On the mechanics of thin films and growing surfaces

Folding drives cortical thickness variations

The cortical thickness is a characteristic biomarker for a wide variety of neurological disorders. While the structural organization of the cerebral cortex is tightly regulated and evolutionarily preserved, its thickness varies widely between 1.5 and 4.5 mm across the healthy adult human brain. It remains unclear whether these thickness variations are a cause or consequence of cortical development. Recent studies suggest that cortical thickness variations are primarily a result of genetic effects. Previous studies showed that a simple homogeneous bilayered system with a growing layer on an elastic substrate undergoes a unique symmetry breaking into a spatially heterogeneous system with discrete gyri and sulci. Here, we expand on that work to explore the evolution of cortical thickness variations over time to support our finding that cortical pattern formation and thickness variations can be explained - at least in part - by the physical forces that emerge during cortical folding. Strikingly, as growth progresses, the developing gyri universally thicken and the sulci thin, even in the complete absence of regional information. Using magnetic resonance images, we demonstrate that these naturally emerging thickness variations agree with the cortical folding pattern in n = 9 healthy adult human brains, in n = 564 healthy human brains ages 7-64, and in n = 73 infant brains scanned at birth, and at ages one and two. Additionally, we show that cortical organoids develop similar patterns throughout their growth. Our results suggest that genetic, geometric, and physical events during brain development are closely interrelated. Understanding regional and temporal variations in cortical thickness can provide insight into the evolution and causative factors of neurological disorders, inform the diagnosis of neurological conditions, and assess the efficacy of treatment options.

Kinetics of surface growth with coupled diffusion and the emergence of a universal growth path

Surface growth by association or dissociation of material on the boundary of a body is ubiquitous in both natural and engineering systems. It is the fundamental mechanism by which biological materials grow, starting from the level of a single cell, and is increasingly applied in engineering processes for fabrication and self-assembly. A significant challenge in modelling such processes arises due to the inherent coupled interaction between the growth kinetics, the local stresses and the diffusing constituents needed to sustain the growth. Moreover, the volume of the body changes not only due to surface growth but also by variation in solvent concentration within the bulk. In this paper, we present a general theoretical framework that captures these phenomena and describes the kinetics of surface growth while accounting for coupled diffusion. Then, by the combination of analytical and numerical tools, applied to a simple growth geometry, we show that the evolution of such growth processes tends towards a universal path that is independent of initial conditions. This path, on which surface growth and diffusion act harmoniously, can be extended to analytically portray the evolution of a body from inception up to a treadmilling state, in which addition and removal of material are balanced.

Symmetry Breaking in Wrinkling Patterns: Gyri Are Universally Thicker than Sulci

Wrinkling instabilities appear in soft materials when a flat elastic layer on an elastic substrate is sufficiently stressed that it buckles with a wavy pattern to minimize the energy of the system. This instability is known to play an important role in engineering, but it also appears in many biological systems. In these systems, the stresses responsible for the wrinkling instability are often created through differential growth of the two layers. Beyond the instability, the upper and lower sides of the elastic layer are subject to different forces. This difference in forces leads to an interesting symmetry breaking whereby the thickness becomes larger at ridges than at valleys. Here we carry out an extensive analysis of this phenomenon by combining analytical, computational, and simple polymer experiments to show that symmetry breaking is a generic property of such systems. We apply our idea to the cortical folding of the brain for which it has been known for over a century that there is a thickness difference between gyri and sulci. An extensive analysis of hundreds of human brains reveals a systematic region-dependent thickness variation. Our results suggest that the evolving thickness patterns during brain development, similar to our polymer experiments, follow simple physics-based laws: Gyri are universally thicker than sulci.

Biological growth in bodies with incoherent interfaces

A general theory of thermodynamically consistent biomechanical–biochemical growth in a body, considering mass addition in the bulk and at an incoherent interface, is developed. The incoherency arises due to incompatibility of growth and elastic distortion tensors at the interface. The incoherent interface therefore acts as an additional source of internal stress besides allowing for rich growth kinematics. All the biochemicals in the model are essentially represented in terms of nutrient concentration fields, in the bulk and at the interface. A nutrient balance law is postulated which, combined with mechanical balances and kinetic laws, yields an initial-boundary-value problem coupling the evolution of bulk and interfacial growth, on the one hand, and the evolution of growth and nutrient concentration on the other. The problem is solved, and discussed in detail, for two distinct examples: annual ring formation during tree growth and healing of cutaneous wounds in animals.

Printing Non-Euclidean Solids

Geometrically frustrated solids with a non-Euclidean reference metric are ubiquitous in biology and are becoming increasingly relevant in technological applications. Often they acquire a targeted configuration of incompatibility through the surface accretion of mass as in tree growth or dam construction. We use the mechanics of incompatible surface growth to show that geometrical frustration developing during deposition can be fine-tuned to ensure a particular behavior of the system in physiological (or working) conditions. As an illustration, we obtain an explicit 3D printing protocol for arteries, which guarantees stress uniformity under inhomogeneous loading, and for explosive plants, allowing a complete release of residual elastic energy with a single cut. Interestingly, in both cases reaching the physiological target requires the incompatibility to have a topological (global) component.

Dimensional, Geometrical, and Physical Constraints in Skull Growth

After birth, the skull grows and remodels in close synchrony with the brain to allow for an increase in intracranial volume. Increase in skull area is provided primarily by bone accretion at the sutures. Additional remodeling, to allow for a change in curvatures, occurs by resorption on the inner surface of the bone plates and accretion on their outer surfaces. When a suture fuses too early, normal skull growth is disrupted, leading to a deformed final skull shape. The leading theory assumes that the main stimulus for skull growth is provided by mechanical stresses. Based on these ideas, we first discuss the dimensional, geometrical, and kinematic synchrony between brain, skull, and suture growth. Second, we present two mechanical models for skull growth that account for growth at the sutures and explain the various observed dysmorphologies. These models demonstrate the particular role of physical and geometrical constraints taking place in skull growth.

Spatial Constraints Also Regulates Final Achene Mass in the Sunflower (Helianthus annuus L.) Capitulum

In capitula of the cultivated sunflower (Helianthus annuus L.) achene size and mass commonly decrease from proximal to distal positions. Temporal limitation of resources of the distal achenes over the proximal ones has been the common explanation for this response. Nevertheless, because the capitulum architecture and expansion dynamics also interacts with achene growth and development, also space exert a coupled effect with resources on achene size along the inflorescence radius. In this work we removed young achenes from different capitulum positions [inner sector (IS) and outer sector (OS)] and applied an artificial restriction to the capitulum/achenes radial expansion. Removal of outer achenes significantly increased the final dry mass of the remnant ones between 17.1 to 27.6%. Removal of inner achenes also produced the same effect but in less magnitude, between 9.3 to 17.9% of the outer ones. The removal of outer achenes with the application of an artificial peripheral constraint did not significantly increase the dry mass of the remnant ones (2.7% of the inner and 7.1% of the control). Percentage of empty achenes significantly diminished in the middle sector (MS) in capitula with the outer achenes removed and in capitula with the outer achenes removed plus a peripheral constraint but in the range of 7.1% (MS achenes) and 2.7% (IS achenes). Percentage of empty achenes of the MS did not change when the outer achenes were removed but was significantly lower when the OS was removed and the peripheral constraint was applied. This results suggest that a part of the reduced growth and development of IS and MS achenes is not only controlled by the competition for resources but also is restricted by space and pressure exerted by the neighboring ones.

Bacillus subtilis Bacteria Generate an Internal Mechanical Force within a Biofilm

A key issue in understanding why biofilms are the most prevalent mode of bacterial life is the origin of the degree of resistance and protection that bacteria gain from self-organizing into biofilm communities. Our experiments suggest that their mechanical properties are a key factor. Experiments on pellicles, or floating biofilms, of Bacillus subtilis showed that while they are multiplying and secreting extracellular substances, bacteria create an internal force (associated with a -80±25 Pa stress) within the biofilms, similar to the forces that self-equilibrate and strengthen plants, organs, and some engineered buildings. Here, we found that this force, or stress, is associated with growth-induced pressure. Our observations indicate that due to such forces, biofilms spread after any cut or ablation by up to 15-20% of their initial size. The force relaxes over very short timescales (tens of milliseconds). We conclude that this force helps bacteria to shape the biofilm, improve its mechanical resistance, and facilitate its invasion and self-repair.

Snap buckling of a confined thin elastic sheet

A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first-order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly unstretchable regime, we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us to resolve the apparent paradox of an unbounded pressure exerted on the external wall by a confined flexible loop.

Systems biology and mechanics of growth

In contrast to inert systems, living biological systems have the advantage to adapt to their environment through growth and evolution. This transfiguration is evident during embryonic development, when the predisposed need to grow allows form to follow function. Alterations in the equilibrium state of biological systems breed disease and mutation in response to environmental triggers. The need to characterize the growth of biological systems to better understand these phenomena has motivated the continuum theory of growth and stimulated the development of computational tools in systems biology. Biological growth in development and disease is increasingly studied using the framework of morphoelasticity. Here, we demonstrate the potential for morphoelastic simulations through examples of volume, area, and length growth, inspired by tumor expansion, chronic bronchitis, brain development, intestine formation, plant shape, and myopia. We review the systems biology of living systems in light of biochemical and optical stimuli and classify different types of growth to facilitate the design of growth models for various biological systems within this generic framework. Exploring the systems biology of growth introduces a new venue to control and manipulate embryonic development, disease progression, and clinical intervention.

The role of mechanics during brain development

Convolutions are a classical hallmark of most mammalian brains. Brain surface morphology is often associated with intelligence and closely correlated to neurological dysfunction. Yet, we know surprisingly little about the underlying mechanisms of cortical folding. Here we identify the role of the key anatomic players during the folding process: cortical thickness, stiffness, and growth. To establish estimates for the critical time, pressure, and the wavelength at the onset of folding, we derive an analytical model using the Föppl-von-Kármán theory. Analytical modeling provides a quick first insight into the critical conditions at the onset of folding, yet it fails to predict the evolution of complex instability patterns in the post-critical regime. To predict realistic surface morphologies, we establish a computational model using the continuum theory of finite growth. Computational modeling not only confirms our analytical estimates, but is also capable of predicting the formation of complex surface morphologies with asymmetric patterns and secondary folds. Taken together, our analytical and computational models explain why larger mammalian brains tend to be more convoluted than smaller brains. Both models provide mechanistic interpretations of the classical malformations of lissencephaly and polymicrogyria. Understanding the process of cortical folding in the mammalian brain has direct implications on the diagnostics of neurological disorders including severe retardation, epilepsy, schizophrenia, and autism.

On the mechanics of growing thin biological membranes

Despite their seemingly delicate appearance, thin biological membranes fulfill various crucial roles in the human body and can sustain substantial mechanical loads. Unlike engineering structures, biological membranes are able to grow and adapt to changes in their mechanical environment. Finite element modeling of biological growth holds the potential to better understand the interplay of membrane form and function and to reliably predict the effects of disease or medical intervention. However, standard continuum elements typically fail to represent thin biological membranes efficiently, accurately, and robustly. Moreover, continuum models are typically cumbersome to generate from surface-based medical imaging data. Here we propose a computational model for finite membrane growth using a classical midsurface representation compatible with standard shell elements. By assuming elastic incompressibility and membrane-only growth, the model a priori satisfies the zero-normal stress condition. To demonstrate its modular nature, we implement the membrane growth model into the general-purpose non-linear finite element package Abaqus/Standard using the concept of user subroutines. To probe efficiently and robustness, we simulate selected benchmark examples of growing biological membranes under different loading conditions. To demonstrate the clinical potential, we simulate the functional adaptation of a heart valve leaflet in ischemic cardiomyopathy. We believe that our novel approach will be widely applicable to simulate the adaptive chronic growth of thin biological structures including skin membranes, mucous membranes, fetal membranes, tympanic membranes, corneoscleral membranes, and heart valve membranes. Ultimately, our model can be used to identify diseased states, predict disease evolution, and guide the design of interventional or pharmaceutic therapies to arrest or revert disease progression.

On the Role of Mechanics in Chronic Lung Disease

Progressive airflow obstruction is a classical hallmark of chronic lung disease, affecting more than one fourth of the adult population. As the disease progresses, the inner layer of the airway wall grows, folds inwards, and narrows the lumen. The critical failure conditions for airway folding have been studied intensely for idealized circular cross-sections. However, the role of airway branching during this process is unknown. Here, we show that the geometry of the bronchial tree plays a crucial role in chronic airway obstruction and that critical failure conditions vary significantly along a branching airway segment. We perform systematic parametric studies for varying airway cross-sections using a computational model for mucosal thickening based on the theory of finite growth. Our simulations indicate that smaller airways are at a higher risk of narrowing than larger airways and that regions away from a branch narrow more drastically than regions close to a branch. These results agree with clinical observations and could help explain the underlying mechanisms of progressive airway obstruction. Understanding growth-induced instabilities in constrained geometries has immediate biomedical applications beyond asthma and chronic bronchitis in the diagnostics and treatment of chronic gastritis, obstructive sleep apnea and breast cancer.

Growing matter: a review of growth in living systems

Living systems can grow, develop, adapt, and evolve. These phenomena are non-intuitive to traditional engineers and often difficult to understand. Yet, classical engineering tools can provide valuable insight into the mechanisms of growth in health and disease. Within the past decade, the concept of incompatible configurations has evolved as a powerful tool to model growing systems within the framework of nonlinear continuum mechanics. However, there is still a substantial disconnect between the individual disciplines, which explore the phenomenon of growth from different angles. Here we show that the nonlinear field theories of mechanics provide a unified concept to model finite growth by means of a single tensorial internal variable, the second order growth tensor. We review the literature and categorize existing growth models by means of two criteria: the microstructural appearance of growth, either isotropic or anisotropic; and the microenvironmental cues that drive the growth process, either chemical or mechanical. We demonstrate that this generic concept is applicable to a broad range of phenomena such as growing arteries, growing tumors, growing skin, growing airway walls, growing heart valve leaflets, growing skeletal muscle, growing plant stems, growing heart valve annuli, and growing cardiac muscle. The proposed approach has important biological and clinical applications in atherosclerosis, in-stent restenosis, tumor invasion, tissue expansion, chronic bronchitis, mitral regurgitation, limb lengthening, tendon tear, plant physiology, dilated and hypertrophic cardiomyopathy, and heart failure. Understanding the mechanisms of growth in these chronic conditions may open new avenues in medical device design and personalized medicine to surgically or pharmacologically manipulate development and alter, control, or revert disease progression.

A novel strategy to identify the critical conditions for growth-induced instabilities

Geometric instabilities in living structures can be critical for healthy biological function, and abnormal buckling, folding, or wrinkling patterns are often important indicators of disease. Mathematical models typically attribute these instabilities to differential growth, and characterize them using the concept of fictitious configurations. This kinematic approach toward growth-induced instabilities is based on the multiplicative decomposition of the total deformation gradient into a reversible elastic part and an irreversible growth part. While this generic concept is generally accepted and well established today, the critical conditions for the formation of growth-induced instabilities remain elusive and poorly understood. Here we propose a novel strategy for the stability analysis of growing structures motivated by the idea of replacing growth by prestress. Conceptually speaking, we kinematically map the stress-free grown configuration onto a prestressed initial configuration. This allows us to adopt a classical infinitesimal stability analysis to identify critical material parameter ranges beyond which growth-induced instabilities may occur. We illustrate the proposed concept by a series of numerical examples using the finite element method. Understanding the critical conditions for growth-induced instabilities may have immediate applications in plastic and reconstructive surgery, asthma, obstructive sleep apnoea, and brain development.