The role of water dynamics in the glymphatic system through a holistic multi-scale mathematical model of the murine extracellular
fluid systems
University of Trento, Italy
Christian Contarino, A. Louveau, S. Da Mesquita, D. Raper, I. Smirnov, N. Agarwal, J. Kipnis and E. F. Toro
XIV Biennial Conference of the Italian Society of Applied and Industrial Mathematics
5/06/2018, Rome
Brain as a sponge
Brain water dynamics?
Louveau et al. 2015, Nature Meningeal lymphatic system drain cerebrospinal fluid
Meningeal lymphatic vessels
• Louveau, A. et al. Structural and functional features of central nervous system lymphatic vessels. Nature 2015
MRI illustration of meningeal lymphatics in human. 3D-rendering of dural lymphatics (green) in a 47 year old woman from skull-stripped subtraction T1-black-blood images.
Meningeal lymphatic vessels
• Absinta, M. et al. Human and nonhuman primate meninges harbor lymphatic vessels that can be visualized noninvasively by MRI. eLife 2017
Glymphatic system
• Louveau, A. et al. Journal of Clinical Investigation 2017 "Understanding the functions and relationships of the glymphatic system and meningeal lymphatics”.
Glymphatic system
• Iliff, J. J. et al. Science Translational Medicine 2012 "A paravascular pathway facilitates CSF flow through the brain parenchyma and the clearance of interstitial solutes, including amyloid β”.
• Jessen, N. A. Neurochemical research 2015 “The glymphatic system: A beginner’s guide”.
Glymphatic system
Driving forces?
Fluid systems
Holistic approach
Multi-scale mathematical model
Zero-dimensional (0D) One-dimensional (1D)
8<
:
@t
A+ @x
q = 0 ,
@t
q + @x
�↵ q
2
A
�+ A
⇢
@x
p = �f
⇢,
A(x, t) q(x, t)V (t)
p = p
✓A
A0
◆+ p
ext
p = p(V ) + pext
dt
V =X
in
qi
�X
out
qi
Mathematical model of the murine extracellular fluid system
• 253 Blood vessels (major arteries and veins) • 112 Lumped parameter models • 161 Connections between lumped parameter models
Hyperbolic Partial Differential Equations (PDEs)
Ordinary Differential Equations (ODEs)
System of 779 equations
+@t
Qk
+ @x
F(Qk
) = S(Qk
),
Mathematical model of the murine extracellular fluid system
d
dtW = G (W,Q1, . . . ,Qn, t)
Mathematical model of the murine extracellular fluid system
Peripheral blood flow
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
80
90
100
110
Pres
sure
[mm
Hg]
-20
0
20
40
60
80
Flow
[mL/
min]
Ascending aorta (1)93.7 mmHg9.65 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
75
80
85
90
95
100
Pres
sure
[mm
Hg]
0
0.5
1
1.5
2
Flow
[mL/
min]
L. subclavian artery II (17)90.0 mmHg0.69 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
80
85
90
95
100
105
Pres
sure
[mm
Hg]
-20
0
20
40
60
Flow
[mL/
min]
Thoracic aorta I (12)93.5 mmHg5.62 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
80
90
100
110
120
Pres
sure
[mm
Hg]
-10
0
10
20
30
Flow
[mL/
min]
Abdominal aorta I (25)93.2 mmHg4.00 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
80
90
100
110
120
Pres
sure
[mm
Hg]
-5
0
5
10
15
Flow
[mL/
min]
Abdominal aorta V (33)92.9 mmHg1.63 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
80
90
100
110
120
Pres
sure
[mm
Hg]
0
1
2
3
4
Flow
[mL/
min]
L. external Iliac artery (50)92.5 mmHg0.67 mL/min
Marjor veins: pressure and flow dynamics Marjor arteries: pressure and flow dynamics
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0.6
0.8
1
1.2
1.4
Pres
sure
[mm
Hg]
0
1
2
3
4
5
Flow
[mL/
min]
R. superior vena cava (84)1.0 mmHg2.10 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0
0.5
1
1.5
Pres
sure
[mm
Hg]
0
5
10
15
Flow
[mL/
min]
Inferior vena cava (85)0.9 mmHg5.22 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0
0.5
1
1.5
2
Pres
sure
[mm
Hg]
-2
0
2
4
6
Flow
[mL/
min]
Inferior vena cava V (93)1.0 mmHg1.82 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0
0.5
1
1.5
2
Pres
sure
[mm
Hg]
-0.5
0
0.5
1
1.5
2
Flow
[mL/
min]
R. external Iliac vein (100)1.1 mmHg0.66 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0.8
1
1.2
1.4
1.6
Pres
sure
[mm
Hg]
0
0.2
0.4
0.6
0.8
Flow
[mL/
min]
Azygos vein (129)1.2 mmHg0.42 mL/min
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0.8
1
1.2
1.4
Pres
sure
[mm
Hg]
0.2
0.4
0.6
0.8
1
1.2
Flow
[mL/
min]
R. subclavian vein III (131)1.2 mmHg0.69 mL/min
41
54
82
Modelled dynamics and mechanisms
• Heart and pulmonary dynamics (Sun et al. 1997, Liang et al. 2009)
• Arterial and venous systems (Müller and Toro 2014)
• Brain and peripheral microcirculation (Müller and Toro 2014)
• Venous valves (Mynard et al. 2012)
• Intracranial Starling resistors (Müller and Toro 2014)
• Cerebrospinal fluid (CSF) dynamics (Linninger et al. 2009, Linninger et al. 2017)
• Modern concept of CSF/ISF dynamics (Oreškovic et al. 2017, Linninger et al. 2017)
• Brain lymphatic drainage• Monroe-Kellie coupling
• Orešković, D. et al. (2017). New concepts of cerebrospinal fluid physiology and development of hydrocephalus. Pediatric Neurosurgery, 52(6), 417–425.
• Linninger, A. A. et al. (2009). A mathematical model of blood, cerebrospinal fluid and brain dynamics. Journal of Mathematical Biology, 59(6), 729–759.
• Linninger, A. A. et al. (2017). Starling forces drive intracranial water exchange during normal and pathological states. Croatian Medical Journal, 58(6), 384–394.
• Sun, Y. et al. (1997). A comprehensive model for right-left heart interaction under the influence of pericardium and baroreflex. American Journal of Physiology, 272(3 Pt 2), H1499–H1515.
• Liang, F. et al. (2009). Multi-scale modeling of the human cardiovascular system with applications to aortic valvular and arterial stenoses. Medical and Biological Engineering and Computing, 47(7), 743–755.
• Mynard, J. P. et al. (2012). A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models. International Journal for Numerical Methods in Biomedical Engineering, 28(6–7), 626–641.
• Müller, L. O., & Toro, E. F. (2014). Enhanced global mathematical model for studying cerebral venous blood flow. Journal of Biomechanics, 47(13), 3361–3372.
CSF absorption by lymphatics
and through arachnoid villi
Brain interactive fluid dynamics
Right ventricleLeft ventricle
Third ventricle
Aqueduct of Sylvius
Fourth ventricle
Cerebral subarachnoid space Superior sagittal sinus
Spinal subarachnoid space
Arterioles Capillaries Venules
Interstitial fluid
Lymphatics
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
Validation of the mathematical model
• Validation with in-vivo intracranial pressure • Validation with SPCP-MR flow measurements • Validation with mouse model of Idiopathic Intracranial Hypertension • Validation with existing literature values
MRI
ISF
In-vivo intracranial pressure
In-vivo mouse model of Idiopathic Intracranial Hypertension
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
LV RV
3V
AoS
4V
Cerebral subarachnoidspace
ISF
0 20 40 60 80 100Reference cardiac cycle [%]
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
Pres
sure
[mmH
g]
Mean reference in vivo pressureMean 0.5 SD in vivo pressureMean SD in vivo pressureComputational result
Computational results vs In-vivo intracranial pressure
Validation with in-vivo intracranial pressure
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
Validation with SPCP-MR flow measurements
0 10 20 30 40 50 60 70 80 90 100Reference cardiac cycle [%]
0
10
20
30
40
50
60
70
80
Flow
[mL/
min
]Ascending aorta (1)
Mean reference MR flow measurementsMean 0.5 SD MR flow measurementsMean SD MR flow measurementsComputational result
0 10 20 30 40 50 60 70 80 90 100Reference cardiac cycle [%]
0
1
2
3
4
5
6
7
Flow
[mL/
min
]
L. common carotid artery (5)
Mean reference MR flow measurementsMean 0.5 SD MR flow measurementsMean SD MR flow measurementsComputational result
0 20 40 60 80 100Reference cardiac cycle [%]
0
0.5
1
1.5
2
2.5
Flow
[mL/
min
]
R. external jugular vein (155)
Mean reference MR flow measurementsMean 0.5 SD MR flow measurementsMean SD MR flow measurementsComputational result
Validation with SPCP-MR flow measurements
Interactive fluid systems
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
Arteries
Veins
Microcirculation
Cerebrospinal fluid
Interstitial fluid
Arteries
Veins
Cerebrospinal fluid
Microcirculation
Interstitial fluid
Monroe-Kellie hypothesisThe cranial compartment is
incompressible and the volume inside the cranium is fixed.
Monroe-Kellie hypothesis: a mathematical model
Monroe-Kellie hypothesis: a mathematical model
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0
0.2
0.4
0.6
0.8
1
Norm
alize
d flo
w [-]
Systo
lic p
eak =
1
Interaction between the four brain fluid system compartmentsAortic flowCranial arterial inflowCranial venous outflowSpinal CSF outflowAqueduct of Sylvius flow
0 0.02 0.04 0.06 0.08 0.1 0.12Time [s]
0
0.2
0.4
0.6
0.8
1
Norm
alize
d vo
lume
[-]Sy
stolic
pea
k = 1
Intracranial volumes interaction: arterial, venous and CSF compartments
Aortic flowIntracranial arterial volumeIntracranial venous volumeIntracranial CSF volume
Brain fluid interaction: arteries, veins and cerebrospinal fluid
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
Interactive fluid systems
• Contarino, C. et al. IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
Idiopathic Intracranial Hypertension
Idiopathic Intracranial Hypertension (IIH)
• Neurological disorder 2 per 100.000 people worldwide • Abnormal increase of the intracranial pressure • Causes Headache, tinnitus, papilledema • 90% suffer from strictures in the transverse sinus (Farb et al. 2003)
Motivation
Can an impairment of
cerebral venous blood outflow
affect
waste product collection in the brain?
Validation with mouse model of Idiopathic Intracranial Hypertension
• Contarino, C. et al. Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
Healthy Bilateral ligation
Validation with mouse model of Idiopathic Intracranial Hypertension
• Contarino, C. et al. Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
Validation with mouse model of Idiopathic Intracranial Hypertension
0 20 40 60 80 100Reference cardiac cycle [%]
4.5
5
5.5
6
6.5
7
7.5
8
Pres
sure
[mm
Hg]
Mean reference in vivo pressureMean 0.5 SD in vivo pressureMean SD in vivo pressureComputational result
LV RV
3V
AoS
4V
Cerebral subarachnoidspace
ISF
Computational results vs in-vivo intracranial pressure
Mouse model with bilateral ligation of petrosquamosus sinuses vs mathematical model
• Contarino, C. et al. Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
Effect of impairment of venous drainage on brain fluid dynamics
-45.
31 %
-5.5
6 %
+74.
38 %
-39.
35 %
21.5
11.0
2.0
29.8
CSF prod. by ISF space
CSF prod. by choroid plexus
CSF abs. by lymphatics
CSF abs. by arachnoid villi0
5
10
15
20
25
30
35
Flow
[L/
h]
Superior sagittal sinus (218)
L. transverse sinus (194)
L. petrosquamosus sinus (188)
L. posterior facial (160)
L. external jugular vein (155)1
2
3
4
5
6
7
Pres
sure
[mmH
g]
+73.
22 %
+159
.38 %
+87.
66 %
-3.1
5 %
-2.2
9 %
3.2
2.1
1.8
1.4 1.4
HealthyBilateral ligation
Arteries
Veins
Microcirculation Cerebrospinal fluid
Interstitial fluid
Lymphatics
• Contarino, C. et al. Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
Impairment of cerebral venous blood outflow
causes a local reduction of ISF efflux and CSF
turnover, potentially leading to
local accumulation of neurotoxins in the brain
• Contarino, C. et al. Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
Special thanks to….
E. F. Toro
Kipnis labJonathan Kipnis
Antoine Louveau Sandro Da Mesquita Dan Raper Jasmin HerzIgor SmirnovRonen WeissTony FilianoGeoffrey Norris Chris OverallAshtyn Smith Andrea Salvador Wendy BakerDylan GoldmanKenneth ViarReinaldo Oria Caroline Addington Zhongxiao Fu
The BIG Center Department for Neuroscience
University of TrentoDepartment of Mathematics
Eleuterio ToroAlberto Valli
Jack Roy Rene
University of Virginia Department of Radiology
Kenneth LiuThomas Buell
University of Virginia Department of Neurological Surgery
Center for Mind/Brain Sciences CIMeCNivedita Agarwal
Adelisa Avezzù Davide Chieco
Special thanks to….
Christian Contarino, Ph.D.,B.Sc, M.Sc, M.Mus. [email protected]
References
• C. Contarino, A. Louveau, S. Da Mesquita, D: Raper, I. Smirnov, N. Agarwal, V. Kurtcuoglu, J. Kipnis and E. F. Toro, Scientific Reports “Heart contraction, Starling forces and cerebrospinal fluid absorption drive the glymphatic system”. In preparation
• C. Contarino, A. Louveau, S. Da Mesquita, D: Raper, I. Smirnov, N. Agarwal, V. Kurtcuoglu, J. Kipnis and E. F. Toro, IJNMB “A holistic multi-scale mathematical model of the murine extracellular fluid systems and study of the brain interactive dynamics”. In preparation
• S. Da Mesquita, A. Louveau, A. Vaccari, I. Smirnov, R. C. Cornelison, K. M. Kingsmore, C. Contarino, S. Onengut-Gumuscu, E. Farber, D. Raper, K. E. Viar, R. D. Powell, W. Baker, N. Dabhi, R. Bai, R. Cao, S. Hu, S. S. Rich, J. M. Munson, M. B. Lopes, C. C. Overall, S. T. Acton and J. Kipnis, Nature “Functional aspects of meningeal lymphatics in aging and Alzheimer’s disease”. Accepted Christian Contarino, Ph.D.,B.Sc, M.Sc, M.Mus.
