Scientific Background

The Need for a Quantitative Perfusion Assessment Tool
 

Angiogenesis plays a critical role in the process of cancer progression.  A tumor is unable to progress beyond its size limit for diffusion of oxygen and nutrients without the formation of new vessels within its microenvironment.  As a tumor grows and develops hypoxic regions, pro-angiogenic factors such as vascular endothelial growth factor (VEGF) are secreted.  VEGF encourages the growth of new vasculature within the tumor microenvironment; enables local invasion, tumor migration, extravasations, and ultimately, metastasis.  Angiogenesis is also a physiologic surrogate of tumor aggressiveness and environmental stress that may influence a tumor's response to treatment.  The availability of small molecule inhibitors directed at angiogenic pathways provides a method to interrupt the angiogenic process.  This has been shown to improve outcomes in tumors such as renal and colorectal cancers [reference].

Dynamic contrast-enhanced CT/MR (DCE-CT/MR) perfusion studies can provide a direct and quantitative assessment of tumor blood flow.  This can potentially be used as a valuable biomarker of tumor response to anti-angiogenic agents as well as conventional therapies.  The analysis of DCE-CT/MR perfusion studies with the derivation of metrics (or parameters) of blood flow is well described in the literature.  The analysis of these studies requires post-imaging processing with specialized software and mathematical techniques; hence, they are computationally demanding and time consuming.  Thus, DCE-CT/MR perfusion studies are a potentially important imaging biomarker, which requires the development of a clinically useful tool for large-scale quantitative assessment.  The purpose of this article is to provide users with a brief overview of the mathematical models used in The DCE Tool.

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Relationship between X-ray attenuation and contrast agent concentration
 

CT is a readily available technology around the world, and the property of x-ray attenuation in either blood or parenchymal tissue is well established.  CT enhancement is the measurement of contrast agent uptake within a region-of-interest (ROI).  It is measured in Hounsfield Units (HU).  A typical CT image consists of 512 X 512 pixel values; these pixel values are linearly related to the HU of each pixel.  ClearCanvas automatically converts pixel values to HU once an image dataset is imported into the workspace.  In addition, The DCE Tool has the capability to convert CT enhancement (HU) to contrast agent concentration (mmol/L or mM).  However, contrast agent concentration is not necessary for the analysis of DCE-CT perfusion studies.
 

Relationship between signal intensity and contrast agent concentration

Gadolinium-based contrast agent is one of the most commonly used contrast agents in DCE-MR studies.  The relationship between signal intensity (or pixel value) and contrast agent concentration cannot be determined directly.  The DCE Tool has the capability to analyze T1-weighted DCE-MR studies by converting signal intensities within ROIs to concentrations (mmol/L or mM) using two established methods - a linear and a non-linear approach (Landis, et al., 2000, and Tofts, 1997).  Working quantities used for T1-weighted DCE-MR signal intensity-concentration conversion are displayed in Table 1.  

 

Table 1. Working Quantities for T1-weighted DCE-MR Perfusion Studies

Quantity

Definition

Unit

α

Flip angle

radian

[Gd]

Gadolinium-based contrast agent concentration

mM or mmol/L

R

Relaxivity

mM-1sec-1

R*1

Relaxation rate (R*1=1/T1)

msec

S

Average signal intensity in a region-of-interest (ROI)

Dimensionless

So

Average signal intensity of ROI in the absence of contrast agent

Dimensionless

S_

Steady-state signal intensity of ROI in the absence of contrast agent

Dimensionless

T1o

T1 of the tissue in the absence of contrast agent

msec

TR

Repetition time

msec

 

Linear relationship between signal intensity and contrast agent concentration

It can be reasonably assumed that signal intensity is linearly proportional to contrast agent concentration.  The relationship between contrast agent concentration, [Gd], and the signal intensity (S) after contrast injection is displayed in equation [1].  The symbols are defined in Table 1.
 

....................[1]
 

Non-linear relationship between signal intensity and contrast agent concentration

The linear relationship between signal intensity and contrast agent concentration is based on two assumptions.  It is assumed that repetition time (TR) is approximately equal to the T1 of the ROI in the absence of contrast agent (T1o), and that the relaxation rate (R*1 or 1/T1) is independent of time.  A non-linear model can be used to avoid these assumptions.  The average intensity of a ROI in the absence of contrast agent (So) can be calculated using equation [2].  (Note: It is assumed that T*2  relaxation time is >> TE or echo time; thus, it is not included in the equations below.)
 

....................[2]
 

The relaxation rate (R*1 or 1/T1) as a function of time can be calculated using equation [3].  

....................[3]

The concentration of contrast agent as a function of time, [Gd(t)], can be defined using equation [4].  

....................[4]
 
 

 

Tracer Kinetic Analysis
 

Ample studies have demonstrated the feasibility of modeling perfusion (i.e. blood flow) in tumors (or tissues) either as a single compartment or two compartments.  Lee, Purdie, and Stewart (2003) and Tofts, et al. (1999) provide a detailed explanation for using compartmental models to approximate physiologically meaningful parameters from a ROI.  General assumptions and equations used in The DCE Tool are based on the models described in Tofts, et al. (1999), which provided a standardization of tracer kinetic parameters.  The DCE Tool is capable of modeling contrast uptake (for both CT and MR) within multiple ROIs using three published models.  These models are: the Standard Tofts, Modified Tofts (Tofts, et al., 1999), and the Adiabatic Tissue Homogeneity Models (St. Lawrence and Lee, 1998a and 1998b).  
 

The Standard and Modified Tofts Model

The Standard and Modified Tofts Models consider the blood plasma (or intravascular space) and the extracellular extravascular space (EES or interstitial space) as two compartments.  Both models assume that a well-mixed tracer exists in each of the compartments in a uniform concentration.  The two models can provide information regarding the distribution of contrast agent across the two spaces (Figure 1).  

 


Figure1: Standard and Modified Tofts Models are used to describe the distribution of blood within a ROI. The whole blood volume (Vb) is not considered in the Standard Tofts Model. The symbols are defined in Table 2.
 

Table 2. Parameter Quantities for both Standard and Modified Tofts Models

Quantity

Definition

Unit

Ca(t)

Arterial concentration as a function of time

HU or mM

Ct(t)

Tissue concentration as a function of time

HU or mM

Hct

Hematocrit value

Dimensionless

Ktrans

Transfer constant from the blood plasma into the EES

mL/g/min

kep

Transfer constant from the EES back to the blood plasma

1/min

t

Onset time of arterial contrast uptake

sec

Vb

Whole blood volume per unit of tissue

mL/g

Ve

Total EES volume (Ve = Ktrans/kep)

mL/g

 

Using the Standard Tofts Model, contrast uptake within a ROI (e.g. a tumor) can be approximated using equation [5].  The symbols are defined in Table 2.  

 

....................[5]

 

Using the MATLAB command "fmincon", The DCE Tool calculates the optimal Ktrans and kep that will result in an optimally fitted contrast enhancement-time curve for the ROI.  Likewise, contrast agent uptake within a ROI can be approximated using the Modified Tofts Model (equation [6]).  This model will generate Ktrans, kep, and Vb as tracer kinetic parameters.  

....................[6]

 

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Semi-quantitative Analysis

 

The DCE Tool will automatically generate a semi-quantitative analysis of your perfusion study.  Three parameters can be calculated using this method, they are the initial area under curve (IAUC), initial slope, peak contrast enhancement and mean squared error.
 

IAUC

The IAUC is calculated by calculating the integral of the contrast-enhancement time curve from the onset time of contrast uptake to 60 seconds after the onset time (i.e. IAUC60).  However, the user can specify the time interval for IAUC calculation (e.g. IAUC30, IAUC90, etc).  The DCE Tool uses the trapeziold rule to calculate the IAUC.
 

Initial Slope

The initial slope is the slope of the contrast enhancement-time curve by calculating the line of best fit from the onset time of contrast agent uptake to 10 seconds after the onset time (by default).  The sum of least-squares method is used to determine the slope of the line of best fit.  However, the user can specify the time interval for initial slope calculation (e.g. 15 sec).
 

Peak

The peak contrast enhancement is simply the absolute maximum contrast enhancement.
 

Mean Squared Error

The mean squared error can be used to estimate the goodness of fit of the model fittings.  It is the sum of squares of the difference between the measured values and the model-fitted values.  

 

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References

Henderson, E., Sykes, J., and Drost, D., Weinmann, H-J., Lee, T-Y. (2000). Simultaneous MRI measurement of blood flow, blood volume, and capillary permeability in mammary tumors using two different contrast agents. Journal of Magnetic Resonance Imaging, 12:991-1003.

Johnson, J.A., and Wilson, T.A. (1966). A model for capillary exchange. American Journal of Physiology, 210:1299-1303. 

Koh, T.S., Zeman, V., Darko, J., et al. (2001). The inclusion of capillary distribution in adiabatic tissue homogeneity model of blood flow. Physics in Medicine and Biology, 46:1519-1538.  

Landis, C.S., Li, X., Telan, F.W., et al. (2000). Determination of the MRI contrast agent concentration time course in vivo following bolus injection: Effect of equilibrium transcytolemmal water exchange. Magnetic Resonance in Medicine, 44:563-574. 

Lee, T.Y., Purdie, T.G., Stewart, E. (2003). CT imaging of angiogenesis. The Quarterly Journal of Nuclear Medicine, 47:171-187. 

St. Lawrence, K.S., Lee, T-Y. (1998a). An adiabatic approximation to the tissue homogeneity model for water exchange in the brain: I. Theoretical derivation. Journal of Cerebral Blood Flow & Metabolism, 18:1365-1377.

St. Lawrence, K.S., Lee, T-Y. (1998b). An adiabatic approximation to the tissue homogeneity model for water exchange in the brain: II. Experimental validation. Journal of Cerebral Blood Flow & Metabolism, 18:1378-1385.  

Tofts, P.S. (1997).  Modeling tracer kinetics in dynamic Gd-DTPA MR imaging. Journal of Magnetic Resonance Imaging, 7:91-101.

Tofts, P.S., Brix, G., Buckley, D.L., et al. (1999). Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusible tracer: Standardized quantities and symbols. Journal of Magnetic Resonance Imaging, 10:223-232.