Mathematical modelling of physiological flows
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1 Mathematical modelling of physiological flows Sarah Waters Oxford Centre for Industrial and Applied Mathematics University of Oxford
2 Research Overview Develop & solve mathematical models to provide insights into biomedical problems Research ranges from classical fluid dynamics problems motivated by physiology to highly interdisciplinary projects addressing key biomedical problems Physiological fluid mechanics, e.g., blood flow, respiratory flows, urology!!!! Tissue engineering, e.g., flow, nutrient & tissue growth in bespoke bioreactors!!!!!!!!!! Tissue biomechanics, e.g., arterial wall mechanics brain folding during development cell motility Close interdisciplinary collaboration with life scientists, clinicians, bioengineers,...
3 Modelling philosophy Experiments Model development Guided by experiment Mathematical model Simplify e.g. exploit disparate length & timescales Stimulate new expts Data for validation & refinement Biological question Simplified model Analysis Numerical solution Model calibration Predictive tool Fundamental understanding Interpretation of biological data Model output
4 Mathematical Medicine and Biology Gene networks; DNA dynamics; cell signalling, adhesion & motility; cell membranes; bacterial infections; tumour growth; angiogenesis; tissue engineering & drug delivery; stem cells; immunology; physiological fluid & solid mechanics; neurodynamics; developmental biology; ecology... Ordinary differential eqns; delay-differential eqns; stochastic differential eqns; partial differential eqns; hybrid multiscale models; free boundary problems... Asymptotic analysis; linear & nonlinear stability theory; similarity solutions; multiple-scales analysis; bifurcation theory; numerical simulations...
5 Case study I: coronary arterial flows Artery wall Lumen Normal artery cross-section Significant atherosclerotic plaque ( Plaque initiation & development correlated with arterial wall shear stress distribution (Caro, Fitzgerald & Schroter 97) Many arteries have significant curvature, e.g., aortic arch has curvature ratio approx /4 (Chang & Tarbell 985) How does curvature influence fluid flow, & hence the likelihood of plaque development?
6 Curved-tube flows L Navier-Stokes equations No-slip boundary conditions R e s a Prescribed axial pressure gradient Straight pipe: R, axial flow u = we s + ( e s) w 2 /R Curved pipe: centrifugal force terms!!! generate cross-sectional flows u = we s + ( e s) Axial Cross-sectional regime: w = O R a /2, = O(), (ar) /2 L < 2 R fully-developed flow: reduce spatial dimension from 3D to 2D centrifugal terms arise at leading-order
7 Model
8 Governing equations
9 Governing parameters
10 Previous work (Siggers & Waters 25, 28)
11 Purely oscillatory pressure gradient (D=) I O I O I O I O I O I O (.) (.) (.) () () () I O I O I O I O I O I O () () () (5) (5) (5) D h 2Rs 3 h cos t How does this bifurcation structure depend on the curvature ratio?
12 Transition between solutions
13 Periodic, symmetric solutions (Branch )
14 Periodic, asymmetric solutions (Branch 2)
15 Periodic, asymmetric solutions (Branch 2)
16 Pulsatile pressure gradient
17 Pulsatile pressure gradient
18 Conclusions
19 Physiological implications & challenges Physiological implications: Finite curvature: significant increase in wall shear stress in at risk regions Spiral flow potentially important for arterial grants Mathematical challenges include: Developing flow; time-dependent geometry; internal blockages Flow-structure interaction: couple fluid mechanics to vessel (patho)physiology Kidney Renal pelvis Uretero-pelvic junction (UPJ) Stent Ureter Bladder (unsteady) Collapsible-tube flows: unsteady finite-re flow + large displacements of a nonaxisymmetrically buckling cylindrical shell Vesico-ureteric junction (VUJ) Urological flows: lubrication theory coupled to non-linear elastic membrane; free boundary, porous media problems for encrusting biofilm (Heil & Waters, J. Fluid Mech., 6, 8; Whittaker et al., a-d, ) (Cummings, Waters, Wattis & Graham, J. Math Biol., 24)
20 Case study II: urological flows Urine produced in kidney flows down ureter & stored in bladder until voiding Kidney Renal pelvis Blockages of ureter, e.g. due to calculi, can result in increased renal pelvic pressure Ureter Renal artery Renal vein Pressure relieved by stent insertion Bladder Kidney Renal pelvis Uretero-pelvic junction (UPJ) Stent Ureter Urethra Complications associated with stent use:!! Reflux (backflow of urine along ureter)!! Infection & encrustation Bladder Vesico-ureteric junction (VUJ) Use combination of mathematical modelling & novel experiments to provide insights
21 Mathematical model Wall of renal pelvis r Wall of ureter Wall of stent gravity A o P k (t ) P s (z, t ) A i A (z, t ) P u (z, t ) P b (t ) z RENAL PELVIS z = (UPJ) URETER BLADDER z = l (VUJ) Renal pelvis: Ureter wall: Permeable stent: Bladder: Elastic bag (Prescribed urine Axisymmetric Nonlinear elastic membrane Rigid, permeable tube Circular cross-section Prescribed pressure flux from kidney) Neglect peristalsis (May be encrusted) (Variable external pressure) Urine: homogeneous, incompressible, Newtonian viscous fluid
22 Governing equations Exploit small aspect ratio to simplify equations (lubrication theory) System reduces to two dimensionless eqns for cross-sectional area & reduced stent pressure where q = D k A t + q z =, 2 p s z 2 d(p s p u ) =, ( ( (A A o ) A + A o 2(A A ) o) pu ln(a/a o ) z + A2 i p s z ), p u = k ) (e γ( A ). A Boundary conditions Initial conditions p s = p u, { p s = p u = p b (t) = A(z, ) = p u = k k q at z =, t { sin 2 (πt) t t > at z =, Dimensionless parameters k Ureter stiffness k k D d Renal pelvis stiffness Pulse duration/ureter relaxation timescale Stent permeability
23 lts Tot.4 Results r values used in the following results are given in 2.3. We illustrate typical flo.2 A(z, t) p (z, t) ong pulse duration (D large) and short pulse duration (D small) respectively. N mensionalisation is such that the dimensionless pulse duration and the dimensionl the prescribed bladder pressure pulse are both. Long pulse τ = 5 cm H2 and τ = s. (a) In this case, D. In fig considering Pb,max cross-sectional area, A(z, t), is shown, and in figure 3b the corresponding press z z ent is illustrated (note that pu may be determined using5: (2.5b)) The Figure Graph. of theprescrib total ure (the pressure profile at z = ) is denoted withduration, the dashed linethe in figure t t τ black. In all figure bladder and renal pelvis reduced pressures are both zero, the ureter is at its natu the lowest curve represents th (a) (b) l area of, and there is no flow. As the reduced bladder pressure increases from ze parameter values for the scena idly inflates and pk increases due to reflux through the UPJ. The size of D is su A(z, t) p (z, t) y uniform state is quickly achieved. s s z t (c) t (d) To mimic shorter pulses we n Short pulse area, the ureter cross-sectional For thisz case, D, and we d pressure increases, a pulse of fl to the downstream end (z =.2
24 Total reflux.2 Total reflux into renal pelvis Total reflux from bladder into ureter.5 Total reflux into renal pelvis Total reflux from bladder into ureter τ Pulse duration a) (b) τ Pulse duration igure 5: Graph of the total reflux into (a) the renal pelvis, and (b) the ureter, against pulse uration, τ. In all the figures, Pb,max increases from cm H 2 to 5 cm H 2. in each figures, Arrow denotes increasing amplitude of bladder pressure pulse he lowest curve represents the lowest value of Pb,max. The large filled black circles indicate the arameter values for the scenarios in figures 3 and 4. (Waters et al., Proc. IMechE H, 28)
25 Results Interrogate model to investigate effect of encrustation, external pressure, stent design etc on fluid flow & reflux Radial coordinate, r Axial coordinate, z Couple fluid mechanical model to model for encrustation (Band et al., J. Math Biol., 29)
26 Conclusions Mathematical modelling provides fundamental insights into complex biological systems Rich, well-defined problems: level of biological understanding enables formulation of physically well-grounded models Exciting research challenges, requiring development & solution of new mathematical models, & leading to quantitative predictions
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