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- Klarbring, Anders, et al.
(author)
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Nutrient modulated structural design with application to growth and degradation
- 2015
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In: Zeitschrift für angewandte Mathematik und Mechanik. - : WILEY-V C H VERLAG GMBH. - 0044-2267 .- 1521-4001. ; 95:11, s. 1323-1334
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Journal article (peer-reviewed)abstract
- Phenomena such as biological growth and damage evolution can be thought of as time evolving processes, the directions of which are governed by descendent of certain goal functions. Mathematically this means using a dynamical systems approach to optimization. We extend such an approach by introducing a field quantity, representing nutrients or other non-mechanical stimuli, that modulate growth and damage evolution. The derivation of a generic model is systematic, starting from a Lyaponov-type descent condition and utilizing a Coleman-Noll strategy. A numerical algorithm for finding stationary points of the resulting dynamical system is suggested and applied to two model problems where the influence of different levels of nutrient sensitivity are observed. The paper demonstrates the use of a new modeling technique and shows its application in deriving a generic problem of growth and damage evolution. (C) 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim
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| 2. |
- Lindström, Stefan, et al.
(author)
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Extension of Murray's law including nonlinear mechanics of a composite artery wall
- 2015
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In: Biomechanics and Modeling in Mechanobiology. - : Springer Berlin/Heidelberg. - 1617-7959 .- 1617-7940. ; 14:1, s. 83-91
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Journal article (peer-reviewed)abstract
- A goal function approach is used to derive an extension of Murray’s law that includes effects of nonlinear mechanics of the artery wall. The artery is modeled as a thin-walled tube composed of different species of nonlinear elastic materials that deform together. These materials grow and remodel in a process that is governed by a target state defined by a homeostatic radius and a homeostatic material composition. Following Murray’s original idea, this target state is defined by a principle of minimum work. We take this work to include that of pumping and maintaining blood, as well as maintaining the materials of the artery wall. The minimization is performed under a constraint imposed by mechanical equilibrium. We derive a condition for the existence of a cost-optimal homeostatic state. We also conduct parametric studies using this novel theoretical frame to investigate how the cost-optimal radius and composition of the artery wall depend on flow rate, blood pressure, and elastin content.
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| 3. |
- Satha, Ganarupan, et al.
(author)
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A goal function approach to remodeling of arteries uncovers mechanisms for growth instability
- 2014
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In: Biomechanics and Modeling in Mechanobiology. - : Springer. - 1617-7959 .- 1617-7940. ; 13:6, s. 1243-1259
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Journal article (peer-reviewed)abstract
- A novel, goal function-based formulation for the growth dynamics of arteries is introduced, and used for investigating the development of growth instability in blood vessels. Such instabilities would lead to abnormal growth of the vessel, reminiscent of an aneurysm. The blood vessel is modeled as a thin-walled cylindrical tube and the constituents that form the vessel wall are assumed to deform together as a constrained mixture. The growth dynamics of the composite material of the vessel wall is described by an evolution equation, where the effective area of each constituent changes in the direction of steepest descent of a goal function. This goal function is formulated in such way that the constituents grow toward a target potential energy and a target composition. The convergence of the simulated response of the evolution equation toward a target homeostatic state is investigated for a range of isotropic and orthotropic material models. These simulations suggest that elastin-deficient vessels are more prone to growth instability. Increased stiffness of the vessel wall, on the other hand, gives a more stable growth process. Another important finding is that an increased rate of degradation of materials impairs growth stability.
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| 4. |
- Satha, Ganarupan
(author)
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Goal Function Approach to Growth and Remodeling of Arteries
- 2014
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Licentiate thesis (other academic/artistic)abstract
- In this thesis we develop a new goal function approach to investigate stability of the growth processes in blood vessels and cost-optimal composition and geometry of these vessels. In the vascular system of a healthy individual, the living composition of the arterial wall must regenerate and remodel continuously during the entire lifetime to maintain itself. In some cases the system destabilizes due to disease, injury or other complex processes. To understand how and when this happens, several mathematical models have been developed. These models have included an evolution equation for mass fractions of the vessel wall, describing how the vessel develops from an actual state to a target state. These works are based on constrained mixture theory (CMT), which takes care of production and removal of arterial constituents. The cost-optimal design of blood vessels has been studied previously by Murray.The aim of this thesis is to contribute to stability analyses of the growth process by formulating a new goal function approach, making it possible to examine under which conditions instability arises. We also aim to analyze changes in the optimum material composition and geometry of the vessel wall, using a more realistic, nonlinear material model.The blood vessel is modeled as a thin-walled tube and the constituents that form the vessel wall are assumed to deform together (CMT). The growth dynamics of the composite material of the vessel wall is described by an evolution equation, where the effective area of each constituent changes in the direction of steepest descent of a goal function. This goal function is formulated in such way that the constituents grow toward a target potential energy and a target composition. The response of the evolution equation is simulated for several dierent material models. These simulations suggest that elastin-decient vessels are more prone to growth instability, but that increased vessel stiness gives a more stable growth process. Another important nding is that an increased rate of degradation of materials impairs growth stability.By extending Murray's law to include effects of nonlinear mechanics of the artery wall and a growth and remodeling mechanism based on CMT, and at the same time having the system satisfy an equilibrium equation, we study cost-optimal compositions and geometries of the vessel wall. This gives new insight into the wall's architecture under optimal conditions.
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