Steven Robert McDougall

Multiscale modelling and nonlinear simulation of vascular tumour growth.

In this article, we present a new multiscale mathematical model for solid tumour growth which couples an improved model of tumour invasion with a model of tumour-induced angiogenesis. We perform nonlinear simulations of the ulti-scale model that demonstrate the importance of the coupling between the development and remodeling of the vascular network, the blood flow through the network and the tumour progression. Consistent with clinical observations, the hydrostatic stress generated by tumour cell proliferation shuts down large portions of the vascular network dramatically affecting the flow, the subsequent network remodeling, the delivery of nutrients to the tumour and the subsequent tumour progression. In addition, extracellular matrix degradation by tumour cells is seen to have a dramatic affect on both the development of the vascular network and the growth response of the tumour. In particular, the newly developing vessels tend to encapsulate, rather than penetrate, the tumour and are thus less effective in delivering nutrients.

Fibroblast migration and collagen deposition during dermal wound healing : mathematical modelling and clinical implications

The extent to which collagen alignment occurs during dermal wound healing determines the severity of scar tissue formation. We have modelled this using a multiscale approach, in which extracellular materials, for example collagen and fibrin, are modelled as continua, while fibroblasts are considered as discrete units. Within this model framework, we have explored the effects that different parameters have on the alignment process, and we have used the model to investigate how manipulation of transforming growth factor-β levels can reduce scar tissue formation. We briefly review this body of work, then extend the modelling framework to investigate the role played by leucocyte signalling in wound repair. To this end, fibroblast migration and collagen deposition within both the wound region and healthy peripheral tissue are considered. Trajectories of individual fibroblasts are determined as they migrate towards the wound region under the combined influence of collagen/fibrin alignment and gradients in a paracrine chemoattractant produced by leucocytes. The effects of a number of different physiological and cellular parameters upon the collagen alignment and repair integrity are assessed. These parameters include fibroblast concentration, cellular speed, fibroblast sensitivity to chemoattractant concentration and chemoattractant diffusion coefficient. Our results show that chemoattractant gradients lead to increased collagen alignment at the interface between the wound and the healthy tissue. Results show that there is a trade-off between wound integrity and the degree of scarring. The former is found to be optimized under conditions of a large chemoattractant diffusion coefficient, while the latter can be minimized when repair takes place in the presence of a competitive inhibitor to chemoattractants.

Mathematical modelling of the influence of blood rheological properties upon adaptative tumour-induced angiogenesis

In this paper, we present a theoretical investigation of the influence of blood flow through a tumour-induced capillary network, whereby the vascular architecture adapts as it grows to the associated haemodynamic forces resulting in what we describe as adaptive tumour-induced angiogenesis (ATIA). The network is generated in response to tumour angiogenic factors (TAFs), which are released from hypoxic cells within a solid tumour. We first describe a refined model for tumour-induced angiogenesis, which aims to describe the capillary growth process at the cellular level by explicitly taking into account the effects of matrix degrading enzymes and the local properties of the host tissue during endothelial cell migration. We then incorporate blood rheological properties into the formulation and investigate the influence of wall shear stress induced by the blood flow during dynamic vascular growth. We then go on to examine a number of feedback mechanisms affecting vascular resistance and network architecture. The mechanisms considered include those proposed by Pries and co-workers [A.R. Pries, T.W. Secomb, P. Gaehtgens, Structural adaptation and stability of microvascular networks: theory and simulation, Am. J. Physiol. Heart Circ. Physiol. 44 (1998) H349-H360; A.R. Pries, B. Reglin, T.W. Secomb, Structural adaptation of microvascular networks: functional roles of adaptative responses, Am. J. Physiol. Heart Circ. Physiol. 281 (2001) H1015-H1025; A.R. Pries, B. Reglin, T.W. Secomb, Structural adaptation of microvascular networks: roles of the pressure response, Hypertension 38 (2001) 1476-1479] and both haemodynamic (non-linear viscosity) and metabolic constraints are taken into account. Subsequent simulations of chemotherapeutic drug perfusion through the system show that vascular adaptation leads to a significant benefit in treatment delivery to the tumour. The results clearly demonstrate that the combined effects of network architecture and vessel compliance should be included in future models of angiogenesis if therapy protocols and treatment efficacy are to be adequately assessed.

The effect of interstitial pressure on tumor growth: coupling with the blood and lymphatic vascular systems

The flow of interstitial fluid and the associated interstitial fluid pressure (IFP) in solid tumors and surrounding host tissues have been identified as critical elements in cancer growth and vascularization. Both experimental and theoretical studies have shown that tumors may present elevated IFP, which can be a formidable physical barrier for delivery of cell nutrients and small molecules into the tumor. Elevated IFP may also exacerbate gradients of biochemical signals such as angiogenic factors released by tumors into the surrounding tissues. These studies have helped to understand both biochemical signaling and treatment prognosis. Building upon previous work, here we develop a vascular tumor growth model by coupling a continuous growth model with a discrete angiogenesis model. We include fluid/oxygen extravasation as well as a continuous lymphatic field, and study the micro-environmental fluid dynamics and their effect on tumor growth by accounting for blood flow, transcapillary fluid flux, interstitial fluid flow, and lymphatic drainage. We thus elucidate further the non-trivial relationship between the key elements contributing to the effects of interstitial pressure in solid tumors. In particular, we study the effect of IFP on oxygen extravasation and show that small blood/lymphatic vessel resistance and collapse may contribute to lower transcapillary fluid/oxygen flux, thus decreasing the rate of tumor growth. We also investigate the effect of tumor vascular pathologies, including elevated vascular and interstitial hydraulic conductivities inside the tumor as well as diminished osmotic pressure differences, on the fluid flow across the tumor capillary bed, the lymphatic drainage, and the IFP. Our results reveal that elevated interstitial hydraulic conductivity together with poor lymphatic function is the root cause of the development of plateau profiles of the IFP in the tumor, which have been observed in experiments, and contributes to a more uniform distribution of oxygen, solid tumor pressure and a broad-based collapse of the tumor lymphatics. We also find that the rate that IFF is fluxed into the lymphatics and host tissue is largely controlled by an elevated vascular hydraulic conductivity in the tumor. We discuss the implications of these results on microenvironmental transport barriers, and the tumor invasive and metastatic potential. Our results suggest the possibility of developing strategies of targeting tumor cells based on the cues in the interstitial fluid.

The impact of wettability on waterflooding: Pore-scale simulation

This paper describes the development and implementation of a pore-scale simulator into which pore-wettability effects have been incorporated. Relative permeability and capillary pressure curves from this steady-state model have been analyzed to allow better interpretation of experimental observations from a microscopic standpoint. The simulated capillary pressure data demonstrate that some standard wettability tests (such as Amott-Harvey and free imbibition) may give misleading results when the sample is fractionally wet in nature. Waterflood displacement efficiencies for a range of wettability conditions have been calculated, and recovery is shown to be maximum when the oil-wet pore fraction approaches 0.5. Furthermore, a novel experimental test is proposed that can be used to distinguish between fractionally wet and mixed-wet porous media. To date, no such satisfactory test exists.

Empirical Measures of Wettability in Porous Media and the Relationship between Them Derived From Pore-Scale Modelling

The wettability of a crude oil/brine/rock system is of central importance in determining the oil recovery efficiency of water displacement processes in oil reservoirs. Wettability of a rock sample has traditionally been measured using one of two experimental techniques, viz. the United States Bureau of Mines and Amott tests. The former gives the USBM index, IUSBM, and the latter yields the Amott–Harvey index, IAH. As there is no well-established theoretical basis for either test, any relationship between the two indices remains unclear.Analytical relationships between IAH and IUSBM for mixed-wet and fractionally-wet media have been based on a number of simplifying assumptions relating to the underlying pore-scale displacement mechanisms. This simple approach provides some guidelines regarding the influence of the distribution of oil-wet surfaces within the porous medium on IAH and IUSBM. More detailed insight into the relationship between IAH and IUSBM is provided by modelling the pore-scale displacement processes in a network of interconnected pores. The effects of pore size distribution, interconnectivity, displacement mechanisms, distribution of volume and of oil-wet pores within the pore space have all been investigated by means of the network model.The results of these analytical calculations and network simulations show that IAH and IUSBM need not be identical. Moreover, the calculated indices and the relationship between them suggest explanations for some of the trends that appear in experimental data when both IUSBM and IAH have been reported in the literature for tests with comparable fluids and solids. Such calculations should help with the design of more informative wettability tests in the future.

Integrated intravital microscopy and mathematical modeling to optimize nanotherapeutics delivery to tumors

Inefficient vascularization hinders the optimal transport of cell nutrients, oxygen, and drugs to cancer cells in solid tumors. Gradients of these substances maintain a heterogeneous cell-scale microenvironment through which drugs and their carriers must travel, significantly limiting optimal drug exposure. In this study, we integrate intravital microscopy with a mathematical model of cancer to evaluate the behavior of nanoparticle-based drug delivery systems designed to circumvent biophysical barriers. We simulate the effect of doxorubicin delivered via porous 1000 x 400 nm plateloid silicon particles to a solid tumor characterized by a realistic vasculature, and vary the parameters to determine how much drug per particle and how many particles need to be released within the vasculature in order to achieve remission of the tumor. We envision that this work will contribute to the development of quantitative measures of nanoparticle design and drug loading in order to optimize cancer treatment via nanotherapeutics.

Saturation-dependencies of three-phase relative permeabilities in mixed-wet and fractionally wet systems

Abstract The saturation-dependency behaviour of three-phase relative permeabilities is of central importance for modelling three-phase displacement processes in porous media. In this paper, and in related work, a method has been developed to determine the saturation-dependencies of three-phase relative permeabilities. This method is suitable for all types of mixed-wet and fractionally wet porous systems that contain clusters of oil-wet and water-wet pores with constant but different oil–water contact angles, reflecting weakly wetted conditions. Based on the classification of all allowed pore occupancies in a completely accessible porous medium, saturation-dependencies of the corresponding relative permeabilities are derived. Furthermore, three-phase relative permeabilities that appear to depend only on their own saturations are either linked to the corresponding two-phase relative permeabilities or it is shown that such a link cannot be established. A comparison has been made with existing relative permeability models with respect to their saturation-dependencies.

The effect of interstitial pressure on therapeutic agent transport: Coupling with the tumor blood and lymphatic vascular systems

Vascularized tumor growth is characterized by both abnormal interstitial fluid flow and the associated interstitial fluid pressure (IFP). Here, we study the effect that these conditions have on the transport of therapeutic agents during chemotherapy. We apply our recently developed vascular tumor growth model which couples a continuous growth component with a discrete angiogenesis model to show that hypertensive IFP is a physical barrier that may hinder vascular extravasation of agents through transvascular fluid flux convection, which drives the agents away from the tumor. This result is consistent with previous work using simpler models without blood flow or lymphatic drainage. We consider the vascular/interstitial/lymphatic fluid dynamics to show that tumors with larger lymphatic resistance increase the agent concentration more rapidly while also experiencing faster washout. In contrast, tumors with smaller lymphatic resistance accumulate less agents but are able to retain them for a longer time. The agent availability (area-under-the curve, or AUC) increases for less permeable agents as lymphatic resistance increases, and correspondingly decreases for more permeable agents. We also investigate the effect of vascular pathologies on agent transport. We show that elevated vascular hydraulic conductivity contributes to the highest AUC when the agent is less permeable, but to lower AUC when the agent is more permeable. We find that elevated interstitial hydraulic conductivity contributes to low AUC in general regardless of the transvascular agent transport capability. We also couple the agent transport with the tumor dynamics to simulate chemotherapy with the same vascularized tumor under different vascular pathologies. We show that tumors with an elevated interstitial hydraulic conductivity alone require the strongest dosage to shrink. We further show that tumors with elevated vascular hydraulic conductivity are more hypoxic during therapy and that the response slows down as the tumor shrinks due to the heterogeneity and low concentration of agents in the tumor interior compared with the cases where other pathological effects may combine to flatten the IFP and thus reduce the heterogeneity. We conclude that dual normalizations of the micronevironment - both the vasculature and the interstitium - are needed to maximize the effects of chemotherapy, while normalization of only one of these may be insufficient to overcome the physical resistance and may thus lead to sub-optimal outcomes.

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the ‘intermediate-wetting’ phase in oil-wet pores as determined by the corresponding contact angles, that is, cos θogw > 0 or cos θogw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the ‘intermediate-wetting’ phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.