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The Physiological Origins ofNon-Linearities in the BOLD Response

Douglas C. NollDouglas C. Noll

Alberto L. Vazquez

Department of Biomedical Engineering

University of Michigan

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OutlineOutline

• 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

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Fitting of EC Model to Duration DataFitting of EC Model to Duration Data

• Single set of model parameters with different Single set of model parameters with different duration stimuli as inputduration stimuli as input

• Model parameters derived from 8 s dataModel parameters derived from 8 s data

2 s2 s 4 s4 s 8 s8 s

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EC Model Shows Same Non-linearitiesEC Model Shows Same Non-linearities

• Comparison of 4 superimposed 2 s stimuli to Comparison of 4 superimposed 2 s stimuli to response to 8 s stimulus.response to 8 s stimulus.

• Actual data and model show non-linear effectsActual data and model show non-linear effects

Superimposed stimuliSuperimposed stimuli

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Fitting of EC Model to Contrast DataFitting of EC Model to Contrast Data

• Single set of model parameters with different Single set of model parameters with different blood flow levels as inputblood flow levels as input

• Model parameters are from 80% contrast dataModel parameters are from 80% contrast data

10% contrast10% contrast 80% contrast80% contrast

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EC Model Shows Same Non-linearitiesEC Model Shows Same Non-linearities

• Comparison of two different contrast stimuli Comparison of two different contrast stimuli normalized to same peak heightnormalized to same peak height

• Actual data and model show non-linear effectsActual data and model show non-linear effects

10%10%

80%80%

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Time-Variant Behavior of fMRI ResponseTime-Variant Behavior of fMRI Response

• Linearity (often means additivity of responses)Linearity (often means additivity of responses)

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

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

– Time between stimuliTime between stimuli

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Non-linearity in the Hemodynamic ResponseNon-linearity in the Hemodynamic Response

• TaskTask– Half visual field alternating checkerboard (8Hz) for a Half visual field alternating checkerboard (8Hz) for a

period of 2speriod of 2s

– TrialTrial

– n-trials = 5n-trials = 5

– Inter-stimulus interval = 10sInter-stimulus interval = 10s

– Inter-trial interval = 90s Inter-trial interval = 90s

ISIISI ITIITI2s2s

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Non-linearity of the Hemodynamic ResponseNon-linearity of the Hemodynamic Response

• AcquisitionAcquisition– General Electric 3.0 Tesla scannerGeneral Electric 3.0 Tesla scanner

– Single-shot EPISingle-shot EPITR = 1000msTR = 1000ms

TE = 25msTE = 25ms

FA = 60degFA = 60deg

– Four coronal slices (3mm, skip 0mm)Four coronal slices (3mm, skip 0mm)

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Responses Differ with Position in SeriesResponses Differ with Position in Series

• Response to the 2nd Response to the 2nd stimulus is:stimulus is:– Delayed in RiseDelayed in Rise

– Delayed in PeakDelayed in Peak

– Lower in AmplitudeLower in Amplitude

– Broader in TimeBroader in Time

• This example is extreme, This example is extreme, but not unique.but not unique.

Response to 2Response to 2ndnd stimulus stimulus

Response to 1Response to 1stst stimulus stimulus

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Non-linearity in the Hemodynamic ResponseNon-linearity in the Hemodynamic Response

EPI DataEPI Data Activation Activation ResponseResponse

Delay in ResponseDelay in ResponseStim. 2 - Stim. 1Stim. 2 - Stim. 1

High intensity responses (probably veins)High intensity responses (probably veins)exhibit largest delaysexhibit largest delays

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Non-linearity in the Hemodynamic ResponseNon-linearity in the Hemodynamic Response

• Plot of response delays Plot of response delays (stimulus 2 - stimulus 1) (stimulus 2 - stimulus 1) vs. percentage signal vs. percentage signal changechange

• Positive correlationPositive correlation– Larger veins usually have Larger veins usually have

largest responseslargest responses

– These also have longest These also have longest delaysdelays

– Implications for modeling Implications for modeling the responsethe response

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Physiologically Relevant ModelPhysiologically Relevant Model

• Expandable compartment model (balloon) model Expandable compartment model (balloon) model of Buxton, et al.of Buxton, et al.

• Increases in blood volume can account for some Increases in blood volume can account for some non-linear behavior (as well as the fMRI response non-linear behavior (as well as the fMRI response undershoot)undershoot)

FFinin FFoutout

OO22

capillariescapillaries venousvenous

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Cascaded Balloon ModelCascaded Balloon Model

......

VV11 VV22 VVnn

FFininFFoutout

OO22

capillariescapillaries venousvenous

• The original model cannot predict our observed The original model cannot predict our observed time-variant behaviortime-variant behavior– Notably, it doesn’t predict a delays for secondary Notably, it doesn’t predict a delays for secondary

stimulistimuli

• New cascaded-compartment model.New cascaded-compartment model.

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Responses in Different CompartmentsResponses in Different Compartments

Compartment 1Compartment 1 Compartment 5Compartment 5

No Delay or Shift in PeakNo Delay or Shift in Peak Delay and Shift in PeakDelay and Shift in Peak

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Comparison to Experimental DataComparison to Experimental Data

Delay in RiseDelay in Rise Shift in PeakShift in Peak Cross-overCross-over

Experimental DataExperimental Data Model PredictionsModel Predictions

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Aspects of Cascaded ModelAspects of Cascaded Model

• The cascaded expandable compartment model will The cascaded expandable compartment model will require one additional parameter (3 or 4 + 1).require one additional parameter (3 or 4 + 1).

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

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ConclusionsConclusions

• The hemodynamic response is quite complexThe hemodynamic response is quite complex

• Physiologically relevant models can predict most Physiologically relevant models can predict most of this complex behaviorof this complex behavior

• There are domains in which the response behaves There are domains in which the response behaves linearly linearly – Linearity greatly eases the analysis and experimental Linearity greatly eases the analysis and experimental

design design

– The models can help establish if linear models will hold The models can help establish if linear models will hold for any given experimentfor any given experiment

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ConclusionsConclusions

• It is also possible to build the non-linear model It is also possible to build the non-linear model directly into the analysisdirectly into the analysis

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

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CommentsComments

• Why do some find mostly linear behavior?Why do some find mostly linear behavior?

• Many task designs reduce the effects of non-Many task designs reduce the effects of non-linearitylinearity– Most block designs with block longer than 4 sMost block designs with block longer than 4 s

– Event-related designs in the steady stateEvent-related designs in the steady state

– Event-related designs that do not allow for blood Event-related designs that do not allow for blood volume changes to return to normal volume changes to return to normal (5 time constants ~ 75 s)(5 time constants ~ 75 s)

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Future WorkFuture Work

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

• Study of non-linearities using flow measuresStudy of non-linearities using flow measures

• Experimental validation of parts of modelExperimental validation of parts of model