Growth Tensor Research Articles

A unified theoretical structure for modeling interstitial growth and muscle activation in soft tissues

The objective of this paper is to develop a new unified theoretical structure for modeling interstitial growth and muscle activation in soft tissues. The model assumes a simple continuum with a single velocity field. In contrast with many other formulations, evolution equations are proposed directly for a scalar measure of elastic dilatation and a tensorial measure of elastic distortional deformation. The evolution equation for elastic dilatation includes a rate of mass supply or removal that controls volumetric growth and causes the elastic dilatation to evolve towards its homeostatic value. Similarly, the evolution equation for elastic distortional deformation includes a rate of growth that causes the elastic distortional deformation tensor to evolve towards its homeostatic value. Specific forms for these inelastic rates of growth and the associated homeostatic values have been considered for volumetric, area and fiber growth processes, as well as for muscle activation. Since the rate of growth appears in the evolution equations and not a growth tensor it is possible to model the combined effects of multiple growth and muscle activation mechanisms simultaneously. Also, robust, strongly objective, numerical algorithms have been developed to integrate the evolution equations.

Growth rate distribution in the forming lateral root of arabidopsis.

Microscopic observations of lateral roots (LRs) in Arabidopsis thaliana reveal that the cross-sectional shape of the organ changes from its basal to its apical region. The founder cells for LRs are elongated along the parent root axis, and thus from the site of initiation the base of LRs resemble an ellipse. The circumference of the apical part of LRs is usually a circle. The objective of this study was to analyse the characteristics of changes in the growth field of LRs possessing various shapes in their basal regions. The LRs of the wild type (Col-0) and two transgenic arabidopsis lines were analysed. On the basis of measurements of the long and short diameters (DL and DS, respectively) of the ellipse-like figure representing the bases of particular LRs, their asymmetry ratios (DL/DS) were determined. Possible differences between accessions were analysed by applying statistical methods. No significant differences between accessions were detected. Comparisons were therefore made of the maximal, minimal and mean value of the ratio of all the LRs analysed. Taking into consideration the lack of circular symmetry of the basal part, rates of growth were determined at selected points on the surface of LRs by the application of the growth tensor method, a mathematical tool previously applied only to describe organs with rotational symmetry. Maps showing the distribution of growth rates were developed for surfaces of LRs of various asymmetry ratios. The maps of growth rates on the surfaces of LRs having various shapes of the basal part show differences in both the geometry and the manner of growth, thus indicating that the manner of growth of the LR primordium is correlated to its shape. This is the first report of a description of growth of an asymmetric plant organ using the growth tensor method. The mathematical modelling adopted in the study provides new insights into plant organ formation and shape.

Mathematical modeling of volumetric material growth

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an elastic tensor, which is characterizing the reorganization of the body. The Blatz-Co hyperelastic constitutive model is adopted for an isotropic body, satisfying convexity conditions (resp. concavity conditions) with respect to the transformation gradient (resp. temperature). The evolution law for the transplant is obtained from the natural assumption that the evolution of the material is independent of the reference frame. It involves a modified Eshelby tensor based on the specific free energy density. The heat flux is dependent upon the transplant. The model consists of the constitutive equation, the energy balance, and the evolution law for the transplant. It is completed by suitable boundary conditions for the displacement, temperature and transplant tensor. The existence of locally unique solutions is obtained, for sufficiently smooth data close to the stable equilibrium. The question of the global existence is examined in the simplified situation of quasistatic isothermal equations of linear elasticity under the assumption of isotropic growth.

A nonlinear finite element model for the stress analysis of soft solids with a growing mass

The paper presents a new finite element (FE) model for the stress analysis of soft solids with a growing mass based on the work of Lubarda and Hoger (2002). Contrary to the traditional numerical methods emphasizing on the influence of growth on constitutive equations, an equivalent body force is firstly detected, which is resulted from the linearization of the nonlinear equation and acts as the driver for material growth in the numerical aspect. In the algorithm, only minor correction on the traditional tangent modulus is needed to take the growth effects into consideration and its objectivity could be guaranteed comparing with the traditional method. To solve the resulted equation in time domain, both explicit and implicit integration algorithms are developed, where the growth tensor is updated as an internal variable of Gauss point. The explicit updating scheme shows higher efficiency, while the implicit one seems to be more robust and accurate. The algorithm validation and its good performance are demonstrated by several two-dimensional examples, including free growth, constrained growth and stress dependent growth.

Modeling of spatial variations of growth within apical domes by means of the growth tensor. II. Growth specified on dome surface

Variations of the elemental relative rate of growth are modeled for parabolic, elliptic and hyperbolic domes of shoot apices by using the growth tensor in a suitable curvilinear coordinate system when the mode of area growth on the dome surface is known. Variations of growth rates within the domes are obtained in forms of computer-made maps for the following variants of growth on the dome surface: (1) constant meridional growth rate, (2) isotropic area growth, (3) anisotropy of area growth which becomes more intensive with increasing distance from the vertex. In variants 1 and 2 a maximum of volumetric growth rate appears in the center of the dome. Such a distribution of growth seems to be unrealistic. However, the corresponding growth tensors are interesting because they can be used in combination with other growth tensors to get the expected minimum volumetric growth rate in the dome center.

Modeling of spatial variations of growth within apical domes by means of the growth tensor. L Growth specified on dome axis

By using the growth tensor and a natural curvilinear coordinate system for description of the distribution of growth in plant organs, three geometric types of shoot apical domes (parabolic, elliptical and hyperbolic) were modeled. It was assumed that apical dome geometry remains unchanged during growth and that the natural coordinate systems are paraboloidal and prolate spheroidal. Two variants of the displacement velocity fields V were considered. One variant is specified by a constant relative elemental rate of growth along the axis of the dome. The second is specified by a rate increasing proportionally with distance from the geometric focus of the coordinate systems (and the apical dome). The growth tensor was used to calculate spatial variations of growth rates for each variant of each dome type. There is in both variants a clear tendency toward lower growth rates in the distal region of the dome. A basic condition for the existence of a tunica is met.

Distribution of linear growth rates in different directions in root apical meristems

Growth of apical meristems in plants may be well described by the growth tensor method. Hejnowicz (Envir. Exp. Bot. 1989, 29) determined growth tensors for roots: one with a minimum and the other with a maximum of the relative elemental growth rate in volume and used them for the description of two types of apices: one with an apical cell and merophytes (I), and the other with files of cells converging towards a quiescent centre, CQ (II). In the present paper the same cases are considered from the point of view of a spatial and directional variation of the relative elemental rate of growth in length, RERG1. Maps of the RERG<sub>1</sub> in two planes: axial and tangential, the latter determined by periclinal-longitudinal (PL) and periclinal-tangential (PT) principal growth directions, are shown. In an apical part of apex i where there is maximum volumetric growth, there also occurs a maximum of RERG<sub>1</sub> for all directions. In regions other than this RERG1 decreases although RERG<sub>1</sub> in the PL direction predominates everywhere. In apex II RERG<sub>1</sub> for all directions has a minimum in CQ and becomes increasingly larger with increasing distance from it - the maximum is in the PL direction in the cylindrical part of the apex. In peripheral parts of both apices, in the place of the root/cap junction, RERG, in the anticlinal direction is significantly small.

Growth and development of shoot apex in barley. III. Study of growth rate variation by means of the growth tensor

Silhouette of the shoot apical dome of barley does not change in time during early generative phase, although the dome height varies cyclically and increases in successive plastochrons. The elongation is limited to the dome region, the young frusta do not elongate. Relative rate of elongation (per plastochron) of the dome as a whole, decreases with increasing dome height. In the protoderm there are packets of cells distinguished by thicker walls. The growth tensor in prolate spheroidal coordinate system was applied to analysis of growth rate distribution within the dome. The meridional displacement velocities on dome surface, which are needed for the analysis, were determined on the basis of cell packet length expressed as a position function. Maximum relative elemental rate of growth occurs in subdistal zone.

Variations of growth in shoot apical domes of spruce seedlings: A study using the growth tensor

Variations of the relative elemental rate of growth within apical domes, for the case when dome geometry changes during development, were modeled. It was ascertained that: 1) the domes of spruce seedlings have a paraboloidal shape; 2) the shape is maintained during growth, but the domes become higher and wider; 3) the relative elemental rate of growth in area on dome surface is isotropic, as indicated by analysis of cell packets in the surface layer. These data were used in modeling by means of the growth tensor and natural coordinate system. Growth of the dome was considered a superposition: 1) of relatively fast steady shape growth, where the isotropy of growth in area on the surface of the dome, was determined, and 2) of relatively slow isogonic growth, which does not disturb the isotropy. The convergent parabolic system was selected as the natural coordinate system. Distributions of the growth rates in the form of computer-made maps for three domes differing in age, were obtained. It appears that the growth rates within the dome are relatively high in the distal part and smaller in the central and peripheral regions. This variation decreases progressively with seedling age when the dome becomes wider. The relative elemental rate of growth in volume, averaged for the whole dome, also decreases.

Trajectories of principal directions of growth, natural coordinate system in growing plant organ

In symplasticly growing organs the principal directions of growth, which are indicated by the eigenvectors of the symmetric part of the growth tensor, can be associated with each positional point and joined up to form a network of orthogonal trajectories, unless the growth is isotropic. The trajectories represent a natural coordinate system suitable for description of growing organs. These trajectories often can be recognized in patterns of nonrandom alignments in the cell wall network: these alignments are normal to anticlinal and periclinal walls. Coordinate systems that fit the trajectories in different types of growing organ are listed.

Spatial variations of growth within domes having different patterns of principal growth directions

Growth rate variations for two paraboloidal domes: A and B, identical when seen from the outside but differing in the internal pattern of principal growth directions, were modeled by means of the growth tensor and a natural coordinate system. In dome A periclinal trajectories in the axial plane were given by confocal parabolas (as in a tunical dome), in dome B by parabolas converging to the vertex (as in a dome without a tunica). Accordingly, two natural coordinate systems, namely paraboloidal for A and convergent parabolic for B, were used. In both cases, the rate of growth in area on the surfaces of domes was assumed to be isotropic and identical in corresponding points. It appears that distributions of growth rates within domes A and B are similar in their peripheral and central parts and different only in their distal regions. In the latter, growth rates are relatively large; the maximum relative rate of growth in volume is around the geometric focus in dome A, and on the surface around the vertex in dome B.

Spatial and Directional Variation of Growth Rates in Arabidopsis Root Apex: A Modelling Study

Growth and cellular organization of the Arabidopsis root apex are investigated in various aspects, but still little is known about spatial and directional variation of growth rates in very apical part of the apex, especially in 3D. The present paper aims to fill this gap with the aid of a computer modelling based on the growth tensor method. The root apex with a typical shape and cellular pattern is considered. Previously, on the basis of two types of empirical data: the published velocity profile along the root axis and dimensions of cell packets formed in the lateral part of the root cap, the displacement velocity field for the root apex was determined. Here this field is adopted to calculate the linear growth rate in different points and directions. The results are interpreted taking principal growth directions into account. The root apex manifests a significant anisotropy of the linear growth rate. The directional preferences depend on a position within the root apex. In the root proper the rate in the periclinal direction predominates everywhere, while in the root cap the predominating direction varies with distance from the quiescent centre. The rhizodermis is distinguished from the neighbouring tissues (cortex, root cap) by relatively high contribution of the growth rate in the anticlinal direction. The degree of growth anisotropy calculated for planes defined by principal growth directions and exemplary cell walls may be as high as 25. The changes in the growth rate variation are modelled.

The tensor-based model of plant growth applied to leaves of Arabidopsis thaliana: A two-dimensional computer model

Plant organs grow in coordinated and continuous way. Such growth is of a tensor nature, hence there is an infinite number of different directions of growth rate in each point of the growing organ. Three mutually orthogonal directions of growth can be recognized in which growth achieves extreme values (principal directions of growth [PDGs]). Models based on the growth tensor have already been successfully applied to the root and shoot apex. This paper presents the 2D model of growth applied to the arabidopsis leaf. The model employs the growth tensor method with a non-stationary velocity field. The postulated velocity functions are confirmed by growth measurements with the aid of the replica method.

Large size lead-free (Na,K)(Nb,Ta)O3 piezoelectric single crystal: growth and full tensor properties

Environmental friendly piezoelectric single crystal, Ta-modified (K,Na)NbO3 with the size of 12 × 11 × 11 mm3, has been successfully grown using the top seeded solution growth technique. This orthorhombic phase (K,Na)(Nb,Ta)O3 single crystal is the largest size to date in KNN-based crystals with homogeneous composition. The large size allowed us to apply the domain engineering technique to further enhance its piezoelectric properties. In addition, a self-consistent complete set of elastic, dielectric and piezoelectric constants for the [001]c poled domain engineered crystal has been measured, which is urgently needed for theoretical studies and simulation designs of electromechanical devices using this lead-free piezoelectric material. The electromechanical coupling factors are very high (k33 = 0.827, kt = 0.646) and the dielectric loss tangent is as low as 0.004 for this lead-free piezoelectric crystal. Such excellent properties make this crystal an excellent candidate to replace lead-containing piezoelectric materials in electromechanical devices.

Principal growth directions in development of the lateral root in Arabidopsis thaliana

During lateral root development a new meristem is formed within the mother root body. The main objective of this work was to simulate lateral root formation in Arabidopsis thaliana and to study a potential role of the principal directions in this process. Lateral root growth is anisotropic, so that three principal directions of growth can be distinguished within the organ. This suggests a tensorial character of growth and allows for its description by means of the growth tensor method. First features of the cell pattern of developing lateral roots were analysed in A. thaliana and then a tensorial model for growth and division of cells for this case was specified, assuming an unsteady character of the growth field of the organ. Microscopic observations provide evidence that the principal directions of growth are manifested at various developmental stages by oblique cell walls observed in different regions of the primordium. Other significant features observed are atypically shaped large cells at the flanks of young apices, as well as distinct boundaries between the mother root and the primordium. Simulations were performed using a model for growth. In computer-generated sequences the above-mentioned features could be identified. An attempt was made to reconstruct the virtual lateral root that included a consideration of the formation of particular tissue types based on literature data. In the cell pattern of the developing lateral root the principal directions of growth can be recognized through occurrence of oblique cell divisions. In simulation the role of these directions in cell pattern formation was confirmed, only when cells divide with respect to the principal directions can realistic results be obtained.

Kinematics of cardiac growth: In vivo characterization of growth tensors and strains

Progressive growth and remodeling of the left ventricle are part of the natural history of chronic heart failure and strong clinical indicators for survival. Accompanied by changes in cardiac form and function, they manifest themselves in alterations of cardiac strains, fiber stretches, and muscle volume. Recent attempts to shed light on the mechanistic origin of heart failure utilize continuum theories of growth to predict the maladaptation of the heart in response to pressure or volume overload. However, despite a general consensus on the representation of growth through a second order tensor, the precise format of this growth tensor remains unknown. Here we show that infarct-induced cardiac dilation is associated with a chronic longitudinal growth, accompanied by a chronic thinning of the ventricular wall. In controlled in vivo experiments throughout a period of seven weeks, we found that the lateral left ventricular wall adjacent to the infarct grows longitudinally by more than 10%, thins by more than 25%, lengthens in fiber direction by more than 5%, and decreases its volume by more than 15%. Our results illustrate how a local loss of blood supply induces chronic alterations in structure and function in adjacent regions of the ventricular wall. We anticipate our findings to be the starting point for a series of in vivo studies to calibrate and validate constitutive models for cardiac growth. Ultimately, these models could be useful to guide the design of novel therapies, which allow us to control the progression of heart failure.

Growing skin: tissue expansion in pediatric forehead reconstruction.

Tissue expansion is a common surgical procedure to grow extra skin through controlled mechanical over-stretch. It creates skin that matches the color, texture, and thickness of the surrounding tissue, while minimizing scars and risk of rejection. Despite intense research in tissue expansion and skin growth, there is a clear knowledge gap between heuristic observation and mechanistic understanding of the key phenomena that drive the growth process. Here, we show that a continuum mechanics approach, embedded in a custom-designed finite element model, informed by medical imaging, provides valuable insight into the biomechanics of skin growth. In particular, we model skin growth using the concept of an incompatible growth configuration. We characterize its evolution in time using a second-order growth tensor parameterized in terms of a scalar-valued internal variable, the in-plane area growth. When stretched beyond the physiological level, new skin is created, and the in-plane area growth increases. For the first time, we simulate tissue expansion on a patient-specific geometric model, and predict stress, strain, and area gain at three expanded locations in a pediatric skull: in the scalp, in the forehead, and in the cheek. Our results may help the surgeon to prevent tissue over-stretch and make informed decisions about expander geometry, size, placement, and inflation. We anticipate our study to open new avenues in reconstructive surgery and enhance treatment for patients with birth defects, burn injuries, or breast tumor removal.

Computational modeling of growth: systemic and pulmonary hypertension in the heart

We introduce a novel constitutive model for growing soft biological tissue and study its performance in two characteristic cases of mechanically induced wall thickening of the heart. We adopt the concept of an incompatible growth configuration introducing the multiplicative decomposition of the deformation gradient into an elastic and a growth part. The key feature of the model is the definition of the evolution equation for the growth tensor which we motivate by pressure-overload-induced sarcomerogenesis. In response to the deposition of sarcomere units on the molecular level, the individual heart muscle cells increase in diameter, and the wall of the heart becomes progressively thicker. We present the underlying constitutive equations and their algorithmic implementation within an implicit nonlinear finite element framework. To demonstrate the features of the proposed approach, we study two classical growth phenomena in the heart: left and right ventricular wall thickening in response to systemic and pulmonary hypertension.

The tensor-based model for growth and cell divisions of the root apex. II. Lateral root formation

In this work, the formation of the virtual lateral root (VLR) is shown. The VLR is formed using the 2D simulation model of growth and cell divisions based on the concept of growth tensor, specified for radish. Growth is generated by the field of growth rates of an unsteady type (GT field). Principal directions of growth (PDGs) are assumed to define the orientation of cell divisions. Temporal sequences of the VLR formation are a result of an application of the GT field to the polygon meshwork representing cell pattern of already initiated primordium. The computer-generated lateral root (LR) develops realistically, and its cell pattern is vivid and similar to that observed in anatomical sections. The real and virtual LRs show similar cellular organization, both originate from a small group of cells situated in two-cell layers of the pericycle and both layers are engaged in the LR development. The LR formation seems to be controlled at the tensor level and individual cells presumably detect PDGs and obey them in the course of the cell divisions. PDGs are postulated to affect the cellular organization of the LR. Using the method of computer simulations, cellular aspects of the LR morphogenesis are discussed.

A generic approach towards finite growth with examples of athlete's heart, cardiac dilation, and cardiac wall thickening

The objective of this work is to establish a generic continuum-based computational concept for finite growth of living biological tissues. The underlying idea is the introduction of an incompatible growth configuration which naturally introduces a multiplicative decomposition of the deformation gradient into an elastic and a growth part. The two major challenges of finite growth are the kinematic characterization of the growth tensor and the identification of mechanical driving forces for its evolution. Motivated by morphological changes in cell geometry, we illustrate a micromechanically motivated ansatz for the growth tensor for cardiac tissue that can capture both strain-driven ventricular dilation and stress-driven wall thickening. Guided by clinical observations, we explore three distinct pathophysiological cases: athlete's heart, cardiac dilation, and cardiac wall thickening. We demonstrate the computational solution of finite growth within a fully implicit incremental iterative Newton–Raphson based finite element solution scheme. The features of the proposed approach are illustrated and compared for the three different growth pathologies in terms of a generic bi-ventricular heart model.