Minimum latency effects for cancer associated with exposures to radiation or other carcinogens

Abstract

Background

In estimating radiation-associated cancer risks a fixed period for the minimum latency is often assumed. Two empirical latency functions have been used to model latency, continuously increasing from 0. A stochastic biologically-based approach yields a still more plausible way of describing latency and can be directly estimated from clinical data.

Methods

We derived the parameters for a stochastic biologically-based model from tumour growth data for various cancers, and least-squares fitted the two types of empirical latency function to the stochastic model-predicted cumulative probability.

Results

There is wide variation in growth rates among tumours, particularly slow for prostate and thyroid cancer and particularly fast for leukaemia. The slow growth rate for prostate and thyroid tumours implies that the number of tumour cells required for clinical detection cannot greatly exceed 10⁶. For all tumours, both empirical latency functions closely approximated the predicted biological model cumulative probability.

Conclusions

Our results, illustrating use of a stochastic biologically-based model using clinical data not tied to any particular carcinogen, have implications for estimating latency associated with any mutagen. They apply to tumour growth in general, and may be useful for example, in planning screenings for cancer using imaging techniques.

Introduction

In performing calculations of population cancer risks a fixed period of minimum latency is often assumed, as for example has been done in evaluations by the United Nations Scientific Committee on the Effects of Atomic Radiation (UNSCEAR) [1] and the International Commission for Radiological Protection [2]. For solid cancer a minimum latency of 5 years is often assumed, while for leukaemia a much shorter period of latency, of 2 years, has generally been used [1], although there have been exceptions [3].

Kocher et al. [4] and Land et al. [5] have proposed an alternative method, in which excess (relative or absolute) risk was assumed to continuously increase from 0 using a latency function, applied as multiplicative adjustment to the excess risk at time since exposure t:

$$F_{\mathrm{latency}}\left(t\right)=\frac{1}{1+\exp \left[-\left(t-\mu \right)/S\right]}$$ 1

where μ is the time since exposure inflection point and S is a shape parameter that determines the steepness of the latency function, with larger values of S representing a slower increase. The latency function having value 0 would correspond to an undetectable tumour, while a value of 1 is one reflects a tumour whose detection is almost certain. This type of latency function has been used in the NCI RADRAT cancer risk calculation software [6]. Ulanowski et al. [7] have proposed a slightly different function to describe latency, in which:

$$F_{\mathrm{latency}}\left(t\right)=\frac{1}{1+\exp \left[-\eta \ln \left(t/\mu \right)\right]}$$ 2

These functions both have a sigmoid form (S-shaped), and are intended to account for a reduced risk during the minimum latency period. The parameter η also determines the steepness of the latency function, with large values implying more rapid changes in latency. The parameter μ, as with the other latency function, determines the inflection point of the latency function. Kocher et al. [4] developed such functions for leukaemia, thyroid and bone cancers and all solid tumours (except thyroid and bone cancer) based on general knowledge from epidemiology and cancer development. The latency function of Ulanowski et al. [7] forces radiation risk to 0 when time since exposure approaches zero, and parameter values were derived for leukaemia and all solid tumours. Illustrative examples of both latency functions are given in Appendix A Fig. A1.

In this paper we explore a biologically-based approach, showing that this may be quite easily derived from clinical data and compare the predictions of this with the latency functions of Kocher et al. [4] and Ulanowski et al. [7].

Methods

A literature review conducted by all authors was used to ascertain tumour growth rates for a number of tumour endpoints, as shown in Table 1. For leukaemia and lung cancer a faster and slower rate are derived, but for other endpoints we generally give just a single estimate, in some cases based on an average of rates from a number of studies. From the doubling time in days t_double given in each report we derive the cell growth rate (/cell/year) via: G = ln[2]/(t_double/365.2425).

Table 1.

Implied rates of malignant cell division derived from tumour doubling times reviewed in the peer-refereed literature.

Endpoint	Subtypes	Malignancy doubling time (days) (range)	Implied malignant cell division rate (G) (/cell/year)	Mean age at diagnosis (range)	Reference	Notes
Leukaemia	Childhood ALL	4 (4–5)	63.292	<15	Frei and Freireich [61]	A summary of a number of clinical studies
	Adult AML	17	14.892	49 (13–83)	Gregory et al. [62]
Lung cancer	Squamous cell carcinoma	88	2.877	NA	Geddes [63]
	Adenocarcinoma	161	1.572
	Squamous cell carcinoma and other non-adenocarcinoma	70	3.617	NA (<65–>65)	Obayashi et al. [64]
	Adenocarcinoma	261	0.970
Breast cancer	All types	141 (46–825)	1.796	50 (29–78)	Ryu et al. [65]	Median doubling time = 141 days
	ER+	241	1.051
	HER2+	162	1.563
	Triple negative	103	2.458
		207.8	1.218	NA (35–74)	Spratt et al. [66]	Doubling times are for the early pre-detection period, but apparently not based on measured doubling times in individual patients, so much as derived from a simple model. Central estimate of doubling time based on geometric mean of limits of range (59, 732 days)
		106.5	2.377	NA (0–89)	Kusama et al. [67]	Data for primary and metastatic breast tumours
		99	2.557	NA (50–69)	Weedon-Fekjær et al. [68]	Model based doubling times, yielding estimate of time taken to go from 10 mm to 20 mm diameter = 1.7 year (SD = 2.2)
Colorectal		130 (53–1570)	1.947	NA (20–90)	Bolin et al. [45]
		211 (112–404)	1.200	80 (73–85)	Burke et al. [46]
		35.5 (17.7–44.4)	7.134	NA (45–85)	Knudsen et al. [47]	Doubling time derived assuming 109 cells required, based on estimates of sojourn time of 4 years (range 2–5) for SimCRC model
Prostate		1674 (730–2200)	0.151	63.7 (49–74)	Egawa et al. [17]	Doubling times among clinically insignificant cases
		2109	0.120	71 (51–83)	Schmid et al. [18]	Doubling times among 27 clinically confined cases
		1309	0.193			Doubling times among 15 non organ-confined cases
		137	1.848		D’Amico and Hanks [42]	Patients given radiotherapy (68–71 Gy)
Hepatocellular carcinoma		140 (65–345)	1.808	NA	Nathani et al. [69]	Meta-analysis of 20 studies. Range is range of study means, mean is pooled mean
Thyroid	Papillary thyroid cancer (PTC)	820.0	0.309	51.1 (42.2–61.0a)	Oh et al. [38]	PTC doubling times among growing tumours
		687.9	0.368	NA (24–79)	Miyauchi et al. [39]
	Medullary thyroid cancer	209.2	1.210	53.5 (9–80)	Kihara et al. [40]	Inferred doubling rate before surgery in hereditary medullary thyroid cancer patients
Head and neck		61 (15–>234)	4.150	59.2 (32–92)	Jensen et al. [70]
Oropharyngeal		96 (21–256)	2.637	68 (47–102)	Waaijer et al. [71]

ALL acute lymphoblastic leukaemia, AML acute myeloid leukaemia, NA not available, ER oestrogen receptor, HER human epidermal growth factor receptor, SD standard deviation, SimCRC Simulation Model of Colorectal Cancer (SimCRC developed by University of Minnesota and Massachusetts General Hospital) (see https://resources.cisnet.cancer.gov/registry/packages/simcrc-minnesota/#summary).

^aInterquartile range.

Description of stochastic biologically-based model

We assume that the first fully malignant cell is subject to a birth-death process, so that the resulting tumour is subject to an exponential growth, with rates of cell division given by G (/cell/year) and competing cell apoptosis or differentiation at rate D (/cell/year). This carries on until the point at which the malignancy is detected clinically. The cell population is modelled through a filtered Poisson process, with each cell in a population at a given time subject to possible cell division into two identical daughter cells, at a rate of G or apoptosis/differentiation at a rate D. Finally, we assume that the probability of clinical detection of malignancy is roughly proportional to the number of malignant cells, given by a proportionality M (/cell/year). This is plausible as one would expect that the larger the tumour, and therefore the more malignant cells, the greater will be the likelihood of clinical detection. The model is shown in schematic form in Appendix A Fig. A2.

In order to estimate the clinical detection rate M we assume a given approximate limiting number of cells for detection N_lim between 10⁶ and 10¹⁰. There are data [8] suggesting that for solid tumours N_lim is about 10⁹, so we take N_lim = 10⁹ as nominal value for many cancers. However, we explored a smaller nominal value of N_lim = 10⁸ as suggested by Del Monte [9], and for prostate and thyroid cancer we used a much smaller value, N_lim = 10⁶ for reasons that we elaborate in the “Results” and “Discussion” sections. We assume that at each approximate limiting cell number the detection rate M is chosen so that, if there were to be precisely that number N_lim of cells, and this was a fixed quantity, the probability of detection of the cancer in a year would be p. This implies that:

$$M=-\ln \left(1-p\right)/N_{\lim }$$ 3

Using p = 0.95 (i.e. 95% probability of detection in a year) we calculate these rates in Appendix A Table A1 for various values of N_lim. We evaluate the probability of clinical detection in relation to time after development of the first malignant cell using the exact stochastic model described in the paper of Little et al. [10], described in outline above, and in more mathematical detail in Appendix A.

In order to derive useful estimates of the number of tumour cells, we give in Appendix A Table A2 the predicted number of tumour cells in various spherical volumes, assuming an optimal packing density predicted by Gauss’ theorem [11], namely ${\mathrm{den}}_{\mathrm{pack}}=100\times \pi /\left(3\sqrt{2}\right)\approx 74.05\%$. Gauss’ model yields the optimal packing density for cells that are assumed to be identically spherical. We also explore the slightly lower predictions of packing density, of about den_pack = 60% estimated by Grantab et al. [12, 13] based on nuclear stained 2D cross sections, as well as somewhat higher estimates, of about den_pack = 95% suggested by Grosser et al. [14]. We compute the volume of a tumour, assuming it to be a sphere of diameter D_tumour, as V_tumour = 4π(D_tumour/2)³/3, the volume of a cell, again assumed spherical of a specified diameter D_cell, as V_cell = 4π(D_cell/2)³/3, and we combine this with the tumour packing density den_pack (%) to obtain the number of cells in the tumour:

$$V_{\mathrm{tumour}}/V_{\mathrm{cell}}\times \left[{\mathrm{den}}_{\mathrm{pack}}/100\right]$$ 4

Statistical model fitting

We evaluated the latency function predicted by the biologically-based model, using expression (A8) in Appendix A, for each tumour endpoint, using the growth rates G given in Table 1 and data on implied detection rates per cell per year M from Appendix A Table A1. In most cases (all except the analyses of Appendix A Fig. A3) we assumed a cell death/differentiation rate D = 0. To these evaluated functions we fitted the two latency functions, as given by Eqs. (1) and (2). The latency functions were fitted via minimisation of the sum of squares of differences between each empirical latency function and the biologically-based model latency (=cumulative probability) at selected time points, summed over particular latency time intervals, using the Excel solver function [15]. For the various tumour endpoints we employed the latency time grid as given by Appendix A Table A3.

Results

Table 1 shows the range of tumour growth rates derived from the clinical data we surveyed. Appendix A Table A4 demonstrates that for breast cancer the implied malignant cell division rate progressively reduces with increasing age. Appendix A Table A2 shows the predicted number of tumour cells in various spherical volumes, assuming various packing densities. As can be seen, assuming a plausible range of detectable tumour sizes ranging from 0.1 cm to 2 cm diameter, and with plausible ranges of cell diameter ranging from 10 × 10⁻⁶ m to 30 × 10⁻⁶ m the estimated numbers of tumour cells range from 2.2 × 10⁴ to 7.6 × 10⁹. All estimates but three (out of 36 possible estimates) are under 10⁹.

The cumulative probability of clinical detection as a function of time from appearance of the first malignant cell predicted by the stochastic biologically-based model is shown in Fig. 1, for leukaemia, and for lung cancer. In each case a faster and slower tumour growth rate is assumed, as given by the parameters in Table 1. In most cases we assume that the cell death/differentiation rate, D, is 0. In Fig. 2 we show the cumulative probability of detection calculated in the same way for a number of other solid cancer sites. This figure illustrates the shorter latency for each cancer site that results from assuming a smaller number of minimal cells, N_lim, required for clinical detection. Figure 2 demonstrates that only for values of limiting cell number N_lim of 10⁶ or less is there an appreciable chance of a prostate tumour being clinically detected within 20 years of the first malignant cell. Likewise, Fig. 2 shows that only for values of limiting cell number N_lim of 10⁶ or so is there an appreciable chance of thyroid cancer within 15 years of the first malignant cell.

Fig. 1.

The cumulative probability of detection of leukaemia, assuming faster- [61] and slower-growing [62] malignancies, also of lung cancer [63, 64], assuming faster- and slower-growing tumours, as given by the parameters of Table 1 and Appendix A Table A1.

Fig. 2. The cumulative probability of detection of various types of solid cancer apart from lung cancer, as given by the parameters of Table 1.

For breast cancer we use the average of the estimates of the cell growth rate G from Ryu et al. [65], Spratt et al. [66], Kusama et al. [67] and Weedon-Fekjær et al. [68] and for colorectal cancer we use the average of the estimates of cell growth rate G from Bolin et al. [45], Burke et al. [46] and Knudsen et al. [47]. In the leftmost panel we assume either the limiting malignancy number of N_lim = 10⁹ as suggested by DeVita et al. [8] with the implied rate of detection of cancer/malignant cell/year of 2.9957 × 10⁻⁹, or the limiting malignancy number of N_lim = 10⁸ as suggested by Del Monte [9] with the implied rate of detection of cancer/malignant cell/year of 2.9957 × 10⁻⁸. In the central and right panels we assume the limiting malignancy number is between N_lim = 10⁶ and N_lim = 10¹⁰ with the implied rate of detection of cancer/malignant cell/year of between 2.9957 × 10⁻⁶ and 2.9957 × 10⁻¹⁰.

Appendix A Fig. A3 shows that when the cell death/differentiation rate D becomes increasingly positive then the asymptotic limit (for large time after the initial tumour cell) to the cumulative probability of clinical detection of tumour is progressively reduced from 1.

Using the stochastic biologically-based model of Appendix A with data derived from Table 1 and Appendix A Table A1, Table 2 shows the least-squares fitted values of the latency functions of Kocher et al. [4] and Ulanowski et al. [7], as given by Eqs. (1) and (2). As can be seen from Figs. 3 and 4, the shapes of these biologically theoretically derived curves are extremely close to those of the least-squares fitted latency functions—indeed the fitted curves are at least for the latency function of Kocher et al. [4] identical (by eye) with those produced by the stochastic biologically-based model, agreeing to within 4.8 × 10⁻⁶. The fit of the latency function of Ulanowski et al. [7] is slightly less good, but agreement remains within 9.6 × 10⁻³. Although we do not give the curves for endpoints other than leukaemia and lung cancer, they also show comparably close fit by the two latency functions. The largest absolute deviation using the latency function of Kocher et al. [4] was for prostate, 5.2 × 10⁻⁶. The largest absolute deviation using the latency function of Ulanowski et al. [7] was for thyroid, 1.6 × 10⁻².

Table 2.

Fitted values of parameters for the latency functions of Kocher et al. [4] and Ulanowski et al. [7] as given by Eqs. (1) and (2), estimated by least squares from stochastic model, with given rates of malignant cell division by endpoint (values from Table 1).

Endpoint	Subtypes	Implied malignant cell division rate (G) (/cell/year)	Latency function of Kocher et al. [4] (see Eq.(1))		Latency function of Ulanowski et al. [7] (see Eq.(2))		Reference
			μ	S	μ	η
Leukaemia	Childhood ALL	63.292	0.376	0.016	0.375	23.806	Frei and Freireich [61]
	Adult AML	14.892	1.499	0.067	1.497	22.361	Gregory et al. [62]
Lung cancer	Squamous cell carcinoma and other non-adenocarcinoma	3.247a	6.407	0.308	6.398	20.840	Geddes [63] and Obayashi et al. [64]
	Adenocarcinoma	1.271a	15.628	0.787	15.602	19.904
Breast cancer		1.987b	10.223	0.503	10.208	20.350	Ryu et al. [65], Spratt et al. [66], Kusama et al. [67] and Weedon-Fekjær et al. [68]
Colorectal		3.427c	6.086	0.292	6.077	20.894	Bolin et al. [45], Burke et al. [46] and Knudsen et al. [47]
Prostate		0.576d	21.110	1.735	21.020	12.209	Egawa et al. [17], D’Amico and Hanks [42] and Schmid et al. [18]
Hepatocellular carcinoma		1.808	11.182	0.553	11.164	22.256	Nathani et al. [69]
Thyroid		0.629e	19.485	1.590	19.403	12.317	Oh et al. [38], Miyauchi et al. [39] and Kihara et al. [40]
Head and neck		4.150	5.072	0.241	5.064	21.085	Jensen et al. [70]
Oropharyngeal		2.637	7.810	0.379	7.798	22.931	Waaijer et al. [71]

In all cases except prostate and thyroid we assume the limiting malignancy number of N_lim = 10⁹ with the implied rate of detection of cancer/malignant cell/year of 2.9957 × 10⁻⁹, but for prostate and thyroid we assume the limiting malignancy number of N_lim = 10⁶ with the implied rate of detection of cancer/malignant cell/year of 2.9957 × 10^–6.

ALL acute lymphoblastic leukaemia, AML acute myeloid leukaemia.

^aMean of values of G derived from studies of Geddes [63] and Obayashi et al. [64].

^bMean of values of G derived from studies of Ryu et al. [65], Spratt et al. [66], Kusama et al. [67] and Weedon-Fekjær et al. [68].

^cMean of values of G derived from studies of Bolin et al. [45], Burke et al. [46] and Knudsen et al. [47].

^dMean of values of G derived from study of Egawa et al. [17] and D’Amico and Hanks [42] and separate estimates for clinically-confined and non-organ confined cases from study of Schmid et al. [18].

^eMean of values of G derived from studies of Oh et al. [38], Miyauchi et al. [39] and Kihara et al. [40].

Fig. 3. The cumulative probability of detection of leukaemia and lung cancer, assuming faster- and slower-growing leukaemia and lung cancer stochastic models, assuming N_lim = 10⁹ cells, with least-squares-fitted latency functions of the type used by Kocher et al. [4] and given by Eq. (1).

The fitted parameter values are as given in Table 2. It should be noted that the stochastic biologically-based model and fitted latency function differ by less than 4.68 × 10⁻⁶ at all time points for leukaemia and 4.78 × 10⁻⁶ at all time points for lung cancer, so that the curves effectively coincide.

Fig. 4. The cumulative probability of detection of leukaemia and lung cancer, assuming faster- and slower-growing leukaemia and lung cancer stochastic models, assuming N_lim = 10⁹ cells, with least-squares-fitted latency functions of the type used by Ulanowski et al. [7] and given by Eq. (2).

The fitted parameter values are as given in Table 2. It should be noted that the stochastic biologically-based model and fitted latency function differ by less than 8.47 × 10⁻³ at all time points for leukaemia and 9.59 × 10⁻³ at all time points for lung cancer.

Discussion

We have shown, based on data assembled from various papers in the literature, that there is wide variation in growth rates among tumours, particularly slow for prostate and thyroid cancer and particularly fast for leukaemia (Table 1). The results of the stochastic biologically-based tumour growth-death model that we use based on this data (Figs. 1 and 2) imply that for prostate and thyroid cancer the latency will be at least 20 years. Our analyses suggest that for prostate and thyroid tumours the number of tumour cells required for clinical detection cannot greatly exceed 10⁶.

We have also shown that the two latency functions of Kocher et al. [4] and Ulanowski et al. [7] for modifying the absolute or relative risk as a function of time after exposure can be made to conform quite well with the predictions of our stochastic growth-death model for most cancers; agreement using both types of function is good, particularly so for the latency function of Kocher et al. [4]. The parameters that we fit for the inflection point μ and shape parameters S and η are cancer-specific and different from those used by Kocher et al. [4] and Ulanowski et al. [7] for all solid tumours as a group (see Appendix A Fig. A1). Even for the same cancer type (e.g., lung), parameters μ, S and η can be very different for different histologic types of cancer (squamous cells vs. adenocarcinoma; Table 2). Several of the studies we survey [16–18] point out that, during the observation period, no tumour growth was observed in certain patients. In the absence of treatment, this indicates that cell division rates are not necessarily constant in time. Often tumours exhibit growth along very thin dendrite-like appendages (e.g. breast cancer) [19], and this growth may not be detected by imaging techniques intended to determine tumour volume doubling time (TVDT). In such cases, reported TVDT would underestimate the true doubling rates leading to an overestimation of the predicted minimum latency.

The minimum latency adjustment function may indeed be related to the cumulative probability of a cancer being discovered by diagnostic techniques and it is reasonable to assume that this probability is associated to the size of the tumour. It is generally accepted that tumour growth is described by sigmoid curves. Such curves can be traced back to that first developed by Gompertz [20]. They were further described by Winsor [21] and fitted with tumour development data in animals (mice, rats, rabbits) by Laird [22, 23] who also attempted to extrapolate them back to one cell. Luebeck et al. [24] used a more biologically motivated framework, somewhat similar to our own. Fakir et al. [25] modelled a potential effect of tumour dormancy on latency of lung adenocarcinoma. Mathematically, it appears that several S-shaped functions, either symmetrical or asymmetrical, can satisfy observations.

Arguably the use of such a function to modulate the minimum latency of cancer after radiation exposure is in some sense plausible, since one would expect that the fact that a single tumour cell has to multiply to a point at which a tumour can be clinically detected will at least set a lower bound on the interval of latency. As such it is arguably more plausible than the sudden increase in risk after the period of latency assumed by UNSCEAR [1] and ICRP [2] in their risk evaluations. That said, it is far from clear what functional form would best describe the gradual increase in risk after exposure. Arguably the stochastic biologically-based model framework we describe is more defensible than the purely empirical functions used by Kocher et al. [4] and Ulanowski et al. [7]. However, there may be other stages between the radiation insult and the time at which the malignant cell population is of sufficient size to be detectable. Conceivably even after the mutational change associated with radiation exposure the affected cells will still have to undergo further mutational changes before becoming malignant. Such changes would be described by multistage cancer models that could be of various parametric forms [24, 26–32]. While these models do furnish a more complete description of the carcinogenic process, they are based on quite specific mathematical formulations, for example of a filtered Poisson process, and they require much more extensive data. Usually these models aim at describing cancer evolution from exposure to the first malignant cell. Most models that have been fitted do not attempt to model the final pure growth stage from the first malignant cell to development of clinical cancer, being the focus of the current work, but assume a fixed latency time. In principle one could fix the parameters of such a model using clinical data of the sort assembled here, but to the best of our knowledge this has not been done. The advantage of the stochastic biologically-based model considered here is that the parameters needed can be directly derived from published clinical data, and assumptions as to the number of detectable cells. It is more likely for leukaemia than for solid cancers that the total number of rate-limiting mutational changes is small, because of the short interval between radiation exposure and clinical detection of excess leukaemia risk, generally assumed to be in the range 2–5 years [1, 33], so that for this type of malignancy it is possible that latency really is determined by the time taken for a single cell to grow to a clinically detectable size. For solid cancers it is generally thought that the interval between radiation exposure and clinically detectable excess risk is rather longer, between 5 and 10 years [1]. There have been instances of excess thyroid risk within 5 years of exposure, in Chornobyl-exposed populations [34, 35], and as such somewhat inconsistent with the much longer minimum latency of 20 years or so implied by our calculations (Fig. 2). However, it is possible the short latency observed in Chornobyl-exposed groups may be connected with the very large thyroid radiation doses received, which can exceed 30 Gy [36, 37]. We discuss below also the implications of a large mutagenic load such as would result from high radiation dose in decreasing observed latency at a population level. It is possible that the thyroid tumour doubling times observed in two of the three studies we use [38, 39] may be too high, because in these datasets they are based on doubling times after a screening-detected tumour. It is notable that these estimates are much higher than the doubling time indicated by Kihara et al. [40] (Table 1). In general one would expect that the time from radiation exposure to clinical detection of a tumour can be decomposed as T = min[T_mal + T_grow, T_dir] where T_mal is the time taken to develop the first fully malignant cell caused by the radiation, T_grow is the time taken to grow from a single malignant cell to a clinically detectable tumour and T_dir is the time taken for partially grown malignancies at the time of radiation exposure to become clinically detectable, for example via a radiation-associated growth promotion effect, as hypothesised by Eidemüller et al. [41]. The minimum value of the latency represents the situation when the radiation exposure produces a malignant cell almost immediately (within a single cell cycle or so) so that T_mal is minimal and we may assume that T_grow >> T_mal. If we can neglect T_dir (implicitly assuming that there are no growth promotion effects) then clearly $P\left[T\le t\right]\le P\left[T_{\mathrm{grow}}\le t\right]$ from which it follows that the tumour growth probability will always dominate the latency probability. It must be borne in mind that the reported doubling times for prostate cancer in the three studies we use [17, 18, 42] are based on doubling of prostate-specific antigen (PSA) levels and not tumour volume or mass doubling, although PSA level is known to correlate with tumour volume [43, 44]. It should also be noted that in the study of Egawa et al. [17] the patients received radical prostatectomy.

Our study has a number of limitations. We do not pretend that the data underlying our derived estimates of tumour growth rates (Table 1) is anything other than illustrative; it does not represent an exhaustive survey of the clinical literature. With the exception of leukaemia and lung cancer, where studies illustrating faster and slower tumour growth rates were chosen, we selected mostly representative groups of studies for a number of major tumour endpoints, and averaged derived tumour growth rates over these studies (see footnotes to Table 2); for head and neck, oropharyngeal and hepatocellular carcinoma only single studies were used. The cancer-specific parameters derived in this study reflect data on TVDT (Table 1) in patients with diagnosed cancers, some of whom may have received treatment that reduced, at least temporarily, the cell division rates, slowing cancer growth. Another general problem with the estimates of tumour growth that we derive is that all are based on observations in clinical series of moderately well-developed tumours, where it is likely that the growth has slowed down. It is possible that the more relevant early growth rates, relating to much smaller populations of cells, could be different from these. For example, for colorectal tumours we have taken the average growth rates derived from rates of tumour growth in two clinical studies, of Bolin et al. [45] and Burke et al. [46] and a model of adenoma growth of Knudsen et al. [47]. The estimate of Burke et al. [46] was derived from doubling times of tumours that were not operated, but managed non-surgically or deemed non-resectable. The estimate from the study of Bolin et al. [45] was based on mean growth rates in colorectal tumours that were already quite large (10–90 mm) at first examination. The ‘matured’ tumours present in both studies might be expected to grow more slowly than early tumours. Arguably the model-based estimate of Knudsen et al. [47] may be a more reliable estimate of early tumour growth rates. By contrast Luebeck et al. [24] estimate a net growth rate of about 2.71/cell/year based on SEER data of colon cancer incidence, slightly lower than our estimate of 3.427/cell/year (Table 1). There is likely to be variability in tumour growth rates, reflecting variability in the sampled population, which we have not attempted to capture. As shown by the data of Appendix A Table A4 tumour growth appears to vary with age for breast cancer; this may be true of other types of cancer also, but we lack data to determine this. Such reductions in growth rates with age imply also that those exposed at younger ages would have reduced latency. It is possible that there are other endogenous variables that will cause growth rates to vary. For example, experimental data suggests that breast cancer latency can be significantly modified by irradiation of the microenvironment [48]. It has been hypothesised that radiation could act to augment clonal growth of an existing latent cancer, leading to latency reduction [41]. The ICRP has highlighted the interplay between latency, the likely number of mutations, and the shape of the dose response curve, pointing out the distinction between endpoints such as leukaemia with curved dose response and very short minimum latency, compared with many solid tumour sites where the number of mutations is probably greater, and the dose response tends to be more linear [49].

Another approach to estimating latency is that adopted by Richardson and Ashmore [50] who fitted a variety of models allowing for smooth variation of latency to Canadian nuclear worker mortality data. They assessed latency for lung cancer, leukaemia and all other cancers. For lung cancer the evidence suggested a latency of about 13 years, although very similar fit was obtained with latencies between 5 and 20 years [50]. For leukaemia and all other cancers latencies of 0 and 5 years were suggested [50]. Daniels et al. [51] assessed latency in the International Nuclear Worker study (INWORKS) mortality data by fitting simple relative risk models with discrete (0/1) latency, with latency estimated using a grid search method; rather large values of latency were estimated for leukaemia, of 19 years, and much shorter estimates for solid cancers, of 3 years. Hauptmann et al. [52] assessed latency in relation to smoking in a German case-control study of lung cancer incidence by fitting cubic B-splines. Hauptmann et al. [52] found suggestions of latency of between 3 and 11 years. However, the evidence for latency is quite weak in all three studies, and assessment is complicated by the fact that latency is bound up with variations in risk with time since exposure.

The results of Appendix A Fig. A3, showing that when the cell death/differentiation rate D becomes increasingly positive then the asymptotic limit (for large time after the initial tumour cell) to the cumulative probability of detection is progressively reduced from 1, is as one would expect. When D is increasingly positive there is an increased chance that a given tumour cell and the cells that grow from it will all eventually die or differentiate, so that the probability of not developing a clinical tumour from a given tumour cell must increase. It is not clear how relevant this phenomenon may be, but there is certainly evidence of periods of growth stagnation among certain cancer types, e.g., breast [17]. Luebeck et al. [24], who developed a stochastic model explicitly accounting for the final growth of the first malignant cell, also discussed the phenomenon of stochastic extinction, but at stages before the final tumour growth stage.

Kocher et al. [4] used excess relative risk (ERR) based risk models from the LSS and other studies, but they applied their latency function to modify both the ERR (multiplicative) and excess absolute risk (EAR) additive projections of risk to the US population. Similarly, Ulanowski et al. [7] have used the specific type of latency function to model both ERR and EAR. For leukaemia and squamous cell lung cancer Fig. 1 demonstrates that the cumulative tumour detection probability operates over a fairly short timescale so that it would not be expected to make much difference which measure of risk (ERR, EAR) is chosen. For cancer sites which have a more extended period over which the tumour is clinically detected, in particular prostate and thyroid cancer (Fig. 2), the cumulative detection probability, which most naturally relates to absolute risk, would be somewhat modified by the likely variations in underlying cancer risk over time [1], so that the latency function that results could be of different form on relative and absolute risk scales.

The minimum latency adjustment functions discussed here are representative for risk of cancer incidence and should not be directly used in calculations of risks of cancer mortality. Persons diagnosed with cancer may survive several years even in the absence of treatment, and survival duration increases with the type and quality of treatment. Given the advances in cancer survival over the last 30 years [53], the percentage of patients with 5 year survival is now for many types of primary cancer appreciably more than 50%, although that is not the case for lung, liver or oesophageal cancer where 5 year survival is still less than 25% [53]. For many types of cancer (but not lung, liver or oesophagus), the minimum latency for cancer mortality is therefore expected to be longer by several years than the minimum latency for cancer incidence. Thus, minimum latency adjustment functions for cancer mortality should be shifted to longer times after exposure than those for cancer incidence.

The results we present relate to any mutagen acting over a fairly short period. In principle the calculations extend straightforwardly to fractionated mutagenic exposures, which would simply result in the superposition of various latency functions, one per exposure. As such the model we present would imply no variation of latency with for example radiation dose or dose rate, as long as the mutagenic increments were over a fairly limited interval. Mutagenic increments spread over a period of many years would result in a spreading out of the latency function, which would increase the interval from the start of exposure to detection of cancer, implying an extension of the latent period. Raabe [54] has argued from a large body of radiological animal data that for localised exposure from internal emitters the extension could be substantial enough to effectively extend the latency beyond the lifespan of the animal. Our results imply that for certain slower-growing tumours such as thyroid and prostate this may happen for any type of protracted mutagenic exposure (Fig. 2). Equally, our model would not imply any variation in the individual person in latency with level of mutagenic load, so that for example the very substantial mutagenic increment associated with radiotherapy would in principle lead to the same latency as a much lower level of insult. However, higher levels of mutagenic load increase the probability of mutation, implying a much larger increment in the excess cancer risk, the detection of which in a population would occur much sooner than a lower level of mutagenic load, implying some effective reduction of latency at a population level. Models have been developed that describe cell development in tissue exposed at high fractionated doses as repeated cell killing and repopulation, effectively accelerating the growth of premalignant cells to cancer [55]. Such processes might result in a reduction of latency. Further research is needed to clarify the effect of radiotherapy exposures on latency.

It should be noted that the rates of normal tissue cell turnover that have been estimated, for example by Tomasetti and Vogelstein [56] are quite different from the rates of tumour growth. We illustrate these in Appendix A Table A5. As can be seen, while for some endpoints (e.g. hepatocellular carcinoma, bone marrow) the rates are quite similar, for many others they differ, by up to an order of magnitude. That this should be so is not altogether surprising, since what drives normal tissue turnover is quite different from what may be driving the turnover in a population of malignant cells.

For all cancers we have generally assumed (e.g. in Table 2) that the limiting minimum number of cells required for detection N_lim = 10⁹, following a suggestion of DeVita et al. [8]. However, some have argued for an estimate of about N_lim = 10⁸, based on arguments about cell geometry and packing density [9], and for this reason results for smaller values of N_lim are also given in Fig. 2. For prostate cancer and thyroid cancer a much smaller value of N_lim = 10⁶ was employed, as a result of the analysis given in Fig. 2. Ultrasound screening of the thyroid can easily detect tumours <5 mm [57], so that based on Appendix A Table A2 it is not implausible that this number of cells can be detected. The results given by Del Monte [9] assumed that the cells of a given diameter were cuboid rather than spherical, which is clearly biologically implausible. Assuming a spherical shape for the cells and the tumours and an optimal packing density the numbers of cells in a tumour would be expected to cover a much wider range, from 2.7 × 10⁴ to 5.9 × 10⁹ (Appendix A Table A2). Arguably these numbers are still too high, since cells would not generally be packed in the optimal 3D hexagonal lattice formation. At least for lung tumours there is data suggesting that numbers could be somewhat larger than this range. Information from clinical data on 21 non-small cell lung tumours at time of radiation treatment supplied to us by Dr Hannes Rennau [58] suggest a mean tumour volume of 187.6 cm³ (range 29.2–630 cm³), implying (with the range of packing densities shown in Appendix A Table A2) a number of lung tumour cells in the range 1.2 × 10⁹–7.2 × 10¹¹.

It is not altogether clear how the radiosensitivity of tissues necessarily relates to the latency of associated tumours. So for example for exposures in childhood thyroid cancer is highly radiogenic, not much less so than leukaemia [1], but the latency would appear (using the data we have assembled) to be much longer than for leukaemia (Table 1 and Figs. 1 and 2). On the other hand prostate cancer is clearly not very strongly radiogenic [1, 59] and the latency that we predict is among the longest (Table 1 and Fig. 2).

In summary we have shown that an S-shaped form of the latency functions proposed by Kocher et al. [4] and Ulanowski et al. [7] are consistent with a more biologically motivated approach presented here. Arguably such an approach should be preferred to one based on a fixed lag time. It is clear that the cumulative probability of tumour detection, which has some bearing on the minimum latency time, is likely to have substantial variation resulting from the differential speed of tumour growth for different organs, indeed within specific sub-types of cancer (Table 1). Our analysis indicates that both prostate and thyroid cancer might have a longer minimum latency time than other cancers (Fig. 2), possibly exceeding 20 years, particularly if the number of limiting cells N_lim is much larger than 10⁶. It is possible that because prostate cancer is generally detected via a prostate antigen test that the number of limiting cells required for detection is somewhat less than other tumours. This may also have a bearing on the weakness of evidence of radiation association for prostate cancer [1]. This topic would benefit from further investigation.

Although latency has been considered in relation to radiation-associated cancer, the results we derive, based on clinical data that is not tied to any particular carcinogen, have implications for latency associated with any carcinogenic mutagen. However, they would not apply to carcinogens that act solely via increases in cell turnover, which are known to be involved in certain tumours [60]. Reported results are applicable for tumour growth in general, and may be useful for example, for planning screening for cancer using medical imaging techniques. Minimum latency times are very relevant for lifetime risk assessments and future research on this topic would be important.

Author contributions

MPL conceived and designed the study, performed the analysis and assembled the first draft. MPL, AIA, ME and JCK performed literature searches and assembled the analytic database. All authors contributed equally to the writing and editing of subsequent drafts. All authors approved the final draft.

Data availability

All data used are given in Table 1, also in Appendix B, and are derived from the peer-reviewed literature.

Code availability

The various calculations used are given in an Excel spreadsheet, provided in Appendix B.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material, including Appendices A and B, and is available at 10.1038/s41416-023-02544-z.