, more curvature is introduced into the best-fit curve by allowing k to increase: see Eq (33). A critical stenosis emerges for Glagov’s data, but not for the PROSPECT trial.