Europ. aT. Cancer Vol. 5, pp. 625--629. Pergamon Press 1969. Printed in Great Britain

Survival Gain and Volume Gain

Mathematical Treatments

Tools in Evaluating

P. CHAHINIAN and L. ISRAEL Centre Hospitalier Universitaire Cochin,

31 rue du Faubourg-Saint-oTacques, Paris XIVe, France

As wB intend to show in the next paper, the way in which screening tests and controlled trials are performed does not meet the require- ments for accurate quantitative evaluations. The present study is devoted to the task of setting up a simple mathematical apparatus allowing such evaluations to be made.

The exponential rate of growth of human solid tumors, first established by Collins [1, 2] on the basis of the experimental work of Mottram [3] has been verified for pulmonary metastases and peripheric bronchogenic carci- nomas by Schwartz [4], Nathan [5], Garland [6], Spratt [7], Weiss [8], Gerstenberg [9], Breur [10], Bru [11], Combes [12] and for cases of other carcinomas by Welin [13], McDonald [14], Spratt [15], Berger [16], Ingleby [17], Philippe [18] and others. We were able to confirm this :model and we discussed the reasons for the apparent discrepancy between the gompertzian model observed in trans- planted tumors and the exponential one ob- served in man during the clinical period [19, 20]. This exponential way of growth leads to some simple algebraic and trigonometric con- siderations which will be discussed around three theoretical models of therapeutic response to chemotherapy.

METHODS Construction of growth curves

The method we used is the graphic one

Submitted for publication 5 June 1969. Accepted 11 August 196!).

proposed by Collins [1, 2]. The mean diameter of a measurable tumor is plotted against time on a semi-logarithmic scale. The curve ob- tained is a straight line, which means that the doubling time is constant. The volume doub- ling time is one-third of the diameter doubling time, and can be read directly from the curve.

Mathematical relations between the volume doubling time and the Slope of the growth curve

The function describing the growth being of the shape:

V= Vo ebt

where V0 is the volume at the time to and b a constant of the system choosing for t the value of the doubling time (T,) gives for b the value:

b= Log 2 T~

Diameters and volumes being bound by the formula:

Vo Z it follows that:

Log d = Log do+ Log 2 3 T ,

or, in terms of decimal logarithms:

log d=log do + 0.3013

3 T ,

625

626 P. Chahinian and L. Israel

or, to a good approximation:

1 log d=log do+

10 T~

Then the function of the growth curve on the semi-logarithmic scale is of the shape:

1 y = l o g do+ ~ t

10 2-2

and the slope of the curve tgO (see Fig. 1) is:

1

10Ts

Graphical expression of the concept survival gain compared to total survival time

Figure 1 shows a theoretical therapeutic growth curve, together with the extrapolated

B

/ I / I

/ t E/__ / s_ _ ] c

/ - -- I a fo

x=t

Fig. 1. Graphic expression of spontaneous doubling time, thera- peutic doubling time, duration of treatment (T), survival gain (S) volume gain (BC). The arrow shows the start of the

treatment.

spontaneous one, in a case of lengthening of 7",. It appears that there is a difference between the total duration of treatment (or the total survival time) T and the benefit induced by this treatment as far as survival and volume are concerned. I f the death were to occur when the diameter of the tumor was the one reached in C, this diameter would have been reached without treatment in E. The survival gain Sg is, in this ease of lengthening, smaller than T.

Mathematical relations between S, and T Considering Fig, 1 again, it appears that the

slope of the therapeutic growth curve, tg~, is:

1 t,13=

10 T' ,

where T' , is the therapeutic doubling time. Trigonometric consideration of the triangles

ADB, BCE and ADC leads to the formula:

S a = T ( 1 - - T'~T~)

So, there is a relationship between the survival gain and the total survival which depends upon the spontaneous and the thera- peutically modified doubling times, which is not surprising. The ratio Sg/T provides a quantitative approach towards the evaluation of the effectiveness of a given treatment in cases of different T,.

Validity of the S, formula in different response patterns

Three different possibilities are outlined in Fig. 2: shrinkage, equilibrium and length- ening. These events are described by the

" 0

o

C t

vl / I ! / /

1[ I Time, t

Treatment T 2[ I 31 J

Fig. 2. Theoretical effects of three different kinds of treatment on the therapeutic growth curve. 1 : shrinkage; 2: equilibrium;

3: lengthening.

different values of the angle a, with B = 0 - - ~ . In the case of the figure the same value for S~ results from three different therapeutic responses showing that treatments with dif- ferent effectiveness and different duration can induce the same Sv. It can be seen that the formula holds true not only when the response is a lengthening or an equilibrium, but also when it has the shape of a shrinkage. In this particular case, T~ must be changed for Tl/, (semi-reduction time) and counted negatively in the formula. This gives for Sg/T a value greater than 1, and T1/, can be read directly on the growth curve in the same manner as T,.

Validity of the S,formula when effectiveness is varying along time

Figure 3 shows a model of such a response

Survgal Gain and Volume Gain: Mathematical Tools in Evaluating Treatments 627

}

/ I

/

//L_.___ , i f

/ •

I l T

t

Fig. 3. Therapeutic growth curve obtained with varying succes- sive effects of the same treatment. Resultant doubling time.

which is in agreement with most clinical curves registered by us. I t can be seen that the vectorial sum of AB, BC, CD and DE is equal to AE. Then the final survival gain can be calculated directly' in E, drawing from E a horizontal line towards the extrapolated spontaneous growtlh curve.

Mathematical evaluation of the volume gain The segment BC in Fig. 1 represents the

volume gain from ~the onset of the treatment to the end point. I f& and VI are the diameter and the volume at C, and d, and V, the diameter and the volume at B, the residual volume in percentage is:

V o=100 .=100 _

The value for BC being &.t~.8, it follows that

R.V.% ---- lOe ~-*so',e

Thus, for a given survival gain, the shorter the doubling time the more important is the volume gain; and, for a given volume gain, the longer the doubting time the more important is the survival gain. In such a case, even a weak therapeutic effect can induce an important survival gain. This formula may be used to estimate which duration of treatment should be

reached in order to produce a given effect, assuming that its effectiveness will be constant. For example, if R .V.% is l%-- i . e , if the reduction in volume induced by the treatment is 99%, the resulting survival gain is:

20T~ Sg--

that is equal to approximately 7 times the doubling time. In tumors with very short doubling times, such a dramatic achievement as far as volume is concerned remains of a somewhat poor value.

DISCUSSION It is not possible to describe here all the

various formulae which may be found in [19]. We shall only underline the following two points:

(a) The calculations discussed here were performed on theoretical models, and it may be difficult in some cases to apply them in human tumors. Most tumors are not measurable. Metastases in different organs can have dif- ferent growth rates and can be reached differently by cytostatic drugs. Furthermore, small tumors may kill due to their site. Never- theless, the formulae developed here may prove of some value to the therapist, as we intend to show in the next paper.

(b) The total survival time from the start of chemotherapy (or, for that matter, after the completion of X-ray therapy or surgery) is of little informational value, since the spontaneous survival time that the patient would have reached without therapy remains unknown. The concept of the survival gain induced by the treatment, the value of which is bound to the total survival time (or the total duration of the treatment) and to the spontaneous doubting time, allows evaluation of treatments to be made accurately even in patients whose tumors have different T~. This may lead to more information being gained from controlled trials.

R E S U M E Les essafi" control& de chimiotMrapie des tumeurs humaines comparent des r&ultats obtenus sur des tumeurs de volume et de vitesse de croissance diff&ents.

Ils gvaluent la survie totale qui confond bgngfice du traitement et le temps pendant lequel le mala& aurait surv&u spontangment.

Les auteurs proposent de remgdier aces insuffsances au moyen de la mesure du gain en survie et en volume. Ces paramktres sont li&, par une formule simple, au temps de double- ment spontang et~ l'effet du traitement sur la croissance. Des exemples sont donngs ~ l'aide de modkles tMoriques de rgponse th&apeutique.

628 P. Chahinian and L. Israel

SUMMARY The usual controls on the effects of chemotherapy on solid human tumors compare the evolution of tumors of different initial volumes and growth rate. They rely on the evaluation of the total survival time, in which the benefit from the treatment and the time of survival without treatment are not distinguishable.

The authors propose a better quantitative appreciation by measuring the gain in survival time and in volume, which are defined in the article. These parameters are connected by simple formulae to the spontaneous doubling time and the effect of the treatment on the growth. Examples are discussed in the light of theoretical models of therapeutic response.

ZUSAMMENFASSUNG Serienuntersuchungen an Tumoren mit chemotherapeutischen Mitteln vergleichen die Entwicklung anf~nglieh verschieden grosser Tumore und ihr Wachstumsverhiiltnis miteinander. Diese Untersuchungen erlauben es die totale Uberlebenszeit, in welcher der Erfolg der Behandlungsmethode und die ~Tberlebenszeit ohne jegliche Behandlung als Parameter eingehen, zu bestimmen.

Die Autoren schlagen in diesem Bericht eine bessere, quantitative Methode vor, die es erlaubt den Gewinn an Uberlebenszeit und die Volumengnderung des Tumors direkt zu bestimmen. Diese beiden Parameter kann man durch einfache Formeln mit der spontanen Verdoppelungszeit, d.h. der Zeit, in der sich der Tumor um das doppelte vergr6ssert h~tte, und der Wirkung der Behandlung auf die Grb'sse des Tumors in Verbindung setzen. Beispiele werden an Hand eines Modells fiir die therapeutische Ansprechwahrscheinlichkeit diskutiert.

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Survival Gain and Volume Gain: Mathematical Tools in Evaluating Treatments 629

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