The LQ model in radiobiology

In addition to the applications of simple ODE models to chemotherapy, exemplified to some extent by the previous section, a formalism equivalent to a pair of ODEs has been used very heavily during the last decade in radiotherapy modeling and in studying damage to cells by ionizing radiation. This formalism is the linear-quadratic (LQ) model [19]. A distinctive feature of radiobiological damage is that damage repair and misrepair occurring during or after irradiation play a key role. The LQ model concisely quantifies the effects of both unrepairable damage and repairable damage susceptible to misrepair.

Ionizing radiation consists of high-energy particles such as photons or ions, whose tracks deposit energy in a cell [20, 21]. Ionizing radiation dose, $D$, is measured in energy per unit mass, with 1 Gy being 1 Joule/kg. The time dependence of dose is much more precisely defined than in chemotherapy, as pharmacokinetic or other accessibility issues do not really arise. A complication in some cases is the stochastic nature of radiation (survey in [20]). However most clinical applications involve sparsely ionizing radiations (such as $\gamma$-rays or hard x-rays) at doses above 1 Gy; in these cases so many independent particles (e.g. photons) strike a cell nucleus that dose, a deterministic quantity by definition, suffices to quantify the insult. We shall here always suppose irradiation occurs within the finite time interval $[0,T]$. $D(t)$ denotes accumulated dose at time $t$. Thus $R (t) \equiv dD/dt$ is radiation dose rate, here assumed smooth.²

The most important radiation damage is to chromatin, e.g. DNA double strand breaks (DSBs). An acute dose of 1 Gy makes many thousands of ionizations in the cell's nucleus, of which a small minority, $\approx$40 in a human cell, quickly induce DSBs. Most DSBs are repaired during the next half hour or so, and a few are misrepaired. Many of the misrepairs involve a binary reaction between two different DSBs [22]. At a typical dose, of several Gy, at least one misrepair usually occurs, which can be enough to kill the cell at the next mitosis. Other kinds of clonogenically lethal lesions, formed "directly" by irradiation and not subject to repair or misrepair (e.g. lethal point mutations) can also be present.

The LQ model has been derived from many different points of view, and is actually more robust than any one of its derivations. It has been applied to cell killing, chromosome aberration formation, mutation, transformation and other endpoints. For brevity we here confine attention to cell killing in the sense of clonogenic inactivation, and to tumor radiotherapy. The applications to endpoints more directly related to carcinogenesis, such as chromosome aberrations, are, broadly speaking, similar. One derivation of the model starts from a standard pair of ODEs [23], as follows. The dependent functions are the average number $U(t)$ of DSBs per cell at time $t$ and the average number $N(t)$ of cells per population (i.e. per tumor or cell culture). The ODEs are:

$$dU(t)/dt = \delta R - (U / \tau ) - \gamma U^2 , \qquad \delta , \tau , \gamma = const. \geq 0 ;$$ (8)

$$dN(t)/dt = -(\alpha R + \kappa U^2 )N ; \qquad \alpha , \kappa = const. \geq 0 , \qquad \alpha \leq \gamma / 2 .$$ (9)

Here $\delta$ is the average number of DSBs induced per unit dose, $\tau$ is a repair time constant, $\gamma$ is a binary reaction rate constant in the sense of mass-action chemical kinetics, $\alpha$ is the average number of unrepairable lethal lesions produced per unit dose directly by the radiation, and $2 \kappa$ is the rate of lethal misrepair of DSBs per DSB pair. In general these constants depend on cell-type, radiation type, experimental conditions, etc. The term involving $\alpha$ is appropriate for radiation killing which is due to "1-track action" (e.g. does not involve interaction of DNA double strand breaks made by two different radiation tracks). These ODEs, involving averages, are only appropriate when either stochastic fluctuations are negligible or Poisson distributions hold for DSBs per cell and for cells per population [24]. Growth of the cell population has been neglected, which is appropriate for short time scales; for longer time scales growth has been incorporated in various ways, most simply by adding a term $\lambda N$ to the right hand side of Eq. 9

It often happens, for doses less than about 5 Gy, that one can approximate by setting $\gamma = 0$ in Eq. 8 (however, the related term involving $\kappa$ may be important in Eq. 9 even when the binary, $\gamma$, term is negligible compared to the other terms in in Eq. 8). If $\gamma = 0$, Eq. 8 can be explicitly integrated. For times greater than the radiation stoppage time $T$ and large compared to the repair time $\tau$, i.e. for $N(\infty)$, which we abbreviate by $N$, inserting the integral of Eq. 8 into 9 and integrating again gives the LQ model. Specifically the result of the two integrations is:

$$\ln (N/N_0) = - \alpha D - \beta G D^2 , \quad \quad \beta \equiv \delta ^2 \kappa \tau /2 .$$ (10)

Here $N_0$ is the number of cells present initially, $D = D(T)$ is total dose, and $G$ is the generalized Lea-Catcheside dose-rate functional, given by [25 - 28]

$$G = 2 \int_0^T f(t) dt \int_0^t e^{ - (t-t ' )/\tau } f(t ' ) d t ' , f(t)= R (t)/D.$$ (11)

Note here that for acute irradiation $G = 1$, but for irradiation prolonged in any way $G < 1$ since the kernel $\exp [- (t-t ' )/\tau] \leq 1$ for $t' \leq t$. Intuitively speaking, $G < 1$ is due to repair which occurs during irradiation or between dose fractions.

The most essential, virtually ubiquitous features characteristic of ionizing radiation damage, captured here in Eqs. 10 and 11, are the following: part of the damage is dose-rate independent and (approximately) linear in dose (term involving $\alpha$); part of the damage corresponds to misrepair of repairable damage (term involving $\beta$); this part gives rise to effects super-linear in dose (here $\beta G D^2$); and this part is decreased by prolonging the dose instead of giving dose all at once. The fact that the LQ model quantifies these normally dominant effects, and, equally important, neglects or slurs over a host of less important effects so that only three adjustable parameters are involved, accounts in large part for its widespread recent popularity. At worst, the model summarizes, in a highly condensed quantitative form, a truly enormous body of radiobiological data and clinical experience.

The detailed properties of the repairable damage component have long been controversial, and a wide variety of ODE models other than Eqs. 8 and 9 have been proposed (survey in [24]). For example, many authors have suggested saturation of repair enzymes rather than binary misrepair as an underlying molecular mechanism [29]. Remarkably, however, first order time-dependent perturbation theory shows that the LQ model, Eqs. 10 and 11, is a low-dose/intermediate-dose approximation to virtually all the different ODE models [30]. There are presumably many different damage repair/misrepair pathways simultaneously operative in a cell, but the LQ model approximates most of them.

The model has become the tool of choice for quantifications of treatment planning in tumor radiotherapy (recent examples include [31 - 35]); often in these applications the model is combined with a tumor growth model, e.g. [31, 36, 37]. The LQ equations, and a wide range of ODE generalizations to approximate such additional aspects as cell cycle kinetics or cell population heterogeneity, are also now heavily used for analyzing cell killing in experimental radiobiology (recent surveys include [38, 39]). The generalizations, e.g. [40], are typically somewhat less practical in that they involve extra adjustable parameters.