IFP could explain the typically ineluctable progression of late stage glioma, which despite maximal safe surgical resection and subsequent chemo-radiotherapy has a median survival of 15 months (with all subgroups combined) ¹. Direct measurement of IFP using a pressure transducer is invasive, and only the measurement at the point of surgical ingress will remain free of bias arising from tissue movement subsequent to incision. Dynamic contrast enhanced (DCE) MRI enables IFP to be inferred non-invasively by modelling the rate of fluid flux from the region of blood brain barrier breakdown ², and utilising the empirically established physical properties of tissue ³. The model requires an accurate characterisation of the rate of flux from the blood into tissue and out again, which is numerically stable and ideally should also account for the preferential directionality of fluid flow, which is obtainable from Diffusion Weighted Images (DWI). This paper defines an approach to achieve this.

A two compartment (plasma and tissue) model has been found to describe the temporal changes of Gadolinium concentration within tissues, , in regions of blood brain breakdown well, if the influx, $$$ K_{in}$$$, and outflux, $$$ k_{out}$$$, constants are allowed to differ as in the extended Tofts model ⁴:

$$\frac{dC_t}{dt} = K_{in}C_p(t) - k_{out}C_t(t) \qquad\qquad\qquad (1) $$

where is the plasma concentration at time t. The flow of fluid through tissue is governed by both hydrostatic pressure (pushing plasma into the interstitium in the arterioles) and osmotic pressure (extracting plasma from interstitium in venules) ⁵, which is mediated by the conductivity of the tissue as expressed by Darcy’s Law ⁶. Where tumours disrupt normal vasculature, the natural equilibrium in pressure is disrupted, resulting in an influx of fluid into the tumour and surrounding regions ³. The fluid flux, $$$ \frac{J_V}{V}$$$, per unit volume from blood into tissue may be computed from the difference between and , after compensating for the osmotic reflection coefficient of Gadolinium, $$$\sigma$$$ ³. Assuming dissipation of excess pressure within the parenchyma is slow, but that steady state has been reached: conservation of mass allows the computation of local pressure, p, from the flux in the region of blood brain barrier breakdown:

$$ \nabla\cdot\frac{J_V}{v}=\nabla\cdot\frac{K_{in}(\mathbf{x}-k_{out}(\mathbf{x}))}{1-\sigma} = \nabla\cdot K\nabla p \qquad\qquad\qquad (2)$$,

where K is the hydraulic conductivity. For a scalar K, $$$\nabla\cdot\nabla$$$ can be encoded as a discrete Laplacian matrix operator and p solved directly. Accounting for local plasma volume, $$$v_p$$$, (1) can be formulated to be optimised over a single non-linear parameter for the time-varying concentration within each voxel, $$$T_i$$$,at location $$$\mathbf{x}_i$$$ over discrete times, $$$t_u$$$:

$$\arg_{k_{out}}\min \arg_{v_p, k_{in}}\min \sum_u \left( T_i(t_u)-[v_p+K_{in}e^{-k_{out}\cdot t}]*C_p(t)\right)^2 \qquad\qquad\qquad (3) $$

while the linear parameters are optimised using a standard non-negative least squares technique ⁷. The optimisation equation (3) was implemented using an open source dynamic analysis platform ⁸. Note that $$$K_{in}$$$ should be inversely scaled to account for plasma fraction, $$$v_p$$$. The hydraulic conductivity, K, is not necessarily isotropic as fluid preferentially flows parallel to white matter tracts ^9,10. If available, DWI data can be incorporated into the matrix operator in (2), where discrete voxels correspond to distinct rows and columns and element ij encodes the relationship between voxels i and j. Only neighbouring voxels, $$$j\in N_i$$$ have non-zero values. Directional preferences of fluid flow can be encoded as reweightings, so if $$$ K_i^{ij}$$$ is the conductivity of voxel i, in the unit direction $$$(x_i-x_j$$$, the Laplacian operator becomes:

$$\nabla\cdot K\nabla = \left\{ \begin{matrix} -\frac{1}{2}\sum_{j\in N_i} K_i^{ij}+K_j^{ij} & i=j \\ \frac{1}{2}(K_i^{ij}+K_j^{ij}) & i\neq j, j \in N_i \\ 0&\textrm{elsewhere}\\ \end{matrix}\right. \qquad\qquad\qquad (4) $$

Fig. 1 shows the pressure map of a glioma patient overlayed on the post-contrast and FLAIR image of the patient. $$$C_p$$$ was estimated from a voxel-cluster in the sagittal sinus of 15x55second DCE. (3) yielded numerically stable influx parameters that varied smoothly over the spatial volume, without regularisation. This was used as an input to compute pressure using (2) and (4).The measurement of IFP using a numerically stable formulation of the extended Tofts model and incorporating the directional preferences of fluid flow has been demonstrated. IFP corresponds to the enhancing region of oedema in the FLAIR image. Future work will explore inclusion of the fact that flow will not traverse sulci, which is expected to further improve the correspondence with the FLAIR image.