Vol. 4, Issue 2, Article 2 Neurographics logo Sanelli, P., Shetty, S. & Lev, M.

II. Theoretical Background

(A) Mathematical Principles: CBF Calculation
*CBF calculation is based on a single-tissue compartment model for non-diffusible tracers (2,3).

*All CT perfusion models are ultimately based on the Fick Principle, or "conservation of flow", where there is a single inflow (artery) and outflow (vein) to the tissue region-of-interest (ROI). This can be mathematically expressed as

dCt(t)/dt = CBF x [Ca(t) - Cv(t)]

Key:

  • dCt(t)/dt - Change in contrast enhancement in tissue per unit time
  • Ca(t) - Contrast concentration in feeding artery as a function of time
  • Cv(t) - Contrast concentration in draining vein as a function of time
Assumptions of single compartment model

  1. Single inflow (arterial supply) to the sampled tissue
  2. Single outflow (venous drainage)
  3. No delay in time from arterial input to tissue enhancement
  4. No dispersion of contrast along varying vascular pathways

(B) Mathematical Principles: CBV Calculation
*For a given voxel of tissue, CBV can be calculated from the area under the contrast concentration versus time curve, Ct(t), as this area is proportional to the fractional vascular volume of the tissue (4). This area is normalized to the contrast concentration of the feeding artery, Ca(t).

*This can be expressed mathematically as

CBV = ∫Ct(t) / ∫Ca(t)

Key:

  • ∫Ct(t) - integral (area) of tissue contrast concentration as a function of time
  • ∫Ca(t) - integral (area) of arterial contrast concentration as a function of time
*If Ct(t) and Ca(t) are both at a constant, steady state level during the scan acquisition - as is typically the case for our stroke CTA source images (CTA-SI) - then fractional CBV can simply be calculated as the ratio of the tissue to arterial Hounsfield opacification (after co-registration and subtraction of the unenhanced from the enhanced scan, resulting in a "pure" contrast image, Fig 9).
Pearls and Pitfalls: Assumptions of tracer kinetic techniques
  1. Arterial and capillary contrast concentration is at a steady state during the CTA portion of the stroke acquisition protocol (5,6). When this assumption is met, then the CTA-source images are blood volume weighted.
  2. The entire bolus of contrast remains in the intravascular compartment, with an intact blood-brain barrier, and no leakage of contrast into the brain parenchyma. If this assumption is not met, CBV is overestimated, and permeability calculations could be performed.

Deconvolution vs. Non-deconvolution-based Methods
*The user should be aware of precisely how CBF, CBV and MTT are calculated using their vendor specific software, with the associated limitations of the method employed.

*Some post-processing software programs simplify the calculation of CBF by assuming that the rapid cine acquisition of perfusion data is completed prior to contrast reaching the intracranial venous system. Although eliminating the venous outflow parameter simplifies the calculation of CBF, this assumption is generally not valid at injection rates lower than 10-20 ml/s.

*Deconvolution-based software employs a more robust model to directly calculate CBF. Of note, the deconvolution technique used by the GE Medical Systems software (General Electric Medical Systems, Milwaukee, WI), for example - unlike that of most commercially available MR perfusion reconstruction software- also includes a correction for the delay (but not dispersion) of contrast that occurs between the peripheral injection of contrast, and it's arrival in the major intracranial vessels. This development makes less critical the choice of arterial input function as ipsi- or contralateral to the side of arterial occlusion in stroke cases. Importantly, deconvolution-based methods are also more quantitatively accurate at lower injection rates (4-7 ml/sec) than are non-deconvolution methods (7). The deconvolution calculation for CBF is mathematically expressed as:

Ct(t) = CBF x [Ca(t) R(t)]

Key:

  • R(t) - Residue function, representing the "ideal" tissue TDC that would theoretically result if the entire contrast bolus could be administered instantaneously into the artery supplying a given brain region.
  • - mathematical "convolution" operator
*The deconvolution equation can be solved to calculate "CBF x R(t)", the "scaled" residue function. CBF is directly proportional to the maximum height of this scaled residue curve; CBV is proportional to the area under the scaled residue curve; MTT can be easily calculated from CBV and CBF using the Central Volume Principle.
Pearls: Benefits of deconvolution-based CTP methods
  1. Deconvolution software permits the acquisition of quantitative CTP datasets at relatively low injection rates (4-7 ml/sec)
  2. Correction for contrast arrival delay (but not dispersion) facilitates choice of either the ipsi- or contralateral arterial input function (AIF) for CTP map reconstruction, relative to location of vascular occlusion (8, 15)
Other Pearls: Deconvolution-based methods
  1. No limiting assumptions regarding venous outflow (8).
  2. Less sensitive to variations in the underlying vascular anatomy than non-deconvolution methods.
  3. Validated for both MR and CT perfusion, in both human and animal models (9,10,11,12,13,14).
  4. Less sensitive to TDC noise than non-deconvolution methods (depending on the specific software implementation) (10).
  5. Commercially available software is easy to operate.

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