Growth Tensor Research Articles

A multiscale model for eccentric and concentric cardiac growth through sarcomerogenesis

We present a novel computational model for maladaptive cardiac growth in which kinematic changes of the cardiac chambers are attributed to alterations in cytoskeletal architecture and in cellular morphology. We adopt the concept of finite volume growth characterized through the multiplicative decomposition of the deformation gradient into an elastic part and a growth part. The functional form of its growth tensor is correlated to sarcomerogenesis, the creation and deposition of new sarcomere units. In response to chronic volume-overload, an increased diastolic wall strain leads to the addition of sarcomeres in series, resulting in a relative increase in cardiomyocyte length, associated with eccentric hypertrophy and ventricular dilation. In response to chronic pressure-overload, an increased systolic wall stress leads to the addition of sacromeres in parallel, resulting in a relative increase in myocyte cross sectional area, associated with concentric hypertrophy and ventricular wall thickening. The continuum equations for both forms of maladaptive growth are discretized in space using a nonlinear finite element approach, and discretized in time using the implicit Euler backward scheme. We explore a generic bi-ventricular heart model in response to volume- and pressure-overload to demonstrate how local changes in cellular morphology translate into global alterations in cardiac form and function.

The insight of mixtures theory for growth and remodeling

The emergence of residual stress as due to growth and remodeling of soft biological tissues is considered in the framework of the mixture theory. The focus is on mixtures composed by one elastic solid component and several fluid ones. It is shown that the standard theory is unable to predict residual stresses unless enriched by a suitable descriptor of growth. Both the introduction of a dependence of the free energy on the density of the solid component and the Kroner–Lee multiplicative decomposition of the gradient of deformation are effective in this respect, with different levels of generality. When adopting a multiplicative decomposition of the tensor gradient of deformation, thermodynamical arguments suggest constitutive laws for the evolution of the growth tensor that point out the role of the concentration of fluid species in driving the emergence of residual stress thanks to inhomogeneous growth.

The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions

Plant organs grow symplastically, i.e. in a continuous and coordinated way. Such growth is of a tensor nature, which is manifested in the property that at every point of the organ three mutually orthogonal principal growth directions (PDG) can be recognized. The PDGs are postulated to affect orientation of cell divisions. This paper shows for the first time the 2D simulation model for growth in which cells divide taking into account the PDGs. The model, conceptually based on the growth tensor (GT), is applied to the root apex of radish, having a quiescent centre (QC). It shows the simulation of how exemplary cell pattern of the real root apex develops in time. The results provide satisfactory description of the root growth. The computer-generated cell pattern is realistic and more or less steady indicating that PDGs are important for growth. Presumably cells detect PDGs and obey them in the course of cell divisions. Computer generated division walls, perpendicular to PDGs, form periclinal and anticlinal zigzags as regular as those observed in microscopic sections. Growth tensor defines a field of growth rates at the organ level. QC, fundamental in this field, determines the group of quiescent initial cells which is, in turn, surrounded by active functional initials, from which all tissues of the root apex originate. The present simulations have shown that stability of generated cell pattern depends on whether the group of the functional initials is permanent; if this is not the case, the cell wall pattern changes in accordance with PDGs.

Application of a changing field of growth rates to a description of root apex formation

Based on the growth tensor method an unsteady field of growth rates for developing root apex is presented. Maps of growth rates distribution as well as simulations in which the field was applied to initially uniform grid of points are presented. In the simulations, the grid undergoes deformation that resembles new root formation in its axial plane. Four variants of field operation on the grid are shown.

Mathematical Modeling of a Solid–Liquid Mixture with Mass Exchange Between Constituents

The mechanical behavior of a mixture composed by an elastic solid and a fluid that exchange mass is investigated. Both the liquid flow and the solid deformation depend on how the solid phase has increased (diminished) its mass, i.e. on the mass conversion between constituents. The model is developed introducing a decomposition of the solid phase deformation gradient. In particular, exploiting the criterion of maximization of the rate of entropy production, we determine an explicit evolution equation for the so-called growth tensor which involves directly the solid stress tensor. An example of a possible choice of the constitutive functions is also presented.

Investigation of midfacial and mandibular morphology: Strain tensor analysis

The method of strain tensor (ST) analysis efficiently demonstrates the morphological differences in skeletal jaw growth and provides more information for orthodontic diagnosis and orthopedic therapy of jaw disharmonies. The purpose of this study was to investigate midfacial and mandibular size and shape differences in a sample of normal Class I molar occlusion and Class III malocclusion. Fifty Chinese boys aged 10–12 years, each occlusal group of 25 subjects, have been selected. The results indicated that statistically significant differences in midfacial and mandibular configurations between Class I occlusion and Class III malocclusion occurred at P< 0.01 levels. The mandibular corpus of Class III malocclusion was larger than that of Class I occlusion and the extension direction of growth tensors was chiefly manifested along the long axis of general mandibular morphology. In contrast, the midfacial size of Class III malocclusion was smaller than that of Class I occlusion and a horizontal reduction was observed at the posterior midface. Shape differences in the midface and mandible manifested for the two occlusal types. The present investigation with a new geometric morphometrics (ST analysis) provides more information and better understanding of the midfacial and mandibular morphology and growth pattern in Class III growing children.

Growth tensor mapping: A rostro-caudal wave of peak growth rates detected in the developing human brain in the first 15 years of life

Growth tensor mapping: A rostro-caudal wave of peak growth rates detected in the developing human brain in the first 15 years of life

Principal directions of growth and the generation of cell patterns in wild-type and gib-1 mutant roots of tomato (Lycopersicon esculentum Mill.) grown in vitro

The patterns of cell growth and division characteristic of the apex of tomato roots grown in vitro were simulated by computer using a growth tensor (GT). The GT was used to clarify the basis of the altered cell patterns found within apices of roots whose gibberellin levels had been depressed by mutation (at the GIB-1 locus) or through application of the gibberellin-biosynthesis inhibitor, 2S,3S paclobutrazol. At the pole of wild-type roots, where the cell files of the cortex converge, there are commonly only one or two tiers of cortical cells sandwiched between the pole of the stele and the cap initials. By contrast, root apices of the gib-1 mutant contain additional tiers in this region. The development of these additional tiers is suppressed when roots of the mutant are grown in the presence of gibberellic acid (GA3), but could be induced in wild-type roots when they are grown in 2S,3S paclobutrazol. The wild-type cell pattern can be simulated using the GT and by the application of appropriate rules that govern cell growth and division. The induced variations in cell pattern are interpreted as being due to displacements, within the apex, of the principal directions of growth (PDGs), which are represented, in part, by the set of periclines and anticlines seen in the cell wall network; these, in turn, are utilized in the specification of the GT. During normal (wild-type) root growth, the PDGs maintain a stable pattern and the corresponding cell pattern is also stable. However, in order to interpret the cellular behaviour found in wild-type roots grown in 2S, 3S paclobutrazol, simulation using the GT shows that, if the pattern of PDGs is destabilized and displaced distally along the root axis, the cell pattern reorganizes into that typical of gib-1 mutant roots. Conversely, the cell pattern of gib-1 roots, which reverts to wild-type upon exposure to GA3, can be simulated if the PDGs are displaced proximally to the inside of the apex whereupon the number of cortical tiers at the root pole decreases. These results suggest a link between endogenous gibberellin level and the specification of the PDGs in the growing tomato root apex. Furthermore, the evidence of cell patterns from gib-1 roots suggests that, in order to achieve stability of PDGs with concomitant stable cellular patterning, an optimal gibberellin level is necessary. In practice, this can be attained by culturing the mutant roots in medium containing 1 μM GA3.

MODELING OF MERISTEMATIC GROWTH OF ROOT APICES IN A NATURAL COORDINATE SYSTEM

A curvilinear, orthogonal coordinate system, which resembles the pattern of periclines and anticlines in the cellular network of root apices, is presented. The system makes possible an analysis of the dynamics of apical growth: the relationship between growth rates and tensile stresses in cell walls. In this paper the coordinate system is used in modeling the growth and cell partitioning in the apical domes. The symplastic growth is described by means of the growth tensor which is assumed to have diagonal form in the system, so its coordinate lines represent the principal directions of growth rate. The coordinate system and the growth tensor in diagonal form assure temporal stability of form and cellular structure of the modeled apex including transition from the dome to the cylindrical part of the apex. The spatial and temporal aspects of the dome part of two types of root apices—one with maximum of volumetric relative growth rate at the center region of the apex, and another with minimum of the rate (quiescent center)—are described. The maximum of the rate at the center results in the cellular pattern with an apical cell and merophytes, the minimum results in ribs of cells (inside the root proper) converging toward the quiescent center.

Modeling of Meristematic Growth of Root Apices in a Natural Coordinate System

A curvilinear, orthogonal coordinate system, which resembles the pattern of periclines and anticlines in the cellular network of root apices, is presented. The system makes possible an analysis of the dynamics of apical growth: the relationship between growth rates and tensile stresses in cell walls. In this paper the coordinate system is used in modeling the growth and cell partitioning in the apical domes. The symplastic growth is described by means of the growth tensor which is assumed to have diagonal form in the system, so its coordinate lines represent the principal directions of growth rate. The coordinate system and the growth tensor in diagonal form assure temporal stability of form and cellular structure of the modeled apex including transition from the dome to the cylindrical part of the apex. The spatial and temporal aspects of the dome part of two types of root apices—one with maximum of volumetric relative growth rate at the center region of the apex, and another with minimum of the rate (quiescent center)—are described. The maximum of the rate at the center results in the cellular pattern with an apical cell and merophytes, the minimum results in ribs of cells (inside the root proper) converging toward the quiescent center.

Modeling the formation of root apices

The formation of new root apices from small groups of cells with different cellular patterns has been simulated using an existing model based on growth tensors. To generate an apex, a steady growth field was used. The pattern of cells evolved to approach the steady state. Two extreme types of progressions have been obtained : one leading to an apex with a single or a few apical cells, and the other to an apex with a quiescent centre. The change of structure while applying a steady growth tensor indicates that development may involve a succession of discrete growth tensors.

Differential growth resulting in the specification of different types of cellular architecture in root meristems

Symplastic growth of plant organs may be described by a continuous growth tensor field. In tensorial analysis of meristems, the trajectoris of periclinal and anticlinal cell walls represent trajectories of the principal directions of growth ( PDGs); this follows from the maintenance of mutual orthogonality between periclinal and anticlinal wall trajectories during growth. Periclinal and anticlinal cell divisions are also oriented in the principal planes of growth. The growth tensor for the root apex is specified in such a way that the principal directions of the tensor fit the pattern of periclinal and anticlinal walls in the apex, and that the grid formed by material particles aligned along PDG trajectories preserve this alignment during growth. Two growth tensors are formulated—one giving a maximum and the other giving a minimum of the volumetric relative elemental growth rate at the region of the initial cell(s). Temporal sequences of deformation of a grid formed by lines coinciding with the principal directions of growth are shown. The formation of cellular patterns in root apices is simulated. Two types of patterns are obtained: one with an apical cell and merophytes, and another with files of cells converging towards a quiescent centre.

Calculation of rates of cell division in shoot meristems with reference to the growth tensor

The theoretical concept of a cell division tensor has some appeal; yet, the actual formulation of such a tensor remains a challenge. It is nonetheless possible, without invoking a cell division tensor, to use the physical components of the growth tensor, along with natural curvilinear coordinate systems, to calculate the theoretical distribution of cell division rates within apical meristems of plant shoots. The patterns of dimensional changes of cells, the distribution of growth rates specified by the growth tensor, and the indicated rates of cell division are described for meristems characterized by steady state fields of cell flux velocities and of mean cell dimensions. These relations are evaluated by means of continuity equations for the fluxes of mean cell dimensions. These methods allow calculation of the relative rates of cell division at specified sites, and within specified orientation classes, as well as of division rates irrespective of orientation.

Growth tensor of plant organs

Analysis of the definition of the relative elemental rate of growth of a line element, RERG1, in a growing plant organ leads to the dyadic ∇V where V is the vector field of displacement velocities of material points in the organ. The components of this dyadic represent physical components of a tensor, which we propose to call the growth tensor. The latter can be derived directly from the definition of RERG1. The growth tensor allows full characterization of the rate of growth in length, area, and volume, as well as rates of angular change between elements, and of vorticity in the growing organ. From the fact that anticlinal and periclinal walls of cells within the organ preserve their orthogonality during growth, we infer that the principal directions of the growth tensor coincide with periclines and anticlines. The definition of the growth tensor based on the dyadic offers an easy way to generate this tensor in different coordinate systems. An example is given of the use of the growth tensor in analyzing elongation growth of a cylindrical plant organ in two alternative modes: with and without rotation of the tip. It is shown that growth by the two modes yields the same relative elemental rates of growth in volume, but that the principal directions of the growth tensors are different. We infer that if growth is a tensorial attribute of an organ, then the controls of growth must also be tensorial attributes. The controlling tensors must have at least as high a rank as the growth tensors, but must be of a higher hierarchical level.