Efficient and Accurate Estimation of Pharmacokinetic Maps from DCE-MRI using Extended Tofts Model in Frequency Domain

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Applying compartmental models pertinent to the pathophysiology of the tissue being studied will enable the quantification of Pharmacokinetic (PK) maps. Integrating time-concentration data into these models in the time domain is the traditional way of determining PK parameters. Although the fitting problem is two-dimensional, this approach suffers from a dimensionality problem, which may be highly computational and lengthy. The Extended Tofts Model in Frequency Domain (ETM-FD) is derived as a convolution-free model from this research's standard comprehensive Tofts Model in Time Domain (ETM-TD). This method decreases the complexity of the curve fitting problem from two to one, minimizes computing time, and allows radiologists to evaluate PK maps within a clinical context. The ETM-FD and ETM-TD procedures were employed to both types of in silico phantoms to verify their resilience, and the Pearson correlation coefficient was calculated to compare with ground truth values. The results indicated that r2 = 1 values were achieved for both ETM-TD and ETM-FD, proving the approach's effectiveness. According to our findings, the ETM-FD took 24–25% less computing time to compute PK maps than the ETM-TD. The computing power of the ETM-FD technique makes it a suitable tool for analyzing three-dimensional and more thorough coverage of the tissue of interest. The ETM-FD technique is especially effective in clinical situations where determining PK maps swiftly and precisely is crucial for patient care. The ETM-FD, which is computationally effective, reliable, and capable of producing precise and real-time PK maps, is a promising method for PK parameter estimation from DCE-MRI data. Our findings demonstrate that this strategy has the potential to assist radiologists in making clinical decisions and enhance patient care. Extracellular Extravascular Space Pharmacokinetic Rate Parameters Perfusion Characteristics and Pharmacokinetic Modeling Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 1 Introduction DCE-MRI is a method that is extensively used to characterize tissues, identify vascularity and tissue perfusion, and track results of therapy for a variety of illnesses, such as heart failure, breast and prostate cancer, renal rejection, and liver cancer [ 1 – 3 ]. This method includes injecting a Contrast Agent (CA) and taking several T1 weighted pictures prior to and following injections to examine CA wash-in and wash-out in the Region of Interest (ROI). By comparing the T1 relaxation times of benign and malignant tissues, the ROI can be identified [ 4 – 5 ]. PK modeling is then used to quantify the kinetics of the CA in both types of tissues, resulting in PK parameters such as K trans , ν e , and ν p that serve as biomarkers for cancer assessment and therapy monitoring. The concentration of the CA is determined by analyzing the relative changes in T1, which can be obtained through voxel wise analysis of the CTCs specific to the tissue ROI. Qualitative, quantitative, and semi-quantitative approaches may be used to assess DCE-MRI data; however, the latter is constrained by a lack of standardization and repeatability [ 5 – 9 ]. Based on the type of pathology and the DCE-MRI data [ 10 – 13 ], it is possible to infer physiological information such as blood flow, tissue permeability, and microvasculature by fitting DCE-MRI data to a variety of pathophysiological models, such as Tofts [ 5 ], Extended Tofts [ 9 ], DP, AATH, Brix, and Shutter Speed. The ETM offers a number of advantages over earlier pharmacokinetic models used to evaluate DCE-MRI data. One of its primary advantages is its simplicity, which requires just the computation of two critical parameters—the transfer constant K trans and the measured percentage of extracellular space ν e . As a result, the ETM may be utilized to analyze DCE-MRI data in both clinical and research settings efficiently. Another advantage of the ETM is its ability to provide data on tissue perfusion, vascular permeability, and extracellular volume fraction all of which are essential markers for evaluating tumor angiogenesis, inflammation, and other physiological processes. This study aims to improve pharmacokinetic (PK) map determination rate and precision using DCE-MRI datasets. The Extended Tofts model is our goal in the time domain. The ETM makes DCE-MRI analyses simpler by determining significant variables, like Ktrans and Ve, that yield information regarding tissue attributes. In contrast, the existing approach to generating PK maps may require a while. This investigation attempts to minimize computation time while conserving accuracy to increase the efficacy of DCE-MRI evaluation for clinical and research utilization. With this development, we can identify significant physiological processes more quickly and precisely, including tumor angiogenesis and resistance. The following is a summary of the main contributions of our work: Faster PK Map Generation: Using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data, we developed an innovative approach for swiftly generating pharmacokinetic (PK) maps that significantly reduces the duration of time needed for computation. Preservation of Accuracy: Our method ensures that PK maps, particularly ones generated from the Extended Tofts model (ETM) in the time domain, continue to have excellent accuracy and dependability despite performance advancements. Clinical Efficiency: The proposed approach reduces the time required for creating the PK map, improving the clinical efficacy of DCE-MRI research. For diseases like cancer, this allows for faster treatment planning and monitoring. Advancements in diagnostic results: Our research advances comprehension of tissue characteristics, such as inflammation and tumor angiogenesis, and enhances our understanding of many different diseases and how they react to treatments. 2 Related Work DCE-MRI, also known as dynamic contrast-enhanced magnetic resonance imaging, is an effective method for evaluating tissue perfusion and vascular characteristics. An improvement to the original Tofts model, the Extended Tofts Model allows for measuring contrast agent concentration in tissue utilizing PK characteristics. The study gains an additional level using the frequency domain technique. The research approach outlined in the paper [ 14 ] serves as a preventative step to defend requests against various risks and unauthorized access attempts. The findings of this research enhance the discussion of security more broadly and emphasize the significance of implementing thorough security measures to protect the confidentiality and integrity of services. According to the study in paper [ 15 ], wavelet transforms have the potential to improve the accuracy of recognition systems. Wavelet transforms may be used to extract features more effectively and represent the traits more precisely, which improves recognition reliability and accuracy. The results indicate that incorporating wavelet-based approaches into recognition systems can increase recognition rates and decrease susceptibility to changes in illumination, position, and other parameters. The authors of the research [16] extract the radiomic characteristics from breast DCE-MRI, which may have the potential to significantly improve the understanding of how pathological alterations in breast cancer patients respond to neoadjuvant chemotherapy treatment. Breast cancer patients may use this technology to have better treatment results, fewer needless procedures, and more individualized care. In this paper [ 17 ], the authors adopt a fully convolutional network (FCN) model with the CTI as one of the inputs for breast lesion segmentation. The suggested technique outperforms current qualitative and numerical methods on an independent public dataset and a private household breast DCE-MRI dataset (TBD In paper [18], the developed approach, a hard-clustering methodology using a Non-dominated Sorting Genetic Algorithm (NSGA-II), is used to segment the breast's 10 Sagittal T2-weighted fat-suppressed DCE-MRI. The comparison analysis with the K-means algorithm was made, and the developed technique improves K-means both computationally and subjectively. In this research, the authors [19] derive five semi-quantitative metrics from image data pertaining to ROI areas. During categorizing, the linear discriminant analysis (LDA) and support vector machine (SVM) had been employed, and the region beneath the receiver's operating characteristic curve (AUC) served to assess the classifier's accuracy. The outcomes suggested that a classifier with training is capable of distinguishing between those who have prostate cancer and those without it. The authors of this paper presented [20], which is the application of a deep learning technique to improve the standard of medical images acquired by DCE-MRI. Mean squared error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) results showed successful communication as well as representation between the generated and original pictures during the general level in addition to particular areas that are relevant. Dynamic contrast-enhanced MRI (DCE-MRI) and Diffusion Weighted Imaging (DWI) serve as input to Convolutional neural network (CNN) models in this article [21]. and an accuracy of 90.8% is achieved by Inception -V3. In paper [22], the authors suggest a unique loss function that enhances the capacity of Generative Adversarial Networks models to acquire the important periodic tissue dynamics in this context, as well as a clinically pertinent measure for assessing efficiency. This forecasted appropriate responses that could potentially utilized to develop rapid diagnostics approaches. In accordance with the research's insights in article [23], the DCE-MRI evaluation procedure increases considerably in both efficiency and accuracy when adaptive models are incorporated. This has facilitated how microvascular properties are assessed in a broad range of tissue types, paving up the door to potentially swifter and more accurate medical diagnosis as well as evaluations. The performance of a new curve-fitting algorithm for determining blood circulation biomarkers in DCE MRI has been evaluated by the authors of article [24], and it was found to be superior in the case of the two-compartment exchange model. For the ETM and the 2CXM, the Bayesian technique revealed considerably fewer non-physiological high-intensity \({\nu }_{e }\) values (p0.0001). 3 Theory Drug dosage and treatment management may be effective when using DCE-MRI for PK mapping. Additionally, the processing time needed for a precise PK map estimation might limit its usage in medical trials. The ETM model is recommended as an alternative to conventional curve fitting techniques because of its promising performance in terms of processing speed and acceptable errors. The applications of the aforementioned approach can improve both the efficacy and accuracy of PK map estimates, which will eventually result in more accurate clinical assessment and treatment planning. 3.1 Extended Tofts Model in Frequency Domain (ETM-FD) Forward The standard method for determining PK maps is based on the Extended Tofts Model in Time Domain (ETM-TD), as demonstrated by Eq. ( 1 ). $$C\left(t\right)={K}^{trans} \times \text{exp}\left(-{K}_{ep }\times t\right) ⨀ Ca\left(t\right)+{\nu }_{p }\times t$$ 1 Where, ⨀ indicates convolution, × denotes multiplication, C(t) is the tissue's concentration of an MR-derived tracer. The concentration of the tracer in blood plasma, Ca(t), may be roughly calculated using an Arterial Input Function (AIF), and t is the time. \({K}^{trans}\) (min-1) is the rate at which the CA moves from plasma into the interstitial space, \({K}_{ep }\) (min-1) is the rate at which the CA moves from the interstitial compartment to the vascular compartment. Pixel-wise kinetic parameters are estimated by fitting data to the model given in (1). ν p depicts the volume in the plasma. Convolution in Eq. ( 1 ) makes it possible to use the Fourier transformation to reduce computing time and effort. Eq. ( 2 ) may be expressed as: $$F\left\{C\left(t\right)\right\}= F\{{K}^{trans} \times exp ({K}_{ep }\times t\left)\right\}\times F\{{\nu }_{p }\times Ca\left(t\right)\}$$ 2 Eq. ( 2 ) can be further simplified as depicted in Eq. ( 3 ) by taking out \(F\) {Ca(t)} as a common term. $$F\left\{C\left(t\right)\right\}= F\{{K}^{trans} \times exp (-{K}_{ep }\times t\left)\right\}\times F\left\{{\nu }_{p }\right)\left\} \right]$$ 3 When the Fast Fourier Transform (FFT) is used, Eq. ( 2 ) may be simplified further by allowing C(t) to equal Q1 and Ca(t) to equal Q2, yielding (4) below: \(F\left\{C\left(t\right)\right\}=Q1\) and \(F\left\{Ca\left(t\right)\right\}=Q2\) , Q1/Q2 = Q (4) which results in (5) below: $$Q= F\{{K}^{trans} \times exp (-{K}_{ep }\times t\left)\right\}\times F\left\{{\nu }_{p }\right)\left\} \right]$$ 5 Applying the inverse fourier transform on each end results in Eq. ( 6 ): $${{F}^{-1}\left(Q\right)= F}^{-1}\left(F\right\{{K}^{trans} \times exp (-{K}_{ep }\times t)\}\times {F}^{-1}(F\left\{{\nu }_{p }\right\})$$ 6 ETM-FD now accepts the final form of Eq. ( 7 ) to be examined for curve fitting of the two kinetic parameters, with the legitimate assumption that only the real component should be fitted, as the RHS must be exclusively real. $$real \left({F}^{-1}\left\{Q\right\}\right)= \{{K}^{trans} \times exp (-{K}_{ep }\times t\left)\right\}\}+ {\nu }_{p }$$ 7 4 Proposed Work The performance is evaluated using QIBA (Quantitative Imaging Biomarkers Alliance) simulated data and randomly simulated data approaches. 4.1 In silico simulation For the simulations, Concentration Time Curves (CTCs) from both randomly generated Concentration Time Curves (CTCs) and CTCs from the Quantitative Imaging Biomarker Alliance (QIBA) were used. This was carried out to make sure that the technique would work for most scenarios and would apply to data from public archives. The ETM-TD and ETM-FD models were used to simulate CTCs in a pathophysiological context. CTCs were simulated using the ETM-TD model using a range of randomized generated values, where 0.01 < K trans < 0.5 min-1, 1 < ν e < 99%, 0.01 < ν p < 0.5 min-1 in order to test the suggested methodology. In the first instance, the ratio of K trans to ν e was calculated in order to estimate the value of K ep . According to published values of K trans (0.01, 0.02, 0.05, 0.1, 0.2, and 0.35 min-1), ν e (0.01, 0.05, 0.1, 0.2, and 0.5), and ν p (0.01, 0.02, 0.05, 0.1, 0.2, and 0.35 min-1), datasets for the QIBA were constructed in the second case. The simulated data was produced with a time preciseness of one second intervals in a time frame lasting 120 seconds, and the simulated values were utilized as the GT values. An Artery Input Function (AIF) curve was built using parameters from references [24] and experimentally derived population averages. The unpredictable least squares method and the trust-region approach were used to fit the simulated CTCs with an AIF to the ETM-TD and ETM-FD models in order to calculate the kinetic variables. By running the code five times for each strategy and calculating the mean and Standard Deviation (SD), the computing time required to calculate these parameters for each randomly generated datasets were assessed. This was done for the randomly produced in silico phantoms using different matrix sizes of 20x20, 30x30, 40x40, 50x50 and 100x100 respectively. The PK parameter estimation periods obtained from the ETM-TD and ETM-FD procedures were compared using a paired t-test. The significance threshold of 0.05 indicated that the variances were statistically meaningful. Additionally, the degree of fit for K trans , K ep , and ν p parameters regarding the GT was evaluated using Pearson's correlation analysis. Matlab software tool was used to develop the ETM-TD and ETM-FD techniques on a windows desktop PC with an Intel Core i5 CPU running at 2.60 GHz and 8GB RAM. To measure computational efficiency, the estimated values of K trans , K ep , and ν p were graphed. Despite the ETM's ability to display K ep values, radiologists are more accustomed to assessing ν e than K ep , thus the values of K trans , ν p , and ν e are depicted as images. This is confirmed further by the fact that these three numbers are connected to one another as a ratio, as shown in Eq. ( 8 ) below: $${v}_{e}=\frac{{K}^{trans}}{{K}_{ep}}$$ 8 5 Simulation Results Employing simulated data gathered by QIBA and randomly simulated data techniques, the performance is assessed. 5.1 In silico simulation data The Pearson correlation study depicted in Fig. 7 and Fig. 8 revealed that the reconstructed PK data had a good correlation with the GT data, with r2 values near to one for both randomly generated and QIBA simulated datasets. Furthermore, the difference maps (difference GT-TD and difference GT-FD) in the fourth and fifth rows have much lower values than their equivalent positions in the GT PK maps, as seen in Fig. 1 - Fig. 6 . The time required for the two techniques is shown in Fig. 9 , and it can be seen that the suggested technique used less time to determine the PK maps than the traditional approach accomplished. Table.1 provides additional support for the effectiveness of the ETM-FD approach by demonstrating the time savings achieved through the use of frequency-based techniques. As the size of the matrix grows, as predicted, the amount of time also improves. Additionally, the statistical results indicate the execution time difference is statistically significant, with a percentage difference ranging from 24–25%. Table 1 Comparison of execution time for ETM-TD and ETM-FD for different matrix dimensions, executed for five times each, analyzed by paired t-test with * indicating P values less than 0.05. Matrix Dimension Time taken (sec) Time difference (min) Percentage difference (%) ETM-TD ETM-FD 10 x 10 0.16 0.12 0.04 25 20 x 20 0.64 0.48 0.16 25 30 x 30 1.45 1.09 0.36 24.82 40 x 40 2.55 1.93 0.62 24.31 50 x 50 4.02 3.05 0.97 24.12 6 Conclusion and Future scope The purpose of this study was to assess the accuracy and computational effectiveness of two methods for implementing the ETM utilising in silico simulation data that was produced randomly and that was QIBA-based. The real component of the final equation only has to be evaluated for curve fitting since the PK parameters are real-valued. The PK parameters are determined using three known parameters (concentration, AIF, and time) in the TM-TD technique, which results in a surface fit. The ETM-FD strategy, in contrast, simplifies the issue by determining the concentration to AIF ratio, as stated in Eq. ( 5 ), leading to a curve fit and streamlining the regression analysis. It is interesting to consider that using the log operator to linearize Eq. ( 6 ) might significantly speed up the existing technique. In this case, the data range would be compressed, which would be undesirable considering the confined dynamic range of the PK parameters, which typically range from 0 to 5 min-1 for K trans , 0 to 1 for ν e , and 0 to 1 for ν p . According to the results, the ETM-FD strategy had less error than the ETM-TD approach, as shown in Fig. 7 and Fig. 8 . Additionally, the results of the pearson correlation analysis demonstrated that both methods were equally successful in calculating the PK parameters. The clinical usefulness of the ETM-FD strategy, which provides a more effective computational solution than the ETM-TD approach, is further supported by these findings. For the real-time visualization of PK maps in a clinical scenario, Table.1 indicates the usage of the ETM-FD strategy saves much more computing time than the ETM-TD approach. The graph also indicates that whenever the size of the matrix increases, so does the execution time. This is critical when analyzing larger ROIs or when a PK map of "normal" regions is required for comparison with the tumor. A more thorough evaluation of the PK behaviour of the whole tumour may be made by radiologists using the TM-FD technique, which can generate real-time PK maps for bigger or 3D ROIs. For estimating PK maps, this work developed a ETM-FD technique as a quicker substitute for the traditional TM-TD approach. It was demonstrated how the ETM-FD approach was faster and more accurate in estimating PK maps than the TM-TD method through in silico simulations used to evaluate the accuracy and performance of the two methods. The results showed an acceleration factor between 24 and 25 percent. Declarations Funding The authors did not receive support from any organization for the submitted work. 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Ioannidis, Katerina Nikiforaki et al., Statistical and spatial correlation between diffusion and perfusion MR imaging parameters: A study on soft tissue sarcomas. European Journal of Medical Physics. (65). pp.59-66. (2019). Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3359472","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":250518678,"identity":"e85976df-c2a6-4608-805e-53eaa95e8377","order_by":0,"name":"KRUTTHIKA HIREBASUR 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College","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Sudhir","middleName":"Kumar","lastName":"Trivedi","suffix":""}],"badges":[],"createdAt":"2023-09-15 17:28:00","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3359472/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3359472/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":46800690,"identity":"db154406-da19-4394-a00b-7a141fc3231a","added_by":"auto","created_at":"2023-11-20 19:42:39","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":257698,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic K\u003csup\u003etrans\u003c/sup\u003e maps of a randomly simulated generated values of size 50 x 50 along with difference with respect to the GT.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/cd32dc1d4161035d4642f48e.png"},{"id":46802414,"identity":"825a4a6c-b477-4e8e-9a73-7913df2b95fd","added_by":"auto","created_at":"2023-11-20 19:50:39","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":268732,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic K\u003csub\u003eep\u003c/sub\u003e maps of a randomly simulated generated values of size 50 x 50 along with difference with respect to the GT.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/64744726c59b6b34fde703c8.png"},{"id":46804394,"identity":"012fa574-dc70-4a15-8f22-61be31e0e8a6","added_by":"auto","created_at":"2023-11-20 20:06:39","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":224767,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic ν\u003csub\u003ep\u003c/sub\u003e maps of a randomly simulated generated values of size 50 x 50 along with difference with respect to the GT\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/0db52c0d2d89dcbde02e2652.png"},{"id":46800686,"identity":"26487bb9-9041-431f-8af7-1257ff870437","added_by":"auto","created_at":"2023-11-20 19:42:38","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":47346,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic K\u003csup\u003etrans\u003c/sup\u003e maps of a QIBA constructed phantom of size 180 x 60 along with difference with respect to the GT\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/71a563400e02e4e8881dc02b.png"},{"id":46800689,"identity":"3d9b8564-1904-4354-b497-6c47910e34a3","added_by":"auto","created_at":"2023-11-20 19:42:39","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":68146,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic K\u003csub\u003eep\u003c/sub\u003e maps of a QIBA constructed phantom of size 180 x 60 along with difference with respect to the GT\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/1b60e27035f6bf47b4fd39ba.png"},{"id":46800688,"identity":"d3f6dc1a-aca3-4a89-bcc6-721ba6d9036f","added_by":"auto","created_at":"2023-11-20 19:42:39","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":45384,"visible":true,"origin":"","legend":"\u003cp\u003ePharmacokinetic ν\u003csub\u003ep\u003c/sub\u003e maps of a QIBA constructed phantom of size 180 x 60 along with difference with respect to the GT\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/f57827319bb0102255ff73c8.png"},{"id":46800687,"identity":"eac53404-4b93-4a80-b0e0-224991b16643","added_by":"auto","created_at":"2023-11-20 19:42:38","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":168053,"visible":true,"origin":"","legend":"\u003cp\u003eScatter plot of randomly simulated data with Pearson correlation coefficient r2 = 1, indicating a perfect positive linear relationship between the two variables for two techniques.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/4402b4666a3f2a2210b4f835.png"},{"id":46800693,"identity":"771dcb89-d14f-44b6-b999-268726e85d72","added_by":"auto","created_at":"2023-11-20 19:42:39","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":174480,"visible":true,"origin":"","legend":"\u003cp\u003eScatter plot of QIBA simulated data with Pearson correlation coefficient r2 = 1, indicating a perfect positive linear relationship between the two variables for two techniques.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/fc2d8b61a716680f22b8eddd.png"},{"id":46803903,"identity":"7986e3f6-e1de-4655-a20f-afcc57efdd25","added_by":"auto","created_at":"2023-11-20 19:58:39","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":138510,"visible":true,"origin":"","legend":"\u003cp\u003eComputational time comparison TD and FD approaches at various matrix sizes.\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/0c4a613c5650751a9ec9b8ed.png"},{"id":48986589,"identity":"13dea5ab-609e-4070-bd4d-e563b375538a","added_by":"auto","created_at":"2023-12-30 01:21:35","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1506704,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3359472/v1/1682adde-1a78-4719-a02f-9419aa6fe9a0.pdf"}],"financialInterests":"","formattedTitle":"Efficient and Accurate Estimation of Pharmacokinetic Maps from DCE-MRI using Extended Tofts Model in Frequency Domain","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eDCE-MRI is a method that is extensively used to characterize tissues, identify vascularity and tissue perfusion, and track results of therapy for a variety of illnesses, such as heart failure, breast and prostate cancer, renal rejection, and liver cancer [\u003cspan additionalcitationids=\"CR2\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. This method includes injecting a Contrast Agent (CA) and taking several T1 weighted pictures prior to and following injections to examine CA wash-in and wash-out in the Region of Interest (ROI). By comparing the T1 relaxation times of benign and malignant tissues, the ROI can be identified [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. PK modeling is then used to quantify the kinetics of the CA in both types of tissues, resulting in PK parameters such as K\u003csup\u003etrans\u003c/sup\u003e, ν\u003csub\u003ee\u003c/sub\u003e, and ν\u003csub\u003ep\u003c/sub\u003e that serve as biomarkers for cancer assessment and therapy monitoring. The concentration of the CA is determined by analyzing the relative changes in T1, which can be obtained through voxel wise analysis of the CTCs specific to the tissue ROI. Qualitative, quantitative, and semi-quantitative approaches may be used to assess DCE-MRI data; however, the latter is constrained by a lack of standardization and repeatability [\u003cspan additionalcitationids=\"CR6 CR7 CR8\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Based on the type of pathology and the DCE-MRI data [\u003cspan additionalcitationids=\"CR11 CR12\" citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], it is possible to infer physiological information such as blood flow, tissue permeability, and microvasculature by fitting DCE-MRI data to a variety of pathophysiological models, such as Tofts [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e], Extended Tofts [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e], DP, AATH, Brix, and Shutter Speed.\u003c/p\u003e \u003cp\u003eThe ETM offers a number of advantages over earlier pharmacokinetic models used to evaluate DCE-MRI data. One of its primary advantages is its simplicity, which requires just the computation of two critical parameters\u0026mdash;the transfer constant K\u003csup\u003etrans\u003c/sup\u003e and the measured percentage of extracellular space ν\u003csub\u003ee\u003c/sub\u003e. As a result, the ETM may be utilized to analyze DCE-MRI data in both clinical and research settings efficiently. Another advantage of the ETM is its ability to provide data on tissue perfusion, vascular permeability, and extracellular volume fraction all of which are essential markers for evaluating tumor angiogenesis, inflammation, and other physiological processes.\u003c/p\u003e \u003cp\u003eThis study aims to improve pharmacokinetic (PK) map determination rate and precision using DCE-MRI datasets. The Extended Tofts model is our goal in the time domain. The ETM makes DCE-MRI analyses simpler by determining significant variables, like Ktrans and Ve, that yield information regarding tissue attributes. In contrast, the existing approach to generating PK maps may require a while. This investigation attempts to minimize computation time while conserving accuracy to increase the efficacy of DCE-MRI evaluation for clinical and research utilization. With this development, we can identify significant physiological processes more quickly and precisely, including tumor angiogenesis and resistance.\u003c/p\u003e \u003cp\u003eThe following is a summary of the main contributions of our work:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eFaster PK Map Generation: Using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) data, we developed an innovative approach for swiftly generating pharmacokinetic (PK) maps that significantly reduces the duration of time needed for computation.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ePreservation of Accuracy: Our method ensures that PK maps, particularly ones generated from the Extended Tofts model (ETM) in the time domain, continue to have excellent accuracy and dependability despite performance advancements.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eClinical Efficiency: The proposed approach reduces the time required for creating the PK map, improving the clinical efficacy of DCE-MRI research. For diseases like cancer, this allows for faster treatment planning and monitoring.\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eAdvancements in diagnostic results: Our research advances comprehension of tissue characteristics, such as inflammation and tumor angiogenesis, and enhances our understanding of many different diseases and how they react to treatments.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e"},{"header":"2 Related Work","content":"\u003cp\u003eDCE-MRI, also known as dynamic contrast-enhanced magnetic resonance imaging, is an effective method for evaluating tissue perfusion and vascular characteristics. An improvement to the original Tofts model, the Extended Tofts Model allows for measuring contrast agent concentration in tissue utilizing PK characteristics. The study gains an additional level using the frequency domain technique.\u003c/p\u003e \u003cp\u003eThe research approach outlined in the paper [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] serves as a preventative step to defend requests against various risks and unauthorized access attempts. The findings of this research enhance the discussion of security more broadly and emphasize the significance of implementing thorough security measures to protect the confidentiality and integrity of services.\u003c/p\u003e \u003cp\u003eAccording to the study in paper [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], wavelet transforms have the potential to improve the accuracy of recognition systems. Wavelet transforms may be used to extract features more effectively and represent the traits more precisely, which improves recognition reliability and accuracy. The results indicate that incorporating wavelet-based approaches into recognition systems can increase recognition rates and decrease susceptibility to changes in illumination, position, and other parameters.\u003c/p\u003e \u003cp\u003eThe authors of the research [16] extract the radiomic characteristics from breast DCE-MRI, which may have the potential to significantly improve the understanding of how pathological alterations in breast cancer patients respond to neoadjuvant chemotherapy treatment. Breast cancer patients may use this technology to have better treatment results, fewer needless procedures, and more individualized care.\u003c/p\u003e \u003cp\u003eIn this paper [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], the authors adopt a fully convolutional network (FCN) model with the CTI as one of the inputs for breast lesion segmentation. The suggested technique outperforms current qualitative and numerical methods on an independent public dataset and a private household breast DCE-MRI dataset (TBD In paper [18], the developed approach, a hard-clustering methodology using a Non-dominated Sorting Genetic Algorithm (NSGA-II), is used to segment the breast's 10 Sagittal T2-weighted fat-suppressed DCE-MRI. The comparison analysis with the K-means algorithm was made, and the developed technique improves K-means both computationally and subjectively. In this research, the authors [19] derive five semi-quantitative metrics from image data pertaining to ROI areas. During categorizing, the linear discriminant analysis (LDA) and support vector machine (SVM) had been employed, and the region beneath the receiver's operating characteristic curve (AUC) served to assess the classifier's accuracy. The outcomes suggested that a classifier with training is capable of distinguishing between those who have prostate cancer and those without it. The authors of this paper presented [20], which is the application of a deep learning technique to improve the standard of medical images acquired by DCE-MRI. Mean squared error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM) results showed successful communication as well as representation between the generated and original pictures during the general level in addition to particular areas that are relevant. Dynamic contrast-enhanced MRI (DCE-MRI) and Diffusion Weighted Imaging (DWI) serve as input to Convolutional neural network (CNN) models in this article [21]. and an accuracy of 90.8% is achieved by Inception -V3. In paper [22], the authors suggest a unique loss function that enhances the capacity of Generative Adversarial Networks models to acquire the important periodic tissue dynamics in this context, as well as a clinically pertinent measure for assessing efficiency. This forecasted appropriate responses that could potentially utilized to develop rapid diagnostics approaches. In accordance with the research's insights in article [23], the DCE-MRI evaluation procedure increases considerably in both efficiency and accuracy when adaptive models are incorporated. This has facilitated how microvascular properties are assessed in a broad range of tissue types, paving up the door to potentially swifter and more accurate medical diagnosis as well as evaluations.\u003c/p\u003e \u003cp\u003eThe performance of a new curve-fitting algorithm for determining blood circulation biomarkers in DCE MRI has been evaluated by the authors of article [24], and it was found to be superior in the case of the two-compartment exchange model. For the ETM and the 2CXM, the Bayesian technique revealed considerably fewer non-physiological high-intensity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\nu }_{e }\\)\u003c/span\u003e\u003c/span\u003e values (p0.0001).\u003c/p\u003e"},{"header":"3 Theory","content":"\u003cp\u003eDrug dosage and treatment management may be effective when using DCE-MRI for PK mapping. Additionally, the processing time needed for a precise PK map estimation might limit its usage in medical trials. The ETM model is recommended as an alternative to conventional curve fitting techniques because of its promising performance in terms of processing speed and acceptable errors. The applications of the aforementioned approach can improve both the efficacy and accuracy of PK map estimates, which will eventually result in more accurate clinical assessment and treatment planning.\u003c/p\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Extended Tofts Model in Frequency Domain (ETM-FD)\u003c/h2\u003e \u003cp\u003eForward The standard method for determining PK maps is based on the Extended Tofts Model in Time Domain (ETM-TD), as demonstrated by Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e).\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$C\\left(t\\right)={K}^{trans} \\times \\text{exp}\\left(-{K}_{ep }\\times t\\right) ⨀ Ca\\left(t\\right)+{\\nu }_{p }\\times t$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere, ⨀ indicates convolution, \u0026times; denotes multiplication, C(t) is the tissue's concentration of an MR-derived tracer. The concentration of the tracer in blood plasma, Ca(t), may be roughly calculated using an Arterial Input Function (AIF), and t is the time. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}^{trans}\\)\u003c/span\u003e\u003c/span\u003e (min-1) is the rate at which the CA moves from plasma into the interstitial space, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({K}_{ep }\\)\u003c/span\u003e\u003c/span\u003e(min-1) is the rate at which the CA moves from the interstitial compartment to the vascular compartment. Pixel-wise kinetic parameters are estimated by fitting data to the model given in (1). ν\u003csub\u003ep\u003c/sub\u003e depicts the volume in the plasma.\u003c/p\u003e \u003cp\u003eConvolution in Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) makes it possible to use the Fourier transformation to reduce computing time and effort. Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) may be expressed as:\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$F\\left\\{C\\left(t\\right)\\right\\}= F\\{{K}^{trans} \\times exp ({K}_{ep }\\times t\\left)\\right\\}\\times F\\{{\\nu }_{p }\\times Ca\\left(t\\right)\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eEq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) can be further simplified as depicted in Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) by taking out \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F\\)\u003c/span\u003e\u003c/span\u003e{Ca(t)} as a common term.\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$F\\left\\{C\\left(t\\right)\\right\\}= F\\{{K}^{trans} \\times exp (-{K}_{ep }\\times t\\left)\\right\\}\\times F\\left\\{{\\nu }_{p }\\right)\\left\\} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhen the Fast Fourier Transform (FFT) is used, Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e) may be simplified further by allowing C(t) to equal Q1 and Ca(t) to equal Q2, yielding (4) below:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(F\\left\\{C\\left(t\\right)\\right\\}=Q1\\)\u003c/span\u003e \u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(F\\left\\{Ca\\left(t\\right)\\right\\}=Q2\\)\u003c/span\u003e\u003c/span\u003e, Q1/Q2\u0026thinsp;=\u0026thinsp;Q (4)\u003c/p\u003e \u003cp\u003ewhich results in (5) below:\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$Q= F\\{{K}^{trans} \\times exp (-{K}_{ep }\\times t\\left)\\right\\}\\times F\\left\\{{\\nu }_{p }\\right)\\left\\} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eApplying the inverse fourier transform on each end results in Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e6\u003c/span\u003e):\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$${{F}^{-1}\\left(Q\\right)= F}^{-1}\\left(F\\right\\{{K}^{trans} \\times exp (-{K}_{ep }\\times t)\\}\\times {F}^{-1}(F\\left\\{{\\nu }_{p }\\right\\})$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eETM-FD now accepts the final form of Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e7\u003c/span\u003e) to be examined for curve fitting of the two kinetic parameters, with the legitimate assumption that only the real component should be fitted, as the RHS must be exclusively real.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$$real \\left({F}^{-1}\\left\\{Q\\right\\}\\right)= \\{{K}^{trans} \\times exp (-{K}_{ep }\\times t\\left)\\right\\}\\}+ {\\nu }_{p }$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e"},{"header":"4 Proposed Work","content":"\u003cp\u003eThe performance is evaluated using QIBA (Quantitative Imaging Biomarkers Alliance) simulated data and randomly simulated data approaches.\u003c/p\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e4.1 In silico simulation\u003c/h2\u003e \u003cp\u003eFor the simulations, Concentration Time Curves (CTCs) from both randomly generated Concentration Time Curves (CTCs) and CTCs from the Quantitative Imaging Biomarker Alliance (QIBA) were used. This was carried out to make sure that the technique would work for most scenarios and would apply to data from public archives. The ETM-TD and ETM-FD models were used to simulate CTCs in a pathophysiological context.\u003c/p\u003e \u003cp\u003eCTCs were simulated using the ETM-TD model using a range of randomized generated values, where 0.01\u0026thinsp;\u0026lt;\u0026thinsp;K\u003csup\u003etrans\u003c/sup\u003e\u0026lt; 0.5 min-1, 1\u0026thinsp;\u0026lt;\u0026thinsp;ν\u003csub\u003ee\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;99%, 0.01\u0026thinsp;\u0026lt;\u0026thinsp;ν\u003csub\u003ep\u003c/sub\u003e\u0026thinsp;\u0026lt;\u0026thinsp;0.5 min-1 in order to test the suggested methodology. In the first instance, the ratio of K\u003csup\u003etrans\u003c/sup\u003e to ν\u003csub\u003ee\u003c/sub\u003e was calculated in order to estimate the value of K\u003csub\u003eep\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eAccording to published values of K\u003csup\u003etrans\u003c/sup\u003e (0.01, 0.02, 0.05, 0.1, 0.2, and 0.35 min-1), ν\u003csub\u003ee\u003c/sub\u003e (0.01, 0.05, 0.1, 0.2, and 0.5), and ν\u003csub\u003ep\u003c/sub\u003e (0.01, 0.02, 0.05, 0.1, 0.2, and 0.35 min-1), datasets for the QIBA were constructed in the second case. The simulated data was produced with a time preciseness of one second intervals in a time frame lasting 120 seconds, and the simulated values were utilized as the GT values. An Artery Input Function (AIF) curve was built using parameters from references [24] and experimentally derived population averages.\u003c/p\u003e \u003cp\u003eThe unpredictable least squares method and the trust-region approach were used to fit the simulated CTCs with an AIF to the ETM-TD and ETM-FD models in order to calculate the kinetic variables. By running the code five times for each strategy and calculating the mean and Standard Deviation (SD), the computing time required to calculate these parameters for each randomly generated datasets were assessed. This was done for the randomly produced in silico phantoms using different matrix sizes of 20x20, 30x30, 40x40, 50x50 and 100x100 respectively.\u003c/p\u003e \u003cp\u003eThe PK parameter estimation periods obtained from the ETM-TD and ETM-FD procedures were compared using a paired t-test. The significance threshold of 0.05 indicated that the variances were statistically meaningful. Additionally, the degree of fit for K\u003csup\u003etrans\u003c/sup\u003e, K\u003csub\u003eep\u003c/sub\u003e, and ν\u003csub\u003ep\u003c/sub\u003e parameters regarding the GT was evaluated using Pearson's correlation analysis.\u003c/p\u003e\u003cp\u003eMatlab software tool was used to develop the ETM-TD and ETM-FD techniques on a windows desktop PC with an Intel Core i5 CPU running at 2.60 GHz and 8GB RAM. To measure computational efficiency, the estimated values of K\u003csup\u003etrans\u003c/sup\u003e, K\u003csub\u003eep\u003c/sub\u003e, and ν\u003csub\u003ep\u003c/sub\u003e were graphed. Despite the ETM's ability to display K\u003csub\u003eep\u003c/sub\u003e values, radiologists are more accustomed to assessing ν\u003csub\u003ee\u003c/sub\u003e than K\u003csub\u003eep\u003c/sub\u003e, thus the values of K\u003csup\u003etrans\u003c/sup\u003e, ν\u003csub\u003ep\u003c/sub\u003e, and ν\u003csub\u003ee\u003c/sub\u003e are depicted as images. This is confirmed further by the fact that these three numbers are connected to one another as a ratio, as shown in Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e8\u003c/span\u003e) below:\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$${v}_{e}=\\frac{{K}^{trans}}{{K}_{ep}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e"},{"header":"5 Simulation Results","content":"\u003cp\u003eEmploying simulated data gathered by QIBA and randomly simulated data techniques, the performance is assessed.\u003c/p\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e5.1 In silico simulation data\u003c/h2\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe Pearson correlation study depicted in Fig.\u0026nbsp;7 and Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e revealed that the reconstructed PK data had a good correlation with the GT data, with r2 values near to one for both randomly generated and QIBA simulated datasets. Furthermore, the difference maps (difference GT-TD and difference GT-FD) in the fourth and fifth rows have much lower values than their equivalent positions in the GT PK maps, as seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e1\u003c/span\u003e- Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eThe time required for the two techniques is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e, and it can be seen that the suggested technique used less time to determine the PK maps than the traditional approach accomplished.\u003c/p\u003e\u003cp\u003eTable.1 provides additional support for the effectiveness of the ETM-FD approach by demonstrating the time savings achieved through the use of frequency-based techniques. As the size of the matrix grows, as predicted, the amount of time also improves. Additionally, the statistical results indicate the execution time difference is statistically significant, with a percentage difference ranging from 24\u0026ndash;25%.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eComparison of execution time for ETM-TD and ETM-FD for different matrix dimensions, executed for five times each, analyzed by paired t-test with * indicating P values less than 0.05.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"5\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eMatrix Dimension\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eTime taken (sec)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eTime difference (min)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003ePercentage difference (%)\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eETM-TD\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eETM-FD\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10 x 10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.04\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e20 x 20\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.48\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.16\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e25\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e30 x 30\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e1.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.09\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.36\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e24.82\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e40 x 40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2.55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.93\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.62\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e24.31\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e50 x 50\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e4.02\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e3.05\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e0.97\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e24.12\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"6 Conclusion and Future scope","content":"\u003cp\u003eThe purpose of this study was to assess the accuracy and computational effectiveness of two methods for implementing the ETM utilising in silico simulation data that was produced randomly and that was QIBA-based. The real component of the final equation only has to be evaluated for curve fitting since the PK parameters are real-valued. The PK parameters are determined using three known parameters (concentration, AIF, and time) in the TM-TD technique, which results in a surface fit. The ETM-FD strategy, in contrast, simplifies the issue by determining the concentration to AIF ratio, as stated in Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e5\u003c/span\u003e), leading to a curve fit and streamlining the regression analysis. It is interesting to consider that using the log operator to linearize Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e6\u003c/span\u003e) might significantly speed up the existing technique. In this case, the data range would be compressed, which would be undesirable considering the confined dynamic range of the PK parameters, which typically range from 0 to 5 min-1 for K\u003csup\u003etrans\u003c/sup\u003e, 0 to 1 for ν\u003csub\u003ee\u003c/sub\u003e, and 0 to 1 for ν\u003csub\u003ep\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003eAccording to the results, the ETM-FD strategy had less error than the ETM-TD approach, as shown in Fig.\u0026nbsp;7 and Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e. Additionally, the results of the pearson correlation analysis demonstrated that both methods were equally successful in calculating the PK parameters. The clinical usefulness of the ETM-FD strategy, which provides a more effective computational solution than the ETM-TD approach, is further supported by these findings.\u003c/p\u003e \u003cp\u003eFor the real-time visualization of PK maps in a clinical scenario, Table.1 indicates the usage of the ETM-FD strategy saves much more computing time than the ETM-TD approach. The graph also indicates that whenever the size of the matrix increases, so does the execution time. This is critical when analyzing larger ROIs or when a PK map of \"normal\" regions is required for comparison with the tumor. A more thorough evaluation of the PK behaviour of the whole tumour may be made by radiologists using the TM-FD technique, which can generate real-time PK maps for bigger or 3D ROIs.\u003c/p\u003e \u003cp\u003eFor estimating PK maps, this work developed a ETM-FD technique as a quicker substitute for the traditional TM-TD approach. It was demonstrated how the ETM-FD approach was faster and more accurate in estimating PK maps than the TM-TD method through in silico simulations used to evaluate the accuracy and performance of the two methods. The results showed an acceleration factor between 24 and 25 percent.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors did not receive support from any organization for the submitted work.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConflicts of interest/competing interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors have no conflicts of interest to declare that are relevant to the content of this article.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics approval\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis material is the authors' own original work, which has not been previously published elsewhere.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eW. Huang et al., Variations of dynamic contrast-enhanced magnetic resonance imaging in evaluation of breast cancer therapy response: A multicenter data analysis challenge. Transl. Oncol. 7(1). Pp.153\u0026ndash;166 (2014).\u003c/li\u003e\n\u003cli\u003eG. Brix et al., Microcirculation and microvasculature in breast tumors: pharmacokinetic analysis of dynamic MR image series. Magn. Reson. Med. 52(2). pp.420\u0026ndash;9 (2004).\u003c/li\u003e\n\u003cli\u003eD. Gianfelice, et al., MR imaging-guided focused ultrasound surgery of breast cancer: Correlation of dynamic contrast-enhanced MRI with histopathologic findings, Breast Cancer Res. Treat, 82(2). pp. 93\u0026ndash;101 (2003).\u003c/li\u003e\n\u003cli\u003eJ. L. Evelhoch et al., Key factors in the acquisition of contrast kinetic data for oncology. J. Magn. Reson. Imaging. 10(3). pp. 254\u0026ndash;9 (1999).\u003c/li\u003e\n\u003cli\u003eN. Hylton et al., Dynamic contrast-enhanced magnetic resonance imaging as an imaging biomarker. Journal of Clinical Oncology. 24(20). pp. 3293\u0026ndash;3298 (2006).\u003c/li\u003e\n\u003cli\u003eX. Li et al., DCE-MRI analysis methods for predicting the response of breast cancer to neoadjuvant chemotherapy: Pilot study findings. Magn. Reson. Med. 4(71), pp.1592\u0026ndash;1602 (2014).\u003c/li\u003e\n\u003cli\u003eG. H. Jahng, et al., Perfusion magnetic resonance imaging: A comprehensive update on principles and techniques. Korean Journal of Radiology. 15(5). pp. 554\u0026ndash;577 (2014).\u003c/li\u003e\n\u003cli\u003eR. J. Gillies et al., Applications of Magnetic Resonance in Model Systems: Tumor Biology and Physiology, Neoplasia. 2(1\u0026ndash;2). pp. 139\u0026ndash;151 (2000).\u003c/li\u003e\n\u003cli\u003eT. Yankeelov et al., Dynamic Contrast Enhanced Magnetic Resonance Imaging in Oncology: Theory, Data Acquisition, Analysis, and Examples. Curr. Med. Imaging Rev. 3(2). pp. 91\u0026ndash;107 (2007).\u003c/li\u003e\n\u003cli\u003eFahmi Khalifa et al.,: Models and methods for analyzing DCE-MRI: a review, Med Phys, 41(12), pp. 8\u0026ndash;9 (2014).\u003c/li\u003e\n\u003cli\u003eNithin N Vajuvalli et al., : Accelerated Pharmacokinetic Map Determination for Dynamic Contrast Enhanced MRI using Frequency-Domain Based Tofts Model, Annu Int Conf IEEE Eng Med Biol Soc, 2404-7 (2014).\u003c/li\u003e\n\u003cli\u003eGabriele Piantadosi et al.,: Data-driven selection of motion correction techniques in breast DCE-MRI, 2015 IEEE International Symposium on Medical Measurements and Applications (MeMeA) Proceedings, pp.07-09 (2015).\u003c/li\u003e\n\u003cli\u003eNikolaos Dikaios et al.,: Direct parametric reconstruction from under sampled (k, t)-space data in dynamic contrast enhancement MRI, 2013 IEEE Nuclear Science Symposium and Medical Imaging Conference, (2013).\u003c/li\u003e\n\u003cli\u003eKrutthika H.K, Nikhila S, Pavitha U.S. Development of CGI based front end design for implementation of security policies and Application layer filtering. International journal on Advanced Computer Theory and Engineering (IJACTE). 2(5) (2013).\u003c/li\u003e\n\u003cli\u003eNikhila S, Pavitha U.S, Krutthika H.K. Face recognition using Wavelet transform. International Journal of Advanced research in Electrical, Electronics and Instrumentation Engineering IJAREEIE. 3(1) (2014).\u003c/li\u003e\n\u003cli\u003ePriscilla Dinkar Moyya; Mythili Asaithambiet al., Extraction of Radiomic Features from Breast DCE-MRI Responds to Pathological Changes in Patients During Neoadjuvant Chemotherapy Treatment. 2019 IEEE International Conference on Imaging Systems and Techniques (IST). (2019).\u003c/li\u003e\n\u003cli\u003eLiang Wang; Haocheng Shen et al., A Clifford Analytic Signal-Based Breast Lesion Segmentation Method for 4D Spatial-Temporal DCE-MRI Sequences. (8). pp. 3901 \u0026ndash; 3910. (2019).\u003c/li\u003e\n\u003cli\u003eKashfia Sailunaz, Tansel \u0026Ouml;zyer et al., A survey of machine learning-based methods for COVID-19 medical image analysis Medical \u0026amp; Biological Engineering \u0026amp; Computing. (61). pp.1257\u0026ndash;1297 (2023).\u003c/li\u003e\n\u003cli\u003ePablo Camp\u0026aacute;z-Usuga; et al., Quality Enhancement of Breast DCE-MRI Images Via Convolutional. Autoencoders. 2021 IEEE 2nd International Congress of Biomedical Engineering and Bioengineering (CI-IB\u0026amp;BI) (2021).\u003c/li\u003e\n\u003cli\u003eSwati Nadkarni; Kevin Noronha. Breast Cancer Detection with Multiparametric Magnetic Resonance Imaging using Deep Learning. 2023 International Conference on Smart Systems for applications in Electrical Sciences (ICSSES). (2023).\u003c/li\u003e\n\u003cli\u003eRub\u0026eacute;n D. Fonnegra; et al., Early-to-Late Prediction of DCE-MRI Contrast-Enhanced Images in Using Generative Adversarial Networks. 2023 IEEE 20th International Symposium on Biomedical Imaging (ISBI). (2023).\u003c/li\u003e\n\u003cli\u003eHassan Bagher-Ebadian, Stephen L. Brown et al., Dynamic contrast enhanced (DCE) MRI estimation of vascular parameters using knowledge-based adaptive models. Scientific Reports. (13) (2023).\u003c/li\u003e\n\u003cli\u003eMikkel B., Anna Tietze, et al., Robust estimation of hemo-dynamic parameters in traditional DCE-MRI models, Conceptualization, Investigation, Methodology, Conceptualization. PLoS One. 14(1). (2019).\u003c/li\u003e\n\u003cli\u003ehttps://dblab.duhs.duke.edu/modules/QIBAcontent/index.php?id=1\u003c/li\u003e\n\u003cli\u003eJesper Folsted Kallehauge, Kari Tanderup et al., Tracer kinetic model selection for dynamic contrast-enhanced magnetic resonance imaging of locally advanced cervical cancer. NACP 2014 and the Turku PET Symposium. 53(8). pp.1064-1072. (2014).\u003c/li\u003e\n\u003cli\u003eSubmit an article Journal homepage\u003c/li\u003e\n\u003cli\u003eM. Venianaki, O. Salvetti et al., Pattern recognition and pharmacokinetic methods on DCE-MRI data for tumor hypoxia mapping in sarcoma. Multimedia Tools and Applications. (77). pp.9417\u0026ndash;9439 (2018).\u003c/li\u003e\n\u003cli\u003eRamesh Paudyal, Yonggang Lu, Vaios Hatzoglou et al., Dynamic contrast-enhanced MRI model selection for predicting tumor aggressiveness in papillary thyroid cancers. NMR in Biomedicine. 33(1). (2019).\u003c/li\u003e\n\u003cli\u003eCarmelo Militello, Leonardo Rundo et al., 3D DCE-MRI Radiomic Analysis for Malignant Lesion Prediction in Breast Cancer Patients. Academic Radiology. 29(6). pp. 830-840. June (2022).\u003c/li\u003e\n\u003cli\u003eGeorgios S. Ioannidis, Katerina Nikiforaki et al., Statistical and spatial correlation between diffusion and perfusion MR imaging parameters: A study on soft tissue sarcomas. European Journal of Medical Physics. (65). pp.59-66. (2019).\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Extracellular Extravascular Space, Pharmacokinetic Rate Parameters, Perfusion Characteristics, and Pharmacokinetic Modeling","lastPublishedDoi":"10.21203/rs.3.rs-3359472/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3359472/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUsing an intravenous contrast agent, dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is a commonly used technique for the non-invasive identification and therapeutic monitoring of various disorders. Applying compartmental models pertinent to the pathophysiology of the tissue being studied will enable the quantification of Pharmacokinetic (PK) maps. Integrating time-concentration data into these models in the time domain is the traditional way of determining PK parameters. Although the fitting problem is two-dimensional, this approach suffers from a dimensionality problem, which may be highly computational and lengthy. The Extended Tofts Model in Frequency Domain (ETM-FD) is derived as a convolution-free model from this research's standard comprehensive Tofts Model in Time Domain (ETM-TD). This method decreases the complexity of the curve fitting problem from two to one, minimizes computing time, and allows radiologists to evaluate PK maps within a clinical context. The ETM-FD and ETM-TD procedures were employed to both types of in silico phantoms to verify their resilience, and the Pearson correlation coefficient was calculated to compare with ground truth values. The results indicated that r2\u0026thinsp;=\u0026thinsp;1 values were achieved for both ETM-TD and ETM-FD, proving the approach's effectiveness. According to our findings, the ETM-FD took 24\u0026ndash;25% less computing time to compute PK maps than the ETM-TD. The computing power of the ETM-FD technique makes it a suitable tool for analyzing three-dimensional and more thorough coverage of the tissue of interest. The ETM-FD technique is especially effective in clinical situations where determining PK maps swiftly and precisely is crucial for patient care. The ETM-FD, which is computationally effective, reliable, and capable of producing precise and real-time PK maps, is a promising method for PK parameter estimation from DCE-MRI data. Our findings demonstrate that this strategy has the potential to assist radiologists in making clinical decisions and enhance patient care.\u003c/p\u003e","manuscriptTitle":"Efficient and Accurate Estimation of Pharmacokinetic Maps from DCE-MRI using Extended Tofts Model in Frequency Domain","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-11-20 19:42:34","doi":"10.21203/rs.3.rs-3359472/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"63135cfb-cbf5-4315-81dc-7adabc00ecd2","owner":[],"postedDate":"November 20th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-12-30T01:13:27+00:00","versionOfRecord":[],"versionCreatedAt":"2023-11-20 19:42:34","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3359472","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3359472","identity":"rs-3359472","version":["v1"]},"buildId":"WrCJVZZCHTDjtuVLN7oU0","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}