Cell invasion, motility, and proliferation level estimate (CIMPLE) maps derived from serial diffusion MR images in recurrent glioblastoma treated with bevacizumab

Abstract

Microscopic invasion of tumor cells and undetected tumor proliferation is the primary reason for a dismal prognosis in glioblastoma patients. Identification and quantification of spatially localized brain regions undergoing high rates of cell migration and proliferation is critical for improving patient survival; however, there are currently no non-invasive imaging biomarkers for estimating proliferation and migration rates of human gliomas in vivo. To accomplish this, we developed CIMPLE (cell invasion, motility, and proliferation level estimates) image maps using serial diffusion MRI scans and a solution to a glioma growth model equation. CIMPLE represent a novel method of quantifying the level of aggressive malignant behavior. In the current pilot study, we demonstrate the utility of CIMPLE maps to predict progression free survival (PFS) and overall survival (OS) in 26 recurrent glioblastoma patients treated with bevacizumab from our Neuro-Oncology database. Voxel-wise estimates of cell proliferation rate predicted spatial regions of contrast enhancement in 35% of patients. A linear correlation was found between the mean proliferation rate and progression-free survival (PFS; P < 0.0001) as well as overall survival (OS; P = 0.0093). Similarly, the mean proliferation rate was able to stratify patients with early and late PFS as well as OS.

Appendix: Solution to the glioma growth equation using estimates of water mobility

Using the glioma growth model developed by Swanson et al. [29, 30], we see the rate of change in glioma cell density is equal to the net invasion of glioma cells plus the net proliferation

$$ \overbrace {{\frac{dc}{dt}}}^{{{\text{Rate}}\;{\text{of}}\;{\text{change}}\;{\text{in }}\;{\text{cell}}\;{\text{density}}}}\,=\,\overbrace {{\nabla \cdot \left( {D\nabla c} \right)}}^{\text{Invasion}}\,+\,\overbrace {\rho \cdot c}^{\text{Proliferation}} $$

(1)

where c is cell density, D is the diffusion coefficient of migrating cells, ρ is the cell proliferation rate, and t is time. Based on evidence of a strong negative correlation between tumor cell density and ADC of water measured using DWI [31–37], ADC can be substituted into Eq. 1 to yield

$$ \overbrace {{ - \frac{d}{dt}{\text{ADC}}}}^{{{\text{Rate}}\;{\text{of}}\;{\text{change}}\;{\text{in}}\;{\text{cell}}\;{\text{density}}}}\,=\,\overbrace {{ - \nabla \cdot \left( {D\nabla {\text{ADC}}} \right)}}^{\text{Invasion}}\,-\,\overbrace {{\rho \cdot {\text{ADC}}}}^{\text{Proliferation}} $$

(2)

where ADC is the apparent diffusion coefficient of water as a three-dimensional scalar field (i.e. ADC image volume acquired using DWI), D is the diffusion coefficient of migrating cells as a three-dimensional scalar field, and ρ is the cell proliferation rate as a three-dimensional scalar field.

An analytical expression for cell diffusion rate, D, and proliferation rate, ρ, can be described by using the Methods of Characteristics [12] to achieve one possible solution:

$$ D = {\frac{{\frac{d}{dt}{\text{ADC}}^{n} - \lambda \frac{d}{dt}{\text{ADC}}^{n - 1} }}{{\nabla^{2} {\text{ADC}}^{n} - \lambda \nabla^{2} {\text{ADC}}^{n - 1} }}} $$

(3)

and

$$ \rho = \rho^{n - 1} = - {\frac{1}{{{\text{ADC}}^{n - 1} }}}\left( {\frac{d}{dt}{\text{ADC}}^{n - 1} - D\nabla^{2} {\text{ADC}}^{n - 1} - \nabla D \cdot \nabla {\text{ADC}}^{n - 1} } \right) $$

(4)

where

$$ \frac{d}{dt}{\text{ADC}}^{n - 1} = {\frac{{{\text{ADC}}^{n - 1} - {\text{ADC}}^{n - 2} }}{{t^{n - 1} - t^{n - 2} }}} $$

(5)

describes the time-rate of change in ADC and

$$ \lambda = {\frac{{{\text{ADC}}^{n} }}{{{\text{ADC}}^{n - 1} }}} $$

(6)

describes the ratio of ADC on the current day n with respect to the previous scan day n−1. Thus, using three ADC maps collected on days t ⁿ, t ⁿ⁻¹, and t ⁿ⁻², the proliferation rate, ρ, and cell motility (diffusion), D, can be directly estimated for the time interval spanned from t ⁿ⁻² to t ⁿ using Eqs. 3–6. Analytical solutions to the glioma growth model were verified in Mathematica v7.01 (Wolfram Mathematica 7.01, Wolfram Research, Inc, Champaign, IL) and are further defined in a previous publication [4].