Daniel Fovargue^{1}, Jack Lee^{1}, Marco Fiorito^{1}, Adela Capilnasiu^{1}, Sweta Sethi^{2}, Stefan Hoelzl^{1}, Jurgen Henk Runge^{1}, Jose de Arcos^{1}, Arnie Purushotham^{3}, Khesthra Satchithananda^{3}, David Nordsletten^{1}, and Ralph Sinkus^{1}

Established tumour treatments include drugs and the emerging class of cell-based therapies. Individual tumours can counteract the delivery of the therapy, for example, they can feature high internal pressures (IFP) caused by blood vessels that are different from normal tissues. Furthermore, a drop in IFP is a well-established marker for successful therapy. Here we demonstrate in simulations, phantom experiments, fatty breast tissue, and a benign breast lesion that tissue biomechanics as quantified via MR-Elastography (MRE) allows in combination with non-linear mechanics to estimate IFP in absolute units non-invasively.

**Methods**

Elevated IFP within the lesion will inflate the tumour
and impose a force onto its surrounding (Fig.1). Consequently, the tumour
environment is stretched by the deformation field D0. Since tissue exhibits
non-linear material properties, the apparent shear modulus will depend upon the
local deformation tensor. Radial shear modulus will decrease due to a
compression, while circumferential shear modulus will increase due to stretch
(Fig.2A). A shear wave that travels through this tumour environment will,
depending on its propagation vector K, sense regionally altered apparent moduli
(Fig.2B). The leading side of the inflated object will exhibit reduced values,
while in the lateral area, increased values compared to the background
stiffness. This has been demonstrated previously^{1} and new improved results
for an inflated balloon inserted within a polymer phantom are shown in Fig.2C,D
confirming this “butterfly pattern”. Thus, MRE is capable of quantifying this
modified mechanical environment in the vicinity of the inflated object.

Our concept of estimating IFP is based upon the idea to virtually deflate the tumour. Mathematically this is analogous to reversing the deformation field D0 in order to recover the state of modulus distribution prior to the stretching effect of IFP. The MRE experiment provides the shear wave field in the inflated state which is under the influence of the non-linear material response. Using these wave fields, we solve the classical linear wave equation whose spatial operators have been scaled according to deformation gradients corresponding to a hypothetical deformation D0*. We consider an idealized radial symmetric deformation mode which is scaled by the “contraction” parameter alpha. The optimal value for alpha is found when the minimum of the objective function (i.e. the variance of the elasticity distribution) is obtained. A value of alpha=0 means that the object has not undergone inflation, while alpha=1 means that the object will collapse to a point if IFP were taken away. Fig.3 shows different initial shapes and sizes of objects subject to a simulated inflation. As clearly visible, the finer shape variations are unimportant when considering the pressure volume relationship.

The validity of this approach is demonstrated via simulation, a polymer phantom with known pressure of the inflated balloon, and the first in-vivo results in fatty breast tissue as well as in a fibroadenoma are presented.

**Results**

**Discussion**

1. Using non-linear tissue biomechanics to infer static forces within tissue: towards quantifying IFP, ISMRM 2017, #0973

2. Interstitial fluid pressure in breast cancer, benign breast conditions, and breast parenchyma. Annals of Surgical Oncology, 1(4):333-338 (1994)

Figure 1. General concept of modified mechanical environment surrounding an
inflated tumour. Our concept is to find the deformation field D0 that occurs
when the tumour is exposed to IFP. In the absence of IFP, it would collapse to
its base state.

Figure 2. Evolution of apparent stiffness in the vicinity of an inflated object
probed by a shear wave that propagates in the direction indicated by the
K-vector (A,B). C: inflated balloon within a non-linear polymer phantom
material and corresponding apparent elasticity map that shows the expected
butterfly pattern (D).

Figure 3. For
simulated tumours of various sizes and shapes (a to i), the pressure – volume
(normalised to the initial volume) curves yield a highly consistent
relationship (j), allowing the inflating pressure to be identified, once alpha
is found. (Shown relationship is for surrounding tissue of fixed shear
modulus).

Figure 4. Row 1: simulation results, row 2: phantom experiments, row 3: in-vivo
breast tissue. Column 1: magnitude image of the object, column 2, example of a
shear wave component, column 3, objective function as a function of contraction
parameter alpha at the inflated object, column 4, objective function as a
function of contraction parameter alpha at a reference point.