# Transfer function analysis

## What is transfer function analysis?

The transfer function of a system is the relationship between the system’s input and output represented in the frequency domain. The technique is commonly used to characterise the autoregulatory function of vascular systems (Zhang et al., 1998, Wittmann et al., 1995).

The toolkit supports the calculation of the transfer function between blood pressure and blood flow velocity (or blood flow) measurements obtained on a beat-by-beat basis.

### What do you need?

• Blood pressure in mmHg.
• Blood flow velocity in cm s-1.

Ensemble currently defaults to blood flow velocity in cm s-1. However, the algorithms are valid for volumetric blood flow measurements as well. When using volumetric flow as the output, please remember to convert your units accordingly (e.g., ml/min).

### Analysis outputs

ParameterDefinition
Blood pressure powerSpectral power for blood pressure across a specified frequency range in mmHg2.
Flow velocity powerSpectral power for blood flow velocity across a specified frequency range in cm s-1.
Transfer functionLinear transfer function between blood pressure [input] and blood flow velocity [output].
VLFVery low frequency range. Default is 0.02-0.07 Hz.
LFLow frequency range. Default is 0.07-0.2 Hz
HFHigh frequency range. Default is 0.2-0.4Hz
CoherenceThe squared coherence function varies between 0 and 1, and is similar to the coefficient of determination in that values approximating zero may indicate a nonlinear relationship, severe extraneous noise in the signals, or no association between signals. A coherence value approaching 1 reflects a strong influence of input on output.
PhaseThe phase reflects the ‘time’ relationships between blood pressure and blood flow velocity across a specified frequency range. Typically a positive value is interpreted as flow leading pressure. The phase is expressed in radians.
GainThe gain reflects the amplitude relationships between blood pressure and blood flow velocity across a specified frequency range. The higher the gain, the greater the influence of a change in input power (blood pressure) is reflected within the output power (flow velocity). The default unit for gain is cm s-1 mmHg-1.
nGainSame as gain except the blood flow velocity values are normalized by dividing beat-to-beat values by the mean value. The unit for normalized gain is % mmHg-1.
Frequency bandsIn physiological research, it is common practice to present spectral power and transfer function coherence, gain, and phase as summary averages across specific frequency bands in Hz.
VLF lowSpecifies the lower cutoff frequency for the VLF band.
VLF highSpecifies the higher cutoff frequency for the VLF band.
LF lowSpecifies the lower cutoff frequency for the LF band.
LF highSpecifies the higher cutoff frequency for the LF band.
HF lowSpecifies the lower cutoff frequency for the HF band.
HF highSpecifies the higher cutoff frequency for the HF band.

### Settings definitions

ParameterDefinition
InputSpecifies the input blood pressure. Default is mean blood pressure (recommended).
OutputSpecifies the output blood flow velocity. Default is mean blood flow velocity (recommended).
Resample FsThe frequency at which beat-beat time series are resampled to achieve equidistant datasets for transfer function analysis. Default is 4 Hz.
Low-pass cutoffCutoff frequency for the low-pass filter. Default is 0.4 Hz.
Averaging methodSpecifies whether coherence is taken into account when calculating band averages. Default is to apply the coherence criterion.
SegmentSpecifies the number of overlapping segments for spectral density estimation using Welch’s method.
OverlapDetermines the size of data overlap. Default is 2 (corresponding to 50% overlap).
alphaThe a priori defined level for statistical significance.
Cursor visibleON shows graph cursors for locating point estimates on graphs.
Default is OFF.
Align cursor to inputON aligns all cursors to the blood pressure (input) cursor.
Default is ON.
ResolutionDenotes the frequency resolution of the power spectrum and transfer function.

### Graph definitions

ParameterDefinitions
Blood pressureInput blood pressure time series.
BP spectrumInput blood pressure power spectrum.
Flow velocityOutput blood flow velocity time series.
Flow velocity spectrumOutput blood flow velocity power spectrum.
CoherenceCoherence function.
GainTransfer function gain in non-normalised and normalised units. Default is non-normalised.
PhaseTransfer function phase.
Impulse responseThe inverse Fourier Transform of the transfer function. The impulse response represents the change in output blood flow velocity in response to a transient change in input blood pressure.

## Technical description

Beat-to-beat blood pressure and flow velocity values are spline interpolated and re-sampled for spectral and transfer function analyses based on the Welch algorithm.  This approach subdivides each selected recording epoch into successive overlapping windows. Data within each window are then linearly detrended and passed through a Hanning window before fast Fourier transform analysis.

The transfer function H (f) between the two signals was calculated as:

H (f) = Sxy(f)/Sxx(f)

where Sxx(f) is the autospectrum of input signal (MAP) and Sxy(f) is the cross-spectrum between the two signals (input blood pressure and output flow velocity).  The transfer function magnitude |H(f)| and phase spectrum |Ф (f)| are obtained from the real part HR(f) and imaginary part HI(f) of the complex transfer function:

|H(f)|=  {[HR(f)]2+[HI(f)]2}

Ф (f) = tan-1[HI(f)/HR(f)]

The squared coherence function MSC (f) was estimated as:

MSC (f) = |Sxy(f)|2/[Sxx(f)Syy(f)],

where Syy(f) is the autospectrum of changes in output signal (MCAv).

### References

Zhang, R., Zuckerman, J. H., Giller, C. A., & Levine, B. D. (1998). Transfer function analysis of dynamic cerebral autoregulation in humans. American Journal of Physiology-Heart and Circulatory Physiology, 274(1), H233–H241.

Wittmann, U., Nafz, B., Ehmke, H., Kirchheim, H. R., & Persson, P. B. Frequency domain of renal autoregulation in the conscious dog. American Journal of Physiology-Renal Physiology, 269(3), F317–F322, 1995.

## Autoregulatory index

Cerebral autoregulation can be evaluated by measuring relative blood flow changes in response to a steady-state change in the blood pressure (static method) or during the response to a rapid change in blood pressure (dynamic method).