Perspectives on biological growth and remodeling

Abstract

The continuum mechanical treatment of biological growth and remodeling has attracted considerable attention over the past fifteen years. Many aspects of these problems are now well-understood, yet there remain areas in need of significant development from the standpoint of experiments, theory, and computation. In this perspective paper we review the state of the field and highlight open questions, challenges, and avenues for further development.

Figure 1

The reference configuration, Ω₀, the current configuration, Ω, and the multiplicative decomposition F = F^eF^g.

Figure 2

Surface plot of the free energy rates associated with tumor growth. All units are Wm⁻³. (a) The rate of change of chemical free energy density stored in the cells. (b) The rate of change of free energy density stored in newly formed cells. (c) The rate of change of free energy density stored in newly-produced ECM. (d) The rate at which free energy density is dissipated into work done as the tumor spheroid grows against stress. (e) The rate at which free energy density is dissipated due to cell motion. (f) The rate of change of free energy density due to glucose consumption. See Narayanan et al., (2010).

Figure 3

Growth stresses in the young sunflower head. (a) The head was cut along two orthogonal lines. The cuts gape widely in the central region while they remain closely appressed in the peripheral region where new organs are being formed. This gaping pattern demonstrates the presence of radial tension and circumferential compression in the head. (b) Typical arcuate crack created when the surface of the head is put in tension. The crack propagates predominantly in the circumferential direction thus releasing the tensile stresses in the radial direction. (c) and (d) The same head before (c) and after (d) reducing the turgor pressure of the cells. The characteristic gaping of the cut has been lost after decreasing the pressure. (In all images, the head is about 5mm in diameter.)

Figure 4

Example of structural optimization and bone remodeling: Typical benchmark problem of the proxima femur. A finite element based smooth topology optimization algorithm can be used to predict the physiological density profile in response to the three most relevant muscle groups activated during abduction, adduction, and the midstance phase of gait. The converged density profile displays the characteristic dense system of compressive trabeculae carrying the load from the superior contact surface to the calcar region, a secondary arc system of trabeculae through the infero-medial joint surface, Ward’s triangle as an area of low density, and a dense cortical shaft.

Figure 5

Example of structural optimization and bone remodeling: Virtual implantation of traditional hip prosthesis. The stiff titanium transfers the joint forces down the distal portion of the implant stem. At the distal tip of the stem, forces are transferred to the outer bone shaft. This triggers the pronounced deposition of bone mass at the distal tip of the prosthesis while the unloaded proximal regions of the femur undergo a severe bone loss that is typically accompanied with aseptic loosening and the need for refixation.

Figure 6

Example of structural optimization and bone remodeling: Novel hip resurfacing technique. To avoid the undesirable long-term effects associated with traditional implants, a novel techniques has been developed that is based on local femor hip resurfacing. The new nail-shaped implant shows a much better ingrowth with an increased density at the medial side. In contrast to the classical implantation technique, the shaft remains virtually unaffected by the treatment.

Figure 7

Example of structural optimization and bone remodeling: Functional adaptation of bone density in the humerus of high performance tennis players. Severe humeral torsion during the serve induces bone remodeling which results in a non-physiological twisted density profile with high density areas wound around the long axis of the bone.

Figure 8

(a) Principal stress directions (segments of black lines) and collagen fiber morphology (segments of red curves). The dashed circle shows a region at the vicinity of the apex where collagen fibers are oriented almost along the direction of the apical ridge resembling a tendon-like structure; taken from Hariton et al. (2007b); (b) Transmission electron photomicrograph of collagen fibrils of an apex adventitia (×40000). Fibrils in the apex region are of uniform size and are co-aligned; taken from Finlay et al. (1998).

Figure 9

Isotropic growth based on phenomenological growth laws (Allen et al., 2001). Patient-specific virtual stent implantation. CT of human aorta, Bezier spline interpolation, solid aorta model, finite element discretization, and simulation of wall growth and in-stent restenosis (from left to right).

Figure 10

Anisotropic growth based on microscopically-motivated growth laws. Ventricular growth and remodeling. Pressure overload-induced hypertrophy in response to aortic stenosis, normal heart, and volume overload-induced dilation in response to myocardial infarction. Sections from Hunter et al. (1999) (left) and finite element simulation (right).

Figure 11

Anisotropic growth based on microscopically-motivated growth laws. Cardiac wall thickening, stress-driven concentric growth, and transmural muscle thickening at constant cardiac size. The concentric growth multiplier gradually increases from 1.00 to 3.00 as the individual heart muscle cells grow concentrically. On the macroscopic scale, cardiac wall thickening manifests itself in a progressive transmural muscle growth to withstand higher blood pressure levels while the overall size of the heart remains constant. Since the septal wall receives structural support through the pressure in the right ventricle, wall thickening is slightly more pronounced at the free wall where the wall stresses are higher.