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. christian.contarino@unitn.it

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.

christian.contarino@unitn.it