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DISEASES AND DISORDERS
Mechanics-driven mechanobiological mechanisms
of arterial tortuosity
Dar Weiss1, Cristina Cavinato1, Authia Gray1, Abhay B. Ramachandra1, Stephane Avril2,
Jay D. Humphrey1*, Marcos Latorre1
Arterial tortuosity manifests in many conditions, including hypertension, genetic mutations predisposing to thoracic
aortopathy, and vascular aging. Despite evidence that tortuosity disrupts efficient blood flow and that it may be
an important clinical biomarker, underlying mechanisms remain poorly understood but are widely appreciated to
be largely biomechanical. Many previous studies suggested that tortuosity may arise via an elastic structural
buckling instability, but the novel experimental-computational approach used here suggests that tortuosity arises
from mechanosensitive, cell-mediated responses to local aberrations in the microstructural integrity of the arterial
wall. In particular, computations informed by multimodality imaging show that aberrations in elastic fiber integ-
rity, collagen alignment, and collagen turnover can lead to a progressive loss of structural stability that entrenches
during the development of tortuosity. Interpreted in this way, microstructural defects or irregularities of the arterial
wall initiate the condition and hypertension is a confounding factor.
INTRODUCTION
Notwithstanding the highly branched morphology of the vascular
tree, most arterial segments are locally circular and relatively straight
under normal conditions. This uniform geometry engenders locally
uniform blood flow that is efficient for fluid transport as well as uni-
form states of flow-induced wall shear stress and pressure-induced
intramural stress consistent with local mechanical homeostasis (1).
Since at least the time of Leonardo da Vinci (1452–1519), however,
it has been known that different conditions give rise to persistent
abnormal bends, twists, turns, and kinks within arteries that are gen-
erally referred to as tortuosity. Advances in modern medical imag-
ing, genetics, and clinical phenotyping continue to reveal more and
more conditions that either cause or are a consequence of arterial
tortuosity. In addition to arterial tortuosity syndrome, which disrupts
a glucose transporter and compromises elastic fibers within the ar-
terial media (2), vascular aging and hypertension and similarly genetic
conditions that predispose to thoracic aortopathies also associate
with arterial tortuosity (3–6). Given the underlying mechanical ba-
sis of circulatory physiology and vessel mechanobiology, it is not
unexpected that arterial tortuosity has been associated with altered
mechanical factors, including changes in blood flow and pressure,
axial tension, and the structural composition of the arterial wall (7,8).
Nevertheless, the biomechanical mechanisms responsible for the
development of tortuosity have remained unclear.
Here, we combine a custom digital imaging correlation method
with optical coherence tomography (OCT) to study in greater detail
the local mechanics of native and tortuous thoracic aorta in a well-
accepted genetically modified mouse model of compromised elastic
fiber integrity. We then use an experimentally informed mechano-
biologically motivated computational model of aortic growth and
remodeling (G&R) to test competing hypotheses on the underlying
mechanisms by which these vessels become tortuous. Although most
previous attention has been directed toward the possibility of an elas-
tic structural buckling instability (7,9), it has been acknowledged
that such an intermittent response does not describe the marked,
persistent tortuosity observed invivo but rather leads to modest de-
flections under physiological pressures (10,11). Our experimental-
computational findings suggest that arterial tortuosity likely arises
via progressive cell-mediated responses to local imperfections and
aberrant microstructural changes that can develop within an aged
or diseased arterial wall, with reduced axial tension and elevated
blood pressure exacerbating these maladaptive responses. The im-
portance of understanding the underlying mechanical and mecha-
nobiological mechanisms of arterial tortuosity is underscored by both
the need to understand better the clinical time course and especially
the desire to use tortuosity as a reliable biomarker for diverse vascu-
lar conditions (6,8).
RESULTS
Absence of fibulin-5 reduces aortic distensibility
and increases tortuosity
Relative to the descending thoracic aorta in age- and sex-matched
wild-type mice, aorta from Fbln5−/− mice exhibit a lower structural
distensibility (i.e., normalized changes in diameter per changes in
pressure) despite modest reductions in biaxial material stiffness
(table S1). They also develop considerable tortuosity invivo (fig.
S1), which persists in situ in the absence of pressurization and invi-
tro whether pressurized or not (Fig.1,AandB, and fig. S2). Re-
constructions of the three-dimensional (3D) geometry of the aorta
invitro (Fig. 1C) reveal further that the degree of tortuosity in-
creases transiently with increases in distending pressure (Fig.1D),
which, as expected, is significantly greater in Fbln5−/− mice com-
pared with wild type (P<0.001 for mixed comparisons between
52- and 20-week-old groups). Quantification of bulk mechanical
properties revealed few overall differences between the two Fbln5−/−
age groups or between the two wild-type groups: Distributions of
circumferential and axial material stiffness were not significantly
different across the four groups (Fig.1,E, F, H,andI), although the
absence of fibulin-5 reduced elastic energy storage capability rela-
tive to control (P<0.01 between Fbln5+/+ and Fbln5−/− groups;
1Department of Biomedical Engineering, Yale University, New Haven, CT, USA. 2Mines
Saint-Etienne, Centre CIS, INSERM, U 1059 Sainbiose University of Lyon, Univ Jean
Monnet, Saint-Etienne, France.
*Corresponding author. Email: jay.humphrey@yale.edu
Copyright © 2020
The Authors, some
rights reserved;
exclusive licensee
American Association
for the Advancement
of Science. No claim to
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Works. Distributed
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License 4.0 (CC BY-NC).
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Fig.1, G andJ), which was exacerbated slightly (not significantly)
by aging from 20 to 52 weeks.
Tortuosity correlates with reduced distensibility
and deformability
Aortic tortuosity in Fbln5−/− aortas correlates well with decreasing
structural distensibility and biaxial deformability (Fig.2), that is, re-
duced values of circumferential and especially axial stretch upon me-
chanical loading. As seen in the figure, the degree of tortuosity also
correlates with reduced energy storage capability (reflecting com-
promised elastic fiber integrity), the key metric that is significantly
different between wild-type and fibulin-5–null aortas. Whereas en-
ergy storage reflects a fundamental mechanical functionality of the
wall (i.e., the normal aorta stores elastic energy during systole and
uses this energy during diastole to work on the blood to augment
flow), circumferential material stiffness appears to be highly mechano-
regulated within the arterial wall (12). There was little correlation
between tortuosity and either circumferential or axial material stiff-
ness (fig. S4), suggesting that the intramural cells were yet able to
sense and regulate the stiffness of the matrix on average.
Multimodality imaging reveals regional differences
in structure and properties in tortuosity
Figure3 contrasts calculated values of circumferential and axial stretch
and similarly biaxial material stiffness within concave and convex re-
gions of the tortuous aorta in Fbln5−/− mice. Notwithstanding con-
siderable variations, circumferential material stiffness was significantly
higher and circumferential stretch was significantly lower on the con-
vex compared with the concave sides of the tortuous aorta. Axial
stretch tended to be significantly greater on the convex side, although
axial stiffness differed little between the two sides. In addition, the
proximal-convex region exhibited significantly higher systolic-to-
diastolic differences in elastically stored energy compared with the
proximal-concave region, while differences in the distal region were
negligible (fig. S5).
Multiphoton microscopy—both second-harmonic generation re-
vealing fibrillar collagen and autofluorescence revealing elastin-
based structures—provided complementary information on local
microstructure. The medial layer appeared nonuniform and scat-
tered in the Fbln5−/− aorta due to agglomerations containing
irregular- shaped elastin (Fig.4A), not compact lamellar structures,
Fig. 1. pDIC data collection and analysis. (A) Representative in situ image (photo credit: Dar Weiss, Yale University) of a tortuous descending thoracic aorta at zero blood
pressure (arrow) in a 20-week-old Fbln5−/− mouse, which was excised and (B) mounted on a custom triple-needle assembly for placement in the panoramic digital image
correlation (pDIC) system (fig. S2). Centerlines were computed for each (C) 3D reconstructed configuration at the sample-specific in vivo axial stretch and (D) used to calculate a
tortuosity index (TI) at each pressure applied in vitro. Using the virtual field-based inverse method, spatial distributions of (E) circumferential and (F) axial material stiffness,
(G) elastic stored energy, and other metrics (fig. S3) were identified and superimposed on the reconstructed reference configuration geometry (at in vivo axial stretch and
80 mmHg) and then unwrapped and plotted in − Z planes around the vessel circumference ( ∈ [− 180,180]) and along its (Z) axial extent. The rightmost edge of each
3D rendering is at about 0°, whereas the leftmost edge is at about ±180°; see the middle panel of Fig. 3 for further clarification of the mapping from 3D to 2D. Edges of the
samples and patches with a poor coefficient of determination were neglected and left blank. Averaged probability density functions are shown in the third row, comparing
the entire distributions of (H) circumferential and (I) axial stiffness as well as (J) stored energy over the four study groups.
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or well-organized elastic fiber networks as in normal vessels; there
were, however, no apparent differences in smooth muscle cell density
between the concave and convex regions (fig. S5). Overall wall thick-
ness was lower in convex compared with concave regions of the tor-
tuous Fbln5−/− aorta (Fig.4B), with loaded thickness significantly less
in 52- than 20-week-old samples (fig. S5). Yet, the convex regions
had a significantly (P<0.05) thicker adventitia than did the concave
regions, with adventitial fraction not different in 20- and 52-week-old
Fbln5−/− samples.
The adventitial layer in the tortuous Fbln5−/− aortas consists of
multiple layers of fibrillar collagen bundles inclined toward a pri-
mary orientation that corresponds, at the invivo axial stretch and
physiological distending pressures, with the overall axial direction
of the vessel (Fig.4A), consistent with previous observations (13).
Although this primary orientation did not differ significantly, upon
pressurization, between the convex and concave regions or between
the 20- and 52-week Fbln5−/− mice, the degree of dispersion of fiber
orientation around the primary direction did vary (Fig.4C), with
collagen fibers dispersed significantly more in convex compared to
concave regions in the 20-week-old Fbln5−/− samples, while this char-
acteristic was inverted in 52-week-old Fbln5−/− samples, with a higher
dispersion of the fibers in concave regions. The orientation disper-
sion increased with pressure in most regions, indicating load-induced
reorientation from axial toward symmetrically diagonal directions
to counter the increasing pressure, as expected. The only case that
showed an opposite trend, on average, was the convex region of 52-week
Fbln5−/− samples.
Computational modeling reveals mechanisms of tortuosity
Data-informed computational simulations enable numerical exam-
inations of competing hypotheses. Simulations confirmed that the
normal aorta is robust against modest local perturbations in material
properties, geometry, and perivascular support (Fig.5), consistent
with the general absence of tortuosity in the wild-type aorta (figs. S1
and S3). Specifically, the central column of Fig.5 shows fully resolved
adaptations of an initially straight cylindrical segment of a normal
aorta having uniform geometry and material properties and subjected
to invivo homeostatic conditions of blood pressure, flow, and axial
stretch plus local perturbations (two sites reflecting those observed in
a representative Fbln5−/− sample) in material properties (modest local
changes in elastin and collagen parameters; top row), perivascular
tethering (mild local “push” and “pull”; middle row), or local geometry
(slight overall undulations superimposed on the straight segment;
bottom row).
In stark contrast, allowing wild-type material properties to de-
generate toward those of the fibulin-5–null aorta (namely, uniformly
decreasing the parameter governing elastic fiber contributions to
overall wall stiffness and evolving the parameter governing the ori-
entation of diagonal families of collagen fibers to reflect an increased
circumferential orientation; table S2) rendered the vessel vulnerable
Fig. 2. Tortuosity correlates with reduced distensibility and deformability. (A) In vitro pressure versus loaded outer diameter curves for the four study groups reveal
structural stiffening in the Fbln5−/− groups. (B) A TI was calculated at each pressure (10 to 140 mmHg) at the specimen-specific in vivo axial stretch. Data are shown as
means ± SEM for each group. Note the marked differences in TI between Fbln5+/+ and Fbln5−/− aortas, with pressure-dependent changes in geometry exemplified above
one TI versus pressure curve. The in vitro measured TI was plotted against the (C) stored energy and (D) circumferential stretch at 120 mmHg and against the (E) in vivo
axial stretch and (F) structural distensibility, defined as (d2 − d1)/d1(P2 − P1), where d denotes diameter and P denotes pressure, with P2 > P1. Each point represents the
mean of the distribution of one sample. Note the significant (P < 0.05) negative correlation between TI and all four metrics. Correlations were evaluated using the Pearson
correlation coefficient, r.
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to local perturbations under isometric, but not isotonic, end condi-
tions (Fig.5). Specifically, degeneration of wild-type properties
tended to relax the invivo axial stress and induce an axial G&R re-
sponse despite the constant blood pressure and flow. Enforcing a
constant axial tension at the ends (isotonic boundary condition, left
column in Fig.5) could progressively restore the level of axial stress,
hence enabling a nearly straight elongation invivo and preventing
the development of a tortuous vessel. Conversely, enforcing a con-
stant end-to-end length (isometric boundary condition; right column)
inevitably allowed the axial unloading to progress via extracellular
matrix turnover in evolving states, with an increasingly lower axial
tension and an increasingly higher lateral force contributing to out-
of-plane deformations and development of tortuosity.
Note, too, that the far-right column in each of the panels in Fig.3
compares mean results from one of these G&R simulations (top right
in Fig.5, for which higher regional differences were observed, con-
sistent with differential growth induced by the nonuniform local per-
turbation in material properties) against the measured regional results
for biaxial stretch and material stiffness from the experiments on
the fibulin-5–null aorta, thus confirming the goodness of the in sil-
ico simulations and, in particular, that mechanobiologically stimulated
turnover of extracellular matrix in evolving mechanical states can
lead to a progressive, persistent tortuosity.
Figure6 highlights one particular progression of tortuosity (absent-
to-severe) for the case of initially perturbed elastin and collagen prop-
erties, with regional values of biaxial wall stress and material stiffness
shown superimposed on the tortuous vessel. The local change in
wall volume (or mass given constant tissue density) is also shown
(bottom row), consistent with the simulated (i.e., emergent, not pre-
scribed) G&R response to the imposed insults. Such simulations can
be stopped at any intermediate time and, with G&R frozen, isochoric
elastic stress-strain responses can be computed for transient changes
in blood pressure and/or axial extension, hence simulating a biaxial
mechanical test. Figure S6 shows such a distension test performed
Fig. 3. pDIC reveals regional differences in structure and properties in tortuosity. Inflection points, defined on the aortic centerline (Fig. 1), were used to divide all
Fbln5−/− samples into four local regions along the long axis of the vessel: proximal (green and yellow), distal (blue and orange), convex (green and orange), and concave
(yellow and blue), each consisting of ~100 elements. Box plots (average denoted by *) compare biaxial material stiffness and stretch at 120 mmHg between convex and
concave regions at proximal and distal ends. Note the markedly higher circumferential (Circ.) stiffness and axial stretch and lower circumferential stretch in convex
regions. The far-right column within each panel (separated by a dotted line and denoted Comp.) shows results from the computational G&R simulations (see below),
confirming the same trends between convex and concave regions. The computational results for the proximal and distal ends are identical and are shown on all panels
for clarity. An overbar denotes statistical significance between regions, with *P < 0.05, **P < 0.01, and ***P < 0.001. Effect sizes are shown in table S3. Last, note that the
intercostal branches are anatomically outside of the four selected regions (black dashed rectangles in the center panel) and were excluded from data analysis. Consistent
with Fig. 1, the rightmost edge of each 3D rendering is at about 0°, whereas the leftmost edge is at about ±180°; see the middle panel for further clarification of the mapping
from 3D to 2D.
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on the partially tortuous artery seen in Fig.6 (second from the right
column) at the invivo value of axial stretch (central column, invivo
systolic pressure). Although the degree of tortuosity is greater at higher
pressures (right column) and less at lower pressures (left column),
marked changes in pressure (from 80 to 160 mmHg) induce only
moderate elastic bends when computed on an otherwise vulnerable
vessel. The local invariance in wall volume (or mass) during tran-
sient loading is also shown (bottom row), consistent with the simu-
lated isochoric elastic response with locally preserved (i.e., from the
previous frozen simulation) material insults.
To explore further the effect of the initial degree of tortuosity on
the subsequent elastic bending response (see Fig.1D), the first row
in fig. S7 shows how a distension test performed on the same vessel
at a time when the degree of growth-induced tortuosity is lower
(Fig.6, second from the left column) generates a much lower effect on
the tortuosity. In stark contrast, fig. S7 (second row) shows that this
modest axial growth–induced tortuous remodeling increases notably
when a moderate increase in systolic pressure up to P=135 mmHg
(<160 mmHg for the elastic distension test) is combined with the
underlying partial change in material properties from normal, hence
reinforcing the idea that tortuosity develops from a progressive
mechanobiological, not just mechanical, structural instability. The
(initially perturbed) normal aorta did not remodel into a tortuous
geometry when subjected to the same increases in pressure, suggest-
ing that persistent hypertension is a risk factor only for previously
compromised vessels.
Last, fig. S7 (last row) shows how, once tortuosity has developed
(see Fig.5, top left), a responsive change in the maximal regional
orientation of diagonal fibers from the convex toward the concave
regions (left to right, simulating the inverted trend for the dispersion
of collagen fibers observed for 52-week-old Fbln5−/− samples rela-
tive to 20-week-old Fbln5−/− samples; see Fig.4C) cannot resolve
the permanent tortuosity of this compromised, yet axially unload-
ed, computational aorta. This last finding suggests that arteries that
have grown and remodeled into a marked tortuous pattern may be
mechanobiologically and structurally stable against possible reverse
remodeling, even if new spatial nonuniformities are inverted with
respect to the (otherwise critical) perturbations that contributed to
dictate the developing arterial shape.
DISCUSSION
It is well known that invitro and exvivo distension tests on isolated
cylindrical segments of even a normal artery can provoke a repro-
ducible, reversible pressure-induced bending that depends on the
length of the specimen, the degree of its fixed axial pre-stretch, and
the magnitude of the pressure applied. Critical distending pressures
that induce such bending are reduced at lower values of axial exten-
sion, and this phenomenon can be understood as an elastic buckling
instability (7). Given the clinical associations of tortuosity with con-
ditions of compromised elastic fiber integrity—aging, arterial tortuosity
syndrome, Loeys-Dietz syndrome, Marfan syndrome, and so forth—
considerable attention to understanding such buckling invitro has
appropriately focused on experimental models of defective, dam-
aged, or degraded elastic fibers. Years ago, Dobrin etal. (14) treated
excised canine carotid arteries with elastase and suggested that loss
of elastin releases axial stress within an artery held at a fixed length,
thus increasing its potential to buckle. Lee etal. (15) confirmed that
Fig. 4. Multiphoton microscopy. (A) 3D images for convex and concave regions of 20- and 52-week-old Fbln5−/− aortas, shown here for selected axial-circumferential
slices at specimen-specific in vivo axial stretch and 80 mmHg. Collagen appeared as undulated bundles with a preferential orientation in the adventitia; elastin-based
structures were irregular and scattered in all regions, as expected with the absence of fibulin-5. (B) Mean thickness of the wall and adventitia obtained from multiple 3D
images (age groups merged) at three physiologic pressures and the sample-specific axial in vivo axial stretch (see fig. S5, age effects). Adventitial thicknesses were nor-
malized by, and thus independent of, overall wall thickness, which was lower in convex than in concave regions, with a higher fraction of adventitia in convex compared
to concave regions and, consequently, the opposite for the medial fraction. (C) Orientation dispersion of collagen fibers determined for concave and convex regions at
three different physiologic pressures and the sample-specific axial in vivo stretch. Higher dispersion manifested in the convex region compared to the concave regions in
20-week Fbln5−/−, while 52-week Fbln5−/− had higher dispersion in concave regions. Aging increased the fiber orientation dispersion. Significance denoted by *P < 0.05.
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critical buckling pressures are lower invitro for elastase-treated than
native porcine carotid arteries, while Luetkemeyer etal. (11) reported
similar findings for elastin haploinsufficient (Eln+/−) murine carotid
arteries. Conversely, increased smooth muscle cell tone can increase
the critical buckling pressure (16), whereas perivascular support can
change the buckling mode (15). Although these studies provide valu-
able insight, there is no clear evidence that such instantaneous (elastic)
buckling occurs invivo; moreover, such buckling resolves instanta-
neously when pressure is lowered, and it has been suggested that this
type of buckling alone cannot explain the marked, persistent tortuosity
observed in patients or mouse models. Even investigators focusing
on elastic buckling instabilities suggest that one should consider ar-
terial G&R as fundamental to the process of tortuosity (10,11,17).
On the basis of clinical observations, many speculate that tortu-
osity develops from the axial growth of an artery in the presence of
geometric constraints, including branches or sites of attachment to
perivascular structures (6,18). Langille and colleagues (18,19) showed
directly in an invivo rabbit model that surgically induced increases
in axial loading of the carotid artery initiate a rapid cell-mediated
G&R response that reduces the axial wall strain back to normal within
1 week while maintaining a straight geometry (captured by our model,
see left column in Fig.5), consistent with exvivo findings (20) and
the concept of mechanical homeostasis (21). In contrast, surgically
induced decreases in axial loading result in marked persistent tortuosit y
and normal cell-mediated responses cannot resolve this tortuosity,
at least over a period of weeks (see Fig.5, right column). These ex-
perimental findings are consistent with clinical inference and early
invitro observations, suggesting that reducing axial tension within
a vessel can lead to persistent tortuosity invivo, although the mech-
anisms remained largely unknown.
The descending thoracic aorta is a common site of tortuosity, es-
pecially in aging (22), with a slight bias toward increased frequency
in females (23). Rather than study sex as a biological variable, we
focused on adult female mice aged 20 or 52 weeks. Moreover, rather
than focusing on elastic buckling of pressurized arteries invitro, we
studied directly both normal and tortuous descending thoracic aortas
using Fbln5+/+ and Fbln5−/− mice from the same breeding colony, the
latter of which have compromised elastic fiber integrity (24), exhib-
ited marked tortuosity invivo (25), and represented a model of ar-
terial aging (26). We used a four-fiber family constitutive relation that
has proven useful in describing the mechanics of murine arteries (24)
and has been independently validated as the best of many common
descriptors, even for describing buckling (11). Notwithstanding the
high fidelity and general utility of biaxial extension-distension tests
Fig. 5. Computational modeling reveals mechanisms of arterial tortuosity. Fully resolved, mechano-adapted, G&R responses (central column)—predicted for an
initially straight cylindrical aortic segment having normal uniform geometric and material properties (ϑ = ϑWT) subjected to in vivo homeostatic conditions of blood
pressure, flow rate, and axial stretch (representing an idealized Fbln5+/+ sample, not shown)—to local insults (two sites reflecting that observed via pDIC in a representa-
tive Fbln5−/− sample) in material properties (representing a modest local loss of elastic fiber integrity and remodeling of diagonal families of collagen toward the circum-
ferential direction, top row), a change in perivascular tethering (mild local push and pull, middle row), or a change in local geometry (prescribed slight overall undulations,
bottom row). Next, for all three types of insults considered here, a marked degeneration in elastin and collagen parameters toward those of the fibulin-5–null aorta
(ϑWT → ϑKO) induces a severe axial growth response under in vivo blood pressure and flow, which leads to a nearly straight lengthening of the aorta for a constant axial
tension prescribed at the ends (isotonic boundary condition, left column) but a markedly tortuous elongation for fixed ends (isometric boundary condition, right column).
See Materials and Methods for definitions of ϑ.
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in characterizing bulk mechanical properties of the murine vasculature
(Fig.2), the uniqueness of our validated panoramic digital image
correlation (pDIC) + OCT inverse approach to quantify local mate-
rial properties (27) revealed marked regional variations in properties
within the tortuous aorta that manifest at major bends (convex versus
concave aspects) therein (Fig.3).
Motivated by these experimental findings, we developed a con-
strained mixture finite element model of aortic G&R to numerically
contrast multiple competing hypotheses. The general theoretical fram e-
work upon which this model rests has proven reliable in describing
diverse vascular adaptations and maladaptations (28) and here pre-
dicted an emergent, progressive, persistent tortuosity similar to that
observed experimentally. The model suggested that a uniform loss
of elastic fiber integrity elicits a G&R response under homeostatic
loading, but this alone does not lead to tortuosity. Rather, an additional
change in material properties (realigned diagonal collagen fibers) is
needed to stimulate a potent axial growth response and associated
axial off-loading that, in combination with an imposed localized per-
turbation in properties, initial geometry, or perivascular tethering,
induces progressive tortuosity if and only if aortic displacements are
fixed at the ends. That is, tortuosity does not develop in an otherwise
vulnerable vessel that is maintained under a constant axial tension
(i.e., with axial strain continuously restored back to normal via G&R
in evolving configurations), which simply elongates while main-
taining a nearly straight geometry. Hence, the current simulations
are consistent with experimental observations from invitro and
invivo tests that loss of axial loading via longitudinal elaboration of
new tissue contributes to the development of tortuosity (19), but
suggest further that anatomical constraints and local structural im-
perfections or loading nonuniformities are critical in driving tortu-
osity when particular material properties differ from normal. Our
findings regarding roles of an increased ratio of circumferential-
to-axial stiffness and axial unloading (Fig.6) are consistent with finite
element simulations of tortuosity in veins (29), although that previ-
ous study called for, but did not include, the essential mechano-
biologically driven G&R response that we included.
Fig. 6. Abnormal axial elaboration of tissue off-loads the vessel and gives rise to tortuosity. Progressive initiation and development of persistent tortuosity (i.e.,
remaining in the absence of loading) for a normal aortic segment (left column; ϑ = ϑWT, including a modest change in local properties at two sites) induced by an increas-
ingly greater loss of elastic fiber integrity ∆ ϑ = ∆ c
e = c
e − c
WT
e combined with a gradual increase in orientation of diagonal fibers of collagen ∆ϑ = ∆0 = 0 − 0WT (left
to right) up to ϑ = ϑKO (right column); see table S2. Shown superimposed on the tortuous vessel are the regional values of biaxial wall stress (first and second rows), ma-
terial stiffness (third and four rows), and change in wall volume (fifth row), the last revealing a gradual increase in mass by deposition along the axial direction (with lumi-
nal radius and wall thickness nearly preserved throughout). The G&R model predicted both an initial axial off-loading (second row) of the nearly straight artery until the
loss of its structural stability (pre-buckling response, ∆ϑ ≲ 0.8 ∙ ∆ϑKO) and a tortuous remodeling that follows the instability (post-buckling response due to matrix turnover
in evolving states, ∆ϑ > 0.8 ∙ ∆ϑKO), with a smooth transition between phases afforded by the initial local perturbation. See Materials and Methods for definitions of ϑ.
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Others have considered G&R effects when performing linearized
stability analyses of arteries. Goriely and Vandiver (17) showed ben-
eficial roles of residual stresses (induced via previous growth) and
axial stretch in preventing elastic buckling of arteries under internal
pressure. Liu etal. (30) analyzed the mechanical stability of collateral
arterioles and concluded that they become prone to buckling due to
a combination of axial and radial growth. Notwithstanding the utility
of these and similar studies to detect elastic bifurcations, which de-
fine critical loads and associated mode shapes under which a tortuous
response may initiate, such linear (pre-buckling) stability analyses
cannot describe the fully nonlinear (post-buckling) G&R response
that may develop after the onset of an initial instability. Further-
more, growth was previously prescribed without connection to the
evolving stress or strain fields, although it is well known that G&R
responses are inherently coupled with the mechanics through the
mechanosensitive responses of the cells of the arterial wall (1).
By contrast, we used a fully nonlinear (geometrically and mate-
rially), mechano-regulated, G&R computational model capable of
describing both the initiation and subsequent development of tor-
tuosity (with a smooth transition under invivo conditions) based on
multiple microstructural hits informed by our experimental study.
It is worth noting that other fully nonlinear frameworks have been
used to study growth-induced structural instabilities involving other
soft tissues, as, for example, cortical folding in the developing brain
(31,32), but not tortuosity in arteries, especially when accounting
for the differential material properties of the different structural
constituents that make up the arterial wall, which proved particu-
larly critical in our constrained mixture model of G&R.
Many conditions that associate with arterial tortuosity clinically
appear to suggest that a loss of elastic fiber integrity alone increases
the possibility of tortuosity (6,23), consistent with invitro studies
showing that elastase treatment and genetic reductions in elastin syn-
thesis decrease the critical pressure at which an elastic buckling in-
stability manifests during invitro distension tests (11,14,15). However,
the invitro determined critical buckling pressure of the carotid ar-
tery was lower in the Eln+/− mouse than in the Fbln5−/− mouse (11),
yet there is no reported invivo tortuosity in Eln+/− mice, which are
hypertensive, compared with the Fbln5−/− mouse, which is normo-
tensive. It is unclear, therefore, whether invitro elastic buckling
studies alone can predict invivo tortuosity. By contrast, the present
experimental-computational study suggests that compromised elastic
fiber integrity represents a critical vulnerability but only when com-
bined with an increased collagen-related circumferential-to-axial
stiffening and axial unloading, fixed displacements at the ends of the
vessel, and localized perturbations from normal, including material
imperfections, nonuniform tethering, or an aberrant local curvature.
Our model allowed each of these conditions to be prescribed conve-
niently and compared directly in sites reflecting those in the tortuous
fibulin-5–null aortas; additional combinations of insults may further
exacerbate the tortuosity.
Regardless, the numerically simulated distension tests revealed
further that the extent of the associated elastically induced bending
is proportional to the underlying, previously manifested, G&R-
induced tortuosity. That is, our findings suggest that the normal aorta,
even in the presence of kinematic constraints and structural imper-
fections, is yet robust against tortuosity and that multiple hits must
coexist for vessels to grow and remodel into a persistent tortuous
geometry. Of course, loss of elastic fiber integrity can alter smooth
muscle cell proliferation (33), and altered axial loading can change
the mitotic axis of the mural cells (34), hence emphasizing the im-
portance of mural composition and loading on the mechanobiology,
not just the mechanics (35).
In good agreement with the G&R simulations and experimentally
observed spatial variations in material properties, our microscopic
observations of vulnerable regions of the aortic wall revealed in-
homogeneities in the adventitial:medial ratio between the convex and
concave regions of the tortuous samples. In particular, the observed
age-related and regional differences in the arrangement of collagen
fibers can be associated with the local variations in material stiffness
observed in both the pDIC + OCT experiments and simulations.
While comparable losses of elastic fiber integrity have been observed
in Fbln5−/− arteries from early stages of maturation (13), reorientations
from an axial to a more diagonal arrangement of collagen fibers in
adult Fbln5−/− mice, due to aging and different collagen fiber organi-
zation as a function of characteristic regions and age, appear related
to the degree of tortuosity and disease progression. In this regard,
the experimentally quantified local microstructural variations due
to arterial aging may suggest, in line with our quantification of the
invitro tortuosity and finite element G&R simulations, a local (in-
complete) restorative process against tortuosity (see fig. S7, third row)
that evolves during aging and that needs to be further investigated.
No model, animal or computational, captures completely the hu-
man condition. We submit, nonetheless, that the Fbln5−/− aorta is a
good invivo model of tortuosity that develops naturally and arises
in the presence of compromised elastic fiber integrity, consistent with
many clinical conditions (4–6,8). Although the pDIC + OCT sys-
tem allowed the first detailed quantification of regional differences
in material properties in a tortuous aorta, the method of cannulation
and lack of perivascular tethering invariably altered the geometry and
wall stress from invivo. Similarly, the computational model cap-
tured many salient features of mechanobiological G&R in arteries,
yet it neglected potentially important effects of regional variations
in flow-induced wall shear stress and fully 3D perivascular tether-
ing, including branch sites. The invivo state is thus more complex
than that considered here. Nevertheless, the simulations delineated
critical differences between uniform and nonuniform geometry, ma-
terial properties, and external loading and similarly between isotonic
and isometric end conditions (Fig.5). Our simulations also distin-
guished two remarkably different competing mechanisms by which
these vessels may become tortuous, namely, a progressive axial grow th–
induced tortuous remodeling under constant pressure (Fig.6) and
a transient pressure–induced tortuous distension with preserved mass
(fig. S6), with only the former leading to a persistent tortuosity as
observed invivo and the latter only manifesting in previously com-
promised vessels (see fig. S7, first row). These predictions are con-
sistent with observations that sustained increases in pressure may
induce extreme tortuosity of the descending thoracic aorta via vessel
wall hypertrophy (36), which, nonetheless, manifested only when
hypertension was superimposed on an otherwise vulnerable, partially
tortuous vessel (fig. S7, second row). We thus submit that this novel
experimental-computational approach reliably uncovered heretofore
unappreciated mechanics-driven mechanobiological instabilities that
contribute to a progressive, persistent tortuosity and can be studied
against additional competing hypotheses in the future.
In summary, there have been multiple calls to consider G&R
processes in the possible emergence and development of tortuosity
(10,11,17,18), but the requisite biomechanical data had not been avail-
able and an appropriate theoretical and computational framework
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9 of 14
capable of describing the nonlinear (including post-buckling) be-
havior of arteries consisting of multiple constituents had not been
used. Both previous limitations were addressed here. Our results
suggest that aortic tortuosity need not arise from an instantaneous
elastic buckling instability. Rather, ubiquitous mechanosensitive
cellular responses to localized microstructural abnormalities can
initiate G&R responses to multiple conspiring insults that collectively
lead to progressive, persistent tortuosity as the vessel attempts to adapt
in the presence of kinematic constraints at branches and sites of strong
perivascular tethering; initial axial unloading and elevated blood
pressure can exacerbate this mechanobiological structural instability.
Our findings are thus consistent with both an invivo study and sug-
gestion that “longitudinal tissue elaboration must first off-load the
axial strain before tortuosity is manifest” (19) and increasing clini-
cal intuition that “vascular elongation between 2 fixed points will
give rise to tortuosity” (6) but bring into focus the probable role of
marked microstructural aberrations and local imperfections. This
interpretation further focuses attention toward the phenotype of the
intramural cells that are responsible for maintaining, remodeling, and
repairing the arterial wall locally, which promises to lead to greater
insight into the potential of tortuosity as a biomarker rather than
simply as a structural anomaly that disturbs blood flow.
MATERIALS AND METHODS
Animals
The Institutional Animal Care and Use Committee of Yale University
approved all animal procedures. Fifteen female wild-type (Fbln5+/+)
and fibulin-5–null (Fbln5−/−) mice were euthanized at 20±0.3 (n=7)
or 52±1.8 (n=8) weeks of age, and the descending thoracic aorta
was excised from the first to the fifth pair of intercostal branches
and prepared for biomechanical testing. These four primary groups
of mice were tested to study the effects of genotype and age on po-
tential aortic tortuosity. Blood pressures were measured 3 to 7 days
before euthanasia using a CODA tail-cuff system (Kent Scientific
Corporation, Torrington, CT); the mice were euthanized via an in-
traperitoneal injection of Beuthanasia-D.
Standard biaxial biomechanical testing
Excess perivascular tissue was removed from the excised aortic seg-
ments, and intercostal branches were ligated individually using single
strands from 7-0 nylon suture. Specimens were then cannulated on
and secured to glass cannulas using 6-0 sutures and then placed within
a custom computer-controlled testing system for biaxial distension-
extension testing, as described previously (37). Briefly, following a
15-min acclimation of the aorta within a Hank’s buffered salt solu-
tion (HBSS) at room temperature (which minimizes contractile tone),
mean arterial pressure, and estimated invivo axial stretch, four pre-
conditioning cycles consisted of pressurization from 10 to 140 mmHg
at the fixed axial stretch. Next, each sample was subjected to cyclic
pressure–diameter tests at each of three fixed axial lengths (in vivo
value and ±5% of this length) and cyclic axial force–length tests at
each of four fixed pressures (10, 60, 100, and 140 mmHg), with a
maximal allowed axial load of 2 to 2.5 g (~20 to 25 mN).
Data from the last cycle of the unloading curve from all seven
protocols were combined and used to determine the best-fit values
of the material parameters in a validated four-fiber family constitu-
tive model to describe the overall bulk passive mechanical behavior,
as described previously (24,37). See table S1.
Optical coherence tomography
Following the distension-extension protocols, the samples were re-
cannulated on a custom blunt-ended triple-needle assembly (using
the same suture locations as in the biaxial test) with the proximal
end of the sample secured to a fixed needle and the distal end to a
sliding one, thus allowing pressurization and axial stretching (fig. S2).
The re-cannulated samples were scanned with a commercially avail-
able OCT system (Thorlabs Inc.) to provide through-the-wall thick-
ness information along the entire circumference and axial length,
with an in-plane resolution of ~7 m. Samples were set at a consistent
invitro reference configuration defined by the specimen-specific
invivo value of axial stretch and a common pressure of 80 mmHg.
Then, 100 cross-sectional images were acquired along the axial length
of the sample (suture-to-suture), yielding an axial spatial resolution
of ~60 to 80 m (depending on sample length). Because of a limited
optical penetration depth, samples were scanned from four differ-
ent rotational views (about the central axes of the vessel) to obtain
sufficient information for a 3D reconstruction.
Multiphoton microscopy
Next, the needle assembly supporting the sample was moved to a
two-photon microscope (Chameleon Vision II, Coherent) with a
titanium-sapphire laser tuned at 840nm and equipped with a water
immersion 20× objective lens [numerical aperture (NA) of 0.95]. The
second-harmonic signal arising from collagen structures was detect-
ed at the wavelength range of 390 to 425 nm, and autofluorescence
of the elastin structures was collected at the range of 500 to 550nm.
3D images represented a volume of 0.05mm3 with an in-plane area
of 500 m × 500 m. The numerical resolution was 0.48 m/pixel,
and the out-of-plane step was 1 m/pixel.
By mounting the needle within a custom-built support, distending
pressure and axial stretch were controlled while the specimen re-
mained in HBSS at room temperature. Moreover, the needle could
be revolved manually about its long axis before a given image acqui-
sition, thus ensuring perpendicularity between the axis of the needle
and that of the objective, which was centered over a specific region
of interest. Again, the sample was preconditioned, this time via five
cycles of pressurization from 10 to 140 mmHg. Next, the distending
pressure was fixed sequentially at 40, 80, and 120 mmHg, each at the
invivo value of axial stretch, thus allowing 3D imaging within mul-
tiple states and multiple (four) regions, focusing in particular on the
two convex and two concave areas of the tortuous samples.
Panoramic digital image correlation
We previously developed a digital image correlation method spe-
cifically for quantifying the 3D geometry and full-field multiaxial
surface strains on small arteries (38). Briefly, following the distension-
extension testing as well as the OCT and multiphoton imaging, the
samples were stained with Evans blue dye and gently airbrushed with
white India ink to form a random characteristic white speckle pattern
on a dark blue background (fig. S2), which does not affect the me-
chanical properties of the arterial wall. The samples (still mounted
on the custom blunt-ended triple-needle assembly) were placed co-
axially (i.e., aligned vertically) within a 45° angle conical mirror (fig.
S2) and again submerged in an HBSS-filled bath at room tempera-
ture. This allowed the speckle pattern to be reflected on the inner surface
of the conical mirror and captured by a single, vertically positioned
digital camera (DALSA Falcon 4M30) from eight symmetric rota-
tional views relative to the central axis of the mirror (fig. S2). Data
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were acquired at multiple quasi-statically loaded combinations:
14 different pressures (10 to 140 mmHg in 10-mmHg increments)
at three different axial stretches (as in the biaxial test). For compu-
tational convenience, the reference configuration was chosen at the
invivo value of axial stretch and a pressure of 80 mmHg.
Last, following all measurements, average values of unloaded wall
thickness were measured from transversely cut rings using a dis-
section microscope, as reported previously (37). These values along
with the OCT-based data of the reference configuration were used for
both the standard biaxial characterization and estimation of the wall
thickness for all reconstructed pDIC surfaces, under the common
assumptions of a thick wall incompressible body and isochoric mo-
tions. See table S1.
OCT and pDIC data analysis
Using custom MATLAB scripts, the many pDIC images were un-
wrapped and a nodal grid of surface positions was defined across
the outer surface of the samples. Serial correlations were performed
over all (neighboring) deformed configurations to compute the full-
field surface deformations at these nodes and to reconstruct the 3D
surface geometry for each deformed configuration. As described
previously (27), the many OCT images (4 views for each of the 100
cross-sectional views) were co-registered to obtain a 3D OCT re-
construction of each aorta using an iterative multistep registration
scheme in 3D Slicer. Once complete, contours of the inner and outer
wall were segmented in each cross-sectional image with co-registration
between the reconstructed pDIC surface and OCT contours, by single
value decomposition, to identify the optimal 3D rigid transforma-
tion, mapping thickness values on the pDIC-derived surfaces. Note
that the local thickness value,hmn, assigned for each discrete patch
and calculated as the distance between the inner and outer wall con-
tours, was not necessarily the precise local thickness of the vessel
itself, as the cross-sectional plane may not always align perfectly nor-
mal to the vessel surface.
For parameter estimation, a cylindrical coordinate system was
defined with Z ∈ [O, L] and ∈ [ − 180,180] coordinates (where L
is the reference length of the sample) and the surface of each configura-
tion was meshed with 40 circumferential (denoted m, 1 ≤ m≤ 40)
by 25 axial (denoted Zn, 1 ≤ n≤ 25) patches (fig. S2), which with the
co-registered OCT images allowed the construction of eight-noded
3D hexahedral elements. All reconstructed configurations were ro-
tated such that intercostal branches were positioned at 90° (Fig.3).
Displacement fields were then derived from the surface deformations,
thus allowing the calculation of the Green strains for each surface
patch at each configuration (35). Further details on the pDIC sys-
tem and analysis can be found elsewhere (27,38). Note that the up-
per and lower 10% of the sample (Zn,1 ≤ n≤ 3 and 23 ≤ n≤ 25)
were excluded from the analysis to eliminate edge effects due to
cannulation and suture ligation.
Inverse characterization of regional mechanical properties
The pDIC-derived point-wise displacement field was assigned to the
40 × 25 outer surface nodes as Dirichlet boundary conditions, and
distending pressure was prescribed as tractions acting normal to each
inner surface node. An incompressible neo-Hookean strain energy
function was first used to solve a finite element model in the open-
source finite element package FEBio (febio.org), thus yielding Green
strains at each Gauss point. Previous experience with murine arteries
showed that a microstructurally motivated constrained mixture mod-
el describes well the constitutive behavior (27). Thus, the passive ma-
terial behavior was then modeled at each Gauss pointk as hyperelastic
with a strain energy function, defined per unit mass, of the form
W k = φ e W
k
e + φ W
k
+ φ z W
k
z + φ d W
k
d (1)
where the elastin-dominated (superscript e) matrix was modeled with
a neo-Hookean strain energy function, and the circumferential (,
muscle and collagen), axial (z, collagen), and diagonal (d, collagen)
contributions of the matrix to the mechanical response were mod-
eled with respective Fung-type exponential strain energy functions
(27). The constituent-specific and (collagen) orientation-specific mass
fractions φe, φ, φz, and φd were assigned on the basis of a previous
histological report of wall composition in the descending thoracic
aorta of Fbln5+/+ and Fbln5−/− mice (overall mass fractions for elas-
tin, e, smooth muscle, m, and collagen, c; table S2) (24) while accoun t-
ing for the respective contribution (, d, and z; table S2) of each
collagen fiber family, which was determined, in combination with
other parameters in table S2, by fits to biaxial data (28).
To calculate the wall stress field at each Gauss point k and the
local material parameters for the mZn elements, the principle of
virtual power was used to achieve inverse characterization. A cost
function Jk involving experimentally measured and theoretically
predicted pressures and axial loads (thin wall assumption) was cal-
culated (27), and all unknown parameters were updated iteratively
until the objective function Jk was minimized (with a stopping cri-
teria of 30s per node and tolerance of 10−6 from one iteration to the
next), signifying final parameter identification. Minimization of Jk
with respect to the unknown linear parameters was achieved via a
non-negative linear least square algorithm and for the remaining
nonlinear parameters via a bounded genetic algorithm. Following
the inverse characterization, the final set of the identified parame-
ters at each element was used to compute the full-field distributions
of various mechanical metrics, in particular, circumferential and axial
material stiffness at the sample-specific invivo axial stretch and sys-
tolic pressure of 120 mmHg, as measured with the tail-cuff system,
and stored energy at either diastole (80 mmHg) or systole (120 mmHg).
For further details on the formulation of the inverse method used
here and discussion on its relevance and uncertainty, see Bersi etal. (27).
Centerlines and tortuosity index
The subroutine vmtkcenterlines, from the open-source Vascular
Modeling Toolkit (vmtk.org), was used to compute centerlines for
each 3D reconstructed configuration. Briefly, source and target seed
points were chosen on the proximal and distal ends of the reconstructed
vascular segment, and centerlines were determined as the weighted
shortest paths traced between those points. To ensure path central-
ity, they were bound to run on the Voronoi diagram of the vessel so
as to minimize the integral of the radius of maximal inscribed spheres
along the path. For further details, see Antiga etal. (39). Sample-specific
centerlines were then used to calculate tortuosity index (TI) as
TI =
L 1 − L 2
─
L 2 (2)
where L1 is the total length of the aortic centerline and L2 is its end-
to-end distance. Values of TI were quantified at each pressure for
the specimen-specific invivo axial stretch for non-tortuous Fbln5+/+
and tortuous Fbln5−/− samples. Note that the tortuous nature of the
latter is characterized by low frequency and high amplitude. It was
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for this reason that this particular measure of tortuosity was adopted,
yet, for completeness, other methods were computed and yielded
similar behavior. Other frequencies and amplitudes exist in other
types of tortuous vessels invivo, but we focused on the descending
thoracic aorta alone.
Last, for the regional data analyses, inflection points (defined as
relative extreme points) on the centerlines were used to divide the
Fbln5−/− samples into four different regions along the long axis of
the vessel (proximal,3 < Z< 14, and distal, 14 ≤ Z< 23) and around
its circumference (convex-concave), with approximately n=100 ele-
ments per region (Fig.3).
Multiphoton image analysis
All acquired 3D images were processed with a custom MATLAB
ImageJ algorithm. Because of the nearly cylindrical shape of the sam-
ples, the structural layers did not reside within circumferential-axial
planes. Thus, images were transformed before analysis: 3D images
were sliced pixel by pixel in the circumferential-radial plane, and
the 2D mid-thickness profile of the wall was used to fit a circle and
obtain the centerline coordinates of the latter. The averaged coordi-
nates of the center were used to perform a Cartesian-to-polar trans-
formation of all circumferential-radial slices. Because the radius of
the cylinder was more than five times the size of the image in the
radial direction, distortions in microstructure due to the transfor-
mation were considered negligible. To capture microstructural metrics
that reflected the aforementioned biomechanical quantities of in-
terest, the analysis focused on layer-specific thickness and in-plane
dispersion of the orientations of the collagen fibers. The layer-specific
thickness was obtained by plotting the mean intensity profile for the
two constituents (collagen and elastin) along the thickness and then
automatically demarcating the adventitial and medial layers via mean
intensity values. The automatic decision algorithm was previously
tuned by using manual selection inputs from multiple 3D images of
murine aortas.
The in-plane dispersion of orientations of the fibers was a mea-
sure of the spreading of the collagen fibers around their primary ori-
entations. Considering previous analyses (40), collagen fibers show
a negligible out-of-plane orientation component. Therefore, the in-
plane distribution of orientations was estimated using a 2D structure
tensor analysis with OrientationJ implemented for ImageJ. Normalized
in-plane distributions of orientation were obtained for all 3D imag-
es and showed the existence of a single in-plane primary orientation
in the axial direction. The distribution was parameterized using a
Von Mises circular probability density function
f(∣, ) = 1
─
2 I 0 () e cos(−) (3)
depending on angle , where I0() is the modified Bessel function of
order 0, is the principal orientation, and is a measure of concen-
tration. The reciprocal 1/ was used for the analysis as a measure of
dispersion of the collagen fiber orientation.
Statistics
Data were analyzed either by comparing the entire distributions
(~800 elements per vessel) of the different mechanical metrics across
the four experimental groups (genotype and age) using a two-way
analysis of variance (ANOVA), followed by Tukey post hoc test for
multiple comparisons, or by comparing different regions (e.g., con-
vex versus concave) within each tortuous sample (Fbln5−/−) using a
one-way ANOVA followed by post hoc Sidak tests. For further in-
terpretation of the regional variations (convex versus concave), Hedge’s
g statistic was used to measure the effect size following the one-way
ANOVA test. Large effect sizes were commonly defined as g>0.8.
Significant differences and effect sizes for the regional analysis are
summarized in table S3. Because of deviations from normality
(Kolmogorov-Smirnov test), differences between TIs among the four
study groups were compared with the nonparametric Kruskall-Wallis
test followed by Dunn’s post hoc test. Differences in the multiphoton
microstructural parameters were assessed with a one-way ANOVA
followed by post hoc Sidak tests. Correlations between TI and the
various mechanical metrics were assessed using the Pearson correla-
tion coefficient, r. For all reported comparisons, a value of P<0.05
was considered significant.
Theoretical G&R framework
The normal thoracic aorta consists of myriad constituents that en-
dow the wall with biomechanical functionality and strength. Most
of these constituents can turn over, which enables the aorta to adapt
its geometry (e.g., caliber and wall thickness) and composition (e.g.,
relative contributions of collagen to elastin), and thus biomechanical
properties, in response to diverse changes in loading (e.g., blood
pressure, flow, or axial stretch) or to actively respond to alterations
that occur in diverse disease processes (e.g., progressive endothelial
dysfunction)—see Humphrey (35) for examples.
Over a period of years, we developed a “constrained mixture model”
of arterial growth (changes in mass) and remodeling (changes in
microstructure) that accounts for different structurally significant
constituents that can have different natural configurations, material
properties, and rates of turnover. This model can thus describe and
predict the evolving composition, geometry, and nonlinear material
properties under many conditions. Briefly, three different constitu-
tive relations (for rates of mass production and removal as well as
the passive multiaxial mechanical properties) must be prescribed for
each of the structurally significant constituents, which, for convenience,
we lump into three groups: elastin-dominated matrix, collagen-
dominated matrix, and smooth muscle cells. Because maximum
smooth muscle tone is reduced in the aortas of Fbln5−/− mice (41)
and invivo tone is even less, we neglected contractility here. Thus,
the passive biomechanical behavior of the aortic wall is given in terms
of the Cauchy stress
(s ) = − p(s ) I + 2
─
J(s) F(s )
∂ W R (s)
─
∂ C(s) F T (s) (4)
where p is a Lagrange multiplier that enforces isochoric motions during
transient loading and C = FTF is the right Cauchy-Green tensor, com-
puted from the deformation gradient F from reference to current
(in vivo) configurations for the mixture, W R = W
R
, that is, the
total stored energy per unit reference volume WR is the sum of the
energies stored in the constituent parts,s is the current G&R time,
and J = det F. For the constituent-specific passive mechanical properti es,
we let
W
R
e (s ) =
R
e (s )
ˆ
W e (s ) =
R
e (s )
c e
─
2 (tr( F eT (s ) F e (s ) ) − 3) (5)
for the elastin-dominated matrix (e), where
R
e = J e represents
its referential mass fraction within the mixture, Fe = FGe is the de-
formation gradient measured from its fixed natural (stress-free)
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12 of 14
configuration, with Ge as a deposition stretch tensor, and ce is a shear
modulus that can be determined from biomechanical testing data.
For the energy stored in multiple fiber families, which turn over
continuously, we let
W
R
c,m (s ) = 1
─
∫−∞
s
m
R
c,m ( ) q c,m (s, )
ˆ
W c,m
(
n()
c,m (s )
)
d (6)
for collagen (c) and passive smooth muscle (m) dominated be-
haviors, where is the mass density of the tissue, m
R
c,m = Jm c,m are
referential mass density production rates, is the time at which the
constituent is produced, and q(s, ) ∈ [0,1] is an exponential decay
function that models removal, where s is the current time of interest.
Last,
n()
c,m (s) is the fiber stretch relative to the associated evolving
natural configurations, including the extent of the respective depo-
sition stretches Gc, m, and
ˆ
W c,m
(
n()
c,m (s )
)
= c
1
c,m
─
4 c
2
c,m
(
exp
[
c
2
c,m
(
(
n()
c,m (s )
)
2 − 1
)
2
]
− 1
)
(7)
with c
j
i material parameters that can be determined from biaxial mechan-
ical tests on excised arteries. Additional details of the constitutive
framework can be found elsewhere (28). However, we emphasize
that our overall approach enforces mechanical equilibrium quasi-
statically (div = 0) throughout G&R, meaning that progressive
processes such as tortuosity are mechanically achievable at every
pseudo- time step and they can thereby persist in the absence of loading
because load-bearing extracellular matrix turns over in the evolving
states.
Model parameters
We used biaxial biomechanical data from a normal descending tho-
racic aorta from a wild-type (Fbln5+/+) mouse to parameterize the
baseline biomechanical model, with additional parameters describ-
ing the time course of a mechano-adaptive G&R response extracted
from Latorre and Humphrey (42). Similarly, a set of passive mechani-
cal parameters was determined from biaxial biomechanical data from
a representative descending thoracic aorta from a fibulin-5–null
(Fbln5−/−) mouse. All parameters are listed in table S2. Note that the
potentially compromised biomechanical response of the fibulin-5–
null mouse does not represent a response that evolved from a normal
wild-type mouse because these mice have a germline mutation and
thus an independent development and maturation processes. Con-
sequently, the fibulin-5 null–related parameters in table S2, which
reveal the main difference in bulk passive mechanical properties
between these two genotypes, provide qualitative insight to guide our
computational modeling of wild-type aortas that may develop tor-
tuosity due to losses in elastic fiber integrity.
Recall, therefore, that in addition to the general risk factor of aging,
compromised elastic fiber integrity (see, e.g., the reduced material
parameter ce and in-plane invivo pre-stretches G
e and G
z
e for fibulin-
5–null mice in table S2) appears to be a major contributor to axial
growth–induced tortuosity (4,6,7,14,23,24). To test numerically
the potential of compromised elastin to induce axial growth and
develop, as a result, tortuous vessels consistent with appropriate ini-
tial perturbations and boundary conditions, we gradually prescribed
a loss of elastic fiber integrity (described by a maximal ~20% reduc-
tion in the value of the elastin-associated material parameter ce; see
table S2) either in isolation or in combination with other potential
contributors consistent with differences observed in table S2. Pre-
liminary simulations revealed that the computed axial growth re-
sponse was notably more pronounced when the prescribed loss of
elastic fiber integrity was combined with a deposition of diagonal
collagen increasingly oriented toward the circumferential direction
(described by a ~30% increase in the value of the collagen-associated
material parameter 0; see table S2). A simultaneous variation of
other parameters (e.g., those in the Fung-type exponentials for smooth
muscle and collagen fibers) either contributed to the axial growth
response to a lesser extent [see Dobrin etal. (14)] or required com-
plex nonlinear relations among their coupled evolutions to yield real-
istic (bounded) G&R responses. Hence, as a first illustrative attempt
to study axial growth with minimal simultaneous changes in pa-
rameters from wild-type to elastic fiber compromised (determined
from respective sets of biaxial biomechanical data), we prescribed
the desired loss of elastic fiber integrity either in isolation or in com-
bination with the aforementioned increased orientation for diagonal
collagen families, with all remaining parameters in our model fixed
to baseline (wild-type) values.
Although it has been suggested that elastic buckling may be an
initiator of arterial tortuosity (10,11), it is not yet clear that the geo-
metrical nonuniformities that arise from this instantaneous buck-
ling (a bifurcation response that takes place over a short time scale)
can result in persistent tortuosity [a gradual remodeling response
that takes place over much longer time scales; (43,44)]. That is, be-
cause a transiently buckled (i.e., bifurcated) vessel adopts a curved
geometry (i.e., secondary solution) that breaks overall axisymmetry,
it is possible that, rather than being an initiator of permanent tortu-
osity, this elastic buckling simply represents an initial (geometrical)
perturbation that enables an axially constrained and prestressed vessel
to develop a growth-induced tortuosity initiated by aging or partic-
ular genetic defects (6). Other structural (geometrical or material)
nonuniformities initially present in the aorta may equally enable a
subsequent tortuous enlargement, as we describe next.
Thus, for purposes of illustration, consider an initially perfectly
straight cylindrical segment of the thoracic aorta of uniform invivo
wall thickness (ho = 0.34m) and luminal radius (ao = 568m) into
which the following one-wave–like asymmetric (i.e., axially and cir-
cumferentially nonuniform, reflecting that observed in a Fbln5−/−
sample) material insult is placed
ϑ( r o , o , z o ) = ϑ WT + 1
─
2 (ϑ − ϑ WT ) + 1
─
2 (ϑ − ϑ WT ) 1
─
2
(
1 − cos 2 z o
─
l o
)
exp
(
−
|
o −
ˆ
─
2
|
2
)
(8)
where ro ∈ [ao, ao + ho], o ∈ [0,2], and zo ∈ [0,2lo] are the radial,
circumferential, and axial coordinates in the reference (homeostat-
ic) configuration, respectively; ϑWT is the baseline [wild-type (WT)]
value of the associated material parameter (ce or 0); and ϑ is the
stimulation driver for the quasi-static G&R response with
ˆ
=
ˆ
1 =
for zo ∈ [0, lo] and
ˆ
=
ˆ
2 = −
ˆ
1 = 0 for zo ∈ [lo,2lo]. We start
our simulations from ϑ = ϑWT, for which ϑ(ro, o, zo) = ϑWT is uni-
form, and then vary ϑ up to ϑ = ϑKO, for which ϑ
(
r o ,
o
cr , z
o
cr
)
= ϑ
(
r o ,
o
tr , z
o
tr
)
= ϑ KO at the crest
(
o
cr = , z
o
cr = l o / 2
)
and trough
(
o
tr = 0,
z
o
tr = 3 l o / 2
)
of the computational aorta. That is, rather than start-
ing our G&R simulations from a geometrical nonuniformity with
uniform material properties (e.g., a buckled artery), we prescribe a
material nonuniformity on a uniform straight artery to enable dif-
ferential (regional) axial growth responses that can potentially induce
Weiss et al., Sci. Adv. 2020; 6 : eabd3574 4 December 2020
SCIENCE ADVANCES | RESEARCH ARTICLE
13 of 14
a mechanobiological structural instability and subsequent tortuosity.
Furthermore, note that we can first prescribe a nonuniform pertur-
bation in parameters on the normal aorta via the third term on the
right-hand side in Eq. 8 and, subsequently, prescribe an incremen-
tal uniform insult via the second term (see Figs.5, first row, and 6).
Alternatively, a uniform change in these elastin-collagen parameters
ϑ ∈ [ϑWT, ϑKO] prescribed on an initially undulated artery (as a result
of nonuniform tethering or geometric imperfections) can, similarly,
give rise to axial growth–induced elongation and tortuosity (Fig.5,
second and third rows).
Here, we use a fast, efficient 3D implementation (28) that is fully
nonlinear (materially and geometrically, properly accounting for finite
strains and rotations) and based on the underlying assumption that
each G&R state is mechanobiologically equilibrated, which holds for
cases wherein the characteristic time scale of the remodeling process
is shorter than the stimulation time scale, that is, fully quasi-static
G&R. This rate-independent formulation eliminates dependence on
G&R time s and allows equilibrium to be enforced invivo in ho-
meostatic grown and remodeled states for given external loads and
boundary conditions. In particular, mechanobiological equilibrium
leads to W
Rh
c,m =
Rh
c,m
ˆ
W c,m ( G
h
c,m ) in Eq. 6, which, in combination with
an equilibrated energy for elastin in Eq. 5, leads to a rule-of-mixtures
relation for the total energy W Rh =
Rh
ˆ
W with evolved constituent
mass fractions and deformations. The Cauchy stress in Eq. 4 also adop ts
a rule-of-mixtures expression in terms of evolved constituent-
specific passive stresses
h =
e,c,m
h
ˆ
h
− p h I (9)
with the Lagrange multiplier ph consistently determined from a mechan o-
biological equilibrium constraint (that enforces a balanced production
and removal of constituents) during the quasi-static G&R evolution.
A consistent linearization of the formulation enables implementa-
tion within a finite element framework, where simultaneous solu-
tion of mechanical and mechanobiological equilibrium can be ensured
efficiently at successive load steps that capture evolving geometries,
compositions, and properties of interest for complex boundary value
problems, including post-buckling–like responses that give rise to
persistent tortuosity.
Last, our nonlinear computational model of the aortic wall allows
us to compute many important biomechanical metrics, including
biaxial (circumferential and axial) wall stretch, stress, and material
stiffness as well as elastic energy storage. The finite element model
for the initially straight cylindrical segment of the thoracic aorta
comprised NrNNz = 1 × 20 × 40 = 800 displacement-based 3D quadratic
elements with full 3 × 3 × 3 Gauss integration and was computed
quasi-statically in a modified open-source code FEBio; see additional
constitutive, algorithmic, and computational details in the study of
Latorre and Humphrey (28).
SUPPLEMENTARY MATERIALS
Supplementary material for this article is available at http://advances.sciencemag.org/cgi/
content/full/6/49/eabd3574/DC1
View/request a protocol for this paper from Bio-protocol.
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Acknowledgments: We thank the staff of the microscopy core at Yale for expert assistance.
Funding: This work was supported by grants from the U.S. NIH (R01 HL105297, P01 HL134605,
and U01 HL142518). Author contributions: D.W. and J.D.H. conceived the project. D.W., C.C.,
and A.B.R. collected the data. D.W., C.C., A.G., and S.A. analyzed the data. M.L., D.W., and J.D.H.
conceived the computational study. M.L. performed the computational simulations. D.W., C.C.,
M.L., and J.D.H. interpreted the results. D.W., M.L., and J.D.H. wrote the manuscript. All authors
revised and approved the final manuscript. Competing interests: The authors declare that
they have no competing interests. Data and materials availability: All data needed to
evaluate the conclusions in the paper are present in the paper and/or the Supplementary
Materials. Additional data related to this paper may be requested from the authors.
Submitted 16 June 2020
Accepted 22 October 2020
Published 4 December 2020
10.1126/sciadv.abd3574
Citation: D. Weiss, C. Cavinato, A. Gray, A. B. Ramachandra, S. Avril, J. D. Humphrey, M. Latorre,
Mechanics-driven mechanobiological mechanisms of arterial tortuosity. Sci. Adv. 6, eabd3574
(2020).
