The scientific value of this article is little but its practical value is great. The literature considers the size of a tumour one of its important characteristics.

I know two methods to specify the size of a tumour:

1. One measurement.

2. Three measurements.

Let us examine these! What do they specify? And is it enough to characterize the tumour geometrically?

I would like to state that I am not a doctor. I am an engineer proficient in certain areas of mathematics, metrology (and medicines but it has no importance here) [1].

1. In the literature – although I haven’t seen it written down – the size of a tumour means its greatest dimension. I also haven’t found any reference as to how it should be measures if we can see the tumour – e.g. through some imaging method.

Doctors, however, make important prognostic statements based on the non-defined size [2], [3] (Serious doubts arise in connection with these. As to the given limit of 5 cm, it is true that “nature does not know the concept of whole numbers.” One can therefore ask: “Why exactly 5 cm, why not 4.8 or 5.1?” It is also very unlikely that this is a sharp limit, that is, if the tumour is 5 cm the patient can live on for years, whereas if it is 5.1 cm they will die soon.)

Putting it mathematically: What is the distance between the furthest points of a closed solid shape?

Even if the shape is well defined with an equation or another method, it would take a considerable amount of time to calculate with a computer (and the program would also be rather complicated). And it, naturally, is not the case with tumours!

An approximate size can be calculated if many photos are taken of the tumour and in each photo the distance of the furthest points are determined (the problem is now simplified to a problem in one plane) and the greatest distance is selected.

As we can see, even this is not a simple task. But how much does it help if it is done?

I show it reduced to a two-dimensional and very much simplified case.

Let all two-dimensional cells be the same size and let them fill the figures completely.

In the first case, let them fill a “tumour” whose size is 1×13 units (Fig. 1).

In the second case the cells make up a tumour of a different shape and 10 units long (Fig. 2).

This way, in the first case, the greatest distance is 13 units, whereas in the second case it is only 10 units. Therefore based on the greatest distance the first “tumour” is larger, even though the first “tumour” contains 13 cells while the second one contains 30, which is more than double that. This way, in real life the latter “tumour” is larger.

Thus it can be seen that the greatest distance says nothing of the real size of the tumour.

2. An ultrasound image and two MR images of a tumour [5 – 7].

According to these, in chronological order, the three measurements of the tumour (also not defined) are the following:

“19X18X7 mm”,

“4,6X2,5X4,3 cm”

“50X58X30 mm”.

Let us first examine the last two.

Using these, if the first one is fixed, we can create series with two elements, each containing three measurements, in three ways:

46X25X43 and 50X58X30,

46X25X43 and 58X30X50,

46X25X43 and 30X50X58.

If it is assumed that in both places the measurements of my tumour were given in the same order (e.g. length, width, height, ignoring the method with which they were determined – I wrote about the problem concerning one measurement in point 1), according to the first series

– in the first direction the tumour slightly grew

– in the second direction the tumour grew considerably

– while in the third direction the tumour shrank considerably,

but according to the second series, it grew in all directions slightly, while the third series is similar to the first but different measurements increased and different ones decreased. Thus it seems that the tumour changed in all kinds of ways.

I assume (but don’t know) that it was not the case, because it was not I who evaluated the images.

What should be done then?

It is not so simple but it can be derived that the greatest measurement should be substituted with an averaged value and the average greatest distances should be weighted and using these we can approximate the diameter of the sphere that has the same volume as the tumour, and this diameter can truly describe the tumour. (For this, however, the volume of the tumour needs to be determined.) This way we get a size measurement.

Let us see a similarly simplified case!

Let us have a different arrangement for the cells (Fig. 3.).

Here both largest diameter is the same as in the second case. The “volume” is the same too but the shape is different.

I recommend three things.

The first is that 3 dimensions should be given everywhere and the order of these should be standardised. The second is that the volume of the tumour should be determined, also in a standardised manner.

It is also just as important, if not more important to specify the proportion of necrotic parts in the tumour. The problem is more complicated mathematically in that the volumes of not one but several, perhaps many solid shapes have to be determined and finally added. The task is also more complicated metrologically because these shapes may be far smaller than the tumour, in which case their dimensions can be determined with far less precision than those of the tumour. (In some tumour types other proportions have to be determined, e.g. in the case of myxoid sarcomas the percentage of all the myxoid parts together. Calculating these can be done in the same way as in the case of necrotic parts.)

And third but most important point is that the direction and magnitude of the change of the tumour should also be determined – I would like to stress it again – with a standardised method. This entails the evaluation of images obtained of the tumour at repeated times – determining the above mentioned things in the case of all images and evaluating the changing tendencies in time.

Let us start working out the method to achieve this as soon as possible!

Dr. Endre Simonyi

*Bibliography:*

*[1] http://www.europeanscience.net/mathematics*

*[2] Soft Tissue Fibroblastic / myofibroblastic tumors Myxofibrosarcoma*

*Reviewer: Annie S. Morrison, M.D. , Editor: Jerad M. Gardner, M.D.*

*Revised: 23 January 2015, last major update April 2013*

*Copyright: (c) 2003-2015, PathologyOutlines.com, Inc.*

*[3] Soft Tissue and Visceral Sarcomas: ESMO Clinical Practice Guidelines*

*Published in 2014 – Ann Oncol (2014) 25 (suppl 3): iii102-iii112.*

*
Authors: The ESMO / European Sarcoma Network Working Group*

*Dr. Endre Simonyi*