A hybrid bioregulatory model of angiogenesis during bone fracture healing

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.     

A cellular automaton model for the migration of glioma cells

We present a study of in vitro cell migration in two dimensions as a first step towards understanding the mechanisms governing the motility of glioma cells. Our study is based on a cellular automaton model which aims at reproducing the kinetics of a lump of glioma cells deposited on a substrate of collagen. The dynamical effects of cell attraction and motion inertia are introduced through adequate automaton rules. We compare the density profiles given by the model to those obtained experimentally. The result of the best fit indicates a substantial cell-cell attraction due to cell-cell communication through gap junctions (or chemotaxis) and negligible inertia effects during migration. Tracking of individual migrating cells indicates highly convoluted cell trajectories.     

On the propagation of a disturbance through a viscous liquid flowing in a distensible tube of appreciable mass

The velocity of propagation of a disturbance wave in a liquid flowing in a distensible tube is computed. The mathematical model is more general than those used in previous analyses: the tube wall properties are realistic; the convective part of the axial inertia forces is taken into account; radial inertia forces of both the fluid and tube wall are present; viscous stresses are present. Four parameters influencing the velocity of propagation are obtained and discussed. Curves are plotted illustrating the effects of the parameters. Contrary to the results of previous analyses, viscous effects are shown to be appreciable in blood flow. It is also shown that radial inertia effects can be important in laboratory set-ups. The material presented in this paper was adapted from the Ph.D. thesis written by the author at Harvard University.     

Midkine-Deficiency Delays Chondrogenesis during the Early Phase of Fracture Healing in Mice

The growth and differentiation factor midkine (Mdk) plays an important role in bone development and remodeling. Mdk-deficient mice display a high bone mass phenotype when aged 12 and 18 months. Furthermore, Mdk has been identified as a negative regulator of mechanically induced bone formation and it induces pro-chondrogenic, pro-angiogenic and pro-inflammatory effects. Together with the finding that Mdk is expressed in chondrocytes during fracture healing, we hypothesized that Mdk could play a complex role in endochondral ossification during the bone healing process. Femoral osteotomies stabilized using an external fixator were created in wildtype and Mdk-deficient mice. Fracture healing was evaluated 4, 10, 21 and 28 days after surgery using 3-point-bending, micro-computed tomography, histology and immunohistology. We demonstrated that Mdk-deficient mice displayed delayed chondrogenesis during the early phase of fracture healing as well as significantly decreased flexural rigidity and moment of inertia of the fracture callus 21 days after fracture. Mdk-deficiency diminished beta-catenin expression in chondrocytes and delayed presence of macrophages during early fracture healing. We also investigated the impact of Mdk knockdown using siRNA on ATDC5 chondroprogenitor cells in vitro. Knockdown of Mdk expression resulted in a decrease of beta-catenin and chondrogenic differentiation-related matrix proteins, suggesting that delayed chondrogenesis during fracture healing in Mdk-deficient mice may be due to a cell-autonomous mechanism involving reduced beta-catenin signaling. Our results demonstrated that Mdk plays a crucial role in the early inflammation phase and during the development of cartilaginous callus in the fracture healing process.     

Murine Model of Wound Healing

Wound healing and repair are the most complex biological processes that occur in human life. After injury, multiple biological pathways become activated. Impaired wound healing, which occurs in diabetic patients for example, can lead to severe unfavorable outcomes such as amputation. There is, therefore, an increasing impetus to develop novel agents that promote wound repair. The testing of these has been limited to large animal models such as swine, which are often impractical. Mice represent the ideal preclinical model, as they are economical and amenable to genetic manipulation, which allows for mechanistic investigation. However, wound healing in a mouse is fundamentally different to that of humans as it primarily occurs via contraction. Our murine model overcomes this by incorporating a splint around the wound. By splinting the wound, the repair process is then dependent on epithelialization, cellular proliferation and angiogenesis, which closely mirror the biological processes of human wound healing. Whilst requiring consistency and care, this murine model does not involve complicated surgical techniques and allows for the robust testing of promising agents that may, for example, promote angiogenesis or inhibit inflammation. Furthermore, each mouse acts as its own control as two wounds are prepared, enabling the application of both the test compound and the vehicle control on the same animal. In conclusion, we demonstrate a practical, easy-to-learn, and robust model of wound healing, which is comparable to that of humans.     

Component inertia modeling of segmental wobbling and rigid masses

A modification to an existing mathematical model is described, which permits the determination of subject-specific inertia parameters for wobbling and rigid masses of female body segments. The model comprises segment-specific soft tissue, bone, and lung components. A total of 59 geometric solids (40 soft tissue, 17 bone, 2 lung) were used to represent the body components. Ninety-five anthropometric measurements were collected from 7 female participants and were used to develop and evaluate the model. The success of the model is evaluated using predicted mass and mass distribution. The overall absolute accuracy in predicted whole body mass was better than 3.0%, with a maximum error of 4.9%. The appropriateness of the cadaver-based density values used in the model is addressed and the accuracy of the component inertia model in relation to uniform density models is discussed. The model offers a novel approach for determining component inertia parameters, which have been used successfully in a wobbling mass model to produce realistic kinetic analyses of drop-landings.     

Does secotioid inertia drive the evolution of false-truffles?

Secotioid inertia is a model implemented to explain the prevalence of highly derived false-truffles with no obvious connection to the Homobasidiomycetes. The model accommodates the apparent lack of epigeous sister taxa for some highly derived hypogeous lineages by assuming that gasteromycetation in some fungi leads to the extinction of their epigeous sister population. The derived state of some hypogeous lineages suggests that they arose early in the evolution of Homobasidiomycetes and that those groups were subject to conditions that favoured hypogeous lineages such that the hypogeous fruit body form became the predominant form for some lineages. The directional selection component of secotioid inertia, termed secotioid drive, led to the extinction of their epigeous sister taxon. Morphological and molecular data from Russulaceae are used to model the evolutionary stages of secotioid inertia. The resulting phylogenetic results are compared with data from the order Leucogastrales, and the genus Destuntzia. The implications of secotioid drive are discussed with reference to gasteromycete phylogenetics, evolution, and conservation. Specifically, secotioid inertia can be used to account for reversals in fruit body morphology and instability in mycorrhizal formation.     

Continuum model of fibroblast-driven wound contraction: inflammation-mediation.

We propose a mathematical model to aid the understanding of how events in wound healing are orchestrated to result in wound contraction. Ultimately, a validated model could provide a predictive means for enhancing or mitigating contraction as is appropriate for managing a particular wound. The complex nature of wound healing and the lack of a modeling framework which can account for both the relevant cell biology and biomechanics are major reasons for the absence of models to date. Here we adapt a model originally proposed by Murray and co-workers to show how cell traction forces can result in spatial patterns of cell aggregates since it offers a framework for understanding how traction exerted by wound fibroblasts drives wound contraction. Since it is a continuum model based on conservation laws which reflect assumed cell and tissue properties, it is readily extended to account for emerging understanding of the cell biology of wound healing and its relationship to inflammation. We consider various sets of assumed properties, based on current knowledge, within a base model of dermal wound healing and compare predictions of the rate and extent of wound contraction to published experimental results.     

Mass, center of mass, and moment of inertia estimates for infant limb segments.

To quantify limb dynamics, accurate estimates are needed of anthropometric inertia parameters (mass, center-of-mass location, and moments of inertia). These estimates, however, are not available for human infants; therefore, the movement dynamics of infants have not been studied extensively. Here, regression equations for the masses, center-of-mass locations, and transverse moments of inertia of upper and lower limb segments (upper arm, forearm, and hand; thigh, leg, and foot) of 0.04 to 1.50 yr old infants are provided. A mathematical model of the human body was used to determine the anthropometric inertia parameters for upper limbs in 44 infants and for lower limbs in 70 infants. Stepwise linear regressions were used to fit the distributions of the anthropometric inertia parameters. The regression equations accounted for significant amounts of the variance (64-98%), and the R2-values compared favorably when our equations were cross-validated. Consequently, these regression equations can provide, for infants of similar ages, reasonable estimates of upper and lower limb anthropometric inertia parameters, suitable for equations of motion in the analysis of limb dynamics in human infants.     

A rat model of diabetic wound infection for the evaluation of topical antimicrobial therapies

Diabetes mellitus is an epidemic multisystemic chronic disease that frequently is complicated by complex wound infections. Innovative topical antimicrobial therapy agents are potentially useful for multimodal treatment of these infections. However, an appropriately standardized in vivo model is currently not available to facilitate the screening of these emerging products and their effect on wound healing. To develop such a model, we analyzed, tested, and modified published models of wound healing. We optimized various aspects of the model, including animal species, diabetes induction method, hair removal technique, splint and dressing methods, the control of unintentional bacterial infection, sampling methods for the evaluation of bacterial burden, and aspects of the microscopic and macroscopic assessment of wound healing, all while taking into consideration animal welfare and the '3Rs' principle. We thus developed a new wound infection model in rats that is optimized for testing topical antimicrobial therapy agents. This model accurately reproduces the pathophysiology of infected diabetic wound healing and includes the current standard treatment (that is, debridement). The numerous benefits of this model include the ready availability of necessary materials, simple techniques, high reproducibility, and practicality for experiments with large sample sizes. Furthermore, given its similarities to infected-wound healing and treatment in humans, our new model can serve as a valid alternative for applied research.     

Inertia and muscle contraction parameters for musculoskeletal modelling of the shoulder mechanism

To develop a musculoskeletal model of the shoulder mechanism, both shoulders of seven cadavers were measured to obtain a complete set of parameters. Using antropometric measurements, the mass and rotational inertia of segments were estimated, followed by three-dimensional measurements of all morphological structures relevant for modelling, i.e. muscle origins and insertions, muscle bundle directions, ligament attachments and articular surfaces; all in relation to selected bony landmarks. Subsequently, muscle contraction parameters as muscle mass and physiological cross-sectional area were measured. The method of data collection and the results for inertia and muscle contraction parameters as prerequisities for modelling are described.     

Intestinal trefoil factor/TFF3 promotes re-epithelialization of corneal wounds

Disorders of wound healing characterized by impaired or delayed re-epithelialization are a serious medical problem. These conditions affect many tissues, are painful, and are difficult to treat. In this study using cornea as a model, we demonstrate the importance of trefoil factor 3 (TFF3, also known as intestinal trefoil factor) in re-epithelialization of wounds. In two different models of corneal wound healing, alkali- and laser-induced corneal wounding, we analyzed the wound healing process in in vivo as well as in combined in vivo/in vitro model in wild type (Tff3(+)(/)(+)) and Tff3-deficient (Tff3(-)(/)(-)) mice. Furthermore, we topically applied different concentrations of recombinant human TFF3 (rTFF3) peptide on the wounded cornea to determine the efficacy of rTFF3 on corneal wound healing. We found that Tff3 peptide is not expressed in intact corneal epithelium, but its expression is extensively up-regulated after epithelial injury. Re-epithelialization of corneal wounds in Tff3(-/-) mice is significantly prolonged in comparison to Tff3(+/+) mice. In addition, exogenous application of rTFF3 to the alkali-induced corneal wounds accelerates significantly in in vivo and in combined in vivo/in vitro model wound healing in Tff3(+/+) and Tff3(-/-) mice. These findings reveal a pivotal role for Tff3 in corneal wound healing mechanism and have broad implications for developing novel therapeutic strategies for treating nonhealing wounds.     

Therapeutic Processes and Perceived Helpfulness of <Emphasis Type="Italic">Dang-Ki</Emphasis> (Chinese Shamanism) from the Symbolic Healing Perspective

This study focuses on the therapeutic process and perceived helpfulness of dang-ki, a form of Chinese shamanistic healing, in Singapore. It aims to understand the healing symbols employed in dang-ki, whether or not patients find them helpful and whether their perceived helpfulness can be explained by the symbolic healing model (Dow, Am Anthropol 88(1):56–69, 1986; Levi-Strauss, Structural anthropology. Basic Books, New York, 1963). Although many researchers have applied this model to explain the efficacy of shamanistic healings, they did not directly provide empirical support. Furthermore, the therapeutic process of a shared clinical reality as proposed by the model may be achievable in small-scale traditional societies that are culturally more homogeneous than in contemporary societies that are culturally more diversified due to globalization and immigration. Patients may hold multidimensional health belief systems, as biomedicine and alternative healing systems coexist. Thus, it would be interesting to see the relevance and applicability of the symbolic healing model to shamanistic healing in contemporary societies. In this study, ethnographic interviews were conducted with 21 patients over three stages: immediately before and after the healing and approximately 1 month later. The dang-ki healing symbols were identified by observing the healing sessions with video recording. Results show that dang-kis normally applied more than one method to treat a given problem. These methods included words, talismans and physical manipulations. Overall, 11 patients perceived their consultations as helpful, 4 perceived their consultations as helpful but were unable to follow all recommendations, 5 were not sure of the outcome because they had yet to see any concrete results and only 1 patient considered his consultation unhelpful. Although the symbolic healing model provides a useful framework to understand perceived helpfulness, processes such as enactment of a common meaning system and symbolic transformation are complex and dynamic, and may be carried over several healing sessions.     

Inter-species investigation of the mechano-regulation of bone healing: comparison of secondary bone healing in sheep and rat

Inter-species differences in regeneration exist in various levels. One aspect is the dynamics of bone regeneration and healing, e.g. small animals show a faster healing response when compared to large animals. Mechanical as well as biological factors are known to play a key role in the process. However, it remains so far unknown whether different animals follow at all comparable mechano-biological rules during tissue regeneration, and in particular during bone healing. In this study, we investigated whether differences observed in vivo in the dynamics of bone healing between rat and sheep are only due to differences in the animal size or whether these animals have a different mechano-biological response during the healing process. Histological sections from in vivo experiments were compared to in silico predictions of a mechano-biological computer model for the simulation of bone healing. Investigations showed that the healing processes in both animal models occur under significantly different levels of mechanical stimuli within the callus region, which could explain histological observations of early intramembranous ossification at the endosteal side. A species-specific adaptation of a mechano-biological model allowed a qualitative match of model predictions with histological observations. Specifically, when keeping cell activity processes at the same rate, the amount of tissue straining defining favorable mechanical conditions for the formation of bone had to be increased in the large animal model, with respect to the small animal, to achieve a qualitative agreement of model predictions with histological data. These findings illustrate that geometrical (size) differences alone cannot explain the distinctions seen in the histological appearance of secondary bone healing in sheep and rat. It can be stated that significant differences in the mechano-biological regulation of the healing process exist between these species. Future investigations should aim towards understanding whether these differences are due to differences in cell behavior, material properties of the newly formed tissues within the callus and/or differences in response to the mechanical environment.     

A point cluster method for in vivo motion analysis: applied to a study of knee kinematics.

A new method for deriving limb segment motion from markers placed on the skin is described. The method provides a basis for determining the artifact associated with nonrigid body movement of points placed on the skin. The method is based on a cluster of points uniformly distributed on the limb segment. Each point is assigned an arbitrary mass. The center of mass and the inertia tensor of this cluster of points are calculated. The eigenvalues and eigenvectors of the inertia tensor are used to define a coordinate system in the cluster as well as to provide a basis for evaluating non-rigid body movement. The eigenvalues of the inertia tensor remain invariant if the segment is behaving as a rigid body, thereby providing a basis for determining variations for nonrigid body movement. The method was tested in a simulation model where systematic and random errors were introduced into a fixed cluster of points. The simulation demonstrated that the error due to nonrigid body movement could be substantially reduced. The method was also evaluated in a group of ten normal subjects during walking. The results for knee rotation and translation obtained from the point cluster method compared favorably to results previously obtained from normal subjects with intra-cortical pins placed into the femur and tibia. The resulting methodology described in this paper provides a unique approach to the measurement of in vivo motion using skin-based marker systems.     

The simulation of aerial movement--II. A mathematical inertia model of the human body

A mathematical inertia model which permits the determination of personalized segmental inertia parameter values from anthropometric measurements is described. The human body is modelled using 40 geometric solids which are specified by 95 anthropometric measurements. A 'stadium' solid is introduced for modelling the torso segments using perimeter and width measurements. This procedure is more accurate than the use of elliptical discs of given width and depth and permits a smaller number of such solids to be used. Inertia parameter values may be obtained for body models of up to 20 segments. Errors in total body mass estimates from this and other models are discussed with reference to the unknown lung volumes.     

Identification of tissue differentiation rates in a mechanobiological model of fracture healing

As a basis for model-based analysis of the processes in secondary fracture healing, a dynamical model is presented that characterises the physiological status in the fracture area by the location-dependent composition of tissues. Five types of tissue are distinguished: connective tissue, cartilage, bone, haematoma and avascular bone. A rule base is given that describes dynamical tissue differentiation processes. The rules consider not only a mechanical stimulus but also osteogenic and a vasculative factors as biological stimuli. Within this model structure, it is possible, e.g., to distinguish intramembranous from endochondral ossification processes. An objective function is introduced to assess accordance between the model-based simulation results and reference healing stages. By minimising this objective function, relevant tissue differentiation rates can be determined. For a reference process of secondary fracture healing it could be shown that the intramembranous ossification rate of 0.313%/day (from connective tissue to bone) is much smaller than the endochondral ossification rate of 1.136%/day (from cartilage to bone). In order to verify the model approach, it is transferred to simulate long bone distraction. Results show that healing patterns of bone distraction can be predicted. Using this method, it is possible to identify model parameters for individual subjects. This will allow a patient-specific analysis of tissue healing processes in future.     

Evolutionary stasis, constraint and other terminology describing evolutionary patterns

Current use of terms to describe evolutionary patterns is vague and inconsistent. In this paper, logical definitions of terms that describe specific evolutionary patterns are proposed. Evolutionary inertia is defined in a manner analogous to inertia in physics. A character in a static state of evolutionary inertia represents evolutionary stasis while a character showing consistent directional evolutionary change represents evolutionary thrust. I argue that evolutionary stasis should serve as the null hypothesis in all character evolution studies. Deviations from this null model consistent with alternative hypotheses (e.g. random drift, adaptation) can then give us insight into evolutionary processes. Failure to reject a null hypothesis of evolutionary stasis should not be used as a serious explanation of data. The term evolutionary constraint is appropriate only when a selective advantage for a character state transition is established but this transition is prevented by specific, identified factors. One type of evolutionary constraint discussed is evolutionary momentum. A final pattern of evolutionary change discussed is closely related to evolutionary thrust and is referred to as evolutionary acceleration. I provide examples of how this set of definitions can improve our ability to communicate interpretations of evolutionary patterns.