Mathematical modeling in wound healing, bone regeneration and tissue engineering

Digital Twins for Tissue Culture Techniques—Concepts, Expectations, and State of the Art

Techniques to provide in vitro tissue culture have undergone significant changes during the last decades, and current applications involve interactions of cells and organoids, three-dimensional cell co-cultures, and organ/body-on-chip tools. Efficient computer-aided and mathematical model-based methods are required for efficient and knowledge-driven characterization, optimization, and routine manufacturing of tissue culture systems. As an alternative to purely experimental-driven research, the usage of comprehensive mathematical models as a virtual in silico representation of the tissue culture, namely a digital twin, can be advantageous. Digital twins include the mechanistic of the biological system in the form of diverse mathematical models, which describe the interaction between tissue culture techniques and cell growth, metabolism, and the quality of the tissue. In this review, current concepts, expectations, and the state of the art of digital twins for tissue culture concepts will be highlighted. In general, DT’s can be applied along the full process chain and along the product life cycle. Due to the complexity, the focus of this review will be especially on the design, characterization, and operation of the tissue culture techniques.

The Wound Healing Index for Predicting Venous Leg Ulcer Outcome

Objective: To develop a venous leg ulcer (VLU) risk stratification system for use in research and clinical practice. Approach: U.S. Wound Registry data were examined retrospectively and assigned an outcome. Bivariate analysis identified significant variables (p < 0.05) that were used to create a multivariable logistic regression model. Ulcers with data for wound area at the first visit before debridement were included in regression analysis, which was based on a 90% development sample. The model was validated on a hold-out 10% data sample. Results: The original dataset included 26,713 VLUs, of which 11,773 ulcers were eligible for preliminary analysis and 10,942 ulcers were eligible for regression analysis. The 90% development model included 9,898 ulcers, of which 7,498 healed (75.8%). The 10% validation sample included 1,044 ulcers, of which 809 healed (77.5%). The following variables significantly predicted healing: number of concurrent wounds of any etiology, wound size, wound age (in days), evidence of bioburden/infection, being nonambulatory, and hospitalization for any reason. Innovation: The VLU Wound Healing Index (WHI) is a comprehensive, validated risk stratification model for predicting VLU healing that incorporates patient- and wound-specific variables. Conclusions: The WHI can identify which VLUs most likely require adjunctive therapies to heal, prioritize referral to venous experts, risk-stratify ulcers to create more generalizable clinical trials and understand the impact of clinical interventions. The Centers for Medicare and Medicaid Services accepts this method for reporting VLU outcome under the Quality Payment Program.

Influences of sodium and glycosaminoglycans on skin oedema and the potential for ulceration: a finite-element approach

Venous ulcers are chronic transcutaneous wounds common in the lower legs. They are resistant to healing and have a 78% chance of recurrence within 2 years. It is commonly accepted that venous ulcers are caused by the insufficiency of the calf muscle pump, leading to blood pooling in the lower legs, resulting in inflammation, skin oedema, tissue necrosis and eventually skin ulceration. However, the detailed physiological events by which inflammation contributes to wound formation are poorly understood. We therefore sought to develop a model that simulated the inflammation, using it to determine the internal stresses and pressure on the skin that contribute to venous ulcer formation. A three-layer finite-element skin model (epidermis, dermis and hypodermis) was developed to explore the roles in wound formation of two inflammation identifiers: glycosaminoglycans (GAG) and sodium. A series of parametric studies showed that increased GAG and sodium content led to oedema and increased tissue stresses of 1.5 MPa, which was within the reported range of skin tissue ultimate tensile stress (0.1-40 MPa). These results suggested that both the oedema and increased fluid pressure could reach a threshold for tissue damage and eventual ulcer formation. The models presented here provide insights to the pathological events associated with venous insufficiency, including inflammation, oedema and skin ulceration.

Effect of ultrasound on bone fracture healing: A computational mechanobioregulatory model

Bone healing process is a complicated phenomenon regulated by biochemical and mechanical signals. Experimental studies have shown that ultrasound (US) accelerates bone ossification and has a multiple influence on cell differentiation and angiogenesis. In a recent work of the authors, a bioregulatory model for providing bone-healing predictions was addressed, taking into account for the first time the salutary effect of US on the involved angiogenesis. In the present work, a mechanobioregulatory model of bone solidification under the US presence incorporating also the mechanical environment on the regeneration process, which is known to affect cellular processes, is presented. An iterative procedure is adopted, where the finite element method is employed to compute the mechanical stimuli at the linear elastic phases of the poroelastic callus region and a coupled system of partial differential equations to simulate the enhancement by the US cell angiogenesis process and thus the oxygen concentration in the fractured area. Numerical simulations with and without the presence of US that illustrate the influence of progenitor cells' origin in the healing pattern and the healing rate and simultaneously demonstrate the salutary effect of US on bone repair are presented and discussed.

Mathematical and computational models for bone tissue engineering in bioreactor systems

Research into cellular engineered bone grafts offers a promising solution to problems associated with the currently used auto- and allografts. Bioreactor systems can facilitate the development of functional cellular bone grafts by augmenting mass transport through media convection and shear flow-induced mechanical stimulation. Developing successful and reproducible protocols for growing bone tissue in vitro is dependent on tuning the bioreactor operating conditions to the specific cell type and graft design. This process, largely reliant on a trial-and-error approach, is challenging, time-consuming and expensive. Modelling can streamline the process by providing further insight into the effect of the bioreactor environment on the cell culture, and by identifying a beneficial range of operational settings to stimulate tissue production. Models can explore the impact of changing flow speeds, scaffold properties, and nutrient and growth factor concentrations. Aiming to act as an introductory reference for bone tissue engineers looking to direct their experimental work, this article presents a comprehensive framework of mathematical models on various aspects of bioreactor bone cultures and overviews modelling case studies from literature.

A Potential Tool for the Study of Venous Ulcers: Blood Flow Responses to Load

Venous ulcers are deep wounds that are located predominantly on the lower leg. They are prone to infection and once healed have a high probability of recurrence. Currently, there are no effective measures to predict and prevent venous ulcers from formation. Hence, the goal of this work was to develop a Windkessel-based model that can be used to identify hemodynamic parameters that change between healthy individuals and those with wounds. Once identified, these parameters have the potential to be used as indicators of when internal conditions change, putting the patient at higher risk for wound formation. In order to achieve this goal, blood flow responses in lower legs were measured experimentally by a laser Doppler perfusion monitor (LDPM) and simulated with a modeling approach. A circuit model was developed on the basis of the Windkessel theory. The hemodynamic parameters were extracted for three groups: legs with ulcers (“wounded”), legs without ulcers but from ulcer patients (“nonwounded”), and legs without vascular disease (“healthy”). The model was executed by two independent operators, and both operators reported significant differences between wounded and healthy legs in localized vascular resistance and compliance. The model successfully replicated the experimental blood flow profile. The global and local vascular resistances and compliance parameters rendered quantifiable differences between a population with venous ulcers and healthy individuals. This work supports that the Windkessel modeling approach has the potential to determine patient specific parameters that can be used to identify when conditions change making venous ulcer formation more likely.

Growth of adult spinal cord in knifefish: Development and parametrization of a distributed model

The study of indeterminate-growing organisms such as teleost fish presents a unique opportunity for improving our understanding of central nervous tissue growth during adulthood. Integrating the existing experimental data associated with this process into a theoretical framework through mathematical or computational modeling provides further research avenues through sensitivity analysis and optimization. While this type of approach has been used extensively in investigations of tumor growth, wound healing, and bone regeneration, the development of nervous tissue has been rarely studied within a modeling framework. To address this gap, the present work introduces a distributed model of spinal cord growth in the knifefish Apteronotus leptorhynchus, an established teleostean model of adult growth in the central nervous system. The proposed model incorporates two mechanisms, cell proliferation by active stem/progenitor cells and cell drift due to population pressure, both of which are subject to global constraints. A coupled reaction-diffusion equation approach was adopted to represent the densities of actively-proliferating and non-proliferating cells along the longitudinal axis of the spinal cord. Computer simulations using this model yielded biologically-feasible growth trajectories. Subsequent comparisons with whole-organism growth curves allowed the estimation of previously-unknown parameters, such as relative growth rates.

Identifying mechanisms driving formation of granuloma-associated fibrosis during Mycobacterium tuberculosis infection

Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB), is a pulmonary pathogen of major global concern. A key feature of Mtb infection in primates is the formation of granulomas, dense cellular structures surrounding infected lung tissue. These structures serve as the main site of host-pathogen interaction in TB, and thus to effectively treat TB we must clarify mechanisms of granuloma formation and their function in disease. Fibrotic granulomas are associated with both good and bad disease outcomes. Fibrosis can serve to isolate infected tissue from healthy tissue, but it can also cause difficulty breathing as it leaves scars. Little is known about fibrosis in TB, and data from non-human primates is just beginning to clarify the picture. This work focuses on constructing a hybrid multi-scale model of fibrotic granuloma formation, in order to identify mechanisms driving development of fibrosis in Mtb infected lungs. We combine dynamics of molecular, cellular, and tissue scale models from previously published studies to characterize the formation of two common sub-types of fibrotic granulomas: peripherally fibrotic, with a cuff of collagen surrounding granulomas, and centrally fibrotic, with collagen throughout granulomas. Uncertainty and sensitivity analysis, along with large simulation sets, enable us to identify mechanisms differentiating centrally versus peripherally fibrotic granulomas. These findings suggest that heterogeneous cytokine environments exist within granulomas and may be responsible for driving tissue scale morphologies. Using this model we are primed to better understand the complex structure of granulomas, a necessity for developing successful treatments for TB.

A Predictive Model for Diabetic Foot Ulcer Outcome: The Wound Healing Index

Objective: To develop a healing index for patients with diabetic foot ulcers (DFUs) for use in clinical practice, research analysis, and clinical trials.

Approach: U.S. Wound Registry data were examined retrospectively and assigned a clear outcome (healed, amputated, etc.). Significant variables were identified with bivariate analyses. A multivariable logistic regression model was created based on significant factors (p < 0.05) and tested on a hold-out sample of data. Out of 13,266 DFUs from the original dataset, 6,440 were eligible for analysis. The logistic regression model included 5,239 ulcers, of which 3,462 healed (66.1%). The 10% validation sample utilized 555 ulcers, of which 377 healed (67.9%).

Results: Variables that significantly predicted healing were as follows: wound age (duration in days), wound size, number of concurrent wounds of any etiology, evidence of bioburden/infection, patient age, Wagner grade, being nonambulatory, renal dialysis, renal transplant, peripheral vascular disease, and patient hospitalization for any reason.

Innovation: We present a validated stratification system, previously described as the Wound Healing Index (WHI), which predicts healing likelihood of patients with DFUs, incorporating patient- and wound-specific variables.

Conclusion: The DFU WHI is a comprehensive and user-friendly validated predictive model for DFU healing. It can risk stratify patients enrolled in clinical research trials, stratify patient data for quality reporting and benchmarking activities, and identify patients most likely to require costly therapy to heal.

A mathematical model for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding

A continuum hypothesis-based model is presented for the simulation of the formation and the subsequent regression of hypertrophic scar tissue after dermal wounding. Solely the dermal layer of the skin is modeled explicitly and it is modeled as a heterogeneous, isotropic and compressible neo-Hookean solid. With respect to the constituents of the dermal layer, the following components are selected as primary model components: fibroblasts, myofibroblasts, a generic signaling molecule and collagen molecules. A good match with respect to the evolution of the thickness of the dermal layer of scars between the outcomes of simulations and clinical measurements on hypertrophic scars at different time points after injury in human subjects is demonstrated. Interestingly, the comparison between the outcomes of the simulations and the clinical measurements demonstrates that a relatively high apoptosis rate of myofibroblasts results in scar tissue that behaves more like normal scar tissue with respect to the evolution of the thickness of the tissue over time, while a relatively low apoptosis rate results in scar tissue that behaves like hypertrophic scar tissue with respect to the evolution of the thickness of the tissue over time. Our ultimate goal is to construct models with which the properties of newly generated tissues that form during wound healing can be predicted with a high degree of certainty. The development of the presented model is considered by us as a step toward their construction.

Computational modeling of bone fracture non-unions: four clinically relevant case studies

The human skeleton has a remarkable regeneration capacity. Nevertheless, 5-10 % of the bone fractures fails to heal and develops into a non-union which is a challenging orthopedic complication requiring complex and expensive treatment. This review paper will discuss four different computational models, each capturing a particular clinical case of non-union: non-union induced by reaming of the marrow canal and periosteal stripping, non-union due to a large interfragmentary gap, non-union due to a genetic disorder [i.e. NF1 related congenital pseudoarthrosis of the tibia (CPT)] and non-union due to mechanical overload. Together, the four computational models are able to capture the etiology of a wide range of fracture non-union types and design novel treatment strategies thereof. Further research is required to corroborate the computational models in both animal and human settings and translate them from bench to bed side.

On the mathematical modeling of wound healing angiogenesis in skin as a reaction-transport process

Over the last 30 years, numerous research groups have attempted to provide mathematical descriptions of the skin wound healing process. The development of theoretical models of the interlinked processes that underlie the healing mechanism has yielded considerable insight into aspects of this critical phenomenon that remain difficult to investigate empirically. In particular, the mathematical modeling of angiogenesis, i.e., capillary sprout growth, has offered new paradigms for the understanding of this highly complex and crucial step in the healing pathway. With the recent advances in imaging and cell tracking, the time is now ripe for an appraisal of the utility and importance of mathematical modeling in wound healing angiogenesis research. The purpose of this review is to pedagogically elucidate the conceptual principles that have underpinned the development of mathematical descriptions of wound healing angiogenesis, specifically those that have utilized a continuum reaction-transport framework, and highlight the contribution that such models have made toward the advancement of research in this field. We aim to draw attention to the common assumptions made when developing models of this nature, thereby bringing into focus the advantages and limitations of this approach. A deeper integration of mathematical modeling techniques into the practice of wound healing angiogenesis research promises new perspectives for advancing our knowledge in this area. To this end we detail several open problems related to the understanding of wound healing angiogenesis, and outline how these issues could be addressed through closer cross-disciplinary collaboration.

A mathematical model for lymphangiogenesis in normal and diabetic wounds

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis) very few have been proposed for the regeneration of the lymphatic network. Moreover, lymphangiogenesis is markedly distinct from angiogenesis, occurring at different times and in a different manner. Here a model of five ordinary differential equations is presented to describe the formation of lymphatic capillaries following a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system is then solved numerically and the results are compared with the available biological literature. Finally, a parameter sensitivity analysis of the model is taken as a starting point for suggesting new therapeutic approaches targeting the enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper understanding of the phenomenon in question, clarifying the main factors involved. In particular, the balance between TGF-β and VEGF levels, rather than their absolute values, is identified as crucial to effective lymphangiogenesis. In addition, the results indicate lowering the macrophage-mediated activation of TGF-β and increasing the basal lymphatic endothelial cell growth rate, inter alia, as potential treatments. It is hoped the findings of this paper may be considered in the development of future experiments investigating novel lymphangiogenic therapies.

Cell based advanced therapeutic medicinal products for bone repair: Keep it simple?

The development of cell based advanced therapeutic medicinal products (ATMPs) for bone repair has been expected to revolutionize the health care system for the clinical treatment of bone defects. Despite this great promise, the clinical outcomes of the few cell based ATMPs that have been translated into clinical treatments have been far from impressive. In part, the clinical outcomes have been hampered because of the simplicity of the first wave of products. In response the field has set-out and amassed a plethora of complexities to alleviate the simplicity induced limitations. Many of these potential second wave products have remained "stuck" in the development pipeline. This is due to a number of reasons including the lack of a regulatory framework that has been evolving in the last years and the shortage of enabling technologies for industrial manufacturing to deal with these novel complexities. In this review, we reflect on the current ATMPs and give special attention to novel approaches that are able to provide complexity to ATMPs in a straightforward manner. Moreover, we discuss the potential tools able to produce or predict 'goldilocks' ATMPs, which are neither too simple nor too complex.

Regenerative orthopaedics: in vitro, in vivo...in silico

In silico, defined in analogy to in vitro and in vivo as those studies that are performed on a computer, is an essential step in problem-solving and product development in classical engineering fields. The use of in silico models is now slowly easing its way into medicine. In silico models are already used in orthopaedics for the planning of complicated surgeries, personalised implant design and the analysis of gait measurements. However, these in silico models often lack the simulation of the response of the biological system over time. In silico models focusing on the response of the biological systems are in full development. This review starts with an introduction into in silico models of orthopaedic processes. Special attention is paid to the classification of models according to their spatiotemporal scale (gene/protein to population) and the information they were built on (data vs hypotheses). Subsequently, the review focuses on the in silico models used in regenerative orthopaedics research. Contributions of in silico models to an enhanced understanding and optimisation of four key elements-cells, carriers, culture and clinics-are illustrated. Finally, a number of challenges are identified, related to the computational aspects but also to the integration of in silico tools into clinical practice.

The role of osteocytes in targeted bone remodeling: a mathematical model

Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. We have aimed to give a simple, yet still tractable, model that remains faithful to the underlying system based on the known literature. This model extends and complements many of the existing mathematical models for bone remodeling, but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to explore theoretically many of the qualitative features of the role of osteocytes in bone biology as presented in recent literature.

A phenomenological model for chemico-mechanically induced cell shape changes during migration and cell-cell contacts

A phenomenological model for the evolution of shape transition of cells is considered. These transitions are determined by the emission of growth-factors, as well as mechanical interaction if cells are subjected to hard impingement. The originality of this model necessitates a formal treatment of the mathematical model, as well as the presentation of elementary cases in order to illustrate the consistence of the model. We will also show some small-scale relevant applications.

Towards a new spatial representation of bone remodeling

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations.

Cervical collagen concentration within 15 months after delivery

OBJECTIVE

Cervical collagen concentration decreases during pregnancy. The increased risk of preterm birth after a short interpregnancy interval may be explained by an incomplete remodeling of the cervix. The objective of this study was to describe the changes in cervical collagen concentration over 15 months after delivery.

STUDY DESIGN

The collagen concentrations were determined in cervical biopsy specimens that were obtained from 15 women at 3, 6, 9, 12, and 15 months after delivery.

RESULTS

The mean cervical collagen concentrations were 50%, 59%, 63%, 65%, and 65% of dry weight (SD, 4.2-6.5). This increase was statistically significant until month 9, but not between months 9 and 12.

CONCLUSION

Low collagen concentrations in the uterine cervix may contribute to the association between a short interpregnancy interval and preterm birth.