Correction to: Mathematical models of neuronal growth.

Correction to: Biomechanics and Modeling in Mechanobiology (2022) 21:89-118 https://doi.org/10.1007/s10237-021-01539-0.

Mechanics reveals the role of peristome geometry in prey capture in carnivorous pitcher plants (Nepenthes).

Carnivorous pitcher plants (Nepenthes) are a striking example of a natural pitfall trap. The trap's slippery rim, or peristome, plays a critical role in insect capture via an aquaplaning mechanism that is well documented. While the peristome has received significant research attention, the conspicuous variation in peristome geometry across the genus remains unexplored. We examined the mechanics of prey capture using Nepenthes pitcher plants with divergent peristome geometries. Inspired by living material, we developed a mathematical model that links the peristomes' three-dimensional geometries to the physics of prey capture under the laws of Newtonian mechanics. Linking form and function enables us to test hypotheses related to the function of features such as shape and ornamentation, orientation in a gravitational field, and the presence of "teeth," while analysis of the energetic costs and gains of a given geometry provides a means of inferring potential evolutionary pathways. In a separate modeling approach, we show how prey size may correlate with peristome dimensions for optimal capture. Our modeling framework provides a physical platform to understand how divergence in peristome morphology may have evolved in the genus Nepenthes in response to shifts in prey diversity, availability, and size.

Mathematical models of neuronal growth.

The establishment of a functioning neuronal network is a crucial step in neural development. During this process, neurons extend neurites-axons and dendrites-to meet other neurons and interconnect. Therefore, these neurites need to migrate, grow, branch and find the correct path to their target by processing sensory cues from their environment. These processes rely on many coupled biophysical effects including elasticity, viscosity, growth, active forces, chemical signaling, adhesion and cellular transport. Mathematical models offer a direct way to test hypotheses and understand the underlying mechanisms responsible for neuron development. Here, we critically review the main models of neurite growth and morphogenesis from a mathematical viewpoint. We present different models for growth, guidance and morphogenesis, with a particular emphasis on mechanics and mechanisms, and on simple mathematical models that can be partially treated analytically.

Virtual preoperative planning of acetabular fractures using patient-specific biomechanical simulation: A case-control study.

INTRODUCTION: The first patient-specific biomechanical model for planning the surgical reduction of acetabular fractures was developed in our institution and validated retrospectively. There are no prior studies showing its effectiveness in terms of reduction quality, operative duration and intraoperative bleeding. Therefore, we performed a case control study aiming to: 1) evaluate the effect of preoperative simulation by patient-specific biomechanical simulator on the operating time and intraoperative bleeding; 2) evaluate the effect of preoperative simulation by patient-specific biomechanical simulator on the quality of reduction. METHOD: All patients operated on between January 2019 and June 2019 after planning by biomechanical simulation were included in this case-control study. Each patient included was matched to 2 controls from our database (2015-2018) according to age and fracture-type. DICOM data were extracted from the preoperative high-resolution scanners to build a three-dimensional model of the fracture by semi-automatic segmentation. A biomechanical model was built to virtually simulate the different stages of surgical reduction. Surgery was then performed according to simulation data. Surgical duration, blood loss, radiological findings and intraoperative complications were recorded, analysed and compared. RESULTS: Thirty patients were included, 10 in the simulation group and 20 in the control group. The two groups were comparable in terms of age, time from accident to surgery, fracture-type and surgical approach. The mean operative time was significantly reduced in the simulation group: 113min±33 (60-180) versus 196min±32 (60-260) (p=0.01). Mean blood loss was significantly reduced in the simulation group: 505mL±189 (100-750) versus 745mL±130 (200-850) (p<0.01). However, no significant difference was found in the radiological results according to Matta's criteria, although an anatomical reduction was obtained for 9 patients in the simulation group (90%) versus 12 patients in the control group (60%) (p=0.26). A postoperative neurological complication was recorded in the control group (sensory deficit of the lateral cutaneous nerve of thigh). CONCLUSION: This study confirms the promising results of preoperative planning in acetabular trauma surgery based on patient-specific biomechanical simulation as well as its feasibility in routine clinical practice. By providing a better understanding of the fracture and its behavior, a reduction in intraoperative bleeding and in operative duration is achieved. LEVEL OF EVIDENCE: III; case-control study.

Theory for Durotactic Axon Guidance.

During the development of the nervous system, neurons extend bundles of axons that grow and meet other neurons to form the neuronal network. Robust guidance mechanisms are needed for these bundles to migrate and reach their functional target. Directional information depends on external cues such as chemical or mechanical gradients. Unlike chemotaxis that has been extensively studied, the role and mechanism of durotaxis, the directed response to variations in substrate rigidity, remain unclear. We model bundle migration and guidance by rigidity gradients by using the theory of morphoelastic rods. We show that, at a rigidity interface, the motion of axon bundles follows a simple behavior analogous to optic ray theory and obeys Snell's law for refraction and reflection. We use this powerful analogy to demonstrate that axons can be guided by the equivalent of optical lenses and fibers created by regions of different stiffnesses.

Planning acetabular fracture reduction using a patient-specific biomechanical model: a prospective and comparative clinical study.

PURPOSE: A simple, patient-specific biomechanical model (PSBM) is proposed in which the main surgical tools and actions can be simulated, which enables clinicians to evaluate different strategies for an optimal surgical planning. A prospective and comparative clinical study was performed to assess early clinical and radiological results. METHODS: From January 2019 to July 2019, a PSBM was created for every operated acetabular fracture (simulation group). DICOM data were extracted from the pre-operative high-resolution CT scans to build a 3D model of the fracture using segmentation methods. A PSBM was implemented in a custom software allowing a biomechanical simulation of the surgery in terms of reduction sequences. From July 2019 to December 2019, every patient with an operated for acetabular fracture without PSBM was included in the standard group. Surgery duration, blood loss, radiological results and per-operative complications were recorded and compared between the two groups. RESULTS: Twenty-two patients were included, 10 in the simulation group and 12 in the standard group. The two groups were comparable regarding age, time to surgery, fracture pattern distribution and surgical approaches. The mean operative time was significantly lower in the simulation group: 113 min ± 33 (60-180) versus 184 ± 58 (90-260), p = 0.04. The mean blood loss was significantly lower in the simulation group, p = 0.01. No statistical significant differences were found regarding radiological results (p = 0.16). No per-operative complications were recorded. CONCLUSION: This study confirms that pre-operative planning in acetabular surgery based on a PSBM results in a shorter operative time and a reduction of blood loss during surgery. This study also confirms the feasibility of PSBM planning in daily clinical routine. LEVEL OF EVIDENCE: II: prospective study.

Multiscale integration of environmental stimuli in plant tropism produces complex behaviors.

Plant tropism refers to the directed movement of an organ or organism in response to external stimuli. Typically, these stimuli induce hormone transport that triggers cell growth or deformation. In turn, these local cellular changes create mechanical forces on the plant tissue that are balanced by an overall deformation of the organ, hence changing its orientation with respect to the stimuli. This complex feedback mechanism takes place in a three-dimensional growing plant with varying stimuli depending on the environment. We model this multiscale process in filamentary organs for an arbitrary stimulus by explicitly linking hormone transport to local tissue deformation leading to the generation of mechanical forces and the deformation of the organ in three dimensions. We show, as examples, that the gravitropic, phototropic, nutational, and thigmotropic dynamic responses can be easily captured by this framework. Further, the integration of evolving stimuli and/or multiple contradictory stimuli can lead to complex behavior such as sun following, canopy escape, and plant twining.

Microtubule-Mediated Wall Anisotropy Contributes to Leaf Blade Flattening.

Plant organs can adopt a wide range of shapes, resulting from highly directional cell growth and divisions. We focus here on leaves and leaf-like organs in Arabidopsis and tomato, characterized by the formation of thin, flat laminae. Combining experimental approaches with 3D mechanical modeling, we provide evidence that leaf shape depends on cortical microtubule mediated cellulose deposition along the main predicted stress orientations, in particular, along the adaxial-abaxial axis in internal cell walls. This behavior can be explained by a mechanical feedback and has the potential to sustain and even amplify a preexisting degree of flatness, which in turn depends on genes involved in the control of organ polarity and leaf margin formation.

Simulating Turgor-Induced Stress Patterns in Multilayered Plant Tissues.

The intertwining between mechanics and developmental biology is extensively studied at the shoot apical meristem of land plants. Indeed, plant morphogenesis heavily relies on mechanics; tissue deformations are fueled by turgor-induced forces, and cell mechanosensitivity plays a major regulatory role in this dynamics. Since measurements of forces in growing meristems are still out of reach, our current knowledge relies mainly on theoretical and numerical models. So far, these modeling efforts have been mostly focusing on the epidermis, where aerial organs are initiated. In many models, the epidermis is assimilated to its outermost cell walls and described as a thin continuous shell under pressure, thereby neglecting the inner walls. There is, however, growing experimental evidence suggesting a significant mechanical role of these inner walls. The aim of this work is to investigate the influence of inner walls on the mechanical homeostasis of meristematic tissues. To this end, we simulated numerically the effect of turgor-induced loading, both in realistic flower buds and in more abstract structures. These simulations were performed using finite element meshes with subcellular resolution. Our analysis sheds light on the mechanics of growing plants by revealing the strong influence of inner walls on the epidermis mechanical stress pattern especially in negatively curved regions. Our simulations also display some strong and unsuspected features, such as a correlation between stress intensity and cell size, as well as differential response to loading between epidermal and inner cells. Finally, we monitored the time evolution of the mechanical stresses felt by each cell and its descendants during the early steps of flower morphogenesis.

Regulation of plant cell wall stiffness by mechanical stress: a mesoscale physical model.

A crucial question in developmental biology is how cell growth is coordinated in living tissue to generate complex and reproducible shapes. We address this issue here in plants, where stiff extracellular walls prevent cell migration and morphogenesis mostly results from growth driven by turgor pressure. How cells grow in response to pressure partly depends on the mechanical properties of their walls, which are generally heterogeneous, anisotropic and dynamic. The active control of these properties is therefore a cornerstone of plant morphogenesis. Here, we focus on wall stiffness, which is under the control of both molecular and mechanical signaling. Indeed, in plant tissues, the balance between turgor and cell wall elasticity generates a tissue-wide stress field. Within cells, mechano-sensitive structures, such as cortical microtubules, adapt their behavior accordingly and locally influence cell wall remodeling dynamics. To fully apprehend the properties of this feedback loop, modeling approaches are indispensable. To that end, several modeling tools in the form of virtual tissues have been developed. However, these models often relate mechanical stress and cell wall stiffness in relatively abstract manners, where the molecular specificities of the various actors are not fully captured. In this paper, we propose to refine this approach by including parsimonious biochemical and biomechanical properties of the main molecular actors involved. Through a coarse-grained approach and through finite element simulations, we study the role of stress-sensing microtubules on organ-scale mechanics.