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Deconvolution-based analysis of CT and MR brain perfusion data is widely used in clinical practice and it is still a topic of ongoing research activities. In this paper, we present a comprehensive derivation and explanation of the underlying physiological model for intravascular tracer systems. We also discuss practical details that are needed to properly implement algorithms for perfusion analysis. Our description of the practical computer implementation is focused on the most frequently employed algebraic deconvolution methods based on the singular value decomposition. In particular, we further discuss the need for regularization in order to obtain physiologically reasonable results. We include an overview of relevant preprocessing steps and provide numerous references to the literature. We cover both CT and MR brain perfusion imaging in this paper because they share many common aspects. The combination of both the theoretical as well as the practical aspects of perfusion analysis explicitly emphasizes the simplifications to the underlying physiological model that are necessary in order to apply it to measured data acquired with current CT and MR scanners.

Tissue perfusion measurement from iodinated contrast agent enhancement on CT scans was first proposed by Axel in 1980 [

With the advent of helical scanners and faster rotating gantries (0.33–0.5 s/rotation) in conjunction with multidetector geometries which provide larger coverage, PCT has now become part of the routine screening for many diseases.

Given the existing developments in perfusion imaging, the purpose of this paper is to focus on a detailed derivation of the theoretical model for deconvolution-based perfusion measurement. While the main equation of this model is well known, its derivation is spread over several publications.

We therefore first present a summary of the derivation, with the aim of fully explaining the parameters and the underlying assumptions that are made. Based on the main equation of the theoretical model, we also present a guideline for the algorithmic implementation of the deconvolution-based perfusion measurement. We discuss robust numerical deconvolution and discuss topics related to data pre-processing, providing references to the literature for each of the special topics. The overall aim of this paper is to provide an understanding of the underlying assumptions of the theoretical model and to show how the (simplified) model can be robustly implemented for clinical image analysis.

Perfusion imaging is most widely used in acute stroke and oncology [

Figure

CT perfusion parameter maps of cerebral blood flow (CBF), cerebral blood volume (CBV), mean transit time (MTT), and time-to-peak (TTP). The ischemic stroke lesion is marked with arrows.

CBF map in mL/100 g/min

CBV map in mL/100 g

MTT map in s

TTP map in s

Blood flow is critical to the functionality of any organ since it provides the essential nutrients and oxygen. In case of flow disruption, the body autoregulates the flow and pressure either by altering blood flow or volume or both. In the brain, there are some fairly well-defined thresholds for the cerebral blood flow in normal, reversibly damaged, and necrotic tissue. The normal value for the cerebral blood flow is between 50 and 60 mL/100 g/min for grey matter [

In the acute stroke setting, conventional CT has been the imaging modality of choice for ruling out intracerebral hemorrhages (ICH). However, overall the sensitivity of CT for stroke detection is 60–65% [

In animal studies, the product of CT cerebral blood volume (CBV) and flow (CBF) from CT measurements was found to have sensitivity of 90.6% and specificity of 93.3% (compared with histological measurements) for discerning ischemic and oligemic tissue [

The gold standard for perfusion CT has been imaging with stable xenon as the contrast agent [

Perfusion imaging in MR can be performed with or without contrast agent [

The aim of this section is to provide a compact outline of both some elementary as well as practically relevant theory of perfusion estimation based on previous work. In particular, we will introduce a theoretical physiological model of tissue perfusion for intravascular tracer systems and present the derivation of a deconvolution-based mathematical approach for the estimation of diagnostically important perfusion parameters. In addition, we will briefly describe alternative methods that do not require deconvolution.

For computing the tissue perfusion, we assume a physiological model of the blood supply to the tissue. Figure

Physiological model of the tissue perfusion. A blood cell can take several paths through the capillary bed. The variables are defined in Table

Once a contrast agent bolus has been injected, it enters the volume

Figure

Examples of the time-concentration curves

Typical time-concentration curves

Zoomed view of (a)

An additional important assumption is that the contrast agent remains in the intravascular space. For our case of cerebral perfusion, it should therefore not cross the blood-brain barrier (BBB). As a consequence, this means that all contrast agents entering from the arterial inlet will eventually leave the volume of interest at the venous outlet. A breakdown of the BBB may occur in tumor patients, in stroke patients, and in patients that suffer from inflammations or infections, for example. In these cases, the methods presented in this paper may lead to inaccurate perfusion estimates and particularly to an overestimation of the blood volume [

Finally, we suppose that the contrast agent mixes perfectly with the blood and that the physical properties of the blood (its flow behavior, in particular) are not influenced by the contrast agent.

As we will see, only knowledge of the functions

As a first diagnostically relevant perfusion parameter, the mean transit time (MTT) of the volume under consideration is defined as the first moment of the probability density function

Furthermore, the residue (or residual) function

The (dimensionless) residue function thus quantifies the relative amount of contrast agent that is still inside the volume

Examples of the distribution function

Distribution

Residue function

Using the parameters defined in Table

Summary of parameters used to derive the indicator-dilution theory and to define clinically relevant tissue perfusion quantities.

Variable | Unit | Description |
---|---|---|

mL | Total volume under consideration | |

mL | Volume of the capillary bed within the volume | |

mL | Volume | |

g/mL | Mean density of the volume | |

g/mL | Mean density of the volume | |

g | Total mass of contrast agent in volume | |

g/mL | Local contrast agent concentration at the arterial inlet, | |

g/mL | Local contrast agent concentration at the venous outlet, | |

g/mL | Average contrast agent concentration in the total volume | |

g/mL | Average contrast agent concentration in the capillary bed, | |

g/mL | Average contrast agent concentration corresponding to | |

mL/s | Volume flow at the arterial inlet and at the venous outlet | |

1/s | Probability density function of the transit times |

We can compute the mass

The contrast agent concentration

Note that throughout this paper, all integrals with infinite integration endpoints shall be interpreted as the limit of the integral when the respective endpoint approaches

From a physiological point of view, it would be more meaningful to normalize CBF by the mass of the volume

The derivation of the indicator-dilution theory in this section was focused on brain perfusion imaging. This theoretical model can be used in stroke patients if the BBB is intact—compare Section

In the context of perfusion measurement, the term recirculation refers to the physiological phenomenon that, due to the patient's cardiac activity, the contrast agent passes through the volume under consideration multiple times. It can easily be shown, however, that there is no need to correct for recirculation when deconvolution methods are applied to determine perfusion parameters [

In (

The flow-scaled residue function

Note that we have assumed that there is a constant

The cerebral blood volume (CBV) corresponding to the tissue volume

A healthy human brain exhibits a CBV of about 4 mL/100 g for grey matter and a CBV of about 2 mL/100 g for white matter [

Note that the definition of CBV that corresponds to the alternative definition of CBF in [

Furthermore, there are references in the literature that suggest measuring the blood volume in units of mL/mL. This alternative dimensionless quantity may therefore be considered as a measure of blood (or vascular) volume fraction. When relating the volume

For the sake of completeness, this section will briefly cover two alternative approaches for CBV and CBF estimation that are practical and relevant, and that do not involve deconvolution operations. Nondeconvolution-based methods for estimating perfusion parameters are also referred to as direct measurement-based approaches [

Firstly, there is an alternative method to compute the blood volume of the tissue volume under consideration [

It is argued in [

Secondly, there is a nondeconvolution-based approach to estimate the blood flow of the tissue volume under consideration; the maximum slope method [

Perfusion parameters that are measured directly using the time-concentration curve. See Sections

An advantage of the maximum slope method is the shorter overall acquisition time. As a downside, however, it requires a faster contrast agent bolus injection rate in order to approximately fulfill the no-venous-outflow condition.

A more comprehensive discussion of the maximum slope method and a comparison with the deconvolution method is presented in [

Besides the aforementioned quantities CBV, CBF, and MTT, there are additional perfusion parameters such as the time-to-peak (TTP) of the time-concentration curve, the maximum contrast agent concentration

Figure

In summary, Table

Summary of perfusion parameters and how these parameters can be estimated using deconvolution-based and nondeconvolution-based methods.

Parameter | w/Deconvolution | w/o Deconvolution |
---|---|---|

CBV | ||

CBF | ||

MTT | see comment in Section | |

TTP | — | |

FM | — |

This section is devoted to the practical computer implementation of algorithms for perfusion image analysis. First, we will discuss the necessary adaptations of the theoretical model from Section

In Section

First, during a standard CT and MR perfusion exam, a volume of interest is scanned and the data is reconstructed on a grid of regularly spaced voxels. In the object domain, each voxel volume

The measured signal (X-ray attenuation or MR relaxation rate) in a voxel is thus a combination of the signals from both the capillary beds as well as the arterial and venous vessels [

The second adaptation of the model concerns the measurement of

This approach leads to a traveling time of the contrast agent bolus from where the AIF is measured to the location of the tissue volume where

The bolus delay has two implications. First, the curve

Second, the flow-scaled residue function

Bolus delay and dispersion may lead to an underestimation of CBF [

In this section, we will discuss the robust numerical solution of the main equation of the indicator-dilution theory—(

Examples of measured time-attenuation curves in perfusion CT in (a) an arterial vessel and (b) in tissue. The time curves have been pre-processed by baseline subtraction and removal of the baseline time frames. The example data is measured at

Arterial time curve

Tissue time curve

We assume that the measured time-attenuation curves can be converted to time-concentration curves using a constant of proportionality of 1 g/mL/HU. Details about the conversion, also discussing perfusion MR data, will be explained in Section

In practice, the time-concentration curves

A standard approach to solve (

However, the least-square solution

As an example, Figure

Least-squares solution vector

In order to get a better understanding of why

SVD analysis of the matrix

It is known from numerical analysis that the discrete Picard condition represents a means to analyze discrete ill-conditioned problems [

To obtain a numerically stable result, a filter is used for regularization. The filter should suppress the influences of small singular values

We will focus on two common definitions of the filter factors

The (absolute) regularization parameter

In order to illustrate the Tikhonov filter factors, Figure

(a) Linear and (b) double logarithmic plot of the Tikhonov filter factor

Linear plot of

Double logarithmic plot of

Interestingly, the solution

Figure

Deconvolution with Tikhonov regularization: (a) Regularized solution

Regularized solution

Dependency of max

The algebraic deconvolution approach from Section

The Fourier transform represents a standard method to solve deconvolution problems [

In contrast to the model-independent deconvolution approaches also model-dependent approaches exist. Model-dependent approaches assume a certain shape of the residue function. For example, in [

Deconvolution using orthogonal polynomials was investigated in [

A comprehensive comparison of all available deconvolution methods has not been carried out yet. The SVD-based deconvolution approach, which is available in several software packages [

Figure

Therefore, an optimal choice of

In [

The L-curve criterion represents a model-independent method to determine

Another method to determine an appropriate regularization parameter is generalized cross-validation as described in [

Furthermore, a parameter estimation method that uses a priori knowledge of the behavior of the residue function was proposed in [

Kudo et al. [

This section gives an overview of pre-processing techniques that can be applied in order to enhance the quality of the estimated perfusion parameters. Pre-processing occurs prior to the deconvolution step which may be implemented as described in Section

A simple, yet mandatory, pre-processing step consists of the conversion to contrast agent concentration values, see Section

The order of the pre-processing steps presented in this section can act as a guideline for their practical implementation. However, a different ordering can of course be reasonable as well. Finally, this overview cannot include all details regarding suitable pre-processing steps. The reader is referred to the available literature for in-depth discussions.

Patient motion (e.g., due to head movement or breathing) can result in a sudden change of the attenuation values at the fixed (stationary) voxel positions. Since this change in the attenuation value is caused by motion and not by contrast agent flow, the computed perfusion values can be severely biased. A practical approach for motion correction is to register all time frames of the reconstructed data set onto the first time frame [

As an alternative to registration, use of groupwise motion correction based on an optimization of a global cost function has been suggested [

A related issue is streak artifact in reconstructed perfusion CT images that are caused by patient motion that occurs while the projection data corresponding to a single time frame is acquired. In perfusion MR images, ghosting artifacts can arise if the patient moves during the data acquisition. These kinds of artifact cannot be corrected by inter-frame motion correction. Instead, dedicated reconstruction algorithms would be required. As a practical alternative, time frames that exhibit severe reconstruction artifacts may simply be removed from the data set (i.e., from the series of successive time frames), which corresponds to the elimination of invalid sampling points of the voxel-specific time-concentration curves.

In the course of a perfusion exam, the measured signal in tissue that is caused by the contrast agent flow can be very low. For the case of perfusion CT, for example, tissue enhancements of less than 10 HU are measured. Hence, noise in the reconstructed images can be of a similar order of magnitude as the signal in tissue itself. Consequently, noise reduction should be taken into consideration in order to improve the accuracy of the estimated perfusion parameters.

Noise reduction can be implemented as a spatial smoothing of the data. Using a basic approach, each time frame can be filtered independently of the other time frames, and linear isotropic filters (e.g., based on a Gaussian filter kernel) may be applied. Alternatively, anisotropic filters that preserve edges and avoid blurring of large vessels can also be employed [

Both linear and nonlinear filtering in the temporal dimension—that is, between successive time frames—represent further methods for noise reduction [

Recently, sophisticated 4D filtering techniques have been proposed that perform filtering in both the spatial and the temporal dimension and that are optimized for perfusion data [

A segmentation of certain anatomic structures in the reconstructed data set can optimize the perfusion image analysis [

Neither for the case of CT imaging nor for the case of MR imaging can the time-concentration curves

In perfusion CT, it is assumed that the (underlying) contrast agent concentration value is proportional to the (measured) X-ray attenuation value [

In perfusion MR, however, the contrast agent concentration value is not proportional to the received signal

Note that if only one time frame is considered as the baseline (i.e., if

Hematocrit (Hct) is a value that describes the proportion of the blood that consists of red blood cells. Hct is higher in arteries than in capillaries. Consequently, the proportion of the plasma in the blood, given by the difference (1-Hct), has a higher value in capillaries than in arteries. Since the contrast agent is distributed in the plasma only, the amount of plasma has a direct influence on the measured Hounsfield value or MR relaxation rate.

If the Hct difference is not corrected, it may bias the absolute quantification of the contrast agent concentration. A constant dimensionless correction factor

The total time for the perfusion image analysis can be shortened and the analysis can be made user independent by an automated estimation of the arterial input function. Several methods have been proposed that detect one global AIF [

An interesting alternative approach is to estimate several local AIFs, which would be better suited to the theoretical model that was introduced in Section

Besides the arterial input function

Due to limited spatial resolution in reconstructed perfusion CT and MR data, the AIF can suffer from partial volume effects [

We have presented an overview of algorithms for the estimation of the most prominent perfusion parameters from CT or MR measurements that play an essential role in the assessment of flow altering diseases such as stroke, for example. In particular, we have emphasized the class of deconvolution-based methods that result from the application of the indicator-dilution theory, which is also derived in detail. Alternative approaches that do not use a deconvolution method are addressed briefly as well. The robust numerical solution of the resulting system of linear equations represents the second major topic of this paper. We have included a detailed discussion regarding the application of the singular value decomposition method as well as the practically relevant introduction of a suitable regularization technique in order to avoid physiologically unrealistic behavior of the estimated solution. Since this paper is intended to provide an introduction both to the underlying theory and to implementation-relevant aspects, we have provided a survey of preprocessing techniques that should be considered when designing a clinically useful tool for CT or MR perfusion analysis.

The novel contribution of this paper is to present the fundamental model, the mathematical deconvolution with regularization, and the practical pre-processing steps in one place. For a thorough understanding of perfusion image analysis, knowledge of all of these aspects is important and we have elaborated several links between these topics.

The matrix

The elements

The elements

The horizontal and vertical lines drawn in (

The authors gratefully acknowledge funding of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in the framework of the German excellence initiative. Financial support was also provided through NIH 1K99EB007676 and the Lucas Foundation. Furthermore, the authors wish to give thanks to T. Struffert, MD, Department of Neuroradiology, Friedrich-Alexander University of Erlangen-Nuremberg, for providing the clinical data.

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