Development & Application of Fluid & Oxygen Transport Models for the Microcirculation Mr Paul Sweeney & Dr Rebecca Shipley, MechEng, UCL Dr Simon Walker-Samuel, Centre for Advanced Biomedical Imaging, UCL Dr Jaime Grutzendler & Dr Robert Hill, Centre for Experimental Neuroimaging, Yale University Background Methods Results Future Development References Simultaneous, in vivo measurement of microvascular (< 200 μm) perfusion and structure is prac4cally infeasible . Thus, understanding the rela4onship between vascular structure and blood transport has yet to be clarified. Computa@onal limita@ons and the size of structural data sets (>10 5 vessel segments) now available mo@vates a con@nuum modelling approach. Here we present a discretecon4nuum mathema4cal model to predict flow and pressure distribu@ons through @ssue by combining a discrete method applied to the branching arterioles and a con@nuum approach applied to the meshlike capillary structure. Case studies: (A) Development of models using a rat mesentery network 1 as a test bed. (B) Study the impact of an ischaemic stroke on @ssue oxygena@on in 3D mouse cortex data 2 . (A) (B) 1. Pries, A. Dept. of Physiology, Charité Universitätsmedizin Berlin. 2. Grutzendler, J. & Hill, R. Centre for Experimental Neuroimaging, Yale University. 3. Shipley, R.J. et al. Spa$al Averaging of Microcirculatory Blood Flow. Math. Med. & Bio. Submi]ed. 4. Secomb, T.W. Green’s Func$on Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks. Annals of Biomed Eng. 2004. Discrete Model • Along with flow and pressure boundary condi@ons, the model takes structural data that has been segmented into a series of cylindrical tubes of constant circular crosssec@on. • At the microvascular scale, blood flow is viscous dominated. Hence, Poiseuille’s Law is valid, where N is the no. of nodes, q j ,M jk and p k are the flow, conductance and fluid nodal pressure of segment j. DiscreteCon4nuum Model 3 • Darcy’s law is describes the coupling between blood velocity, u, and pressure, p, with the aid of κ, the permeability of the capillary network to fluid transport. • Bloody supply into the capillaries is represented by influx condi@ons at point sources represen@ng connec@ons between arterioles and capillaries. Oublow to the venules is accounted for by a constant drainage term, β, chosen to conserve mass. • Conserva@on of mass yields where p v is constant venous pressure, N t is the no. of @ssue points and C j are source strengths. Green’s Func4on Method for Oxygen Delivery 4 • Oxygen is bound to RBCs, dissolved in plasma and diffuses into @ssue then metabolised by cells. • Oxygen sources represent blood vessels and a set of discrete oxygen sinks represent @ssue. • Using Fick’s law of diffusion and conserva4on of mass, the Green’s func@on, G(x;x*), for a given domain may be defined as the PO 2 at a point x resul@ng from a unit point source at x* is the solu@on to where D and α are oxygen diffusivity and solubility. UCL MECHANICAL ENGINEERING q j = M jk p k k ∈ N ∑ ∇⋅ u = − κ ∇ 2 p = − β ( p − p v ) + C j ( x ) δ ( x − x j ) j =1 N t ∑ D α∇ 2 G = −δ ( x − x*) Figure 1. Rat mesentery (a) Discrete modelling of fluid pressure (mean segment pressure mmHg). (b) Discrete con@nuum model predic@ons of the pressure profiles in both the arteriolar network and capillary domain. Case Study (A) – Rat Mesentery • Discrete Model – max. blood pressure was 81.56 mmHg along with a min. of 13.8 mmHg (venous oublow pressure, p v ). Mean capillary pressure was 26.23 mmHg. • DiscreteCon4nuum Model – capillary permeability, κ, was chosen by comparing metrics for known pressure and flow condi@ons in the discrete model. • % errors in the mean and standard devia@on of the source pressures was within 10%. • % errors in the mean and standard devia@on of source flows was less successful, at ~40%. Case Study (B) – Mouse Cortex • Oxygen Delivery – simula@ons were run on both a healthy and ischaemic stroke induced network to compare PO 2 levels. • Healthy – mean & max. PO 2 of 32.11 and 93.27 mmHg with a SD of 3.72. • Stroke – mean & max. PO 2 of 29.44 and 71.39 mmHg and a SD of 3.43. • A clear shim in PO 2 can be seen (Fig. 2) when a stroke is induced, indica@ng an increase in hypoxia. Figure 2. Mouse Cortex (a) Tissue PO 2 levels in (i) Healthy (blue) & (ii) stroke induced (red) simula@ons (b) O 2 delivery to healthy @ssue (mmHg). • Extend to 3D domain and incorporate discrete venular network. • Apply DiscreteCon@nuum model to mouse cortex. • Study effects of microvascular blockages on PO 2 transport. • Incorporate inters44al flow to predict 4ssuescale fluid and drug transport in porous and healthy blood vessels.
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Development & Application of Fluid & Oxygen Transport Models for the Microcirculation Mr Paul Sweeney & Dr Rebecca Shipley, MechEng, UCL Dr Simon Walker-Samuel, Centre for Advanced Biomedical Imaging, UCL Dr Jaime Grutzendler & Dr Robert Hill, Centre for Experimental Neuroimaging, Yale University
Background
Methods
Results
Future Development
References
Simultaneous, in vivo measurement of microvascular (< 200 μm) perfusion and structure is prac4cally infeasible. Thus, understanding the rela4onship between vascular structure and blood transport has yet to be clarified. Computa@onal limita@ons and the size of structural data sets (>105 vessel segments) now available mo@vates a con@nuum modelling approach. Here we present a discrete-‐con4nuum mathema4cal model to predict flow and pressure distribu@ons through @ssue by combining a discrete method applied to the branching arterioles and a con@nuum approach applied to the mesh-‐like capillary structure. Case studies: (A) Development of models using a rat mesentery network1 as a
test bed. (B) Study the impact of an ischaemic stroke on @ssue oxygena@on
in 3D mouse cortex data2.
(A) (B)
1. Pries, A. Dept. of Physiology, Charité Universitätsmedizin Berlin.
2. Grutzendler, J. & Hill, R. Centre for Experimental Neuroimaging, Yale University.
3. Shipley, R.J. et al. Spa$al Averaging of Microcirculatory Blood Flow. Math. Med. & Bio. Submi]ed.
4. Secomb, T.W. Green’s Func$on Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks. Annals of Biomed Eng. 2004.
Discrete Model
• Along with flow and pressure boundary condi@ons, the model takes structural data that has been segmented into a series of cylindrical tubes of constant circular cross-‐sec@on.
• At the microvascular scale, blood flow is viscous dominated. Hence, Poiseuille’s Law is valid,
where N is the no. of nodes, qj, Mjk and pk are the flow, conductance and fluid nodal pressure of segment j.
Discrete-‐Con4nuum Model3
• Darcy’s law is describes the coupling between blood velocity, u, and pressure, p, with the aid of κ, the permeability of the capillary network to fluid transport.
• Bloody supply into the capillaries is represented by influx condi@ons at point sources represen@ng connec@ons between arterioles and capillaries. Oublow to the venules is accounted for by a constant drainage term, β, chosen to conserve mass.
• Conserva@on of mass yields
where pv is constant venous pressure, Nt is the no. of @ssue points and Cj are source strengths.
Green’s Func4on Method for Oxygen Delivery4
• Oxygen is bound to RBCs, dissolved in plasma and diffuses into @ssue then metabolised by cells.
• Oxygen sources represent blood vessels and a set of discrete oxygen sinks represent @ssue.
• Using Fick’s law of diffusion and conserva4on of mass, the Green’s func@on, G(x;x*), for a given domain may be defined as the PO2 at a point x resul@ng from a unit point source at x* is the solu@on to
where D and α are oxygen diffusivity and solubility.
UCL MECHANICAL ENGINEERING
qj = M jk pkk∈N∑
∇⋅u = −κ ∇2p = −β(p − pv )+ Cj (x)δ (x − x j )j=1
Nt
∑
Dα∇2G = −δ (x − x*)
Figure 1. Rat mesentery -‐ (a) Discrete modelling of fluid pressure (mean segment pressure -‐ mmHg). (b) Discrete-‐con@nuum model predic@ons of the pressure profiles in both the arteriolar network and capillary domain.
Case Study (A) – Rat Mesentery
• Discrete Model – max. blood pressure was 81.56 mmHg along with a min. of 13.8 mmHg (venous oublow pressure, pv). Mean capillary pressure was 26.23 mmHg.
• Discrete-‐Con4nuum Model – capillary permeability, κ, was chosen by comparing metrics for known pressure and flow condi@ons in the discrete model.
• % errors in the mean and standard devia@on of the source pressures was within 10%.
• % errors in the mean and standard devia@on of source flows was less successful, at ~40%.
Case Study (B) – Mouse Cortex
• Oxygen Delivery – simula@ons were run on both a healthy and ischaemic stroke induced network to compare PO2 levels.
• Healthy – mean & max. PO2 of 32.11 and 93.27 mmHg with a SD of 3.72.
• Stroke – mean & max. PO2 of 29.44 and 71.39 mmHg and a SD of 3.43.
• A clear shim in PO2 can be seen (Fig. 2) when a stroke is induced, indica@ng an increase in hypoxia.
Figure 2. Mouse Cortex -‐ (a) Tissue PO2 levels in (i) Healthy (blue) & (ii) stroke induced (red) simula@ons (b) O2 delivery to healthy @ssue (mmHg).
• Extend to 3D domain and incorporate discrete venular network.
• Apply Discrete-‐Con@nuum model to mouse cortex. • Study effects of microvascular blockages on PO2 transport. • Incorporate inters44al flow to predict 4ssue-‐scale fluid
and drug transport in porous and healthy blood vessels.