A three-dimensional, discrete-continuum model of blood pressure in microvascular networks.

We present a 3D discrete-continuum model to simulate blood pressure in large microvascular tissues in the absence of known capillary network architecture. Our hybrid approach combines a 1D Poiseuille flow description for large, discrete arteriolar and venular networks coupled to a continuum-based Darcy model, point sources of flux, for transport in the capillary bed. We evaluate our hybrid approach using a vascular network imaged from the mouse brain medulla/pons using multi-fluorescence high-resolution episcopic microscopy (MF-HREM). We use the fully-resolved vascular network to predict the hydraulic conductivity of the capillary network and generate a fully-discrete pressure solution to benchmark against. Our results demonstrate that the discrete-continuum methodology is a computationally feasible and effective tool for predicting blood pressure in real-world microvascular tissues when capillary microvessels are poorly defined.

A hybrid neural ordinary differential equation model of the cardiovascular system.

In the human cardiovascular system (CVS), the interaction between the left and right ventricles of the heart is influenced by the septum and the pericardium. Computational models of the CVS can capture this interaction, but this often involves approximating solutions to complex nonlinear equations numerically. As a result, numerous models have been proposed, where these nonlinear equations are either simplified, or ventricular interaction is ignored. In this work, we propose an alternative approach to modelling ventricular interaction, using a hybrid neural ordinary differential equation (ODE) structure. First, a lumped parameter ODE model of the CVS (including a Newton-Raphson procedure as the numerical solver) is simulated to generate synthetic time-series data. Next, a hybrid neural ODE based on the same model is constructed, where ventricular interaction is instead set to be governed by a neural network. We use a short range of the synthetic data (with various amounts of added measurement noise) to train the hybrid neural ODE model. Symbolic regression is used to convert the neural network into analytic expressions, resulting in a partially learned mechanistic model. This approach was able to recover parsimonious functions with good predictive capabilities and was robust to measurement noise.

Formation of vascular-like structures using a chemotaxis-driven multiphase model.

We propose a continuum model for pattern formation, based on the multiphase model framework, to explore in vitro cell patterning within an extracellular matrix (ECM). We demonstrate that, within this framework, chemotaxis-driven cell migration can lead to the formation of cell clusters and vascular-like structures in 1D and 2D respectively. The influence on pattern formation of additional mechanisms commonly included in multiphase tissue models, including cell-matrix traction, contact inhibition, and cell-cell aggregation, are also investigated. Using sensitivity analysis, the relative impact of each model parameter on the simulation outcomes is assessed to identify the key parameters involved. Chemoattractant-matrix binding is further included, motivated by previous experimental studies, and found to reduce the spatial scale of patterning to within a biologically plausible range for capillary structures. Key findings from the in-depth parameter analysis of the 1D models, both with and without chemoattractant-matrix binding, are demonstrated to translate well to the 2D model, obtaining vascular-like cell patterning for multiple parameter regimes. Overall, we demonstrate a biologically-motivated multiphase model capable of generating long-term pattern formation on a biologically plausible spatial scale both in 1D and 2D, with applications for modelling in vitro vascular network formation.

Reconstructing microvascular network skeletons from 3D images: What is the ground truth?

Structural changes to microvascular networks are increasingly highlighted as markers of pathogenesis in a wide range of disease, e.g. Alzheimer's disease, vascular dementia and tumour growth. This has motivated the development of dedicated 3D imaging techniques, alongside the creation of computational modelling frameworks capable of using 3D reconstructed networks to simulate functional behaviours such as blood flow or transport processes. Extraction of 3D networks from imaging data broadly consists of two image processing steps: segmentation followed by skeletonisation. Much research effort has been devoted to segmentation field, and there are standard and widely-applied methodologies for creating and assessing gold standards or ground truths produced by manual annotation or automated algorithms. The Skeletonisation field, however, lacks widely applied, simple to compute metrics for the validation or optimisation of the numerous algorithms that exist to extract skeletons from binary images. This is particularly problematic as 3D imaging datasets increase in size and visual inspection becomes an insufficient validation approach. In this work, we first demonstrate the extent of the problem by applying 4 widely-used skeletonisation algorithms to 3 different imaging datasets. In doing so we show significant variability between reconstructed skeletons of the same segmented imaging dataset. Moreover, we show that such a structural variability propagates to simulated metrics such as blood flow. To mitigate this variability we introduce a new, fast and easy to compute super metric that compares the volume, connectivity, medialness, bifurcation point identification and homology of the reconstructed skeletons to the original segmented data. We then show that such a metric can be used to select the best performing skeletonisation algorithm for a given dataset, as well as to optimise its parameters. Finally, we demonstrate that the super metric can also be used to quickly identify how a particular skeletonisation algorithm could be improved, becoming a powerful tool in understanding the complex implication of small structural changes in a network.

A variable heart rate multi-compartmental coupled model of the cardiovascular and respiratory systems.

Current mathematical models of the cardiovascular system that are based on systems of ordinary differential equations are limited in their ability to mimic important features of measured patient data, such as variable heart rates (HR). Such limitations present a significant obstacle in the use of such models for clinical decision-making, as it is the variations in vital signs such as HR and systolic and diastolic blood pressure that are monitored and recorded in typical critical care bedside monitoring systems. In this paper, novel extensions to well-established multi-compartmental models of the cardiovascular and respiratory systems are proposed that permit the simulation of variable HR. Furthermore, a correction to current models is also proposed to stabilize the respiratory behaviour and enable realistic simulation of vital signs over the longer time scales required for clinical management. The results of the extended model developed here show better agreement with measured bio-signals, and these extensions provide an important first step towards estimating model parameters from patient data, using methods such as neural ordinary differential equations. The approach presented is generalizable to many other similar multi-compartmental models of the cardiovascular and respiratory systems.

Mathematical modelling with Bayesian inference to quantitatively characterize therapeutic cell behaviour in nerve tissue engineering.

Cellular engineered neural tissues have significant potential to improve peripheral nerve repair strategies. Traditional approaches depend on quantifying tissue behaviours using experiments in isolation, presenting a challenge for an overarching framework for tissue design. By comparison, mathematical cell-solute models benchmarked against experimental data enable computational experiments to be performed to test the role of biological/biophysical mechanisms, as well as to explore the impact of different design scenarios and thus accelerate the development of new treatment strategies. Such models generally consist of a set of continuous, coupled, partial differential equations relying on a number of parameters and functional forms. They necessitate dedicated in vitro experiments to be informed, which are seldom available and often involve small datasets with limited spatio-temporal resolution, generating uncertainties. We address this issue and propose a pipeline based on Bayesian inference enabling the derivation of experimentally informed cell-solute models describing therapeutic cell behaviour in nerve tissue engineering. We apply our pipeline to three relevant cell types and obtain models that can readily be used to simulate nerve repair scenarios and quantitatively compare therapeutic cells. Beyond parameter estimation, the proposed pipeline enables model selection as well as experiment utility quantification, aimed at improving both model formulation and experimental design.

Local Administration of Minocycline Improves Nerve Regeneration in Two Rat Nerve Injury Models.

Peripheral nerve injuries are quite common and often require a surgical intervention. However, even after surgery, patients do not often regain satisfactory sensory and motor functions. This, in turn, results in a heavy socioeconomic burden. To some extent, neurons can regenerate from the proximal nerve stump and try to reconnect to the distal stump. However, this regenerating capacity is limited, and depending on the type and size of peripheral nerve injury, this process may not lead to a positive outcome. To date, no pharmacological approach has been used to improve nerve regeneration following repair surgery. We elected to investigate the effects of local delivery of minocycline on nerve regeneration. This molecule has been studied in the central nervous system and was shown to improve the outcome in many disease models. In this study, we first tested the effects of minocycline on SCL 4.1/F7 Schwann cells in vitro and on sciatic nerve explants. We specifically focused on the Schwann cell repair phenotype, as these cells play a central role in orchestrating nerve regeneration. Finally, we delivered minocycline locally in two different rat models of nerve injury, a sciatic nerve transection and a sciatic nerve autograft, demonstrating the capacity of local minocycline treatment to improve nerve regeneration.

Micro to macro scale analysis of the intact human renal arterial tree with Synchrotron Tomography.

Background: The kidney vasculature is exquisitely structured to orchestrate renal function. Structural profiling of the vasculature in intact rodent kidneys, has provided insights into renal haemodynamics and oxygenation, but has never been extended to the human kidney beyond a few vascular generations. We hypothesised that synchrotron-based imaging of a human kidney would enable assessment of vasculature across the whole organ. Methods: An intact kidney from a 63-year-old male was scanned using hierarchical phase-contrast tomography (HiP-CT), followed by semi-automated vessel segmentation and quantitative analysis. These data were compared to published micro-CT data of whole rat kidney. Results: The intact human kidney vascular network was imaged with HiP-CT at 25 µm voxels, representing a 20-fold increase in resolution compared to clinical CT scanners. Our comparative quantitative analysis revealed the number of vessel generations, vascular asymmetry and a structural organisation optimised for minimal resistance to flow, are conserved between species, whereas the normalised radii are not. We further demonstrate regional heterogeneity in vessel geometry between renal cortex, medulla, and hilum, showing how the distance between vessels provides a structural basis for renal oxygenation and hypoxia. Conclusions: Through the application of HiP-CT, we have provided the first quantification of the human renal arterial network, with a resolution comparable to that of light microscopy yet at a scale several orders of magnitude larger than that of a renal punch biopsy. Our findings bridge anatomical scales, profiling blood vessels across the intact human kidney, with implications for renal physiology, biophysical modelling, and tissue engineering.

Perspectives on optimizing local delivery of drugs to peripheral nerves using mathematical models.

Drug therapies for treating peripheral nerve injury repair have shown significant promise in preclinical studies. Despite this, drug treatments are not used routinely clinically to treat patients with peripheral nerve injuries. Drugs delivered systemically are often associated with adverse effects to other tissues and organs; it remains challenging to predict the effective concentration needed at an injured nerve and the appropriate delivery strategy. Local drug delivery approaches are being developed to mitigate this, for example via injections or biomaterial-mediated release. We propose the integration of mathematical modeling into the development of local drug delivery protocols for peripheral nerve injury repair. Mathematical models have the potential to inform understanding of the different transport mechanisms at play, as well as quantitative predictions around the efficacy of individual local delivery protocols. We discuss existing approaches in the literature, including drawing from other research fields, and present a process for taking forward an integrated mathematical-experimental approach to accelerate local drug delivery approaches for peripheral nerve injury repair. This article is categorized under: Neurological Diseases > Molecular and Cellular Physiology Neurological Diseases > Computational Models Neurological Diseases > Biomedical Engineering.

A combined experimental and computational framework to evaluate the behavior of therapeutic cells for peripheral nerve regeneration.

Recent studies have explored the potential of tissue-mimetic scaffolds in encouraging nerve regeneration. One of the major determinants of the regenerative success of cellular nerve repair constructs (NRCs) is the local microenvironment, particularly native low oxygen conditions which can affect implanted cell survival and functional performance. In vivo, cells reside in a range of environmental conditions due to the spatial gradients of nutrient concentrations that are established. Here we evaluate in vitro the differences in cellular behavior that such conditions induce, including key biological features such as oxygen metabolism, glucose consumption, cell death, and vascular endothelial growth factor secretion. Experimental measurements are used to devise and parameterize a mathematical model that describes the behavior of the cells. The proposed model effectively describes the interactions between cells and their microenvironment and could in the future be extended, allowing researchers to compare the behavior of different therapeutic cells. Such a combinatorial approach could be used to accelerate the clinical translation of NRCs by identifying which critical design features should be optimized when fabricating engineered nerve repair conduits.

A mathematical investigation into the uptake kinetics of nanoparticles in vitro.

Nanoparticles have the potential to increase the efficacy of anticancer drugs whilst reducing off-target side effects. However, there remain uncertainties regarding the cellular uptake kinetics of nanoparticles which could have implications for nanoparticle design and delivery. Polymersomes are nanoparticle candidates for cancer therapy which encapsulate chemotherapy drugs. Here we develop a mathematical model to simulate the uptake of polymersomes via endocytosis, a process by which polymersomes bind to the cell surface before becoming internalised by the cell where they then break down, releasing their contents which could include chemotherapy drugs. We focus on two in vitro configurations relevant to the testing and development of cancer therapies: a well-mixed culture model and a tumour spheroid setup. Our mathematical model of the well-mixed culture model comprises a set of coupled ordinary differential equations for the unbound and bound polymersomes and associated binding dynamics. Using a singular perturbation analysis we identify an optimal number of ligands on the polymersome surface which maximises internalised polymersomes and thus intracellular chemotherapy drug concentration. In our mathematical model of the spheroid, a multiphase system of partial differential equations is developed to describe the spatial and temporal distribution of bound and unbound polymersomes via advection and diffusion, alongside oxygen, tumour growth, cell proliferation and viability. Consistent with experimental observations, the model predicts the evolution of oxygen gradients leading to a necrotic core. We investigate the impact of two different internalisation functions on spheroid growth, a constant and a bond dependent function. It was found that the constant function yields faster uptake and therefore chemotherapy delivery. We also show how various parameters, such as spheroid permeability, lead to travelling wave or steady-state solutions.

Modelling-informed cell-seeded nerve repair construct designs for treating peripheral nerve injuries.

Millions of people worldwide are affected by peripheral nerve injuries (PNI), involving billions of dollars in healthcare costs. Common outcomes for patients include paralysis and loss of sensation, often leading to lifelong pain and disability. Engineered Neural Tissue (EngNT) is being developed as an alternative to the current treatments for large-gap PNIs that show underwhelming functional recovery in many cases. EngNT repair constructs are composed of a stabilised hydrogel cylinder, surrounded by a sheath of material, to mimic the properties of nerve tissue. The technology also enables the spatial seeding of therapeutic cells in the hydrogel to promote nerve regeneration. The identification of mechanisms leading to maximal nerve regeneration and to functional recovery is a central challenge in the design of EngNT repair constructs. Using in vivo experiments in isolation is costly and time-consuming, offering a limited insight on the mechanisms underlying the performance of a given repair construct. To bridge this gap, we derive a cell-solute model and apply it to the case of EngNT repair constructs seeded with therapeutic cells which produce vascular endothelial growth factor (VEGF) under low oxygen conditions to promote vascularisation in the construct. The model comprises a set of coupled non-linear diffusion-reaction equations describing the evolving cell population along with its interactions with oxygen and VEGF fields during the first 24h after transplant into the nerve injury site. This model allows us to evaluate a wide range of repair construct designs (e.g. cell-seeding strategy, sheath material, culture conditions), the idea being that designs performing well over a short timescale could be shortlisted for in vivo trials. In particular, our results suggest that seeding cells beyond a certain density threshold is detrimental regardless of the situation considered, opening new avenues for future nerve tissue engineering.

Selectively Cross-Linked Tetra-PEG Hydrogels Provide Control over Mechanical Strength with Minimal Impact on Diffusivity.

Synthetic hydrogels formed from poly(ethylene glycol) (PEG) are widely used to study how cells interact with their extracellular matrix. These in vivo-like 3D environments provide a basis for tissue engineering and cell therapies but also for research into fundamental biological questions and disease modeling. The physical properties of PEG hydrogels can be modulated to provide mechanical cues to encapsulated cells; however, the impact of changing hydrogel stiffness on the diffusivity of solutes to and from encapsulated cells has received only limited attention. This is particularly true in selectively cross-linked "tetra-PEG" hydrogels, whose design limits network inhomogeneities. Here, we used a combination of theoretical calculations, predictive modeling, and experimental measurements of hydrogel swelling, rheological behavior, and diffusion kinetics to characterize tetra-PEG hydrogels' permissiveness to the diffusion of molecules of biologically relevant size as we changed polymer concentration, and thus hydrogel mechanical strength. Our models predict that hydrogel mesh size has little effect on the diffusivity of model molecules and instead predicts that diffusion rates are more highly dependent on solute size. Indeed, our model predicts that changes in hydrogel mesh size only begin to have a non-negligible impact on the concentration of a solute that diffuses out of hydrogels for the smallest mesh sizes and largest diffusing solutes. Experimental measurements characterizing the diffusion of fluorescein isothiocyanate (FITC)-labeled dextran molecules of known size aligned well with modeling predictions and suggest that doubling the polymer concentration from 2.5% (w/v) to 5% produces stiffer gels with faster gelling kinetics without affecting the diffusivity of solutes of biologically relevant size but that 10% hydrogels can slow their diffusion. Our findings provide confidence that the stiffness of tetra-PEG hydrogels can be modulated over a physiological range without significantly impacting the transport rates of solutes to and from encapsulated cells.

Design considerations for engineering 3D models to study vascular pathologies in vitro.

Many cardiovascular diseases (CVD) are driven by pathological remodelling of blood vessels, which can lead to aneurysms, myocardial infarction, ischaemia and strokes. Aberrant remodelling is driven by changes in vascular cell behaviours combined with degradation, modification, or abnormal deposition of extracellular matrix (ECM) proteins. The underlying mechanisms that drive the pathological remodelling of blood vessels are multifaceted and disease specific; however, unravelling them may be key to developing therapies. Reductionist models of blood vessels created in vitro that combine cells with biomaterial scaffolds may serve as useful analogues to study vascular disease progression in a controlled environment. This review presents the main considerations for developing such in vitro models. We discuss how the design of blood vessel models impacts experimental readouts, with a particular focus on the maintenance of normal cellular phenotypes, strategies that mimic normal cell-ECM interactions, and approaches that foster intercellular communication between vascular cell types. We also highlight how choice of biomaterials, cellular arrangements and the inclusion of mechanical stimulation using fluidic devices together impact the ability of blood vessel models to mimic in vivo conditions. In the future, by combining advances in materials science, cell biology, fluidics and modelling, it may be possible to create blood vessel models that are patient-specific and can be used to develop and test therapies. STATEMENT OF SIGNIFICANCE: Simplified models of blood vessels created in vitro are powerful tools for studying cardiovascular diseases and understanding the mechanisms driving their progression. Here, we highlight the key structural and cellular components of effective models and discuss how including mechanical stimuli allows researchers to mimic native vessel behaviour in health and disease. We discuss the primary methods used to form blood vessel models and their limitations and conclude with an outlook on how blood vessel models that incorporate patient-specific cells and flows can be used in the future for personalised disease modelling.

Insights into the design of spray systems for cell therapies for retinal disease using computational modelling.

Chronic eye diseases are the main cause of vision loss among adults. Among these, retinal degenerative diseases affect millions of people globally, causing permanent loss of cells and organ dysfunction. Despite recent progress in developing stem cell therapies for retinal diseases, methods for delivery remain an area of intense research. Aerosol technology is a promising technique with the potential to spray cells evenly and directly across the retinal surface, promoting cell attachment and survival. Here we implement mathematical modelling of the spraying process to develop organ-specific spraying parameters in this therapeutic scenario. Firstly, we characterise the rheological parameters for a typical hydrogel used for spraying cells. These parameters are then integrated into a 3D computational model of an adult human eye under realistic surgical conditions. Simulation results provide quantitative relationships between the volume flow rate of the cell-laden hydrogel, external pressure needed for aerosolization, angle of the spraying, and properties of the cell delivery. An experimental assessment is also carried out to explore the impact of spraying under the regimes identified by the computational model on cell viability. This is the first stage towards using computational models to inform the design of spray systems to deliver cell therapies onto the human retina.

A four-compartment multiscale model of fluid and drug distribution in vascular tumours.

The subtle relationship between vascular network structure and mass transport is vital to predict and improve the efficacy of anticancer treatments. Here, mathematical homogenisation is used to derive a new multiscale continuum model of blood and chemotherapy transport in the vasculature and interstitium of a vascular tumour. This framework enables information at a range of vascular hierarchies to be fed into an effective description on the length scale of the tumour. The model behaviour is explored through a demonstrative case study of a simplified representation of a dorsal skinfold chamber, to examine the role of vascular network architecture in influencing fluid and drug perfusion on the length scale of the chamber. A single parameter, P, is identified that relates tumour-scale fluid perfusion to the permeability and density of the capillary bed. By fixing the topological and physiological properties of the arteriole and venule networks, an optimal value for P is identified, which maximises tumour fluid transport and is thus hypothesised to benefit chemotherapy delivery. We calculate the values for P for eight explicit network structures; in each case, vascular intervention by either decreasing the permeability or increasing the density of the capillary network would increase fluid perfusion through the cancerous tissue. Chemotherapeutic strategies are compared and indicate that single injection is consistently more successful compared with constant perfusion, and the model predicts optimal timing of a second dose. These results highlight the potential of computational modelling to elucidate the link between vascular architecture and fluid, drug distribution in tumours.

A hybrid discrete-continuum approach for modelling microcirculatory blood flow.

In recent years, biological imaging techniques have advanced significantly and it is now possible to digitally reconstruct microvascular network structures in detail, identifying the smallest capillaries at sub-micron resolution and generating large 3D structural data sets of size >106 vessel segments. However, this relies on ex vivo imaging; corresponding in vivo measures of microvascular structure and flow are limited to larger branching vessels and are not achievable in three dimensions for the smallest vessels. This suggests the use of computational modelling to combine in vivo measures of branching vessel architecture and flows with ex vivo data on complete microvascular structures to predict effective flow and pressures distributions. In this paper, a hybrid discrete-continuum model to predict microcirculatory blood flow based on structural information is developed and compared with existing models for flow and pressure in individual vessels. A continuum-based Darcy model for transport in the capillary bed is coupled via point sources of flux to flows in individual arteriolar vessels, which are described explicitly using Poiseuille's law. The venular drainage is represented as a spatially uniform flow sink. The resulting discrete-continuum framework is parameterized using structural data from the capillary network and compared with a fully discrete flow and pressure solution in three networks derived from observations of the rat mesentery. The discrete-continuum approach is feasible and effective, providing a promising tool for extracting functional transport properties in situations where vascular branching structures are well defined.

Mechanical Response of Neural Cells to Physiologically Relevant Stiffness Gradients.

Understanding the influence of the mechanical environment on neurite behavior is crucial in the development of peripheral nerve repair solutions, and could help tissue engineers to direct and guide regeneration. In this study, a new protocol to fabricate physiologically relevant hydrogel substrates with controlled mechanical cues is proposed. These hydrogels allow the analysis of the relative effects of both the absolute stiffness value and the local stiffness gradient on neural cell behavior, particularly for low stiffness values (1-2 kPa). NG108-15 neural cell behavior is studied using well-characterized collagen gradient substrates with stiffness values ranging from 1 to 10 kPa and gradient slopes of either 0.84 or 7.9 kPa mm-1 . It is found that cell orientation is influenced by specific combinations of stiffness value and stiffness gradient. The results highlight the importance of considering the type of hydrogel as well as both the absolute value of the stiffness and the steepness of its gradient, thus introducing a new framework for the development of tissue engineered scaffolds and the study of substrate stiffness.

Reproducibility of an airway tapering measurement in computed tomography with application to bronchiectasis.

We propose a pipeline to acquire a scalar tapering measurement from the carina to the most distal point of an individual airway visible on computed tomography (CT). We show the applicability of using tapering measurements on clinically acquired data by quantifying the reproducibility of the tapering measure. We generate a spline from the centerline of an airway to measure the area and arclength at contiguous intervals. The tapering measurement is the gradient of the linear regression between area in log space and arclength. The reproducibility of the measure was assessed by analyzing different radiation doses, voxel sizes, and reconstruction kernel on single timepoint and longitudinal CT scans and by evaluating the effect of airway bifurcations. Using 74 airways from 10 CT scans, we show a statistical difference, p = 3.4 × 10 - 4 , in tapering between healthy airways ( n = 35 ) and those affected by bronchiectasis ( n = 39 ). The difference between the mean of the two populations is 0.011 mm - 1 , and the difference between the medians of the two populations was 0.006 mm - 1 . The tapering measurement retained a 95% confidence interval of ± 0.005 mm - 1 in a simulated 25 mAs scan and retained a 95% confidence of ± 0.005 mm - 1 on simulated CTs up to 1.5 times the original voxel size. We have established an estimate of the precision of the tapering measurement and estimated the effect on precision of the simulated voxel size and CT scan dose. We recommend that the scanner calibration be undertaken with the phantoms as described, on the specific CT scanner, radiation dose, and reconstruction algorithm that are to be used in any quantitative studies.