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					    The Physiological Origins of
Non-Linearities in the BOLD Response




             Douglas C. Noll
            Alberto L. Vazquez

    Department of Biomedical Engineering
           University of Michigan

                                           Noll
                     Outline
•   Study of Linearity in the BOLD Response
•   Expandable Compartment Model(s)
•   Study of Time-Invariance in the BOLD Response
•   Cascaded Expandable Compartment Model
•   Comments and Future Work




                                                    Noll
   Fitting of EC Model to Duration Data
• Single set of model parameters with different
  duration stimuli as input
• Model parameters derived from 8 s data




  2s                     4s                       8s
                                                       Noll
  EC Model Shows Same Non-linearities
• Comparison of 4 superimposed 2 s stimuli to
  response to 8 s stimulus.
• Actual data and model show non-linear effects




                        Superimposed stimuli      Noll
   Fitting of EC Model to Contrast Data
• Single set of model parameters with different
  blood flow levels as input
• Model parameters are from 80% contrast data




    10% contrast                 80% contrast     Noll
  EC Model Shows Same Non-linearities
• Comparison of two different contrast stimuli
  normalized to same peak height
• Actual data and model show non-linear effects

                       80%




                        10%                       Noll
Time-Variant Behavior of fMRI Response
• Linearity (often means additivity of responses)

• Time-invariance (a second and necessary
  condition for the convolution model)

• We examined the responses to stimuli with
  manipulations of:
   – Time preceding initial stimulus in a series
   – Time between stimuli


                                                    Noll
Non-linearity in the Hemodynamic Response
• Task
   – Half visual field alternating checkerboard (8Hz) for a
     period of 2s
   – Trial              2s
             ISI                ITI


   – n-trials = 5
   – Inter-stimulus interval = 10s
   – Inter-trial interval = 90s

                                                              Noll
Non-linearity of the Hemodynamic Response
• Acquisition
   – General Electric 3.0 Tesla scanner
   – Single-shot EPI
      TR = 1000ms
      TE = 25ms
      FA = 60deg
   – Four coronal slices (3mm, skip 0mm)




                                            Noll
 Responses Differ with Position in Series
• Response to the 2nd        Response to 1st stimulus
  stimulus is:
   –   Delayed in Rise
   –   Delayed in Peak
   –   Lower in Amplitude
   –   Broader in Time

• This example is extreme,
  but not unique.



                             Response to 2nd stimulus
                                                        Noll
Non-linearity in the Hemodynamic Response
   EPI Data       Activation     Delay in Response
                  Response        Stim. 2 - Stim. 1




              High intensity responses (probably veins)
                        exhibit largest delays
                                                  Noll
Non-linearity in the Hemodynamic Response
• Plot of response delays
  (stimulus 2 - stimulus 1)
  vs. percentage signal
  change

• Positive correlation
   – Larger veins usually have
     largest responses
   – These also have longest
     delays
   – Implications for modeling
     the response


                                            Noll
      Physiologically Relevant Model

              O2
        Fin                            Fout


      capillaries                   venous

• Expandable compartment model (balloon) model
  of Buxton, et al.
• Increases in blood volume can account for some
  non-linear behavior (as well as the fMRI response
  undershoot)
                                                      Noll
                       Cascaded Balloon Model
  capillaries                                                        venous
      O2
Fin                                                                           Fout
                                                   ...

                        V1                  V2                       Vn

           • The original model cannot predict our observed
             time-variant behavior
              – Notably, it doesn’t predict a delays for secondary
                stimuli
           • New cascaded-compartment model.
                                                                          Noll
  Responses in Different Compartments
   Compartment 1              Compartment 5




                            Delay and Shift in Peak
No Delay or Shift in Peak
                                                 Noll
  Comparison to Experimental Data
Experimental Data             Model Predictions




           Delay in Rise   Shift in Peak Cross-over
                                                  Noll
         Aspects of Cascaded Model
• The cascaded expandable compartment model will
  require one additional parameter (3 or 4 + 1).

• This additional parameter might be indicative of
  distance in the vasculature.




                                                     Noll
                     Conclusions
• The hemodynamic response is quite complex

• Physiologically relevant models can predict most
  of this complex behavior

• There are domains in which the response behaves
  linearly
   – Linearity greatly eases the analysis and experimental
     design
   – The models can help establish if linear models will hold
     for any given experiment
                                                                Noll
                   Conclusions
• It is also possible to build the non-linear model
  directly into the analysis

• Parameters might tell not only where activation
  occurs, but might be used to discriminate between
  signals from distal and proximal veins




                                                      Noll
                      Comments
• Why do some find mostly linear behavior?

• Many task designs reduce the effects of non-
  linearity
   – Most block designs with block longer than 4 s
   – Event-related designs in the steady state
   – Event-related designs that do not allow for blood
     volume changes to return to normal
     (5 time constants ~ 75 s)


                                                         Noll
                  Future Work
• Modifications to Buxton’s model (notably the
  transformation to MR signal parameters)

• Study of non-linearities using flow measures

• Experimental validation of parts of model




                                                 Noll

				
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