Joint estimation of Arterial Input Function and tracer kinetic parameters from undersampled DCEMRI
Abstract
Dynamically Contrast Enhanced Magnetic Resonance Imaging (DCEMRI) is a relatively new modality that helps in both preclinical tumor modelling as well as clinical oncology. It requires acquisition of MR images before and during the administration of contrast agent (CA) or tracer. Further quantitative analysis of the acquired data provides information for diagnosis and monitoring response to medication. In DCEMR images we need high spatial and temporal resolution. However, a tradeoff exists between the two. In order to obtain a large spatial coverage at high temporal resolution, undersampling technique is a feasible option. This brings out the need for a good reconstruction technique to make up the loss of data in the kspace. AIF and TK parameters can be jointly estimated from the undersampled kspace data directly without reconstructing the image of the tissue. This work aims at the comparison of L_{p}norm regularisation schemes (p≤1) and L_{2}norm regularisation scheme to enforce model consistency and compare the accuracy and the reconstruction time of the two methods with 20x, 40x, 60x, 100x and 120x undersampled data.
Keywords: T1 weighted image, kspace, Arterial Input Function, tracer kinetic parameter
Abbreviations
DCEMRI  Dynamic Contrast Enhanced Magnetic Resonance Imaging 
CA  Contrast Agent 
AIF  Arterial Input Function 
TK  Tracer Kinetic 
ROI  Region Of Interest 
patAIF  patient specific Arterial Input Function 
popAIF  population average Arterial Input Function 
INTRODUCTION
Background
A tumor, however benign it might be, cannot depend on diffusion metabolics for its proliferation. Thus, it starts developing its own vasculature which is known as angiogenesis. These new vessels, unlike the vasculature of normal healthy tissues, are leaky, missing epithelial cells and have blind ends. DCEMRI exploits these differences to characterise the neovasculature, indirectly the tumor, as they appear as a bright spot due to leakage of the CA.
The conventional Nyquist sampling is unable to simultaneously provide the adequate temporal resolution and spatial coverage. DCEMR images acquired at Nyquist rate can provide around 510 slices in the time span of flow of CA. To achieve the required temporal resolution at Nyquist rate, spatial resolution can be coarsened, but this results in a compromise with the quality of the image and also hinders the clinical evaluation of the image. Thus techniques involving undersampling and reconstruction have been developed to simultaneously obtain the required spatial coverage and temporal resolution. A good reconstruction model uses the prior information provided by standard modules on the tracer kinetics in capillaries while complying to the model consistency constraints and data fidelity. Early work on reconstruction was based on reconstructing the MR image from the undersampled k,tspace and then applying TK modelling software to obtain the high resolution TK maps based on the reconstructed images. However, newer studies have been focusing on directly using the TK models to reconstruct the undersampled k,tspace data. These studies have shown to allow higher undersampling rates.
Statement of the Problems
Studies have been using L_{2}norm regularisation to enforce model consistency. Recent studies have also suggested the possible use of L_{p}norm (p≤1) regularisation methods. This review draws an efficient comparison between the two methods of regularisation and also tries to compare the existing backward model methods i.e. Patlak model and eTofts model.
LITERATURE REVIEW
Dynamically Contrast Enhanced (DCE)  MRI
To support the DCEMRI data quantitative analysis, compartment models are used. Human body can be represented as one or more 'compartments' in which the CA dynamically flows. A compartment is a bound space that a CA can occupy and whose volume remains constant on the time scale of the experiment. In each compartment, it is assumed that, the CA is uniformly distributed throughout, without any time delay.
A more realistic model is the two compartment model which considers the extracellular intravascular space i.e. the blood plasma to be the central compartment and the extracellular extravascular space as the peripheral compartment.
Arterial Input Function
Obtaining the Arterial Input Function (AIF) is by far the most technically demanding process in the acquisition of DCEMRI. The AIF is the estimate of the concentration of CA in the blood plasma as a function of time, C_{p}(t). There are three main types of obtaining the AIF for data analysis.
The first method involves introducing an arterial catheter into the subject and measuring the concentration of CA during the imaging process. This method is highly accurate but at the same time it is invasive and has very poor temporal resolution.
Patient specific AIF can be obtained from the DCEMRI data set itself. The signal intensity changes, due to the passing CA, in blood plasma and the surrounding tissues is simultaneously measured and later converted to intravascular concentration levels of CA using methods discussed later. The advantage of this method is that AIF is obtained at an individual basis while being completely noninvasive.
The third method, population average AIF, assumes that the AIF is similar for similar subjects. The AIF is measured for a small cohort of subjects and the same is assumed to be the valid AIF for the later subjects also. The merit of this method is its simplicity in both acquisition and analysis. The major drawback is that the changes introduced by the pathology under study are ignored.
Backward model
The concentration change at every pixel can be used to measure the flow of CA at that pixel. There are various models that can be used to measure the flow of CA at a pixel. Some of the most common ones are the Patlak model and the eTofts model.
Patlak model
The Patlak model is a linear model which follows the two compartment theory.
where C_{t} is the total CA concentration, V_{p} is the fractional plasma volume per tissue volume, C_{p} is the AIF and K^{trans} is the forward transfer model. Since this is a linear model, concentration can be retrieved by taking the pseudo inverse.
eTofts model
The extended Tofts model assumes a term K_{ep} which accounts for the backward flow of fluids into capillaries.
Since, eTofts model is nonlinear in nature, a simple pseudoinverse technique is not sufficient to obtain concentration levels back. An iterative algorithm can be used to solve this model fitting:
As the name suggests, Forward model is essentially the reverse operation of Backward model, i.e., the concentration maps are derived from the TK parametric maps.
METHODOLOGY
Concepts
Model consistency constraint
Guo et. al, 2016 and Megha propose jointly estimating contrast concentration versus time images (C) and TK parameter maps (θ) from undersampled data (y) by solving the least square problem:
The first term, L_{2} norm, represents data consistency, i.e., C should be consistent with the measured data y by signal equation (ψ), undersampling mask (U), fourier transform (F) and sensitivity encoding (E). S_{0} is the first time frame image which is fully sampled. S_{0} is usually obtained before administration of the CA. The second term, L_{P} norm (Pant et. al., 2014) represents the model consistency constraint, where, C must be consistent with the modelling TK parameter maps i.e. Patlak or eTofts in this scenario.
This review is on joint estimation of AIF and TK parameters. AIF can be incorporated in equation (5),
we solve each variable alternatively as shown in (8), (n^{th} iteration):
equation (8) is backward TK modelling, which also estimates patient specific AIF. For the estimation of patAIF, the arterial ROI must be identified.
Methods
The aim of this project was to do a comparative analysis of the methods proposed by Guo et. al., and Megha Goel, i.e., L_{2} norm regularisation and L_{p} norm regularisation (p≤1) in model consistency constraints using both Patlak and eTofts model.
Dataset
The data used in this project is the Invivo dataset, which has been made available by Guo et. al, 2016. This consists of nine fully sampled brain tumor DCE MR images of a male patient of age 65 years diagnosed with glioblastoma. The parameters of this data set are as follows: field of view = 22 x 22x 4.2 cm^{3}, spatial resolution = 0.9 x 1.3 x 7.0 mm^{3}, temporal resolution = 5s, 50 time frames, eight receiver coils, flip angle = 15°, echo time = 1.3ms, repetition time = 6 ms, DEPOSIT1 has been performed before DCEMRI, with flip angle of 2°, 5°, and 10° to estimate precontrast T_{1} and M0 maps. The contrast agent that has been used is gadobenate dimeglumine.
The data was retrospectively undersampled using randomised Golden angle Ratio at R= 20x, 40x, 80x, 100x and 120x. Gaussian noise was added to the image. For patient specific AIF, the ROI was detected and AIF was calculated by applying the forward model to the fully sampled image S_{0}. A modified version of Conjugate gradient method proposed by Pant et. al., 2014 was used to to optimize the function.
RMSE
Root Mean Square Error is used as the figure of merit in this review to compare the different methods and models used. The K^{trans} maps obtained by reconstruction of the undersampled data is quantitatively compared with the K^{trans} maps obtained from the fully sampled S_{0} pixel by pixel, in the ROI.
Normalised RMSE is calculated as
RMSE was also calculated for the AIF that was estimated in the models that used patAIF. The AIF that was estimated was quantitatively compared with the AIF obtained from the fully sampled S_{0}, for every time frame.
RESULTS AND DISCUSSION
Fig 4 shows the K^{trans} maps of the ROI for L_{1} and L_{0.7} norms using Patlak model and L_{2} using eTofts.
Fig 5 and table 1 show the RMSE values for the undersampling values 20x, 40x, 60x, 100x and 120x which were performed for Patlak and eTofts model using L_{2}norm, L_{1}norm and L_{0.7}norm regularisation on model consistecy constraint. As seen on the graph, L_{0.7} norm regularisation and L_{1}norm patspecific have performed fairly well even at 120x undersampling when compared to the other methods.
Model/Undersampling rate  20  40  60  100  120 
L2 Patlak patspecific  0.1677  0.2041  0.2419  0.2962  0.2755 
L2 Patlak popavg  0.362  0.3502  0.3895  0.4105  0.4114 
L1 Patlak patspecific  0.1037  0.1349  0.1553  0.2011  0.2548 
L0.7 Patlak patient specific  0.1023  0.1356  0.1601  0.2102  0.2789 
L2 eTofts Popavg  0.2786  0.2881  0.2442  0.2482  0.2568 
eTofts has maintained a steady average RMSE of 0.26 throughout all the undersampling rates. Whereas, L_{2}norm regularisation of Patlak model using popAIF has performed poorly having an average of 0.38 RMSE worsening with progressing undersampling rates. L_{1} and L_{0.7} norm regularisations have performed exceptionally well with negligible error rates.
Fig 6 depicts the graph of AIF values for the 50 slices of images in the data for L0.7, L1 and L2 regularisations at R=60x undersampling. As the value of p in L_{p}norm reduces, there is a significant rise in accuracy in the estimation of the AIF peak. Conversely, as p value increases, the average estimation has improved of the AIF after the 15th slice. Hence, it can be said that a tradeoff exists between the two in terms of estimation of AIF.
Fig7 shows the RMSE values of AIF estimation for different undersampling rates. As seen, L_{1} norm is the least erroneous of the three. Although, for this range of undersampling rates, all three methods have minuscule error rates.
As suggested by Khalifa et al., 2014, we can observe the simplicity of Patlak model as it omits reverse vascular transfer coefficient, K_{ep}. This highly simplifies the model and makes it linear thus facilitating the easy extraction of concentration values from TK parameters. At the same time eTofts model, even though slightly more accurate than Patlak in some cases, fails to perform adequately due to its acceptance of popAIF and less flexibility in comparison with Patlak model. For all the above reasons, the Patlak model has evolved to be more popular in the quantitative analysis of DCEMRI than eTofts model. Studies also claim that the eTofts method sometimes produces drastic errors as opposed to Patlak model.
CONCLUSION
We explored the implementation of the proposed methods of joint estimation of AIF and TKparameters, weighing them based on their RMS errors. While L_{2}norm regularisation of Patlak model failed to perform on heavily undersampled data, the same on eTofts model seemed to show stable performance but with low accuracy. As we moved to L_{p}norm (p≤1) , the accuracy visibly improved in TK parameter estimation, but at the cost of increased error in AIF estimation.
Methods which strike a balance between the L_{2} and L_{p} norms incorporating the strengths of each and also reaching the optimum solution quicker are worth exploring. The scarcity of DCEMRI datasets that provide complete AIF information is a major setback in this regard.
Bibliography
References

Thomas Yankeelov, John Gore, 2007, Dynamic Contrast Enhanced Magnetic Resonance Imaging in Oncology:Theory, Data Acquisition,Analysis, and Examples, Current Medical Imaging Reviews, vol. 3, no. 2, pp. 91107

Yi Guo, Sajan Goud Lingala, Yinghua Zhu, R. Marc Lebel, Krishna S. Nayak, 2016, Direct estimation of tracerkinetic parameter maps from highly undersampled brain dynamic contrast enhanced MRI, Magnetic Resonance in Medicine, vol. 78, no. 4, pp. 15661578

Jeevan K. Pant, WuSheng Lu, Andreas Antoniou, 2014, New Improved Algorithms for Compressive Sensing Based on $\ell_{p}$ Norm, IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 61, no. 3, pp. 198202

Fahmi Khalifa, Ahmed Soliman, Ayman ElBaz, Mohamed Abou ElGhar, Tarek ElDiasty, Georgy Gimel'farb, Rosemary Ouseph, Amy C. Dwyer, 2014, Models and methods for analyzing DCEMRI: A review, Medical Physics, vol. 41, no. 12, pp. 124301
Source

Fig 1: http://sites.google.com/site/bmed360

Fig 2: Yankeelov, Thomas E., and John C. Gore. "Dynamic contrast enhanced magnetic resonance imaging in oncology: theory, data acquisition, analysis, and examples." Current medical imaging reviews 3.2 (2007): 91107.

Fig 3: Yankeelov, Thomas E., and John C. Gore. "Dynamic contrast enhanced magnetic resonance imaging in oncology: theory, data acquisition, analysis, and examples." Current medical imaging reviews 3.2 (2007): 91107.
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