A Pulsatile Cerebrospinal Fluid Model for Hydrocephalus Kathleen Wilkie Workshop on Brain Biomechanics: Mathematical Modelling of Hydrocephalus and Syringomyelia Centre for Mathematical Medicine Fields Institute Friday July 27, 2007
Collaborators This work was done in collaboration with ◮ Prof. S. Sivaloganathan ◮ Prof. G. Tenti ◮ Dr. J.M. Drake ◮ Dr. A. Jea, and was supported by NSERC. [Medtronic, Inc. 2007]
Overview ◮ Recent research by Egnor et al. [2001] and others suggest that CSF pulsations may be an important factor in the pathogenesis of hydrocephalus.
Overview ◮ Recent research by Egnor et al. [2001] and others suggest that CSF pulsations may be an important factor in the pathogenesis of hydrocephalus. ◮ The goal is to determine if these pulsations are mechanically relevant to the development of hydrocephalus.
Overview ◮ Recent research by Egnor et al. [2001] and others suggest that CSF pulsations may be an important factor in the pathogenesis of hydrocephalus. ◮ The goal is to determine if these pulsations are mechanically relevant to the development of hydrocephalus. ◮ My tools include a one-compartment CSF model and a poroelastic thick-walled cylinder brain parenchyma model.
Overview ◮ Recent research by Egnor et al. [2001] and others suggest that CSF pulsations may be an important factor in the pathogenesis of hydrocephalus. ◮ The goal is to determine if these pulsations are mechanically relevant to the development of hydrocephalus. ◮ My tools include a one-compartment CSF model and a poroelastic thick-walled cylinder brain parenchyma model. ◮ The poroelastic model provides a time- and space-dependent analysis of the pulsations which demonstrate the mechanical effects the pulsations have on the parenchyma.
The One-Compartment CSF Model This is an extension of the one-compartment model described in Sivaloganathan et al. [1998]. Elastic�Walls CSF Compartment CSF�Formation CSF�Absorption By the principle of conservation of mass, assuming CSF to be incompressible, the governing equation can be written as: � rate of volume � rate of CSF � rate of CSF � � � = . (1) − change in time formation absorption
The One-Compartment CSF Model Since intracranial volume depends on pressure, V ( t ) = V ( P ( t )) , and in mathematics we write, � rate of volume � = d V d t = d V d P d t = C ( P ) d P (2) d t , change in time d P where C ( P ) is the compliance function.
The One-Compartment CSF Model The rate of CSF formation is assumed to be in the following form: � rate of CSF � constant rate of � pulsatile rate of � � � = + formation CSF formation CSF formation I ( e ) + a sin 2 ( ω t ) , = (3) f where a is the displacement of CSF due to blood flow and ω is the angular frequency of the heart beat.
The One-Compartment CSF Model The rate of CSF formation is assumed to be in the following form: � rate of CSF � constant rate of � pulsatile rate of � � � = + formation CSF formation CSF formation I ( e ) + a sin 2 ( ω t ) , = (3) f where a is the displacement of CSF due to blood flow and ω is the angular frequency of the heart beat. Finally, experimental evidence has shown that � rate of CSF � = 1 ( P ( t ) − P ss ) , (4) absorption R a where R a is the resistance to CSF flow and P ss is the sagittal sinus pressure.
The One-Compartment CSF Model Putting all of this together gives a differential equation describing the pressure in the CSF model: C ( P ) d P d t + 1 ( P ( t ) − P ss ) = I ( e ) + a sin 2 ( ω t ) . (5) f R a I will consider two cases: 1. the simple case, when compliance is constant: C ( P ) = C 0 , and 1 2. when compliance fits the experimental data: C ( P ) = kP .
Case 1. CSF Model with Constant Compliance The differential equation d t + 1 d P ( P ( t ) − P ss ) = I ( e ) + a sin 2 ( ω t ) , C 0 (6) f R a together with the initial condition P ( t = 0) = P 0 , has the solution aR a 4 ω 2 τ 2 � � e − t P 0 − R a I ( e ) 0 P ( t ) = − P ss − τ 0 f 2(1 + 4 ω 2 τ 2 0 ) + P ss + 1 aR a R a I ( e ) + 2 aR a − f � 1 + 4 ω 2 τ 2 2 0 aR a � ω t − 1 � sin 2 2 tan − 1 (2 ωτ 0 ) + (7) , � 1 + 4 ω 2 τ 2 0 where τ 0 = C 0 R a is the characteristic time.
Case 1. CSF Model with Constant Compliance Looking at the oscillating term, (remember τ 0 = C 0 R a ) aR a � ω t − 1 � sin 2 2 tan − 1 (2 ωτ 0 ) , � 1 + 4 ω 2 τ 2 0 1 ◮ if C 0 R a << 2 ω then the resulting phase shift is π 4 , 1 ◮ if C 0 R a = 2 ω then the resulting phase shift is π 8 , and ◮ if 1 C 0 R a >> 2 ω then the resulting phase shift is 0, i.e. the CSF pulsations are synchronous with the forcing.
Pressure using Data from Shapiro [1979] 16.2 16.1 Pressure [mm Hg] 16.0 15.9 15.8 0 1 2 3 4 5 6 Time [sec] Case 1. Simulations Typical values of the parameters for a normal adult are [Shapiro et al. 1979]: ◮ P ss = 12 . 2 mm Hg ◮ R a = 2 . 8 mm Hg/ml/min ◮ C 0 = 0 . 85 ml/mm Hg. Also chosen were ◮ I ( e ) = 0 . 35 ml/min, f ◮ ω = 140 π rad/min, and ◮ a = 2 ml/min.
Case 1. Simulations Typical values of the Pressure using Data from Shapiro [1979] 16.2 parameters for a normal adult are [Shapiro et al. 1979]: 16.1 ◮ P ss = 12 . 2 mm Hg ◮ R a = 2 . 8 mm Hg/ml/min Pressure [mm Hg] 16.0 ◮ C 0 = 0 . 85 ml/mm Hg. Also chosen were 15.9 ◮ I ( e ) = 0 . 35 ml/min, f 15.8 ◮ ω = 140 π rad/min, and 0 1 2 3 4 5 6 Time [sec] ◮ a = 2 ml/min. Using these values, the model predicts pressure pulsations that would not be visible on typical ICP measurements.
Pressure in Synchrony with Arterial Forcing 20 18 16 14 Pressure [mm Hg] 12 10 8 0 1 2 3 4 5 6 Time [sec] Case 1. Simulations Using I ( e ) = 0 . 35 ml/min, f P ss = 10 mm Hg, and ω = 140 π rad/min, and requiring that the: ◮ pressure pulsations have peak-to-peak amplitude of 5 mm Hg, ◮ mean CSF pressure is 13 . 5 mm Hg, and ◮ the phase shift is zero (i.e. synchrony exists)
Case 1. Simulations Using I ( e ) Pressure in Synchrony with Arterial Forcing = 0 . 35 ml/min, 20 f P ss = 10 mm Hg, and 18 ω = 140 π rad/min, and 16 requiring that the: ◮ pressure pulsations have 14 Pressure [mm Hg] peak-to-peak amplitude of 12 5 mm Hg, 10 ◮ mean CSF pressure is 8 13 . 5 mm Hg, and 0 1 2 3 4 5 6 ◮ the phase shift is zero Time [sec] (i.e. synchrony exists) results in a waveform consistent with experiments and values of: ◮ R a = 2 . 86 mm Hg/ml/min ◮ C 0 = 3 . 98 · 10 − 6 ml/mm Hg, and ◮ a = 1 . 75 ml/min.
Case 1. Simulations Time changes in the amplitude of the pulsatile CSF formation rate ( a ), the base CSF formation rate ( I ( e ) ), and the resistance to f CSF absorption ( R a ) may help explain the appearance of plateau or B waves observed in patients with hydrocephalus. Example of Production Rate Amplitude Increasing to Cause Appearance of a B-wave 30 20 Pressure [mm Hg] 10 0 0 5 10 15 Time [min]
Case 2. CSF Model with Experimental Compliance In 1978, Marmarou et al. determined that the pressure-volume relationship is exponential, implying that compliance is of the form C = 1 kP , where ln 10 is known as the pressure-volume index ( PVI ). k Pressure-Volume Curve Compliance Function 80 50 70 40 60 50 30 Pressure 40 Compliance 20 30 20 10 10 0 0 0 5 10 15 20 0 5 10 15 20 Volume Pressure [mm Hg]
Case 2. CSF Model with Experimental Compliance The governing differential equation now becomes 1 d t + 1 d P ( P ( t ) − P ss ) = I ( e ) + a sin 2 ( ω t ) , (8) f kP ( t ) R a which is a Riccati equation with solution P 0 e k ( I ( e ) + Pss Ra + a 2 ) t − k 4 ω a sin(2 ω t ) f P ( t ) = (9) . � t 0 e k ( I ( e ) + Pss Ra + a 2 ) s − k 4 ω a sin(2 ω s ) d s 1 + k P 0 f R a
Case 2. Simulations Using parameter values of ◮ R a = 2 . 8 mm Hg/ml/min [Shapiro 1979] ◮ I ( e ) = 0 . 35 ml/min f ml − 1 [Shapiro 1979] ◮ k = 2 . 3026 25 . 9 ◮ a = 1 . 75 ml/min Pressure With Pulsations (a=1.75) 20 the compliance of the compartment is approximately 18 0 . 8 ml/mm Hg which is too large to allow 16 Pressure [mm Hg] pulsations with a peak-to-peak 14 amplitude of 5 mm Hg. 12 10 0 5 10 15 20 Time [min]
Case 2. Simulations To decrease compliance, the pressure-volume index must decrease. Choosing PVI = 0 . 002 ml gives k = 1151 . 3 ml − 1 which corresponds to a compliance of approximately C = 6 . 4 · 10 − 5 ml/mm Hg. Setting a = 1 . 75 ml/min results in a presure profile with peak-to-peak amplitude of about 5 mm Hg. Pressure With Pulsations (a=1.75) 18 17 16 15 Pressure [mm Hg] 14 13 12 11 10 0 1 2 3 4 5 6 Time [sec]
Conclusions from the One-Compartment Model ◮ Using experimentally determined parameter values, the model does not predict experimentally observed pressure pulsations.
Conclusions from the One-Compartment Model ◮ Using experimentally determined parameter values, the model does not predict experimentally observed pressure pulsations. ◮ Using much smaller values of compliance, the model accurately predicts experimentally observed pressure pulsations.
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