Modelling cardiovascular physiology
Fantastic
voyage (1966)
Dr Emma Chung (emlc1@le.ac.uk)
University of Leicester
Department of Cardiovascular Sciences
Distance dramatically slows diffusion
The rate of diffusive transport is
important, because nutrient
delivery needs to keep up with
metabolic demand
Einstein’s theory of Brownian
Motion (1905)
tx
Example
Distance (x) Time (t)
Capillary wall
1.0 m
510-6 s
Capillary to cell
10 m
0.05 s
Skin to artery wall
1 mm
9.26 min
Left ventricular wall
1 cm
15.4 h
Table showing diffusion times for glucose. (NB. for glucose in water at
37C D=0.9 10-5, and for oxygen in water D=310-5 cm2s-1)
2
2D
D is the solute coefficient,
x is distance and t is time
Most cells lie between
10 and 20 m of the
nearest capillary
Evolutionary biology
Very simple organisms like the
hydra exchange oxygen and
nutrients purely by diffusion
In larger organisms
the organs are
supplied via a
‘closed
cardiovascular
system’ with
varying numbers of
hearts!
Many molluscs and invertebrates
have an ‘open circulatory system’
(e.g. Snails)
3
Vasculogenesis in the embryo
The initial ‘setting up’ of the blood vessels is dictated by the genes.
At day 21 of embryogenesis, the heart begins beating and vascular remodelling
starts to occur.
Pressure, flow patterns, shear stress and metabolic demand then interact with
genetic and environmental factors to produce the mature vasculature.
Optimising the delivery of fluid by branching
5
Branching angle (Murray’s law 1926)
Optimal branching angle depends on fluid
dynamical efficiency at the bifurcation.
Branches with smaller diameters make
larger branching angles
Murray's law (1926)
The hydraulic conductance
per blood volume of the
cardiovascular system is
maximized when the cube
of the radius of a parent
vessel equals the sum of
the cubes of the radii of the
daughters.
‘Small side branch’
Coronary arteries
• Genetic code
• Physiology
• Fluid dynamical efficiency
AIM: Simulate optimised 3D
vascular
trees
for
any
arbitrary tissue or organ
For inclusion in simulations of
blood flow and autoregulation
Medical applications include
the design of 3D printed
artificial organs (e.g. for
kidney replacement)
3D bio-printing of replacement organs
Making a bit of me: A machine that prints organs is coming to market
The Economist, Feb 18th 2010
“the company expects that within
five years, once clinical trials are
complete, the printers will produce
blood vessels for use as grafts in
bypass surgery. With more research
it should be possible to produce
bigger, more complex body parts.
Because the machines have the
ability to make branched tubes, the
technology could, for example, be
used to create the networks of
blood vessels needed to sustain
larger printed organs, like kidneys,
livers and hearts.”
- Keith Murphy, Organovo's Chief
Executive
Mini-human brains only 4 mm wide grown from stem cells
“The mini-brains can not
grow larger than a few
millimetres because of a
lack of blood supply”
Lancaster et al. Cerebral organoids model human brain
development and microcephaly. Nature 501, 373–379, 2013
Constrained Constructive Optimisation (CCO)
The current standard
technique for generating
arterial trees is called
constrained constructive
optimisation
In silico vascular tree
grown using constrained
constructive optimisation
(optimised using genetic
algorithms)
Jonathan Keelan, Jim Hague and Emma Chung
Coronary arteries
• Anatomy of the major
coronary arteries is
similar between
individuals
• Large trunk vessels asymmetric branching
• Only fine vessels
venture into the tissue
Modelling the coronary vasculature - CCO
CCO growth of the first
few layers of coronary
arteries to supply a
‘myocardial substrate’
simulating the left atrium
Jonathan Keelan, Jim Hague and Emma Chung
•
•
•
Too symmetrical
Too many ‘constraints’ required
(especially for hollow organs)
No guarantee that the final solution is
globally optimised
Simulated annealing (SA)
Simulated Annealing (SA) techniques can be used to obtain a good
approximation to the global optimum of a given function in a
large search space.
Modelling the coronary vasculature – Simulated Annealing
Coronary arterial trees
grown using Simulated
Annealing to achieve
global optimisation
Only need to define:
• Geometry and
metabolic demand
of the tissue
substrate
• Position of the start
nodes
• Number of nodes
Looks much better, but how can we make a
quantitative comparison with real coronary arteries?
• Vessels above a
certain size were
not allowed to
penetrate the tissue
Strahler (stream) Order
Within the Strahler Order scheme the lowest Strahler order numbers
correspond to the smallest arterioles and the largest numbers to major vessels.
Vessels of the same Strahler Order in the simulated coronary arterial trees
were grouped.
The average length, diameter and branching properties were then compared
to existing morphological data from Kassab et al. 1993
Results of comparison with pig coronary arteries
Vessel diameter
Vessel length
Artefact
from early
termination
Simulated annealing provided an almost perfect match to pig morphological
data for vessel lengths and diameters.
Kassab et al. Morphometry of pig coronary arterial trees. American Journal of Physiology,
265(1):H350{65, 1993
Results of comparison with pig coronary arteries
Ratio of diameter of smallest
daughter (DS) to parent
vessel (DP).
For the major arteries there tends to be a major trunk vessel of similar size
to the parent vessel with a small side branch. As vessels become smaller
(i.e. For lower Strahler orders) the branches become more symmetric.
Modelling the cerebral arteries
“The brain is the most complex
structure in the universe”
• Very high density of vessels Grey matter makes up less
than 1% of the mass of the
body but accounts for nearly
20% of total oxygen
consumption
• Two types of tissue substrate
(grey and white matter)
• Fluctuating metabolic
demand
• Complex geometry with
boundaries
• Position of the start nodes??
Anatomy varies between individuals
ACAs
MCA
BA
Posterior
communicating
arteries
VA
Relative sizes of the vessels
There is no typical ‘textbook’ circle of Willis
Variations in the circle of
Willis identified by Kleiss
(1941).
Wax models and static phantoms of the circulation
18th century Wax
model of the arteries
from La Specola
museum, Florence,
Polymer cast of the
cerebral circulation
from Gunter von
Hagens’ “Body
Worlds” exhibition.
Scanning electron
microscopy of a corrosion
cast of the cerebral
microcirculation. (La
Torre 1998)
With modern imaging methods we can create patient
specific models
Vascular phantom fabrication
Rapid
prototyping
CAD
package
Virtual
patient
simulation
Imaging data
Laboratory vascular
replica
Vascular phantom
• Programmable pump
unit
• Injection ports
• Silicon model of the
major arteries
• Distal resistances
controlled by length and
diameter of outlet tubing
• Reservoir containing a
Blood Mimicking Fluid
(BMF)
Electrical circuit analogue approaches
V=IR
Electrical circuit analogue
‘Idra’
‘Electra’
Study of cerebral hemodynamics by analogic models Fasano et al. (1966)
Electrical circuit
models
R
8l
a 4
Perfusion territories supplied by the circle of Willis
Arterial Spin Label imaging with MRI
Hendrikse et al.
Cerebral autoregulation
Bayesian analysis of cerebral blood
flow autoregulatory mechanisms for
personalised medicine.
Computational fluid dynamics
Computational Fluid Dynamics
(CFD) simulations of blood flow
require a well-defined mesh to
model anatomically realistic
vasculature,and are good at
modelling complex
haemodynamics.
BUT... they struggle to model
solid-fluid interactions:
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•
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Courtesy of Quang Long (Brunel)
Interactions between bloodflow and vessel wall motion
Cardiac valve motion
Thrombus and plaque
formation
Particulate nature of the blood
Motion of objects in the flow
Do emboli ‘go with the flow’?
Results:
25%
1000 m emboli: 98:2
500 m emboli: 93:7
200 m emboli: 89:11
MCA:ACA flow: 75:25
75%
http://news.bbc.co.uk/1/hi/health/8516802.stm
High-speed video recorded at 300 fps and slowed to 30 fps (1 cardiac
cycle every 10 s)
Chung et al. Preferred embolus trajectory through a replica of the major cerebral arteries
Stroke 2010;41:647-652
Preferred embolus trajectory
75
30.2
2
8.4
Watershed
regions
0
3.4
10%
20%
5
18
30%
18
40
40%
Emboli
100
100
Flow
Emboli are more likely to
come to rest at the tips of the
major arteries and branches
Brain injury after cardiac surgery
Patient 13
Patient 19
Patient 63
Before
surgery
6-8 weeks
after
surgery
Subtraction
image
Registration and
subtraction of ‘before’ and
‘after’ 3-T MRI FLAIR
images to distinguish
between ‘old’ and ‘new’
lesions
Brain injury after cardiac surgery
New lesions following surgery are relatively minor compared to the number
and volume of pre-existing lesions due to chronic cardiovascular disease.
Brain injury after cardiac surgery
Composite image of new lesions observed in 24 patients after cardiac surgery
Smoothed-particle hydrodynamics (or: How we learned to
stop worrying (about meshes) and love particles)
Computational fluid dynamics
involves building a mesh and
watching how the fluid flows past
the mesh.
Smoothed Particle Hydrodynamics
(SPH) is a mesh-free Lagrangian
method (where the coordinates
move with the fluid)
Smoothed-particle hydrodynamics
Tom Hands, 4th year
Physics project
Monte-Carlo simulation of bubbles moving through the
cerebral vasculature
Capillary mesh ~ 6 m
12 m
n=3
n=5
n=1
β = 2.3
Major arteries > 1 mm
1 mm
History of flow in end arterioles.
Fully blocked clusters are shown
in black and free flow in white.
Featured article: Hague JP, Banahan C and Chung EML. Modelling of impaired cerebral blood
flow due to gaseous emboli. Phys. Med. Biol. 2013 58:4381-4394
Monte-Carlo simulation of bubbles moving through the
cerebral vasculature
Timing and diameters of bubbles entering the middle cerebral arteries in a 55 year
old patient during combined atrial valve replacement and coronary artery bypass
graft surgery. Monte-Carlo simulations are used to estimate the potential impact of
bubbles on MCA blood flow.
Funders:
Collaborators:
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Jim Hague (Open University)
Mark Horsfield (Leicester)
Kumar Ramnarine (UHL)
Ronney Panerai (Leicester)
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Engineering and Physical Sciences
Research Council (EPSRC)
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British Heart Foundation
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Wellcome Trust
Post-docs:
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NIHR Leicester Cardiovascular
Biomedical Research Unit
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TH Wathes Foundation
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Nihon Kohden (Japan)
Caroline Banahan (UHL)
Baris Kanber (Leicester)
PhD students:
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Nikil Patel (Leicester)
David Marshall (Leicester)
Jonathan Keelan (Open University)
Undergraduate students:
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Tom Hands
Katie Masters
Thank you!