Cancer math project

Status: in development, February, 2013.


Math helps. This project is dedicated for problems handled by math and software which arise while cancer treatment. The journey of math support starts from interpretation and filtering of tomography data, followed by multiple tasks of screening and visualization, distance-area-volume estimation, optimization of parameters for gamma-knives, modeling and prototyping.


Materials below are structured in context of that or another processing layer. There are next layers of data processing:

  1. Data gathering (mechanics, energy selection, geometry defined by scanner maker)
  2. Reconstruction (by measured discreet data, scalable raster model reconstructed)
  3. Filtering (noise, smoothing, contrast)
  4. Post-reconstruction processing: anti-artifact methods (more advanced filtering) and image normalization (rotation, scaling)
  5. Volumetric modeling and visualization
  6. Segmentation and vectorization (subparts selection, contours selection)
  7. Estimation of distances, area, volume
  8. Optimization of trajectories of gamma-rays
  9. Tumor growth prediction

Data gathering

Scanners. The devices are different. They are eyes of computers. Some of them allows to see what human eye sees (3D-scanners), some of them gather data of object shape, another ones may gather color data as well. Another devices allows to see deeper (4D/5D scanners like ultrasound, MRI and CT) and named as tomography (sometimes internal vision (intervision)).

The ways how the devices gather information about objects are so different and demonstrate how far and how smart could be engineer idea: active and passive, different geometries of measurements, homemade and industrial etc.1)

We would concentrate our attention on MRI and CT scanners. The devices are gathering information in slices (“tomos” means slice from Greek). And at each section scanner does finite number of measurements that allows then to build an image of the slice.

The colors at this image are basically linked to density of a tissue. Hounsfield scale defines density of water as 0 Hounsfield unit (HU), density of air as -1000 HU, muscles - +40 HU, bones - +400 HU.2).

Geometry of measurements could be different: some scanners take measurements by circle trajectory , another ones - by fan, some devices are linear.3).

As you may see the first task that occurs is reconstruction of solid image from such a discrete measurements.


Algorithms work with discreet data. The same story is with visualization of images. It is common way to distribute images as 2D-arrays of pixels. 2D-array is a discreet grid of a picture human eye could see. This representation has height and width, representing number of rows and columns in the grid. Thus the grid is rectangular.

Getting rectangular grid for intervision images is not practical, mentioning that millions and billions of measurements are required. For tomography imaging W x H x S measurements of energy change are required. Mentioning that width and height would start from 128 x 128 to 1024 x 1024 voxels and say that slices width could be from 1 mm to 10 cm and average body length is near 170 cm, then we see that entire body scan requires from 2.780M to 1.170B of measurements.

Thus measurement speed optimization takes part at every step of intervision. It starts from mechanics and selection of the grid of measurements.

The grids are different starting from a 2D-schemas like circle, fan, combined circle and fan, projected to a line or arc and ending by 3D-schemas like sphere and cone.

Back-projection by Fourier transform etc.

Volumetric Images

Pixel is a unit of a flat image. Voxel (volume pixel or volumetric picture element) is one for volumetric images:

Rotating Images

  1. Rotate 2D image by given center and angle
  2. Rotate volumetric image by given center and angles

Symmetry of Images

  1. Find center and angle maximizing symmetry of the image
  2. Find center and angles maximizing symmetry of the volume


Next stage is filtering and removing of artifacts.


Next stage is filtering and removing of artifacts.

Volumetric modeling and visualization

At this point we are getting a set of slices that describe patient. However it is not convenient for some purposes to look at flat images, so number of application layer tasks arise. Such as 3D/4D/5D modeling and further visualization of the models, search for anomalies (tumors), auto selection of tumors and auto-segmentation, anatomical lengths measurement, tumor/organ area ration calculation, tumor/organ volume ratio calculation etc.

Segmentation and vectorization

Estimation of distances, area, volume

Optimization of trajectories of gamma-rays

When patient model is reconstructed, and tumor is located. One of possible applications is gamma-knives 4). Quite important is to find optimal ratio between accuracy of treatment and radiation which a patient gets 5).

A Tumor Growth Prediction

Math models for tumor grow prediction. That's pretty actual - see two links at press.

Perfect system. It takes data in DICOM/Analyze format (set of flat images). Searches for anomalies. Segments tumor and not tumor. Outlines contour.

Allows to setup parameters of growth. Shows it.

Correctness. Compares modeled growth with actual growth.

Risk Factors


  1. Приставка П. О. Поліноміальні сплайни при обробці даних. – Д.: ДГУ, 2004. – 41-48
  2. Cтаття “Масштабування цифрових зображень”, П.О.Приставка, 17с., 2006 рік
  3. Стаття “Низькочастотні, високочастотні та контрастні фільтри”, П.О.Приставка, 11с., 2006 рік




Artificially Created Organs

Another Applications of Tomography

Data Formats




Tumor Growth Prediction



Editing Formulas

Cancer Statistics