398
7 Remodeling
a r = 1.2 d
−1
a θ = 0.6 d
−1
a τ = 1.2 d
−1
σ 0 = 160 kPa
τ f 0 = 1.5 Pa
b τ = 2 d
−1
b K = 0.2 d
−1
K 0 = 0.9.
(a) Determine σ m
θ , σ a
θ , and σ c
θ in terms of the total stretch ratio λ θ , growth
ratios G n
i , contraction ratio K θ , and collagen deposition stretch ratio λ c
0 .
(b) For the given pressure P 0 and flow rate Q 0 , compute the deformed radius
and wall thickness under homeostatic conditions. These values can guide
your selection of a 0 and h 0 to start the computation.
(c) Write a system of four equations to solve for λ θ (t), G m
θ (t), G m
r (t), and
K θ (t).
(d) Write a code to compute the temporal behavior of the model for specified
blood pressure and flow rate.
(e) Compute the solution for constant pressure and flow, i.e., P (t) = P 0 and
Q(t) = Q 0 . Plot the total and partial wall stresses, the fluid shear stress,
and the growth and contraction ratios as functions of time for 0 ≤ t ≤
20 days. Do the same for the deformed radius a, wall thickness h, and
the ratio a/ h. If things are working correctly, the solution should reach
a homeostatic state within this time. Compare solutions with contraction
turned off (K θ = 1) and turned on.
(f) Repeat part (e) for double pressure (P = P 0 , Q = 0) and double
flow rate (Q = Q 0 , P = 0), beginning at the homeostatic state
(approximately t 0 = 10 days). Also, plot P (t)/P 0 and Q(t)/Q 0 . Explain
your results.
References
Baek S, Rajagopal KR, Humphrey JD (2005) Competition between radial expansion and thickening in the enlargement of an intracranial saccular aneurysm. J Elasticity 80:13–31
Baek S, Rajagopal KR, Humphrey JD (2006) A theoretical model of enlarging intracranial fusiform
aneurysms. J Biomech Eng 128:142–149
Bovendeerd PH (2012) Modeling of cardiac growth and remodeling of myofiber orientation. J
Biomech 45:872–881
Chanet S, Martin AC (2014) Mechanical force sensing in tissues. In: Progress in molecular biology
and translational science, vol 126. Elsevier, New York, pp 317–352
Chen K, Vigliotti A, Bacca M, McMeeking RM, Deshpande VS, Holmes JW (2018) Role of
boundary conditions in determining cell alignment in response to stretch. Proc Natl Acad
Sci U S A 115:986–991
Chow MJ, Turcotte R, Lin CP, Zhang Y (2014) Arterial extracellular matrix: a mechanobiological
study of the contributions and interactions of elastin and collagen. Biophys J 106:2684–2692
Coravos JS, Mason FM, Martin AC (2017) Actomyosin pulsing in tissue integrity maintenance
during morphogenesis. Trends Cell Biol 27:276–283
Cowin SC (2001) The false premise of Wolff’s law. In: Cowin SC (ed) Bone biomechanics
handbook. CRC Press, Boca Raton, pp 30-31-15
7 Remodeling
a r = 1.2 d
−1
a θ = 0.6 d
−1
a τ = 1.2 d
−1
σ 0 = 160 kPa
τ f 0 = 1.5 Pa
b τ = 2 d
−1
b K = 0.2 d
−1
K 0 = 0.9.
(a) Determine σ m
θ , σ a
θ , and σ c
θ in terms of the total stretch ratio λ θ , growth
ratios G n
i , contraction ratio K θ , and collagen deposition stretch ratio λ c
0 .
(b) For the given pressure P 0 and flow rate Q 0 , compute the deformed radius
and wall thickness under homeostatic conditions. These values can guide
your selection of a 0 and h 0 to start the computation.
(c) Write a system of four equations to solve for λ θ (t), G m
θ (t), G m
r (t), and
K θ (t).
(d) Write a code to compute the temporal behavior of the model for specified
blood pressure and flow rate.
(e) Compute the solution for constant pressure and flow, i.e., P (t) = P 0 and
Q(t) = Q 0 . Plot the total and partial wall stresses, the fluid shear stress,
and the growth and contraction ratios as functions of time for 0 ≤ t ≤
20 days. Do the same for the deformed radius a, wall thickness h, and
the ratio a/ h. If things are working correctly, the solution should reach
a homeostatic state within this time. Compare solutions with contraction
turned off (K θ = 1) and turned on.
(f) Repeat part (e) for double pressure (P = P 0 , Q = 0) and double
flow rate (Q = Q 0 , P = 0), beginning at the homeostatic state
(approximately t 0 = 10 days). Also, plot P (t)/P 0 and Q(t)/Q 0 . Explain
your results.
References
Baek S, Rajagopal KR, Humphrey JD (2005) Competition between radial expansion and thickening in the enlargement of an intracranial saccular aneurysm. J Elasticity 80:13–31
Baek S, Rajagopal KR, Humphrey JD (2006) A theoretical model of enlarging intracranial fusiform
aneurysms. J Biomech Eng 128:142–149
Bovendeerd PH (2012) Modeling of cardiac growth and remodeling of myofiber orientation. J
Biomech 45:872–881
Chanet S, Martin AC (2014) Mechanical force sensing in tissues. In: Progress in molecular biology
and translational science, vol 126. Elsevier, New York, pp 317–352
Chen K, Vigliotti A, Bacca M, McMeeking RM, Deshpande VS, Holmes JW (2018) Role of
boundary conditions in determining cell alignment in response to stretch. Proc Natl Acad
Sci U S A 115:986–991
Chow MJ, Turcotte R, Lin CP, Zhang Y (2014) Arterial extracellular matrix: a mechanobiological
study of the contributions and interactions of elastin and collagen. Biophys J 106:2684–2692
Coravos JS, Mason FM, Martin AC (2017) Actomyosin pulsing in tissue integrity maintenance
during morphogenesis. Trends Cell Biol 27:276–283
Cowin SC (2001) The false premise of Wolff’s law. In: Cowin SC (ed) Bone biomechanics
handbook. CRC Press, Boca Raton, pp 30-31-15