On the mechanics of continua with boundary energies and growing surfaces

Wrinkling and restabilization of a hyperelastic PDMS membrane at finite strain

Wrinkles are commonly observed in uniaxially stretched hyperelastic membranes and eventually disappear with the increase of stretching. The widely used scheme at present assumes the material parameters to be empirical values and straightforwardly considers some constitutive models to explore the wrinkling and restabilization behavior. However, this simple treatment may cause deviation from experiment by ignoring the applicability of the models and the authenticity of the input parameters, prompting us to report based on realistic material parameters. This paper presents an experimental, theoretical and numerical investigation on the wrinkling and restabilization behavior of hyperelastic materials. By fitting experimental stress-strain curves of PDMS films, we confirm that the 3-term Ogden model bears a closer resemblance to the experimental data than the widely used neo-Hookean, Mooney-Rivlin, and Arruda-Boyce models under certain circumstances. The simulation results indicate that different constitutive models quantitatively affect the critical buckling strain, wrinkling amplitudes, and restabilization points. Furthermore, the isolated central bifurcation point solved by Koiter stability theory agrees well with the simulation and experimental results. A 3D phase diagram of stability boundaries was established to gain a comprehensive insight into the effects of geometric parameters (length, width, and thickness) on wrinkling.

Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localized) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved ‘two-phase’ state. How the localized instability initiates and evolves into the final ‘two-phase’ state is still a matter of contention, and this is the focus of the current study. Through a weakly nonlinear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localized bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localization. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by finite-element method simulations.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

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.

The origin of compression influences geometric instabilities in bilayers

Geometric instabilities in bilayered structures control the surface morphology in a wide range of biological and technical systems. Depending on the application, different mechanisms induce compressive stresses in the bilayer. However, the impact of the chosen origin of compression on the critical conditions, post-buckling evolution and higher-order pattern selection remains insufficiently understood. Here, we conduct a numerical study on a finite-element set-up and systematically vary well-known factors contributing to pattern selection under the four main origins of compression: film growth, substrate shrinkage and whole-domain compression with and without pre-stretch. We find that the origin of compression determines the substrate stretch state at the primary instability point and thus significantly affects the critical buckling conditions. Similarly, it leads to different post-buckling evolutions and secondary instability patterns when the load further increases. Our results emphasize that future phase diagrams of geometric instabilities should incorporate not only the film thickness but also the origin of compression. Thoroughly understanding the influence of the origin of compression on geometric instabilities is crucial to solving real-life problems such as the engineering of smart surfaces or the diagnosis of neuronal disorders, which typically involve temporally or spatially combined origins of compression.

Wrinkling patterns in soft shells

Curvature plays an important role in the morphological evolution of soft shells under stretch. Here, through a combination of experiment, theory and simulation, we investigate the behavior of a hemispherical soft shell subject to an increasing outward point force at its pole. In contrast to an inward point force inducing a polygonal pattern of buckling in the shell, we observe a four-stage morphological transition and symmetry breaking under an increasing outward point force. The shell undergoes axisymmetric deformation around its pole and then buckles into a non-axisymmetric shape with a number of shallow wrinkles emanating from the pole, followed by the emergence of crater-like deep crumples and ultimately a transformation into a wrinkled pseudocone. Our theoretical analysis and numerical simulations yield the critical conditions for the morphological transitions at each stage of deformation and reveal the underlying interplays between elastic bending and stretching energies and the curvature of the shell.

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.

Wrinkling instabilities in soft bilayered systems

Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

A closed form large deformation solution of plate bending with surface effects

We study the effect of surface stress on the pure bending of a finite thickness plate under large deformation. The surface is assumed to be isotropic and its stress consists of a part that can be interpreted as a residual stress and a part that stiffens as the surface increases its area. Our results show that residual surface stress and surface stiffness can both increase the overall bending stiffness but through different mechanisms. For sufficiently large residual surface tension, we discover a new type of instability - the bending moment reaches a maximum at a critical curvature. Effects of surface stress on different stress components in the bulk of the plate are discussed and the possibility of self-bending due to asymmetry of the surface properties is also explored. The results of our calculations provide insights into surface stress effects in the large deformation regime and can be used as a test for implementation of finite element methods for surface elasticity.

How the embryonic chick brain twists

During early development, the tubular embryonic chick brain undergoes a combination of progressive ventral bending and rightward torsion, one of the earliest organ-level left-right asymmetry events in development. Existing evidence suggests that bending is caused by differential growth, but the mechanism for the predominantly rightward torsion of the embryonic brain tube remains poorly understood. Here, we show through a combination of in vitro experiments, a physical model of the embryonic morphology and mechanics analysis that the vitelline membrane (VM) exerts an external load on the brain that drives torsion. Our theoretical analysis showed that the force is of the order of 10 micronewtons. We also designed an experiment to use fluid surface tension to replace the mechanical role of the VM, and the estimated magnitude of the force owing to surface tension was shown to be consistent with the above theoretical analysis. We further discovered that the asymmetry of the looping heart determines the chirality of the twisted brain via physical mechanisms, demonstrating the mechanical transfer of left-right asymmetry between organs. Our experiments also implied that brain flexure is a necessary condition for torsion. Our work clarifies the mechanical origin of torsion and the development of left-right asymmetry in the early embryonic brain.

Elastosis during airway wall remodeling explains multiple co-existing instability patterns

Living structures can undergo morphological changes in response to growth and alterations in microstructural properties in response to remodeling. From a biological perspective, airway wall inflammation and airway elastosis are classical hallmarks of growth and remodeling during chronic lung disease. From a mechanical point of view, growth and remodeling trigger mechanical instabilities that result in inward folding and airway obstruction. While previous analytical and computational studies have focused on identifying the critical parameters at the onset of folding, few have considered the post-buckling behavior. All prior studies assume constant microstructural properties during the folding process; yet, clinical studies now reveal progressive airway elastosis, the degeneration of elastic fibers associated with a gradual stiffening of the inner layer. Here, we explore the influence of temporally evolving material properties on the post-bifurcation behavior of the airway wall. We show that a growing and stiffening inner layer triggers an additional subsequent bifurcation after the first instability occurs. Evolving material stiffnesses provoke failure modes with multiple co-existing wavelengths, associated with the superposition of larger folds evolving on top of the initial smaller folds. This phenomenon is exclusive to material stiffening and conceptually different from the phenomenon of period doubling observed in constant-stiffness growth. Our study suggests that the clinically observed multiple wavelengths in diseased airways are a result of gradual airway wall stiffening. While our evolving material properties are inspired by the clinical phenomenon of airway elastosis, the underlying concept is broadly applicable to other types of remodeling including aneurysm formation or brain folding.

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.

Patient-Specific Airway Wall Remodeling in Chronic Lung Disease

Chronic lung disease affects more than a quarter of the adult population; yet, the mechanics of the airways are poorly understood. The pathophysiology of chronic lung disease is commonly characterized by mucosal growth and smooth muscle contraction of the airways, which initiate an inward folding of the mucosal layer and progressive airflow obstruction. Since the degree of obstruction is closely correlated with the number of folds, mucosal folding has been extensively studied in idealized circular cross sections. However, airflow obstruction has never been studied in real airway geometries; the behavior of imperfect, non-cylindrical, continuously branching airways remains unknown. Here we model the effects of chronic lung disease using the nonlinear field theories of mechanics supplemented by the theory of finite growth. We perform finite element analysis of patient-specific Y-branch segments created from magnetic resonance images. We demonstrate that the mucosal folding pattern is insensitive to the specific airway geometry, but that it critically depends on the mucosal and submucosal stiffness, thickness, and loading mechanism. Our results suggests that patient-specific airway models with inherent geometric imperfections are more sensitive to obstruction than idealized circular models. Our models help to explain the pathophysiology of airway obstruction in chronic lung disease and hold promise to improve the diagnostics and treatment of asthma, bronchitis, chronic obstructive pulmonary disease, and respiratory failure.

Mechanics of the brain: perspectives, challenges, and opportunities

The human brain is the continuous subject of extensive investigation aimed at understanding its behavior and function. Despite a clear evidence that mechanical factors play an important role in regulating brain activity, current research efforts focus mainly on the biochemical or electrophysiological activity of the brain. Here, we show that classical mechanical concepts including deformations, stretch, strain, strain rate, pressure, and stress play a crucial role in modulating both brain form and brain function. This opinion piece synthesizes expertise in applied mathematics, solid and fluid mechanics, biomechanics, experimentation, material sciences, neuropathology, and neurosurgery to address today’s open questions at the forefront of neuromechanics. We critically review the current literature and discuss challenges related to neurodevelopment, cerebral edema, lissencephaly, polymicrogyria, hydrocephaly, craniectomy, spinal cord injury, tumor growth, traumatic brain injury, and shaken baby syndrome. The multi-disciplinary analysis of these various phenomena and pathologies presents new opportunities and suggests that mechanical modeling is a central tool to bridge the scales by synthesizing information from the molecular via the cellular and tissue all the way to the organ level.

Period-doubling and period-tripling in growing bilayered systems

Growing layers on elastic substrates are capable of creating a wide variety of surface morphologies. Moderate growth generates a regular pattern of sinusoidal wrinkles with a homogeneous energy distribution. While the critical conditions for periodic wrinkling have been extensively studied, the rich pattern formation beyond this first instability point remains poorly understood. Here we show that upon continuing growth, the energy progressively localizes and new complex morphologies emerge. Previous studies have often overlooked these secondary bifurcations; they have focused on large stiffness ratios between layer and substrate, where primary instabilities occur early, long before secondary instabilities emerge. We demonstrate that secondary bifurcations are particularly critical in the low stiffness ratio regime, where the critical conditions for primary and secondary instabilities move closer together. Amongst all possible secondary bifurcations, the mode of period-doubling plays a central role - it is energetically favorable over all other modes. Yet, we can numerically suppress period-doubling, by choosing boundary conditions, which favor alternative higher order modes. Our results suggest that in the low stiffness regime, pattern formation is highly sensitive to small imperfections: surface morphologies emerge rapidly, change spontaneously, and quickly become immensely complex. This is a common paradigm in developmental biology. Our results have significantly applications in the morphogenesis of living systems where growth is progressive and stiffness ratios are low.

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.

A mechanical model predicts morphological abnormalities in the developing human brain

The developing human brain remains one of the few unsolved mysteries of science. Advancements in developmental biology, neuroscience, and medical imaging have brought us closer than ever to understand brain development in health and disease. However, the precise role of mechanics throughout this process remains underestimated and poorly understood. Here we show that mechanical stretch plays a crucial role in brain development. Using the nonlinear field theories of mechanics supplemented by the theory of finite growth, we model the human brain as a living system with a morphogenetically growing outer surface and a stretch-driven growing inner core. This approach seamlessly integrates the two popular but competing hypotheses for cortical folding: axonal tension and differential growth. We calibrate our model using magnetic resonance images from very preterm neonates. Our model predicts that deviations in cortical growth and thickness induce morphological abnormalities. Using the gyrification index, the ratio between the total and exposed surface area, we demonstrate that these abnormalities agree with the classical pathologies of lissencephaly and polymicrogyria. Understanding the mechanisms of cortical folding in the developing human brain has direct implications in the diagnostics and treatment of neurological disorders, including epilepsy, schizophrenia, and autism.

Modeling of elasto-capillary phenomena

Surface energy is an important factor in the deformation of fluids but is typically a minimal or negligible effect in solids. However, when a solid is soft and its characteristic dimension is small, forces due to surface energy can become important and induce significant elastic deformation. The interplay between surface energy and elasticity can lead to interesting elasto-capillary phenomena. We present a finite-element-based numerical simulation capability for modeling these effects in a static, implicit framework. We demonstrate the capacity of the simulation capability by examining three elasto-capillary problems: (i) wetting of an elastic hemispherical droplet on a substrate, (ii) cavitation of an elastomer, and (iii) the Rayleigh-Plateau instability in soft elastic filaments.

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.

On the mechanics of thin films and growing surfaces

Many living structures are coated by thin films, which have distinct mechanical properties from the bulk. In particular, these thin layers may grow faster or slower than the inner core. Differential growth creates a balanced interplay between tension and compression and plays a critical role in enhancing structural rigidity. Typical examples with a compressive outer surface and a tensile inner core are the petioles of celery, caladium, or rhubarb. While plant physiologists have studied the impact of tissue tension on plant rigidity for more than a century, the fundamental theory of growing surfaces remains poorly understood. Here, we establish a theoretical and computational framework for continua with growing surfaces and demonstrate its application to classical phenomena in plant growth. To allow the surface to grow independently of the bulk, we equip it with its own potential energy and its own surface stress. We derive the governing equations for growing surfaces of zero thickness and obtain their spatial discretization using the finite-element method. To illustrate the features of our new surface growth model we simulate the effects of growth-induced longitudinal tissue tension in a stalk of rhubarb. Our results demonstrate that different growth rates create a mechanical environment of axial tissue tension and residual stress, which can be released by peeling off the outer layer. Our novel framework for continua with growing surfaces has immediate biomedical applications beyond these classical model problems in botany: it can be easily extended to model and predict surface growth in asthma, gastritis, obstructive sleep apnoea, brain development, and tumor invasion. Beyond biology and medicine, surface growth models are valuable tools for material scientists when designing functionalized surfaces with distinct user-defined properties.