Age-Specific Characteristics and Coupling of Cerebral Arterial Inflow and Cerebrospinal Fluid Dynamics

The objective of this work is to quantify age-related differences in the characteristics and coupling of cerebral arterial inflow and cerebrospinal fluid (CSF) dynamics. To this end, 3T phase-contrast magnetic resonance imaging blood and CSF flow data of eleven young ( years) and eleven elderly subjects ( years) with a comparable sex-ratio were acquired. Flow waveforms and their frequency composition, transfer functions from blood to CSF flows and cross-correlations were analyzed. The magnitudes of the frequency components of CSF flow in the aqueduct differ significantly between the two age groups, as do the frequency components of the cervical spinal CSF and the arterial flows. The males' aqueductal CSF stroke volumes and average flow rates are significantly higher than those of the females. Transfer functions and cross-correlations between arterial blood and CSF flow reveal significant age-dependence of phase-shift between these, as do the waveforms of arterial blood, as well as cervical-spinal and aqueductal CSF flows. These findings accentuate the need for age- and sex-matched control groups for the evaluation of cerebral pathologies such as hydrocephalus.


Nominal cardiac cycle length
The influence of extending diastole to normalize the volunteers' cardiac cycle length can be shown via the example of a periodic square pulse wave [1] with pulse duration of 2ϕ and period of ωT, see (1) This periodic square pulse can be rendered comparable to the temporal course of a subject's cardiac cycle by taking T sys (systole) constant and T variable, corresponding to different CCs. An additional similarity between the real flow curves and the square pulse is the high signal amplitude during T sys and the small amplitude during the rest of the period.
= a 0 + a 1 cos(ωt) + a 2 cos(2ωt) + + • • • + a n cos(nωt), where a 0 , a 1 , . . . , a n are the Fourier coefficients. For the periodic square pulse wave f (ωt) (1) with |f | = hT sys T , the resulting Fourier coefficients are The frequency component a 0 corresponds to the normalized net flow. In this special case, the nonnormalized net flow is since for this particular periodic square pulse wave |f (ωt)| = f (ωt).  We show 1) that the frequency components a 1 . . . a n depend on the period length T (CC) as well as on the pulse width T sys (systole); 2) that even if the systoles of two subjects are equal, but the CCs are not, the amplitudes of the frequency components will be different; and 3) the spacings of the frequency components depend on T. For these reasons, it is necessary to normalize the heart rates if frequencies are analyzed as a mean of several subjects.
The extension of the diastolic period has the effect that the amplitudes of the frequency components are scaled uniformly. This effect would also be observed if the same subject, instead of having his or her period extended, would be scanned at the corresponding lower heart rate. Additionally, frequency patterns' characteristics reported in [2][3][4] are comparable to our homogenized and pooled results (Fig.  2 in main article), which indicates that the data were rendered comparable without changing their characteristics.

Transfer function identification
A parameter estimation method [5] was applied to the input-output data of one CC. The insufficient frequency content of the signal impeded the use of a nonparametric identification method, e.g. calculating the TF by dividing each amplitude of the output by the corresponding input frequency component, since it would result in an unduly noisy function where the magnitude and phase patterns would not be detectable. A linear fifth-order model was analyzed by the prediction-error identification method (PEM) which, using a least-squares estimation, identifies the model parameters by minimizing the difference between model output and measured data. The input-output data was first preprocessed by detrending, i.e. by the elimination of offsets and drifts. Subsequently, the influence of the input to the output was characterized without considering the actual data level. Flow measurements of one volunteer were repeated and the identified models validated with the data of the second scan. The models were simulated with the input from the respective validation data, while the corresponding outputs (ŷ) were compared to the measured data (y).
The fit was calculated according to (11), with y being the mean of the measured output.
The fit of the identified models for the arterial-to-cervical flow was 80% and for the arterial-toaqueductal transmission 95%.
Nonlinear Hammerstein-Wiener and nonlinear ARX models were also tested, validated and compared to the linear models. The non-linear models gave a poorer fit than the linear ones and most identified non-linear models became instable during the validation process.