Dynamic PET, in contrast to static PET, can identify temporal variations in the radiotracer concentration. Mathematical modeling of the tissue of interest in dynamic PET can be simplified using compartment models as a linear system where the time activity curve of a specific tissue is the convolution of the tracer concentration in the plasma and the impulse response of the tissue containing kinetic parameters. Since the arterial sampling of blood to acquire the value of tracer concentration is invasive, blind methods to estimate both blood input function and kinetic parameters have recently drawn attention. Several methods have been developed, but the effect of accuracy of the estimated blood function on the estimation of the kinetic parameters is not studied. In this paper, we present a method to compute the error in the kinetic parameter estimates caused by the error in the blood input function. Computer simulations show that analytical expressions we derive are sufficiently close to results obtained from numerical methods. Our findings are important to observe the effect of the blood function on kinetic parameter estimation, but also useful to evaluate various blind methods and observe the dependence of kinetic parameter estimates to certain parts of the blood function.

Positron emission tomography (PET) is a functional imaging modality to observe the physiological processes in the body. To conduct a PET scan, positron-emitting radioisotopes, as a tracer, are injected into the living subject (usually into blood circulation). When a positron encounters and annihilates an electron, it emits two gamma rays in reverse directions which will be sensed at two detectors at roughly the same time. Hence it is possible to locate the source along the line of response using a scanner around the subject. The data from the detector is then used to reconstruct an image of the subject [

Temporal variation of the tracer concentration can be obtained through dynamic imaging so that the physiological function of the subject can be tracked more accurately. Such a kinetic approach is commonly used in other imaging modalities (e.g., dynamic contrast enhanced MRI [

The differential equations that describe the FDG three-compartment model (Figure

Three compartments used to model the transfer of the tracer between physical compartments and chemical states.

Estimation of the kinetic parameters

Since we have discrete measurements at times

We can estimate the kinetic parameters and the blood function by minimizing the following cost function:

Several methods for blind kinetic parameter estimation has been proposed, but no study has shown the effect of the errors in the estimated blood on the estimation of kinetic parameters except for our preliminary work [

Although, these errors in the parameters can be found performing a separate optimization for each of the error combinations in the blood that we want to study, this is a very time-consuming method, considering that the optimization procedure is iterative. This is especially important when we are interested in pixel by pixel kinetic parameter estimation, and/or when the space of the erroneous blood functions we want to analyze is large. Based on the results of this paper, the optimization needs to be performed only once, and the error propagation can be calculated very fast based on this single optimization.

Our major contribution in this paper is to construct a mathematical model to derive the error in the kinetic parameter estimates caused by the error in estimation of the blood input function. Our results are conceptually important to observe the effect of the error in the blood function on kinetic parameter estimation, and also practically useful to evaluate various blind methods and observe the dependence of kinetic parameter estimates to certain parts of the blood function such as the peak and tail part. In Section

In this section, we explain how we can calculate the errors in the kinetic parameters for three-compartment tissue modeling due to the error in the blood function. We assume that a unique solution for the blood estimates, at least locally, exists. The estimates of the kinetic parameters

The first step can be performed by using the chain rule and implicit function theorem [

For a solution

Thus we derive the derivative expression that we are interested in,

This completes the calculation of the errors in

The detailed computation of our mathematical model can be seen in the appendix.

We apply the proposed mathematical model to simulated dynamic PET data with

The input blood function.

We simulate a noisy TAC value as follows:

The observed noisy TAC for background, liver, and tumor.

We have performed two sets of experiments. For these three experiments, we set an error margin for

First, we test the mathematical model in one dimension and assume that one of the 19 samples of

Comparison between the estimated

For several of the blind methods, the error in the blood function is not usually confined in a single sample. To illustrate this fact, we have performed a simple simulation where we have estimated the blood function with three different noise realizations. Figure

Three compartments used to model the transfer of the tracer between physical compartments and chemical states.

Figures

Comparison between the estimated

Comparison between the estimated

The following items summarize the effect of the blood function error to the final kinetic parameter estimates.

For Case I, when all 19 samples have the same error rate from −20% to 20%, we can see that the changes in

For Case II, we observe that

For Case III, we observe a different effect; the error in the tail part affects the estimation of kinetic parameter

In addition to the figures, Table

The percentage error in estimated

Error rate of | Percentage error in parameters | |||

Case I | ||||

10% | 10% | |||

20% | 20% | |||

Case II | ||||

10% | 10% | 12.5% | 4% | |

20% | 20% | 25% | 8% | |

Case III | ||||

10% | 10% | 5% | ||

20% | 20% | 10% |

Our conclusion of the relation between the blood function error and the error on the kinetic parameters can be summarized as follows:

We can see that the error in the initial peak of the blood input function affects the estimation of kinetic parameter

And the error in the tail part affects the estimation of kinetic parameter

If the overall blood input function has almost the same error rate, we find that the error in the parameters

Table

The percentage error in estimated

Error rate of Cp | Percentage error in parameters | |||

Case I | ||||

10% | 10% | |||

20% | 20% | |||

Case II | ||||

10% | 10% | 7.5% | 25% | |

20% | 20% | 15% | 50% | |

Case III | ||||

10% | 9% | 30% | ||

20% | 18% | 60% |

These simulation results show that the derived expressions provide a very accurate approximation of the errors in the kinetic parameters, and several useful observations related to the effect of the blood function on the kinetic parameter estimates can be made.

Blind identification is recently studied to estimate the kinetic parameters for the compartment without a known blood input function since the arterial sampling of blood input function involves vital risks, requires trained personnel, is not comfortable for the patient in clinical applications, and is difficult to perform in small animals. There are several solutions for blind identification: maximum likelihood methods, cross-relation method, mixture analysis method, and factor analysis of dynamic structures.

Despite several blind methods, how an erroneous blood function affects the final parameter values has not been studied. In this paper, we have derived mathematical expressions that quantify how errors in the blood estimate propagate into errors in the kinetic parameter estimates. Our model is constructed on the base of the implicit function theorem and the Runge-Kutte methods. We first derive the derivative of the kinetic parameters with respect to blood input function at a fixed point. Then we implement the Runge-Kutte approximation to calculate the accumulated errors affected by the gradually increased error in blood input function. The accuracy of the mathematical model can be modified by adjusting the number of steps in the Runge-Kutte approximation as desired.

Computer simulations show that the proposed mathematical model can yield accurate estimates of the errors in

The developed method can quantify the errors in the kinetic parameters for different error combinations in the blood function, without having to perform optimization for each of the error cases to be analyzed. Instead of iteratively optimizing the result of the error using the old method, we can use this analytical method to derive the ultimate error step by step. This would be computationally prohibitive especially for pixel by pixel kinetic parameter estimation, and large ranges of blood error to be analyzed. Future work includes generalization to estimation based on the sinogram instead of reconstructed TAC's and application to real PET data.

For a solution

We can write these four equations in matrix form

In this paper, we modify regular Runge-Kutte methods [

Let us define