Incorporating contrast agent diffusion into the analysis of DCE-MRI data


  • Martin Pellerin,

    1. Centre d'imagerie moléculaire de Sherbrooke and Département de médecine nucléaire et de radiobiologie, Université de Sherbrooke, Sherbrooke, Québec, Canada
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  • Thomas E. Yankeelov,

    1. Institute of Imaging Science, Vanderbilt University, Nashville, Tennessee, USA
    2. Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, Tennessee, USA
    3. Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee, USA
    4. Department of Biomedical Engineering, Vanderbilt University, Nashville, Tennessee, USA
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  • Martin Lepage

    Corresponding author
    1. Centre d'imagerie moléculaire de Sherbrooke and Département de médecine nucléaire et de radiobiologie, Université de Sherbrooke, Sherbrooke, Québec, Canada
    • Centre d'imagerie moléculaire de Sherbrooke, Département de médecine nucléaire et de radiobiologie, Université de Sherbrooke, 3001, 12e Avenue Nord, Sherbrooke, Québec, J1H 5N4, Canada
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Standard two-compartment pharmacokinetic models that describe the dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) time course of gadolinium diethylenetriamine pentaacetic acid (Gd-DTPA) concentration in a tissue do not account for the passive diffusion of contrast agent (CA) from a well-perfused to a less vascularized region. Even when the arterial input function (AIF) is perfectly known, the standard Tofts model returns inaccurate values of Ktrans (mean absolute relative difference [ARD] of 43%) from realistic simulated data where a well-defined delineation exists between a well-perfused and a poorly vascularized region. This contribution proposes a diffusion-perfusion (DP) model in which diffusion of a low molecular weight CA is incorporated in the standard two-compartment Tofts model. The proposed DP model reliably retrieved the values of Ktrans and ve (mean ARD of 16% and 17%, respectively) from simulated data. On mouse adenocarcinoma xenograft data showing evidence of CA diffusion, the standard model returned unphysical values of ve in the tumor core whereas the proposed DP model found values that were in the physical range (0 < ve < 1) throughout the tissue. In addition, Ktrans distributions from the DP model more closely corresponded to the observed sharp delineation between highly and poorly perfused areas observed in the mouse tumors. Magn Reson Med, 2007. © 2007 Wiley-Liss, Inc.

The exchange of an inert gas within a tissue was described in 1951 using a pharmacokinetic model to extract physiologically relevant parameters from time-dependent measurements of tracer concentration (1). Many authors have since adapted this early model to the specific requirements of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) experiments to relate the enhancement pattern of a tissue to the underlying exchange between blood plasma and the extravascular extracellular space (2–5). In the clinic, DCE-MRI has shown potential as a powerful tool to characterize the microvascular environment in tissue lesions as diverse as multiple sclerosis and tumors (6, 7). Recent improvements to pharmacokinetic models accounted for additional phenomena, such as the variation in shutter speed limits associated with contrast agent (CA) concentration (8, 9). Those studies have shown that neglecting the limits of the transcytolemmal and transendothelial water exchanges can lead to erroneous estimations of standard model parameters. Recently, the tedious problem of evaluating the arterial input function (AIF) has been circumvented by the use of reference region models developed by us and others (10–14). To our knowledge, the possibility that a CA can passively diffuse from a well-perfused region to a less vascularized region has not been addressed, although diffusion is known to occur within tumors (15).

Tumor tissues show clear signs of heterogeneity in their perfusion pattern (16, 17). They are often characterized by a well perfused rim surrounding a poorly vascularized, possibly necrotic core (18, 19). In DCE-MRI experiments, this translates into a rapid and intense signal enhancement at the rim followed by a delayed enhancement of the core, which may be the result of CA diffusion from the rim to the core. A common assumption made by all current models is that DCE-MRI data can be analyzed on a voxel-by-voxel basis, which inherently neglects diffusion of low molecular weight CA. The mathematical models used with DCE-MRI experiments could potentially be improved to obtain a better representation of the underlying physiological processes. A better understanding of the microvascular environment of tumors could impact the treatment protocols using agents delivered via the blood circulation, such as chemotherapy agents. In this case, high transcapillary exchange rates commonly found in tumors are expected to result in an increased concentration of the agent. In addition, antiangiogenic treatments attempt to compromise the neovascular system (20, 21), although an effective treatment may actually require the reestablishment of a functioning microvascular system (22). Finally, a high perfusion also correlates with a larger oxygen concentration in tumor tissue (23), which increases the radiosensitivity of a tumor (24).

In this contribution, we propose a diffusion-perfusion (DP) model in which CA diffusion is explicitly taken into account and included in the standard Tofts model (6). The overall goals of this effort are: 1) to build a pharmacokinetic model that incorporates the effects of CA diffusion; 2) to test the model in simulations and to assess accuracy and precision of the model when experimental noise levels are present; and 3) to apply the model in experimental studies of mice with MC7-L1 mammary carcinomas. The comparison of the DP model with the standard model exposes limitations of the standard model.


Proposed DP model

Using the standardized symbolic notation described by Tofts et al. (6), the variation of CA concentration within a voxel is described by a standard two-compartment model such that:

equation image(1)

where Ci,j(t) are the elements of the matrix C representing the tissue concentration in mM, Cp(t) is the plasma concentration of CA in mM, Ktrans is the transcapillary transfer rate in min−1, and ve is the extravascular extracellular volume fraction. The value of i and j range from 1 to m and 1 to n, respectively, where m is the number of frequency encoding steps and n is the number of phase encoding steps in the image acquisition matrix.

The variation of CA concentration exchanged by diffusion caused by concentration gradients from one voxel of dimension a × a × e to its immediate neighbor in 2D can be approximated by Eq. [2]:

equation image(2)

where D is the matrix containing the diffusion coefficients of the CA for each voxel, V = a2e is the volume of a voxel and S = ae is the surface between two neighboring voxels of the same slice. This description of CA diffusion assumes instantaneous mixing of the CA within a voxel and keeps the overall quantity of CA constant. Since the surface between voxels of adjacent slices in 2D experiments is usually much smaller than between voxels of the same slice, the error associated to a 2D description of CA diffusion is small. Nevertheless, this description of the diffusion process is readily expandable to 3D at the expense of computation time. A complete derivation of Eq. [2] can be found in the Appendix.

Incorporating both the effects of active delivery (perfusion, Eq. [1]) with passive delivery (diffusion, Eq. [2]) yields the proposed DP model in 2D (Fig. 1):

equation image(3)

where the subscript N stands for Neighbor. More explicitly, expanding Eq. [3] yields:

equation image(4)

To simplify the notation and computation, the matrix C of size m × n is expressed as a vector C̄ of length mn by the following transformation:

equation image(5)

Equation [4] can be rewritten:

equation image(6)

In this form, K̄ and V̄ are diagonal matrices of size mn × mn with the values of Kmath image and 1/vei,j on their diagonal, respectively. The column vector filled with ones has a length mn. D̄ is a nearly diagonal sparse matrix of size mn × mn. From Eq. [4], the elements of D̄ are given by:

equation image
equation image(8)
equation image

Terms associated with a voxel located outside the tissue must be set to 0. This scheme prevents CA from leaking outside the tissue and so formulates the boundary conditions.

Figure 1.

Schematic representation of the proposed DP model. The extracellular compartment exchanges CA with the blood plasma compartment (perfusion) and its four immediate neighbors within the same slice (diffusion).

Using a short time step, Δt, during which only a small relative amount of CA is exchanged by diffusion between neighboring voxels, the first-order solution to Eq. [6] is:

equation image(9)

where I stands for the identity matrix of size mn × mn. This approximation is valid when the amount of CA transferred from one voxel to another by diffusion does not exceed a few percent per Δt.

The temporal resolution of conventional DCE-MRI images ranges from a few to 10s of seconds. This resolution might be too low for the voxel size used in small animal scanners. For example, given a diffusion coefficient of 1 × 10−3 mm2/s and t = 60 s, the average displacement of a free particle in 2D is equation image ≈ 0.5 mm. This is larger than the voxel dimension (≈0.25 mm) in these scanners, and this requires additional “diffusion steps” to be calculated at a time interval shorter than the experimental temporal resolution. However, a smaller Δt implies more computation steps and this lengthens the evaluation of C̄(t). As a compromise, 10 “CA exchange steps” between the plasma compartment and the extravascular extracellular space are considered between two successive images and the time resolution of the “diffusion steps” is set close to 1 s. For example, if images are acquired with a temporal resolution of 50 s, exchange of CA is computed every 5 s and the diffusion of CA is computed every 1 s. The AIF is interpolated to match the “exchange steps” time resolution (e.g., 5 s in the example). In this case Eq. [9] takes the form:

equation image(10)

where Δt′ = Δt/b and b is an integer set so that Δt′ is closest to 1 s. These additional diffusion steps do not increase considerably the overall computation time since the term to the power of b only needs to be evaluated once.

To reduce the size of the solution space and to prevent overparameterization, the assumption that D is known for every voxel is made. As a first-order approximation, it is set equal to the apparent diffusion coefficient (ADC) of water as determined from diffusion-weighted spin-echo images, which will not be valid for nondiffusible CA. For the DP model to be applied, the ADC value for every voxel must be available. The DP model has the same two free parameters per voxel as the standard Tofts model: Ktrans and ve. D is not used as a free parameter in this work.

Clearly, a voxel-by-voxel fit is no longer adequate since intervoxel exchange of CA through diffusion requires fitting all voxels simultaneously. This increases the complexity of the fitting problem. For example, if the values of Ktrans and ve are each discretized over 27 = 128 values, each voxel has 214 different solutions. For a tumor covering 25 × 25 voxels, then a voxel-by-voxel approach needs to find the best among the 214 solutions for 625 uncoupled problems but the DP model needs to find the best among the (214)625 solutions.



When considering the diffusion of CA between voxels, a complete image must be fitted as a whole using a 2D CA diffusion scheme, thereby increasing the complexity of the optimization problem. Furthermore, the “landscape” of the solution space is filled with countless local minima, which prevent the use of standard techniques (e.g., steepest descent). An algorithm based on stochastic search methods is essential to efficiently converge to the optimal solution.

In the present study, the DP model is coupled to a simulated annealing search algorithm (see25. Genetic algorithms were also tested but were found to be inefficient when using serial computing. The simulated annealing algorithm searches randomly through the solution space, hopping from one solution to another using the Metropolis criteria (26). Given a function to be minimized, F(S), henceforth called the cost function, and T, a control parameter, the transition from the current solution Si to a randomly chosen neighboring Si′ is ruled as follows: if F(Si′) < F(Si) then Si′ becomes the current solution, otherwise Si′ has a probability of becoming the current solution given by P(Si,Si′,T) = exp(–(F(Si′) – F(Si))/T). As will be detailed below, convergence is attained by lowering in steps the control parameter T after a fixed number of transitions have occurred. If T is lowered slowly enough, the algorithm will converge to a solution close to the absolute minimum. This search method is heuristic since the true solution is approached more closely as the search time increases.

The size of the solution space can be decreased by an appropriate discretization of the possible parameter values. The values of Ktrans are coded in binary over 6 bits (Ktrans ∈ {0, 1, 2, …, 63}/100 [min−1]) and ve over 7 bits (ve ∈ {0, 1, 2, …, 127}/128). This range of values encompasses those reported by other groups, and the interval between two values is smaller than errors reported (27, 28). The neighborhood of a solution is chosen to include the solutions differing by only one bit in one parameter in only one voxel. The number of transitions needed before stepping down the control parameter is fixed to one-half the size of the neighborhood as suggested in Ref.25.

The descent of the control parameter T has the form: T(n + 1) = ϵT(n) where ϵ is a parameter associated to the speed of the descent and is in the range [0,1]. The initial value of T is set to allow the transition to any solution, better or worse. The main advantage of this approach is that the algorithm becomes insensitive to initial guesses.

The choice of the cost function has a critical importance to determine the direction of the search. We found that a standard sum of squared differences leads to erroneous solutions with too low values of Ktrans and too high values of ve. This problem is solved by considering only the part of the CA time course most sensitive to variations in Ktrans and ve. Figure 2 shows that the effect of varying Ktrans is most significant in the initial uptake of CA. On the other hand, the effect of varying ve is found in the “steady state” and “washout” occurring later after CA injection. Similar conclusions were reached in Ref.29. The cost function used for the simulations and real data fits is thus chosen to be:

equation image(11)

where S is the solution to be evaluated, Ap is a weighting factor, and p ranges from one to the number of images. In all cases, Ap = 2 for the first two images after CA injection, one for the last two images, and zero otherwise. This function was found to be reliable in retrieving values of Ktrans and ve used to generate simulated data. The choice of using only part of the data to evaluate the cost function of a solution increases the sensitivity to noise. However, this approach prevents the algorithm from converging prematurely to “wrong” answers.

Figure 2.

Simulated data show the influence of Ktrans and ve on the shape of the CA concentration time course. a:Ktrans values range from 0.05 min−1 to 0.50 min−1 by 0.05 increments while ve is set to 0.1. b:ve values range from 0.1 to 1.0 by 0.1 increments while Ktrans is set to 0.1 min−1. The curves were calculated using the standard Tofts model and a simulated AIF.


The model and the simulated annealing algorithm were coded in Matlab™, compiled using the Matlab™ compiler, and exported to computation nodes. Computation was executed on a fraction of the serial supercomputer form the Université de Sherbrooke (Mammouth-Série, 872 nodes, Intel P4 with 2 GB of RAM per node).

Simulated Data

Two sets of simulated tissue concentration time courses, both including the effects of diffusion, have been produced from known Ktrans, ve, and D maps and the DP model. These will serve to expose some limitations of the standard model and to evaluate the performance of the simulated annealing algorithm. The first case shows a circular tumor having Ktrans values of 0.1 min−1 in the rim and 0 in the core and uniform ve and D values of 0.2 and 0.001 mm2/s, respectively (Fig. 3). The second case uses more realistic Ktrans, ve, and D maps with somewhat less severe topological differences (see Fig. 7, below).

Figure 3.

Simulation of a circular tumor. a: Concentration time course calculated from known Ktrans, ve, and D maps and the DP model. b: Known Ktrans and ve maps referred to as the true values and the parameters maps returned by the DP model, the simulated annealing algorithm with the diffusion scheme turned off and the standard Tofts model. Removing the diffusion scheme from the DP model yields inaccurate values of Ktrans and ve, similar to those from the standard model. [Color figure can be viewed in the online issue, which is available at]

An AIF was created using a box-shaped CA plasmatic input convoluted with a biexponential clearance (30) (Fig. 4). To assess the effects of noise on the result from the DP model, noise was added to the second set of tissue concentration and to the AIF with an SD equal to a fraction {0%, 1%, 2.5%, and 5%} of the highest concentration in each compartment, respectively.

Figure 4.

AIF used for all the simulations.

Animal Model

The proposed DP model was also tested against real tumor data taken from mouse breast carcinomas (MC7-L1) (31). Subcutaneous tumors were inoculated (injection of 107 cells) in two Balb/c mice on the hind limb. Imaging was performed between four and eight weeks after implantation.

MRI Protocol

Imaging was performed with a Varian 7T small animal scanner using a 40-mm Millipede™ probe and using either sets of gradient coils (210/120 mm with 30 G/cm or 120/60 mm with 100 G/cm). Mice were anesthetized using a mix of 1.5% to 2% isoflurane in oxygen with the flow set at 2 liter/min and were restrained to prevent movement during the experiment. A precontrast T1-map was calculated from a set of T1-weigthed multiple flip angle gradient echo images (TR = 100 or 200 ms, TE = 2.4 ms, α = {10°, 20°, 25°, 30°, 35°, 50°}, FOV = 32 × 32 mm2, data matrix = 128 × 128, number of averages [NA] = 4, slices = 10, slice thickness = 1.5 mm). The same geometrical prescription is used for all imaging sequences. The surface between slices is only 8.3% of the surface between neighboring voxels, thus justifying a 2D analysis.

A total of 40 consecutive sets of images were acquired with the same parameters and α fixed at 30°. The time resolution of those images were 51 s and 102 s for TR = 100 ms and 200 ms, respectively. A bolus of gadolinium diethylenetriamine pentaacetic acid (Gd-DTPA) (Magnevist™; Berlex) was injected intravenously (i.v.) via the tail vein after the second set of images (1.3 mmol/kg). The first 13 sets were used for fitting with the DP model to reduce computation time, which is proportional to the number of images considered.

The ADC for each voxel was determined from a diffusion-weighted spin-echo pulse sequence using b-values ranging from 2 to 492 s/mm2 (TR = 2000 ms, TE = 25 ms, FOV = 32 × 32 mm2, data matrix = 128 × 128, NA = 1, slices = 10, slice thickness = 1.5 mm, Gdiff = {2, 10, 20, 30} G/cm in the readout direction, δ = 3.5 ms, Δ = 9 ms).

Studies have shown that the use of a generic AIF is not appropriate for individual estimation of CA uptake in tumors (32). Since it is difficult to accurately determine the AIF in individual animals, we derived the AIF from a reference region (RR) as suggested in Ref.10 using Ktrans,RR = 0.1 min−1 and ve,RR = 0.1 (12). The standard Tofts model was used with the same reference region for comparison with the DP model on animal data (12).



Limitations of the standard model are illustrated by simulating results from a simple circular tumor. Figure 3a depicts the concentration time course for this simulated tumor where an early enhancement in the periphery is observed. With time, the CA reaches the core of the tumor by diffusion. Ktrans is zero in the core of the tumor such that only diffusion can result in a nonzero concentration in this location. Both the standard Tofts model and the DP model were used to fit this dataset using the same AIF (Fig. 4). The Ktrans map from the standard Tofts model appears “averaged” over the entire tissue and while the ve values at the periphery are reasonable, unphysical values (ve > 1) related to fit failures are obtained in the core. Regarding Ktrans, we find that in a heterogeneous tissue, voxels with a high true value of Ktrans receive a large amount of CA but lose some of this amount by diffusion to neighboring voxels that have a lower CA concentration because of their lower value of Ktrans, thus effectively smoothing the distribution of CA. As a consequence, the standard Tofts model is smoothing Ktrans of these voxels to depict their time courses. In other words, the standard Tofts model will provide a good fit for the data, but the values of Ktrans will be lower than the true value for the voxels supplying CA to their neighbors and Ktrans will be higher than the true value for those being supplied by diffusion. In addition, to account for the significant amount of CA that reaches the core and the very slow uptake for those voxels, the standard model has to assign very low (but nonzero) Ktrans values and unphysically high ve values. None of these problems can be related to the AIF in this simulated experiment for which it was known exactly and where diffusion was explicitly added. Indeed, scaling the AIF limits the extent of the fit failure in the core but it also alters the correct ve estimations in the rim (data not shown). Ignoring CA diffusion thus leads to erroneous estimations of Ktrans and ve, even when the AIF is perfectly known.

The Ktrans and ve maps obtained with the DP model are very close to the true values and deviations are related to the stochastic search method. The convergence of the DP model toward the true solution shows that it is not subject to overparameterization. Figure 3 shows the parameter maps returned by the simulated annealing algorithm when the diffusion coefficient is set to 0. It can be seen that in this case the Ktrans map is smoothed much like the one from the standard Tofts model and that ve is overestimated in the core of the tumor where the model failed to fit the data well.

Using the second, more realistic, simulated tumor CA concentration time course, we evaluated the effect of changing the value of ϵ on the evolution of the cost function during the fit. Figure 5 shows the value of the time variation of the mean cost for four different values of ϵ. Only a fraction of the original dataset is displayed for clarity. Each point represents the mean values of 10 runs, every one of which converged to a different solution. Those results exhibit the characteristic behavior of heuristics; if the control parameter T is dropped too quickly (low value of ϵ) the algorithm converged rapidly to a suboptimal solution. Figure 5 shows that the descent was too fast for ϵ = 0.5 and the algorithm converged to a local minimum far from the optimal solution. A better solution (i.e., with a lower cost) was obtained using a larger value of ϵ (ϵ = 0.95). It is likely that the mean value of the cost function would have been even lower for ϵ = 0.99, but convergence was not yet reached after 120 h.

Figure 5.

Convergence of the DP model coupled to the simulated annealing algorithm fitting noiseless simulated data. The mean value of the cost function is plotted as a function of fitting time for different values of ϵ. The initial rise found for all values of ϵ indicates the algorithm is not sensitive to the initial guess. The accuracy of the final solution is related to the mean cost found for the lower plateau. More accurate values can be found for higher ϵ, at the expense of a longer computation time.

The initial guess used by the algorithm was produced by a random distribution of Ktrans and ve values with a mean value of 0.10 min−1 and 0.20 and an SD of 0.05 min−1 and 0.05, respectively. However, the initial rise of the mean cost value demonstrates that the algorithm is insensitive to the initial guess since it originally accepts numerous poorer solutions.

The performance of the DP model was evaluated by computing the absolute relative difference (ARD) between fitted and true Ktrans and ve values (Fig. 6). The results for two different values of ϵ are displayed as histograms of the fraction of voxels and their ARD for both Ktrans and ve on Fig. 6a and b, respectively. It can be observed that the highest value of ϵ (the slowest descent) leads to more accurate solutions; i.e., more voxels have a smaller difference compared to the true values. Indeed, for ϵ = 0.95, a difference below 20% of the true value is found for 49% of the Ktrans values and 77% of ve values. Recalling that the increments in Ktrans and ve are 0.01 and 0.0078, respectively, a difference of 10% from a value of 0.1 represents a difference close to a single increment. Nevertheless, the accuracy of the solutions can be improved by averaging the results from multiple independent computations (Fig. 6c and d). Using 10 calculations and ϵ = 0.95, 73% of the Ktrans values are now within 20% of the true value. However, it can be noticed that even if 77% of the ve values are still within 20% of the true values, it appears that averaging somewhat reduced the accuracy of this parameter. Each individual run had a similarly low mean cost value, but Ktrans and ve were averaged on a voxel-by-voxel basis. Thus, if a particular voxel in any of the 10 runs did not converge to the true value, the average value for that voxel is likely to be affected. Overall, averaging multiple runs gives more accurate results and this approach is well suited for adapting simulated annealing to parallel computing (25).

Figure 6.

Histograms of the number of voxels plotted against their ARD to the true parameter values used to create the simulated concentration data. a,b:Ktrans and ve, respectively, for a single run of the algorithm. c,d:Ktrans and ve, respectively, for the mean solution taken from 10 runs. e,f:Ktrans and ve, respectively, for both the DP model (ϵ = 0.95, means of 10 runs) and the standard Tofts model.

Next, the performance of the DP model was compared to the standard Tofts model using the same simulated data for an average of 10 runs and ϵ = 0.95. Using the DP model, 73% of the Ktrans and 77% of the ve values were within 20% of the true values (mean ARD from the DP model is 16% for Ktrans and 17% for ve), compared to 25% and 97% for the standard model (mean ARD from the standard model is 43% for Ktrans and 4% for ve) (Fig. 6e and f). We attribute this poor performance of the standard model in retrieving the real Ktrans values by the observation that diffusion of CA leads to “smoothed” concentration maps leading to erroneous Ktrans values in the standard model in which diffusion is ignored. This can be seen on Fig. 7, in which the Ktrans map returned by the Tofts model is a smoothed version of the true distribution. Conversely, the DP model preserved the sharpness of the Ktrans distribution. We recall that the values of Ktrans were still relatively uniform throughout the tissue in this case and there was no necrotic center, where the true value of Ktrans would be approximately zero. Thus, the Tofts model yielded accurate values of ve, which are mostly influenced by the “steady state” concentration found when the tissue reaches equilibrium with the blood plasma concentration several minutes after CA injection (see Fig. 2).

Figure 7.

Results of the DP model (ϵ = 0.95, means of 10 runs) and the standard Tofts model. Results show that the simulated annealing algorithm coupled to the DP model can retrieve the true values of both parameters. The use of the standard Tofts model returns excellent maps of ve but somewhat less accurate, smoothed values of Ktrans. [Color figure can be viewed in the online issue, which is available at]

Figure 8 shows the effect of adding noise (0–5% of the maximum concentration) on the accuracy of the solutions returned by the DP model. Ktrans is mostly influenced by the variations in concentration immediately after injection of CA (Fig. 2). Noise added to these concentrations is therefore likely to severely affect the fitted Ktrans values. On the other hand, ve regulates the concentration reached in the “steady state” during which the concentration in the extravascular extracellular space follows the plasma concentration. It is not surprising then that this parameter is less sensitive to random variations around this slowly varying concentration. For comparison, data acquired from mice with a 7T small animal scanner display a noise level lower than 1% of the maximum concentration.

Figure 8.

Effect of noise (0–5% of the maximum concentration) on the accuracy of the parameter values returned by the DP model (ϵ = 0.95, means of 10 runs). Histograms show the number of voxels plotted against their ARD to the true parameter values. Noise was added to both the simulated concentration data and the AIF. Ktrans shows the greatest sensitivity to noise (a) while ve is less affected (b).

Experimental Data

The top panels of Figs. 9 and 10 show examples of xenograft tumors displaying evidence of CA diffusion from the rim to the core. Both datasets were fitted with the DP model and the standard Tofts model using the same reference region (12). Since all voxels are fitted as a whole in the DP model, voxels having an odd behavior can hinder the convergence of the algorithm. Hence voxels with irregular R1 time courses (R1 decreasing dramatically after CA injection, unphysical R1 values, etc.) related to partial filling effects and susceptibility artifacts or voxels for which no values of ADC could be calculated were removed from the data sets used with the DP model. This was necessary only for the second data set (Fig. 10), in which 18% of the voxels were removed. A total of 96% of those voxels were found at the edge of the tissue where the standard model failed to fit the data and returned unreasonably high values of Ktrans and ve (Fig. 10). Erroneous parameter values would be obtained from these voxels independently of the model used. Removing these peripheral voxels from the analysis is not expected to alter the description of the underlying physiological process of the neighboring voxels.

Figure 9.

Top panel: CA concentration calculated from DCE-MRI images of a mouse breast carcinoma xenograft. Imaging time resolution was 102 s. Bottom panel: Ktrans and ve maps returned by the DP model (means of 50 runs, ϵ = 0.9) and the standard Tofts model. The data used by the DP model has been downsampled to reduce computation time. A sharp delineation observed in the dynamic images is clearly seen in the Ktrans map from the DP model, but not from standard voxel-by-voxel approach. ve values from the DP model are reasonable in the core of the tumor, as opposed to unphysical values from the standard model.

Figure 10.

Top panel: CA concentration taken from DCE-MRI images of a mouse breast carcinoma xenograft. Imaging time resolution is 51 s. Bottom panel: Ktrans and ve maps returned by the DP model (means of 10 runs, ϵ = 0.9) and the standard Tofts model. The standard model yielded unphysical ve values (ve > 1) for 40% of the retained tumor voxels (excluding the rim) as opposed to the results from the DP model.

Looking at the images of the concentration as a function of time, a clear delineation is apparent between the rim and the core of the tumor. The concentration in the rim rises much more quickly than in the core. Those enhancement patterns have been reported by other groups (18–19) and have been correlated with microvessel density measurements (33). This delineation is not clear on the parameter maps from the standard Tofts model. However, the Ktrans maps from the DP model show the expected sharp distribution. Furthermore, we note that the range of Ktrans values returned by the DP model is larger than that obtained from the standard model. This is because the standard model “averages” the Ktrans values of neighboring voxels. This situation is similar to that discussed above for the simulated data.

For both datasets, the standard model returned unphysical values of ve (i.e., ve > 1) in the core of the tumors (reaching 40% of the retained voxels of the tumor in Fig. 10). This observation can also be made on our simulations (Fig. 3), and is explained by the slow uptake of CA by diffusion. In that case, the voxel-by-voxel approach assigns unrealistic ve values to voxels that accumulate a significant amount of CA even though their Ktrans value is negligible (34). The DP model returned more sensible values in the core of the tumors. While it is beyond the scope of this contribution to confirm these values correspond to the underlying physiology of the tumor, the fact that the values do not approach the maximum of the possible ve values is certainly encouraging.

An error in the estimation of the AIF through the reference region scheme could certainly affect the values of both parameters. However, changes in the AIF used would affect the parameters of every voxel such that lowering the values of ve in the core would result in exceedingly low values in the rim.


We proposed a DP pharmacokinetic model in which diffusion of a low molecular weight CA from one voxel to its neighbors was explicitly taken into account and incorporated into the standard Tofts model. We have exposed the limitations of the standard model to describe a concentration time course in which significant CA diffusion is present, which is expected to be the case for most tumors having well vascularized and poorly perfused regions in close proximity. We showed that the proposed DP model could reliably retrieve both Ktrans and ve from simulated data (mean ARD of 16% for Ktrans and 17% for ve) even in the presence of realistic noise. When applied to the analysis of mice breast carcinoma xenograft data, the model outperformed the voxel-by-voxel analysis of the standard Tofts model by yielding sharper Ktrans distributions in accordance with what could be observed in the time-varying concentration data. The DP model also yielded ve maps that did not contain unrealistic values. This is in contrast to the standard model that cannot describe the slow uptake of CA in a poorly perfused, potentially necrotic, region.

This first attempt to incorporate CA diffusion into a pharmacokinetic model used several approximations and the following refinements will be examined. The diffusion coefficient was approximated to the ADC of water obtained from diffusion-weighted images. This approximation is not crude, but a more realistic value for the diffusion coefficient of the CA could be used, although this would imply either that this value can be computed on a voxel-by-voxel basis or that it could be scaled to that of water. Nevertheless, we propose that adding diffusion with a potential error on the diffusion coefficient is probably better than neglecting diffusion, which is shown to lead to marked errors in the estimated parameters. Attention must also be paid to the determination of the ADC since it has been shown that it is dependent on the range of b-values used in the imaging protocol (35). The model uses a 2D description of the diffusion process that can readily be expanded to 3D at the expense of computation time. It should be noted that using smaller voxels will actually increase the effect of CA diffusion such that the model we propose is predicted to be even more useful using this geometry. One of the limitations of the DP model is the heavy computation power needed to carry out the stochastic search. The use of more efficient coding languages and refinement in the elaboration of the algorithm could increase the speed of the algorithm (Using C instead of Matlab™). We have already undertaken this task and preliminary results indicate at least a 10-fold reduction in computation time. A thorough study of the best parameters (initial temperature, descent rate, and neighborhood) that shape the simulated annealing algorithm could lead to more optimized use of the computation time. The details of the cost function could also be modified to take advantage of the complete dataset.

Our results indicated that a larger range of Ktrans values was obtained by the DP model whereas the standard model appeared to smooth these values. Beyond providing potentially more physiologically relevant information on the tumor microenvironment, this larger range is anticipated to provide better diagnostic information based on this parameter. Finally, the DP and nearly all other DCE-MRI pharmacokinetic models require a histology-based validation.


We thank Mélanie Archambault for expert animal handling, Céléna Dubuc, Simon Authier, and Véronique Dumulon for the preparation of the animal models, Luc Tremblay for technical assistance, and David Sénéchal for helpful discussions. M.P. is supported by a student stipend from Natural Sciences and Engineering Research Council (NSERC), T.E.Y. is supported by National Institutes of Health (NIH)/National Institute of Biomedical Imaging and Bioengineering (NIBIB) (1 K25 EB005936-01), and M.L. is the Canada Research Chair in Magnetic Resonance Imaging.


Diffusion at the Surface of a Voxel

The variation of the amount of CA Q(t) present in a voxel from diffusion to neighboring voxels can be described by the continuity equation:

equation image(A1)


equation image

is the flux of CA. The amount of CA is given by

equation image(A2)

where Ce(

equation image

,t) and C(

equation image

,t) are the CA concentration in the extravascular extracellular space and the CA concentration in the tissue, respectively, and V is the voxel volume. The concentrations are related by the extravascular extracellular volume fraction ve such that C(

equation image


equation image


equation image

). The “〈 〉” symbols express the mean value of the quantity. Equation [A1] becomes:

equation image(A3)

Applying Fick's first law of diffusion yields:

equation image(A4)

where D(

equation image

) is the diffusion coefficient of CA. The integral of the last equation can be calculated over the subsurfaces separating one voxel from its neighbors. This leads to:

equation image(A5)
equation image(A6)

where (

equation image

) is the oriented surface of one interface and the mean value of the expression between brackets is taken over the interface. Since D(

equation image

) and ▿(C(

equation image


equation image

)) are not expected to vary significantly along the interface of two voxels, the mean value of their product can be approximated by the product of their extrapolated values at the interface.

equation image(A7)

Going from a continuous to a discrete formulation, Eq. [A7] becomes:

equation image(A8)

where a is the length of one side of a square prism voxel.