^{1,a)}, Benjamin R. Galloway

^{2}, Adele P. Peskin

^{3}, Claus P. Heussel

^{4}and Joseph J. Chen

^{5}

### Abstract

**Purpose:**

The authors investigate the extent to which Response Evaluation Criteria in Solid Tumors (RECIST) can predict tumor volumes in ideal geometric settings and using clinical data.

**Methods:**

The authors consider a hierarchy of models including uniaxial ellipsoids, general ellipsoids, and composites of ellipsoids, using both analytical and numerical techniques to show how well RECIST can predict tumor volumes in each case. The models have certain features that are compared to clinical data.

**Results:**

The principal conclusion is that a change in the reported RECIST value needs to be a factor of at least 1.2 to achieve a 95% confidence that one ellipsoid is larger than another assuming the ratio of maximum to minimum diameters is no more than 2, an assumption that is reasonable for some classes of tumors. There is a significant probability that RECIST will select a tumor other than the largest due to orientation effects of nonspherical tumors: in previously reported malignoma data, RECIST would have selected a tumor other than the largest in 9% of the cases. Also, the widely used spherical model connecting RECIST values for a single tumor to volumes overestimates these volumes.

**Conclusions:**

RECIST imposes a limit on the ability to determine tumor volumes, which is greater than the limit imposed by modern medical computed tomography machines. It is also likely the RECIST limit is above natural biological variability of stable lesions. The authors recommend the study of such natural variability as a fruitful avenue for further study.

BRG was supported by a Summer Undergraduate Research Fellowship at the National Institute of Standards and Technology. ZHL and Steven Grantham have a US Patent pending based on the work in Ref. 22.

I. INTRODUCTION

II. RECIST WITH ELLIPSOIDS

III. DISCUSSION AND CONCLUSIONS

## Figures

Probability density from Eq. (8) for randomly oriented prolate ellipsoids with major radius *c* and minor radius *a* as shown in the figure. The vertical line corresponds to the diameter for a sphere with the volume *π*/6. Each curve diverges at its upper bound but is finite at the lower bound.

Probability density from Eq. (8) for randomly oriented prolate ellipsoids with major radius *c* and minor radius *a* as shown in the figure. The vertical line corresponds to the diameter for a sphere with the volume *π*/6. Each curve diverges at its upper bound but is finite at the lower bound.

For randomly oriented prolate ellipsoids with given *c*/*a* ratios, (a) the mean RECIST *d* and (b) its standard deviation. The probability distribution is given in Eq. (8) and plotted in Fig. 1. See Eq. (5) for the oblate case.

For randomly oriented prolate ellipsoids with given *c*/*a* ratios, (a) the mean RECIST *d* and (b) its standard deviation. The probability distribution is given in Eq. (8) and plotted in Fig. 1. See Eq. (5) for the oblate case.

Logarithm (base 10) of RECIST diameter in millimeters vs logarithm of volume in cubic millimeters for 5000 ellipsoids with random orientations, sizes, and *a*:*b*:*c* ratios restricted between 1 and *ɛ*. Upper lines are best fit; lower lines correspond to spheres.

Logarithm (base 10) of RECIST diameter in millimeters vs logarithm of volume in cubic millimeters for 5000 ellipsoids with random orientations, sizes, and *a*:*b*:*c* ratios restricted between 1 and *ɛ*. Upper lines are best fit; lower lines correspond to spheres.

Logarithm of RECIST diameter in millimeters vs logarithm of volume in cubic millimeters for (a) liver malignoma (Ref. 18) and (b) nasopharyngeal carcinoma (Ref. 19). Upper lines are best fit; lower lines correspond to spheres. In (a), black points correspond to baseline data and green points correspond to follow up data.

Logarithm of RECIST diameter in millimeters vs logarithm of volume in cubic millimeters for (a) liver malignoma (Ref. 18) and (b) nasopharyngeal carcinoma (Ref. 19). Upper lines are best fit; lower lines correspond to spheres. In (a), black points correspond to baseline data and green points correspond to follow up data.

Probability densities of residuals of the fits in Fig 3.for *ɛ* = 1.1, 2, and 16, and a histogram of empirical data from Heußel *et al.* (Ref. 18). Normal distributions with the same standard deviations are plotted with dashed lines.

Probability densities of residuals of the fits in Fig 3.for *ɛ* = 1.1, 2, and 16, and a histogram of empirical data from Heußel *et al.* (Ref. 18). Normal distributions with the same standard deviations are plotted with dashed lines.

(a) Probabilities of two sampled ellipsoids with random orientations, random *a:b:c* ratios, and random volumes, to have the relationship V_{2} > V_{1} given the ratio of their measured RECIST diameters. Curves are plotted in order for *ɛ* = 1.1, 1.5, 2, 4, 8, 16, and 8000. Horizontal lines represent the 5% and 95% probabilities. Vertical lines represent the RECIST criteria for PR and PD. The full range of Δ in this plot represents 1/2 ≤ *d* _{2}/*d* _{1} ≤ 2. (b) Probabilities for all pairs of tumors measured in a previous study of liver malignoma (Ref. 18) fit to *ɛ* = 5 of the series in (a). (c) Relative frequencies of observed malignoma pairs with (left) V2 < V1 and (right) V2 > V1.

(a) Probabilities of two sampled ellipsoids with random orientations, random *a:b:c* ratios, and random volumes, to have the relationship V_{2} > V_{1} given the ratio of their measured RECIST diameters. Curves are plotted in order for *ɛ* = 1.1, 1.5, 2, 4, 8, 16, and 8000. Horizontal lines represent the 5% and 95% probabilities. Vertical lines represent the RECIST criteria for PR and PD. The full range of Δ in this plot represents 1/2 ≤ *d* _{2}/*d* _{1} ≤ 2. (b) Probabilities for all pairs of tumors measured in a previous study of liver malignoma (Ref. 18) fit to *ɛ* = 5 of the series in (a). (c) Relative frequencies of observed malignoma pairs with (left) V2 < V1 and (right) V2 > V1.

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