Capturing the Dynamics of a Hybrid Multiscale Cancer Model with a Continuum Model

Cancer is a complex disease involving processes at spatial scales from subcellular, like cell signalling, to tissue scale, such as vascular network formation. A number of multiscale models have been developed to study the dynamics that emerge from the coupling between the intracellular, cellular and tissue scales. Here, we develop a continuum partial differential equation model to capture the dynamics of a particular multiscale model (a hybrid cellular automaton with discrete cells, diffusible factors and an explicit vascular network). The purpose is to test under which circumstances such a continuum model gives equivalent predictions to the original multiscale model, in the knowledge that the system details are known, and differences in model results can be explained in terms of model features (rather than unknown experimental confounding factors). The continuum model qualitatively replicates the dynamics from the multiscale model, with certain discrepancies observed owing to the differences in the modelling of certain processes. The continuum model admits travelling wave solutions for normal tissue growth and tumour invasion, with similar behaviour observed in the multiscale model. However, the continuum model enables us to analyse the spatially homogeneous steady states of the system, and hence to analyse these waves in more detail. We show that the tumour microenvironmental effects from the multiscale model mean that tumour invasion exhibits a so-called pushed wave when the carrying capacity for tumour cell proliferation is less than the total cell density at the tumour wave front. These pushed waves of tumour invasion propagate by triggering apoptosis of normal cells at the wave front. Otherwise, numerical evidence suggests that the wave speed can be predicted from linear analysis about the normal tissue steady state.     

Individual-based and continuum models of growing cell populations: a comparison

In this paper we compare two alternative theoretical approaches for simulating the growth of cell aggregates in vitro: individual cell (agent)-based models and continuum models. We show by a quantitative analysis of both a biophysical agent-based and a continuum mechanical model that for densely packed aggregates the expansion of the cell population is dominated by cell proliferation controlled by mechanical stress. The biophysical agent-based model introduced earlier (Drasdo and Hoehme in Phys Biol 2:133-147, 2005) approximates each cell as an isotropic, homogeneous, elastic, spherical object parameterised by measurable biophysical and cell-biological quantities and has been shown by comparison to experimental findings to explain the growth patterns of dense monolayers and multicellular spheroids. Both models exhibit the same growth kinetics, with initial exponential growth of the population size and aggregate diameter followed by linear growth of the diameter and power-law growth of the cell population size. Very sparse monolayers can be explained by a very small or absent cell-cell adhesion and large random cell migration. In this case the expansion speed is not controlled by mechanical stress but by random cell migration and can be modelled by the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) reaction-diffusion equation. The growth kinetics differs from that of densely packed aggregates in that the initial spread, as quantified by the radius of gyration, is diffusive. Since simulations of the lattice-free agent-based model in the case of very large random migration are too long to be practical, lattice-based cellular automaton (CA) models have to be used for a quantitative analysis of sparse monolayers. Analysis of these dense monolayers leads to the identification of a critical parameter of the CA model so that eventually a hierarchy of three model types (a detailed biophysical lattice-free model, a rule-based cellular automaton and a continuum approach) emerge which yield the same growth pattern for dense and sparse cell aggregates.     

Simulation of vessel morphogenesis using cellular automata

We present a cellular automaton model, including lateral inhibition of an autocatalytic morphogen, as well as a genetic switch that differentiates tissue into substrate-depleting vessels. This model yields isotropic morphogenesis, including: dichotomous and lateral branching, blind vessel ends, and closed loops due to anastosmosis. The algorithm consists of a list of simple rules describing the essential biophysical features, permitting comfortable programming and fast computations. Depending on the choice of the substrate s, the model is applicable to leaf veins (s is auxin), insect trachea (s is CO2) or neovascularization (s is an angiogenesis factor). Sequential addition of rules can be correlated to evolutionary steps in leaf morphogenesis.     

A model of growing vascular structures

Increasing attention is being paid to the configuration and development of vascular structures and their possible correlations with physiological events. The study of angiogenesis in normal and pathological states as well as in the embryo and adult has provided new insights into the mechanism of vessel growth and organization of the vasculature. Various mathematical branching models have been developed. These constructions are mainly geometrical and only involve a branching phenomenon. We propose the use of a deterministic non-linear model based on physiological laws and hydrodynamics. Growth, branching and anastomosis, the three actual main events occurring in vascular growth, are included in this model. Space growth, including cells and vessels, is defined by a decreasing transformation. Space density and the length of new sprouts are controlled by a set of parameters. The conditions on these parameters are well established, which allows the production of realistic patterns.     

Computational Screening of Tip and Stalk Cell Behavior Proposes a Role for Apelin Signaling in Sprout Progression

Angiogenesis involves the formation of new blood vessels by sprouting or splitting of existing blood vessels. During sprouting, a highly motile type of endothelial cell, called the tip cell, migrates from the blood vessels followed by stalk cells, an endothelial cell type that forms the body of the sprout. To get more insight into how tip cells contribute to angiogenesis, we extended an existing computational model of vascular network formation based on the cellular Potts model with tip and stalk differentiation, without making a priori assumptions about the differences between tip cells and stalk cells. To predict potential differences, we looked for parameter values that make tip cells (a) move to the sprout tip, and (b) change the morphology of the angiogenic networks. The screening predicted that if tip cells respond less effectively to an endothelial chemoattractant than stalk cells, they move to the tips of the sprouts, which impacts the morphology of the networks. A comparison of this model prediction with genes expressed differentially in tip and stalk cells revealed that the endothelial chemoattractant Apelin and its receptor APJ may match the model prediction. To test the model prediction we inhibited Apelin signaling in our model and in an in vitro model of angiogenic sprouting, and found that in both cases inhibition of Apelin or of its receptor APJ reduces sprouting. Based on the prediction of the computational model, we propose that the differential expression of Apelin and APJ yields a “self-generated” gradient mechanisms that accelerates the extension of the sprout.     

Angiogenic morphogenesis driven by dynamic and heterogeneous collective endothelial cell movement

Angiogenesis is a complex process, which is accomplished by reiteration of modules such as sprouting, elongation and bifurcation, that configures branching vascular networks. However, details of the individual and collective behaviors of vascular endothelial cells (ECs) during angiogenic morphogenesis remain largely unknown. Herein, we established a time-lapse imaging and computer-assisted analysis system that quantitatively characterizes behaviors in sprouting angiogenesis. Surprisingly, ECs moved backwards and forwards, overtaking each other even at the tip, showing an unknown mode of collective cell movement with dynamic 'cell-mixing'. Mosaic analysis, which enabled us to monitor the behavior of individual cells in a multicellular structure, confirmed the 'cell-mixing' phenomenon of ECs that occurs at the whole-cell level. Furthermore, an in vivo EC-tracking analysis revealed evidence of cell-mixing and overtaking at the tip in developing murine retinal vessels. In parametrical analysis, VEGF enhanced tip cell behavior and directed EC migration at the stalk during branch elongation. These movements were counter-regulated by EC-EC interplay via γ-secretase-dependent Dll4-Notch signaling, and might be promoted by EC-mural cell interplay. Finally, multiple regression analysis showed that these molecule-mediated tip cell behaviors and directed EC migration contributed to effective branch elongation. Taken together, our findings provide new insights into the individual and collective EC movements driving angiogenic morphogenesis. The methodology used for this analysis might serve to bridge the gap in our understanding between individual cell behavior and branching morphogenesis.     

Investigating CTL Mediated Killing with a 3D Cellular Automaton

Cytotoxic T lymphocytes (CTLs) are important immune effectors against intra-cellular pathogens. These cells search for infected cells and kill them. Recently developed experimental methods in combination with mathematical models allow for the quantification of the efficacy of CTL killing in vivo and, hence, for the estimation of parameters that characterize the effect of CTL killing on the target cell populations. It is not known how these population-level parameters relate to single-cell properties. To address this question, we developed a three-dimensional cellular automaton model of the region of the spleen where CTL killing takes place. The cellular automaton model describes the movement of different cell populations and their interactions. Cell movement patterns in our cellular automaton model agree with observations from two-photon microscopy. We find that, despite the strong spatial nature of the kinetics in our cellular automaton model, the killing of target cells by CTLs can be described by a term which is linear in the target cell frequency and saturates with respect to the CTL levels. Further, we find that the parameters describing CTL killing on the population level are most strongly impacted by the time a CTL needs to kill a target cell. This suggests that the killing of target cells, rather than their localization, is the limiting step in CTL killing dynamics given reasonable frequencies of CTL. Our analysis identifies additional experimental directions which are of particular importance to interpret estimates of killing rates and could advance our quantitative understanding of CTL killing.     

Complex cell rearrangements during intersegmental vessel sprouting and vessel fusion in the zebrafish embryo

The formation of intersegmental blood vessels (ISVs) in the zebrafish embryo serves as a paradigm to study angiogenesis in vivo. ISV formation is thought to occur in discrete steps. First, endothelial cells of the dorsal aorta migrate out and align along the dorsoventral axis. The dorsal-most cell, also called tip cell, then joins with its anterior and posterior neighbours, thus establishing a simple vascular network. The vascular lumen is then established via formation of vacuoles, which eventually fuse with those of adjacent endothelial cells to generate a seamless tube with an intracellular lumen. To investigate the cellular architecture and the development of ISVs in detail, we have analysed the arrangement of endothelial cell junctions and have performed single cell live imaging. In contrast to previous reports, we find that endothelial cells are not arranged in a linear head-to-tail configuration but overlap extensively and form a multicellular tube, which contains an extracellular lumen. Our studies demonstrate that a number of cellular behaviours, such as cell divisions, cell rearrangements and dynamic alterations in cell-cell contacts, have to be considered when studying the morphological and molecular processes involved in ISV and endothelial lumen formation in vivo.     

A multiphase model describing vascular tumour growth

In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.     

A Cell-Based Model of Extracellular-Matrix-Guided Endothelial Cell Migration During Angiogenesis

Angiogenesis, the formation of new blood vessels sprouting from existing ones, occurs in several situations like wound healing, tissue remodeling, and near growing tumors. Under hypoxic conditions, tumor cells secrete growth factors, including VEGF. VEGF activates endothelial cells (ECs) in nearby vessels, leading to the migration of ECs out of the vessel and the formation of growing sprouts. A key process in angiogenesis is cellular self-organization, and previous modeling studies have identified mechanisms for producing networks and sprouts. Most theoretical studies of cellular self-organization during angiogenesis have ignored the interactions of ECs with the extra-cellular matrix (ECM), the jelly or hard materials that cells live in. Apart from providing structural support to cells, the ECM may play a key role in the coordination of cellular motility during angiogenesis. For example, by modifying the ECM, ECs can affect the motility of other ECs, long after they have left. Here, we present an explorative study of the cellular self-organization resulting from such ECM-coordinated cell migration. We show that a set of biologically-motivated, cell behavioral rules, including chemotaxis, haptotaxis, haptokinesis, and ECM-guided proliferation suffice for forming sprouts and branching vascular trees.     

A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis

This work describes the first cell-based model of tumor-induced angiogenesis. At the extracellular level, the model describes diffusion, uptake, and decay of tumor-secreted pro-angiogenic factor. At the cellular level, the model uses the cellular Potts model based on system-energy reduction to describe endothelial cell migration, growth, division, cellular adhesion, and the evolving structure of the stroma. Numerical simulations show: 1), different tumor-secreted pro-angiogenic factor gradient profiles dramatically affect capillary sprout morphology; 2), average sprout extension speeds depend on the proximity of the proliferating region to the sprout tip, and the coordination of cellular functions; and 3), inhomogeneities in the extravascular tissue lead to sprout branching and anastomosis, phenomena that emerge without any prescribed rules. This model provides a quantitative framework to test hypotheses on the biochemical and biomechanical mechanisms that control tumor-induced angiogenesis.     

IGF-binding proteins 3 and 4 are regulators of sprouting angiogenesis

Purpose

We have previously identified insulin-like growth factor 2 (IGF2) and insulin-like growth factor 1 receptor (IGF1R) as essential proteins for tip cell maintenance and sprouting angiogenesis. In this study, we aim to identify other IGF family members involved in endothelial sprouting angiogenesis.

Methods

Effects on sprouting were analyzed in human umbilical vein endothelial cells (HUVECs) using the spheroid-based sprouting model, and were quantified as mean number of sprouts per spheroid and average sprout length. RNA silencing technology was used to knockdown gene expression. Recombinant forms of the ligands (IGF1 and IGF2, insulin) and the IGF-binding proteins (IGFBP) 3 and 4 were used to induce excess effects. Effects on the tip cell phenotype were analyzed by measuring the fraction of CD34⁺ tip cells using flow cytometry and immunohistochemistry in a 3D angiogenesis model. Experiments were performed in the presence and absence of serum.

Results

Knockdown of IGF2 inhibited sprouting in HUVECs, in particular when cultured in the absence of serum, suggesting that components in serum influence the signaling of IGF2 in angiogenesis in vitro. We then determined the effects of IGFBP3 and IGFBP4, which are both present in serum, on IGF2-IGF1R signaling in sprouting angiogenesis in the absence of serum: knockdown of IGFBP3 significantly reduced sprouting angiogenesis, whereas knockdown of IGFBP4 resulted in increased sprouting angiogenesis in both flow cytometry analysis and immunohistochemical analysis of the 3D angiogenesis model. Other IGF family members except INSR did not affect IGF2-IGF1R signaling.

Conclusions

Serum components and IGF binding proteins regulate IGF2 effects on sprouting angiogenesis. Whereas IGFBP3 acts as co-factor for IGF2-IGF1R binding, IGFBP4 inhibits IGF2 signaling.

     

Rat mesentery exteriorization: a model for investigating the cellular dynamics involved in angiogenesis

Microvacular network growth and remodeling are critical aspects of wound healing, inflammation, diabetic retinopathy, tumor growth and other disease conditions. Network growth is commonly attributed to angiogenesis, defined as the growth of new vessels from pre-existing vessels. The angiogenic process is also directly linked to arteriogenesis, defined as the capillary acquisition of a perivascular cell coating and vessel enlargement. Needless to say, angiogenesis is complex and involves multiple players at the cellular and molecular level. Understanding how a microvascular network grows requires identifying the spatial and temporal dynamics along the hierarchy of a network over the time course of angiogenesis. This information is critical for the development of therapies aimed at manipulating vessel growth. The exteriorization model described in this article represents a simple, reproducible model for stimulating angiogenesis in the rat mesentery. It was adapted from wound-healing models in the rat mesentery, and is an alternative to stimulate angiogenesis in the mesentery via i.p. injections of pro-angiogenic agents. The exteriorization model is attractive because it requires minimal surgical intervention and produces dramatic, reproducible increases in capillary sprouts, vascular area and vascular density over a relatively short time course in a tissue that allows for the two-dimensional visualization of entire microvascular networks down to single cell level. The stimulated growth reflects natural angiogenic responses in a physiological environment without interference of foreign angiogenic molecules. Using immunohistochemical labeling methods, this model has been proven extremely useful in identifying novel cellular events involved in angiogenesis. Investigators can readily correlate the angiogenic metrics during the time course of remodeling with time specific dynamics, such as cellular phenotypic changes or cellular interactions.     

Evolution of angiogenesis

Endothelial cells, which are the main agents of the angiogenic process in vertebrates, are lacking in the vessels of invertebrates. These are limited by the basement membranes of epithelial or myoepithelial cells. This fact leads to the questions of how vessels grow in invertebrates and how vertebrate angiogenesis evolved. We herein review the knowledge available about vascular growth in invertebrates. The cases described include the ascidian Botryllus, the annelid Hirudo and the squid Idiosepius. All these processes of vascular growth in invertebrates show substantial differences with the vertebrate angiogenesis, although the signalling system mediated by VEGF and its receptor VEGFR seems to be involved in all cases. However, VEGF signalling is used by many processes of cell directional migration, and it cannot be considered as a hallmark of angiogenesis. We also describe the close similarity between the molecular control of the endothelial angiogenesis and the branching morphogenesis of the tracheal system of insects. In both cases, the process is regulated by hypoxia and activates a leading tip cell which inhibits responsiveness of the adjacent cells through a Delta/Notch signalling pathway. We suggest that endothelial angiogenesis in vertebrates arose through cooption of this hypoxia-sensing mechanism by replacing the FGF/FGFR axis of insects by a VEGF/VEGFR-mediated system, and adding a second layer of control of the endothelial state (quiescent or activated) mediated by angiopoietins and Tie receptors. This evolutionarily new control mechanism of endothelial angiogenesis establishes an endothelial/perivascular cell crosstalking which does not exist in invertebrates.     

2-Methoxycinnamaldehyde inhibits tumor angiogenesis by suppressing Tie2 activation

Blood vessels are mainly composed of intraluminal endothelial cells (ECs) and mural cells adhering to the ECs on their basal side. Immature blood vessels lacking mural cells are leaky; thus, the process of mural cell adhesion to ECs is indispensable for stability of the vessels during physiological angiogenesis. However, in the tumor microenvironment, although some blood vessels are well-matured, the majority is immature. Because mural cell adhesion to ECs also has a marked anti-apoptotic effect, angiogenesis inhibitors that destroy immature blood vessels may not affect mature vessels showing more resistance to apoptosis. Activation of Tie2 receptor tyrosine kinase expressed in ECs mediates pro-angiogenic effects via the induction of EC migration but also facilitates vessel maturation via the promotion of cell adhesion between mural cells and ECs. Therefore, inhibition of Tie2 has the advantage of completely inhibiting angiogenesis. Here, we isolated a novel small molecule Tie2 kinase inhibitor, identified as 2-methoxycinnamaldehyde (2-MCA). We found that 2-MCA inhibits both sprouting angiogenesis and maturation of blood vessels, resulting in inhibition of tumor growth. Our results suggest a potent clinical benefit of disrupting these two using Tie2 inhibitors.     

Contact-inhibited chemotaxis in de novo and sprouting blood-vessel growth

Blood vessels form either when dispersed endothelial cells (the cells lining the inner walls of fully formed blood vessels) organize into a vessel network (vasculogenesis), or by sprouting or splitting of existing blood vessels (angiogenesis). Although they are closely related biologically, no current model explains both phenomena with a single biophysical mechanism. Most computational models describe sprouting at the level of the blood vessel, ignoring how cell behavior drives branch splitting during sprouting. We present a cell-based, Glazier-Graner-Hogeweg model (also called Cellular Potts Model) simulation of the initial patterning before the vascular cords form lumens, based on plausible behaviors of endothelial cells. The endothelial cells secrete a chemoattractant, which attracts other endothelial cells. As in the classic Keller-Segel model, chemotaxis by itself causes cells to aggregate into isolated clusters. However, including experimentally observed VE-cadherin-mediated contact inhibition of chemotaxis in the simulation causes randomly distributed cells to organize into networks and cell aggregates to sprout, reproducing aspects of both de novo and sprouting blood-vessel growth. We discuss two branching instabilities responsible for our results. Cells at the surfaces of cell clusters attempting to migrate to the centers of the clusters produce a buckling instability. In a model variant that eliminates the surface-normal force, a dissipative mechanism drives sprouting, with the secreted chemical acting both as a chemoattractant and as an inhibitor of pseudopod extension. Both mechanisms would also apply if force transmission through the extracellular matrix rather than chemical signaling mediated cell-cell interactions. The branching instabilities responsible for our results, which result from contact inhibition of chemotaxis, are both generic developmental mechanisms and interesting examples of unusual patterning instabilities.     

The potential for intercellular mechanical interaction: simulations of single chondrocyte versus anatomically based distribution

Computational studies of chondrocyte mechanics, and cell mechanics in general, have typically been performed using single cell models embedded in an extracellular matrix construct. The assumption of a single cell microstructural model may not capture intercellular interactions or accurately reflect the macroscale mechanics of cartilage when higher cell concentrations are considered, as may be the case in many instances. Hence, the goal of this study was to compare cell-level response of single and eleven cell biphasic finite element models, where the latter provided an anatomically based cellular distribution representative of the actual number of cells for a commonly used \(100 \, \upmu \hbox {m}\) edge cubic representative volume in the middle zone of cartilage. Single cell representations incorporated a centered single cell model and eleven location-corrected single cell models, the latter to delineate the role of cell placement in the representative volume element. A stress relaxation test at 10% compressive strain was adopted for all simulations. During transient response, volume- averaged chondrocyte mechanics demonstrated marked differences (up to 60% and typically greater than 10%) for the centered single versus the eleven cell models, yet steady-state loading was similar. Cell location played a marked role, due to inhomogeneity of the displacement and fluid pressure fields at the macroscopic scale. When the single cell representation was corrected for cell location, the transient response was consistent, while steady-state differences on the order of 1–4% were realized, which may be attributed to intercellular mechanical interactions. Anatomical representations of the superficial and deep zones, where cells reside in close proximity, may exhibit greater intercellular interactions, but these have yet to be explored.     

Exclusion processes on a growing domain

A discrete model provides a useful framework for experimentalists to understand the interactions between growing tissues and other biological mechanisms. A cellular automata (CA) model with domain growth, cell motility and cell proliferation, based on cellular exclusion processes, is developed here. Average densities can be defined from the CA model and a continuum representation can be determined. The domain growth mechanism in the CA model gives rise to a Fokker-Planck equation in the corresponding continuum model, with a diffusive and a convective term. Deterministic continuum models derived from conservation laws do not include this diffusive term. The new diffusive term arises because of the stochasticity inherited from the CA mechanism for domain growth. We extend the models to multiple species and investigate the influence of the flux terms arising from the exclusion processes. The averaged CA agent densities are well approximated by the solution of nonlinear advection-diffusion equations, provided that the relative size of the proliferation processes to the diffusion processes is sufficiently small. This dual approach provides an understanding of the microscopic and macroscopic scales in a developmental process.     

Incomplete and transitory decrease of glycolysis

During vessel sprouting, a migratory endothelial tip cell guides the sprout, while proliferating stalk cells elongate the branch. Tip and stalk cell phenotypes are not genetically predetermined fates, but are dynamically interchangeable to ensure that the fittest endothelial cell (EC) leads the vessel sprout. ECs increase glycolysis when forming new blood vessels. Genetic deficiency of the glycolytic activator PFKFB3 in ECs reduces vascular sprouting by impairing migration of tip cells and proliferation of stalk cells. PFKFB3-driven glycolysis promotes the tip cell phenotype during vessel sprouting, since PFKFB3 overexpression overrules the pro-stalk activity of Notch signaling. Furthermore, PFKFB3-deficient ECs cannot compete with wild-type neighbors to form new blood vessels in chimeric mosaic mice. In addition, pharmacological PFKFB3 blockade reduces pathological angiogenesis with modest systemic effects, likely because it decreases glycolysis only partially and transiently.     

A multiscale computational model of arterial growth and remodeling including Notch signaling

Blood vessels grow and remodel in response to mechanical stimuli. Many computational models capture this process phenomenologically, by assuming stress homeostasis, but this approach cannot unravel the underlying cellular mechanisms. Mechano-sensitive Notch signaling is well-known to be key in vascular development and homeostasis. Here, we present a multiscale framework coupling a constrained mixture model, capturing the mechanics and turnover of arterial constituents, to a cell–cell signaling model, describing Notch signaling dynamics among vascular smooth muscle cells (SMCs) as influenced by mechanical stimuli. Tissue turnover was regulated by both Notch activity, informed by in vitro data, and a phenomenological contribution, accounting for mechanisms other than Notch. This novel framework predicted changes in wall thickness and arterial composition in response to hypertension similar to previous in vivo data. The simulations suggested that Notch contributes to arterial growth in hypertension mainly by promoting SMC proliferation, while other mechanisms are needed to fully capture remodeling. The results also indicated that interventions to Notch, such as external Jagged ligands, can alter both the geometry and composition of hypertensive vessels, especially in the short term. Overall, our model enables a deeper analysis of the role of Notch and Notch interventions in arterial growth and remodeling and could be adopted to investigate therapeutic strategies and optimize vascular regeneration protocols.