Research projects and publications

Multiscale analysis of nutrient uptake by plant roots covered with hairs

My PhD research was part of the FutureRoots project at the University of Nottingham with an aim of improving crop performance by redesigning root system architecture. Broadly, I modelled plant nutrient and water uptake at various spatial scales using (mostly) continuum approaches, with a particular emphasis on uptake by root hairs. Resulting (reaction-convection-diffusion) equations were solved using a combination of asymptotic (multiscale homogenisation, matched asymptotic expansions) and numerical (finite element, finite difference) methods (1). In particular, I extended the existing homogenization framework from "A dynamic model of nutrient uptake by root hairs." by Leitner et al, to account for the fact that the root hairs are thin in comparison with a typical inter-hair distance (2) and also for the curvature of the root system (3).

Supervisors: 

Profs John King and Markus Owen (The University of Nottingham)

Close collaborators: 

Dr Mariya Ptashnyk (Heriot Watt)

Publications:

Microvascular blood flow and tissue oxygenation

Local oxygen levels strongly impact proper functioning of living tissues and even the efficacy of anti-cancer treatments, such as radiotherapy and chemotherapy. As red blood cells (RBCs) contain haemoglobin that carries oxygen molecules, a detailed knowledge of their distribution is required to predict tissue oxygenation with a satisfactory precision. This distribution in turn is a result of complex microvascular blood rheology including the laws governing RBCs partition at diverging bifurcations (haematocrit splitting, HS) and the formation of a cell-free layer (CFL) near the vessel wall. Existing HS rules apply whenever the flow upstream from the bifurcation is fully established, which is not true for very short vessels as often found in tumour microenvironments. With Prof Miguel Bernabeu (The University of Edinburgh) and other co-workers, we extended a standard HS model to account for these effects, using the CFL recovery as observed using detailed (lattice Boltzmann) numerical simulations. I implemented this new model at a network scale using Microvessel Chaste and arrived at significantly more heterogeneous tissue oxygen distributions compared to the standard model (1). Furthermore, the width of the CFL depends on various geometrical and rheological parameters - I developed simple formulae describing these dependencies and showed that these are in good agreement with existing in vivo data (2). I further analysed the data from my experimental collaborators on blood vessel pruning induced by irradiation during radiotherapy treatment, aiming to find key determinants of network perfusion post-irradiation. With Vedang Narain (University of Oxford), we extended the capabilities of Microvessel Chaste to mimic vessel pruning and compute network perfusion in synthetic networks. We determined under what conditions vessel pruning results in increased perfusion and also developed methods to assess the extent of blood flow re-routing (3). Tumour microenvironment often contains compressed vessels due to abnormal tumour cell proliferation which also affects HS and in turn the distribution of tissue oxygen. In close collaboration with Dr Romain Enjalbert (University of Edinburgh) we are currently preparing a manuscript elucidating how distinct transport mechanisms of oxygen and T-DM1 (antibody–drug conjugate used in breast cancer treatment) impact drug efficacy in tumours with compressed vessels. For this purpose, I developed a new reduced-order model of drug delivery exploiting distinct timescales of the governing processes.

Compressibility effects in filtration

Pressure gradients that cause a fluid to flow through a porous medium also deform the solid matrix of this medium. This in turn affects local permeability and thus, the flow through and deformation of such porous medium are (both-ways) coupled. The problem can be modelled using the well-studied framework of Biot's poroelasticity. The first analytic solutions for a steady one-dimensional flow, using simple constitutive assumptions relating local permeability and strain, were given in "Steady flow in porous, elastically deformable materials." by Parker et al. With appropriate nondimensionalization and assuming a linear relationship between the local permeability and strain, this model is governed by a single dimensionless parameter γ, which measures how sensitive the medium permeability is to the applied strain. I applied such model to steady filtration, aiming to find material properties that optimize operating flux and power use while avoiding filter shutdown. I further modified the existing model to allow for non-uniform initial (pre-deformation) permeability distributions and found an initial distribution maximizing the flux for a given applied pressure (1). Then, I applied the same ideas to cake filtration. This problem is inevitably dynamic (the cake is growing) and is governed by two sensitivities - that of the filter and that of the cake. I derived an explicit filtration law which allows one to determine whether the filter or the cake shuts down first and found optimal material properties under industrially relevant operating conditions (2).

Discrete and continuum modelling in biomechanics

Currently, I am working on discrete and continuum models for mechanical response of crosslinked protein networks subjected to various external loadings. I first studied pre-stressed cytoskeletal networks (actin, vimentin) subjected to local deformations aiming to mimic intracellular transport. Such a discrete model with arbitrary microscale nonlinearity, modelling (axial) force-extension behaviour for a single filament segment, can be upscaled to a continuum problem formulated within the framework of nonlinear elasticity. Moreover, in the small-deformations limit, I found an analytical approximation to the stress and strain fields near a small (transported) object, predicting the net force required to generate a given deformation and elucidating the dependency on microscale properties of the network. I then demonstrated good agreement between the discrete, continuum and analytical solutions (1). Led by Dr. Namshad Thekkethil, we then extended this (elastic) model to include visco-elastic relaxation due to the unbinding kinetics of crosslinkers as well as poro-elastic effects resulting from the cytosolic flow which yields the equilibration of the cytosolic (pore) pressure (2). I further worked with Prof. Anna Pandolfi (Politecnico di Milano) on discrete and continuum mechanics of collagen networks in cornea subjected to intra-ocular pressure and linked the observed microscale architectural and mechanical changes that accompany the onset of keratoconus – a degenerative disease of the cornea – with the corresponding macroscale change in the corneal shape (3). Led by Dr Yangkun Du, we also studied indentation of second-order hyperelastic materials and derived analytical solutions for parabolic and quartic surfaces to mimic a spherical indenter which agree well with finite element simulations (4).