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.

Detection of SARS-CoV-2 Delta Variant (B.1.617.2) in Domestic Dogs and Zoo Tigers in England and Jersey during 2021.

Reverse zoonotic transmission events of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) have been described since the start of the pandemic, and the World Organisation for Animal Health (WOAH) designated the detection of SARS-CoV-2 in animals a reportable disease. Eighteen domestic and zoo animals in Great Britain and Jersey were tested by APHA for SARS-CoV-2 during 2020-2023. One domestic cat (Felis catus), three domestic dogs (Canis lupus familiaris), and three Amur tigers (Panthera tigris altaica) from a zoo were confirmed positive during 2020-2021 and reported to the WOAH. All seven positive animals were linked with known SARS-CoV-2 positive human contacts. Characterisation of the SARS-CoV-2 variants by genome sequencing indicated that the cat was infected with an early SARS-CoV-2 lineage. The three dogs and three tigers were infected with the SARS-CoV-2 Delta variant of concern (B.1.617.2). The role of non-human species in the onward transmission and emergence of new variants of SARS-CoV-2 remain poorly defined. Continued surveillance of SARS-CoV-2 in relevant domestic and captive animal species with high levels of human contact is important to monitor transmission at the human-animal interface and to assess their role as potential animal reservoirs.

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.

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.

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.

Characterisation of a Live-Attenuated Rabies Virus Expressing a Secreted scFv for the Treatment of Rabies.

Rabies virus (RABV) causes possibly the oldest disease and is responsible for an estimated >59,000 human fatalities/year. Post exposure prophylaxis (PEP), the administration of vaccine and rabies immunoglobulin, is a highly effective tool which is frequently unavailable in RABV endemic areas. Furthermore, due to the constraints of the blood-brain barrier, current PEP regimes are ineffective after the onset of clinical symptoms which invariably result in death. To circumvent this barrier, a live-attenuated recombinant RABV expressing a highly RABV-neutralising scFv antibody (62-71-3) linked to the fluorescent marker mCherry was designed. Once rescued, the resulting construct (named RABV-62scFv) was grown to high titres, its growth and cellular dissemination kinetics characterised, and the functionality of the recombinant 62-71-3 scFv assessed. Encouraging scFv production and subsequent virus neutralisation results demonstrate the potential for development of a therapeutic live-attenuated virus-based post-infection treatment (PIT) for RABV infection.

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.

Taiwan Bat Lyssavirus: In Vitro and In Vivo Assessment of the Ability of Rabies Vaccine-Derived Antibodies to Neutralise a Novel Lyssavirus.

Rabies is a neglected tropical disease. The prototype virus, the rabies virus, still causes tens of thousands of human fatalities annually. Rabies is one member of the genus Lyssavirus. The burden of other lyssaviruses is unclear. The continued emergence of novel lyssaviruses means that assessment of vaccine efficacy against these viruses is critical, as standard rabies vaccines are not efficacious against all lyssaviruses. Taiwan bat lyssavirus (TWBLV) was first reported in 2018 following isolation from Japanese house bats. Since the initial detection and genetic characterisation, no attempts have been made to antigenically define this virus. Due to the inaccessibility of the wildtype isolate, the successful generation of a live recombinant virus, cSN-TWBLV, is described, where the full-length genome clone of the RABV vaccine strain, SAD-B19, was constructed with the glycoprotein of TWBLV. In vitro and in vivo characterization of cSN-TWBLV was undertaken and demonstrated evidence for cross-neutralisation of cSN-TWBLV with phylogroup I -specific sera and rabies virus standard sera. For neutralisation equivalent to 0.5 IU/mL of WHO and World Organisation of Animal Health (WOAH) sera against CVS, 0.5 IU/mL of WOAH sera and 2.5 IU/mL of WHO sera were required to neutralise cSN-TWBLV. In addition, specific sera for ARAV and EBLV-1 exhibited the highest neutralising antibody titres against cSN-TWBLV, compared to other phylogroup I-specific sera.

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.

Challenges and opportunities of integrating imaging and mathematical modelling to interrogate biological processes.

Advances in biological imaging have accelerated our understanding of human physiology in both health and disease. As these advances have developed, the opportunities gained by integrating with cutting-edge mathematical models have become apparent yet remain challenging. Combined imaging-modelling approaches provide unprecedented opportunity to correlate data on tissue architecture and function, across length and time scales, to better understand the mechanisms that underpin fundamental biology and also to inform clinical decisions. Here we discuss the opportunities and challenges of such approaches, providing literature examples across a range of organ systems. Given the breadth of the field we focus on the intersection of continuum modelling and in vivo imaging applied to the vasculature and blood flow, though our rationale and conclusions extend widely. We propose three key research pillars (image acquisition, image processing, mathematical modelling) and present their respective advances as well as future opportunity via better integration. Multidisciplinary efforts that develop imaging and modelling tools concurrently, and share them open-source with the research community, provide exciting opportunity for advancing these fields.

Infectious droplet exposure is an inefficient route for SARS-CoV-2 infection in the ferret model.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the causative agent of coronavirus disease 2019 (COVID-19) in humans, has a wide host range, naturally infecting felids, canids, cervids, rodents and mustelids. Transmission of SARS-CoV-2 is universally accepted to occur via contact with contaminated secretions from the respiratory epithelium, either directly or indirectly. Transmission via droplet nuclei, generated from a cough or sneeze, has also been reported in several human and experimental animal scenarios. However, the role of droplet transmission at the human-animal interface remains to be fully elucidated. Here, the ferret infection model was used to investigate the routes of infection for the SARS-CoV-2 beta variant (B.1.351). Ferrets were exposed to droplets containing infectious SARS-CoV-2, ranging between 4 and 106 µm in diameter, simulating larger droplets produced by a cough from an infected person. Following exposure, viral RNA was detected on the fur of ferrets, and was deposited onto environmental surfaces, as well as the fur of ferrets placed in direct contact; SARS-CoV-2 remained infectious on the fur for at least 48 h. Low levels of viral RNA were detected in the nasal washes early post-exposure, yet none of the directly exposed, or direct-contact ferrets, became robustly infected or seroconverted to SARS-CoV-2. In comparison, ferrets intranasally inoculated with the SARS-CoV-2 beta variant became robustly infected, shedding viral RNA and infectious virus from the nasal cavity, with transmission to 75 % of naive ferrets placed in direct contact. These data suggest that larger infectious droplet nuclei and contaminated fur play minor roles in SARS-CoV-2 transmission among mustelids and potentially other companion animals.

Serum Neutralization Profiles of Straw-Colored Fruit Bats (Eidolon helvum) in Makurdi (Nigeria), against Four Lineages of Lagos Bat Lyssavirus.

Lagos bat lyssavirus (LBV) comprising four lineages (A, B, C and D) can potentially cause the fatal disease rabies. Although LBV-B was initially isolated in Nigeria in 1956, there is no information on LBV lineages circulating in Nigeria. This study was undertaken for the first time to measure the neutralizing antibodies against four lineages of LBVs in straw-colored fruit bats (Eidolon helvum) in Makurdi, Nigeria. Serum samples (n = 180) collected during two periods (November 2017-March 2018 and November 2018-March 2019) from terminally bled bats captured for human consumption were tested using a modified fluorescent antibody virus neutralization (mFAVN) assay. A high proportion of bat sera (74%) neutralized at least one lineage of LBV (with reciprocal titers from 9 to >420.89) and most of them neutralized LBV-A (63%), followed by LBV-D (49%), LBV-C (45%) and LBV-B (24%). The majority of positive sera (75%, n = 100) neutralized multiple LBV lineages while the remaining 25% (n = 33) neutralized only a single lineage, i.e., LBV-A (n = 23), LBV-D (n = 8) and LBV-C (n = 2). None exclusively neutralized LBV-B. The results suggest that exposure to LBV is common in E. helvum and that LBV-A (but not LBV-B) is likely to be circulating in this region of Nigeria.

Renewed Public Health Threat from Emerging Lyssaviruses.

Pathogen discovery contributes to our knowledge of bat-borne viruses and is linked to the heightened interest globally in bats as recognised reservoirs of zoonotic agents. The transmission of lyssaviruses from bats-to-humans, domestic animals, or other wildlife species is uncommon, but interest in these pathogens remains due to their ability to cause an acute, progressive, invariably fatal encephalitis in humans. Consequently, the detection and characterisation of bat lyssaviruses continues to expand our knowledge of their phylogroup definition, viral diversity, host species association, geographical distribution, evolution, mechanisms for perpetuation, and the potential routes of transmission. Although the opportunity for lyssavirus cross-species transmission seems rare, adaptation in a new host and the possibility of onward transmission to humans requires continued investigation. Considering the limited efficacy of available rabies biologicals it is important to further our understanding of protective immunity to minimize the threat from these pathogens to public health. Hence, in addition to increased surveillance, the development of a niche pan-lyssavirus vaccine or therapeutic biologics for post-exposure prophylaxis for use against genetically divergent lyssaviruses should be an international priority as these emerging lyssaviruses remain a concern for global public health.

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.

Feasibility of Noninvasive Positive Pressure Ventilation in the Treatment of Oxygen-Dependent COVID-19 Patients in Peru.

Intensive care is expensive, and availability is limited. Low- and middle-income countries in particular have struggled to cope with the large influx of critically ill patients during the COVID-19 pandemic. Noninvasive respiratory support devices delivering continuous positive airways pressure (CPAP) require less resource and staff expertise compared with invasive mechanical ventilators and can be routinely used outside of intensive care units. This study assessed the use of the UCL-Ventura Wayrachi CPAP device in hospitalized patients with COVID-19 in Peru. A secondary analysis of data collected for a feasibility study commissioned by the Peruvian Ministry of Health was conducted. Data were collected from three hospitals, including patient demographics, clinical data, and outcomes. Forty-five patients were enrolled from July 16 to September 1, 2020. Eight patients (18%) were intolerant of the CPAP mask. Of the remainder, 18 (48.7%) improved and were discharged from hospital after 6 days. Eight (21.6%) died while on CPAP and 11 (29.7%) were eventually intubated, of whom two died. In total, 27 (60%) survived to hospital discharge. Participating physicians noted the device was easy to use and provided patient benefit, though voiced concerns about the strain on hospital oxygen supplies. In conclusion, the UCL Ventura Wayrachi CPAP device proved feasible in COVID-19 patients in Peru, and offered a bridging therapy for patients who required a ventilator when none were available.

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.