Theoretical derivation and clinical dose-response quantification of a unified multi-activation (UMA) model of cell survival from a logistic equation

Objective: To theoretically derive a unified multiactivation (UMA) model of cell survival after ionising radiation that can accurately assess doses and responses in radiotherapy and X-ray imaging. Methods: A unified formula with only two parameters in fitting of a cell survival curve (CSC) is first derived from an assumption that radiation-activated cell death pathways compose the first- and second-order reaction kinetics. A logit linear regression of CSC data is used for precise determination of the two model parameters. Intrinsic radiosensitivity, biologically effective dose (BED), equivalent dose to the traditional 2 Gy fractions (EQD2), tumour control probability, normal-tissue complication probability, BED50 and steepness (Γ50) at 50% of tumour control probability (or normal-tissue complication probability) are analytical functions of the model and treatment (or imaging) parameters. Results: The UMA model has almost perfectly fit typical CSCs over the entire dose range with R2≥0.99. Estimated quantities for stereotactic body radiotherapy of early stage lung cancer and the skin reactions from X-ray imaging agree with clinical results. Conclusion: The proposed UMA model has theoretically resolved the catastrophes of the zero slope at zero dose for multiple target model and the bending curve at high dose for the linear quadratic model. More importantly, it analytically predicts dose–responses to various dose–fraction schemes in radiotherapy and to low dose X-ray imaging based on these preclinical CSCs. Advances in knowledge: The discovery of a unified formula of CSC over the entire dose range may reveal a common mechanism of the first- and second-order reaction kinetics among multiple CD pathways activated by ionising radiation at various dose levels.


INTRODUCTION
For over a century, radiobiologists, medical physicists, and clinicians who use ionising radiation for medical imaging and therapy, have been searching for a simple formula to accurately describe the intrinsic radiosensitivity of cell lines to various doses or the cell survival curves (CSCs). A single-hit multi target (MT) model by Lea to describe the shoulder of the HeLa CSC to X-rays, where m is the number of hits per target site. Both the single-hit and multiple-hit MT models have the zero slope at the zero dose that conflicts with constant slopes of observed CSCs to high linear energy transfer (LET) radiations. A two-component MT model by to define a non-zero initial slope by the third parameter k 1 which has also complicated its theoretical interpretation and clinical application. A linear-quadratic (LQ) model 5 of y = −ln ( S ) = αD + βD 2 for the yield of chromosomal aberrations has a theoretical derivation by Kellerer and Rossi 6 to correlate the linear and quadratic terms to the intra-and intertrack actions, respectively. The target size is estimated by Zaider and Rossi 7 to be the size of DNA-helix. A molecular theory of cell survival by Chadwick and Leenhouts 8 has also concluded "that the induction of DNA double strand breaks (DSB) should be linear-quadratic". The LQ model became the clinical standard in assessing radiosensitivity and radiocurability in radiology and radiotherapy with its conspicuous advantages of simple formulation and plausible interpretations but the bending curve discrepancy from the straight portion of many CSCs at high doses has never been forgotten. An early attempt is the lethal and potential lethal (LPL) damage repairing model https:// doi. org / 10. 1259/ bjro. 20210040 Objective: To theoretically derive a unified multi activation (UMA) model of cell survival after ionising radiation that can accurately assess doses and responses in radiotherapy and X-ray imaging. Methods: A unified formula with only two parameters in fitting of a cell survival curve (CSC) is first derived from an assumption that radiation-activated cell death pathways compose the first-and second-order reaction kinetics. A logit linear regression of CSC data is used for precise determination of the two model parameters. Intrinsic radiosensitivity, biologically effective dose (BED), equivalent dose to the traditional 2 Gy fractions (EQD2), tumour control probability, normal-tissue complication probability, BED 50 and steepness (Γ50) at 50% of tumour control probability (or normal-tissue complication probability) are analytical functions of the model and treatment (or imaging) parameters.
by Curtis 9 to extrapolate a CSC as an exponentially straight line at high dose but its four parameters and five assumptions prohibits it from clinical usage. Recently, advances of hypofractionated radiotherapy have encouraged investigators to develop an effectively linear-quadratic-linear (LQL) model by methods of: (1) changing the β parameter as a function of time and repairing-rate constants 10 ; (2) adding a function of dose shift and constants 11 ; or (3) combining the LQ model at the low-dose domain, the MT model at the highdose domain, and a transition dose point between the two dose domains for a universal survival curve (USC) model. 12,13 However, these LQL models are either inconvenient to use clinically as they involve complicated functions and difficult to explain theoretically for mechanism changes at different dose levels, or potentially uncertain to correlate outcomes with more parameters. A unified model of CSCs over the entire dose range with a common mechanism is desired for clinical applications and theoretical extensions to very low and high doses that have perplexed us for a long time. 14,15

METHODS AND MATERIALS
Derivation of a unified formula There are millions of possible chemical reactions from thousands of different molecules in a cell but only these sped-up by available protein enzymes takes place at measureable rates. Radiationinduced DNA damages and free radicals may trigger some biochemical reactions that would lead cell death (CD) through some CD pathways (or cell dying processes) with distinct morphologies or molecular mechanisms. 16 The radiationinduced reactions are assumed to be the first-and second-order reaction kinetics in the cellular scale. Based on the evidences that ionising radiation produces free radicals and DNA damages at a rate proportional to the dose rate, e.g. ~40 DSBs/Gy from X-ray radiation, the first-order reactions would depopulate the cells of interest at a rate of γḊN , where γ, Ḋ and N are the apparent firstorder-reaction activation constant per unit of dose, dose rate and number of cells in the region of interest, respectively. The exponentially straight CSC to α particles is an example of CD through the first-order cellular kinetics. The second-order reactions would depopulate the cells at a rate of −dS/S 2 = δNodD , where δ is the apparent second-order-reaction activation constant per unit of N per unit of dose. The second-order kinetics has dominated the renaturation of isolated fragments from mammalian DNA. 17 The combination of the first-and second-order reactions in the dose domain results where the ratio of γ/δ is the "carrying capacity" in the dose domain in unit of N. Eq. (1) is similar to the logistic growth of cells 18 but converted the CD in the dose domain.  (2) where n = γ γ+δN o is the ratio of the first-order reaction rate of γḊ to the total reaction rate of ( γ + δN oḊ ). The γ, δ and n are assumed to be dose-independent in the derivation.
A modified logit linear regression is useful to precisely determine the parameters γ and n in the unified formula. A parameterised logit function of ln

Model interpretations and definition of intrinsic radiosensitivity
The unified formula of Eq. (2) represents a CSC over the entire dose range that is derived from the apparent first-and secondorder reaction kinetics for the cells of interest undergoing through multiple CD pathways activated by ionising radiation. Thus, it is a unified multi activation (UMA) model. A positive γ would result a net CD (S < 1) from the first-order reactions while a negative γ may result a net cell growth (S > 1) or CD (S < 1) depending upon the n. There are five mathematical deductions for variations of n.
allows only γ < 0 and S ∈ . Formula (A5) γ δN o = n 1−n > 0 requires that γ and δ have the same sign. A negative γ with n ∈ ( 0, 1 ) is impossible to yield a negative survival of S = n n−1 < 0 at any large doses. Positive γ with n ∈ ( 0, 1 ) have been obtained for typical CSCs with inverted shoulders. Hyperfractionated radiotherapy is desired for tumour cell lines with inverted shoulders. 19 BJR|Open Original research: The theory and application of a UMA model of cell survival That is CD or cell growth from the first-order reactions only.
1−n < −1 requires that γ and δ have opposite signs. Typical tumour CSCs to X-rays have γ > 0 for CD through the first-order reactions and δ < 0 for cell growth or damage repairing through the second-order reactions. If there were a γ < 0 for cell growth from the firstorder reactions and δ > 0 for CD from the second-order reactions, there would be a net cell growth with S ∈ In summary, excluding the initial condition of S(0) = 1, γ > 0 exists at n > 0 with S < 1; γ = 0 exists at n = 0 with δ > 0 and S = 1/(δN o D + 1) < 1; γ < 0 mayt exist at n < 0 with S ∈ ( n n−1 , 1 The UMA model predicts that it is possible for S > 1 with γ < 0 and n ≥ 1 at all dose levels of a CSC. But S > 1 has only been observed on some low dose points of a few CSCs and the UMA model fitting of these CSCs have resulted γ > 0 and the regression residuals on these low doses are within their experimental errors. Thus, positive γ is presumably used for all clinical applications of the UMA model. The unified formula can analytically quantify the intrinsic radiosensitivity (RS) of any cell lines from the negative derivative of natural log survival to the dose as The initial RS at the zero dose is γ/n, corresponding to the α parameter in the LQ model. At a very low dose of γD ≪ 1 , RS approximates to γ/ . At a high dose of γD ≫ 1 , RS approximates to the γ and cell survival approximates the straight line of S ≈ ne γD that is the same as the MT model with γ = 1/D o but the γ and n are theoretically the first-order-reaction activation constant and the ratio of the first-order-reaction rate to the total reaction rate, respectively. The UMA model describes CSCs over the entire dose range with one mechanism and have resolved the catastrophes of zero slope at zero dose from MT models and of the bending curve at high dose from LQ model. Since RS changes with dose, it is reasonable to explore the new UMA model for global radiosensitivity (GRS) of a cell line to a type of radiation. For which, the reciprocal of the mean inactivation dose introduced by Fertil et al 20 is applied for The dose equivalent to 2 Gy fractions (EQD2) has been previously derived 19 as For a tumour or an organ consisting heterogeneous cell lines, the dominate radioresistant cell line to the tumour and radiosensitive cell line to the organs at risk are selected for computing the tumour control probability (TCP) and normal tissue complication probability (NTCP) at a follow-up time, respectively. Poisson distribution to approximate the binomial distribution of death or survival of cells gives where the target cell number (or concentration) N o can be derived from clinical data, e.g. SBRT of early stage lung cancer having N o from 10 4 to 10 6 . 21 Note that NTCP is defined in the same way as that of TCP because they share the same sigmoid shape of dose-responses but having different N o and RS. An example of the NTCP calculation for skin reactions will be provided in the result section.
The total dose to achieve 80% TCP or 20% NTCP can be estimated by TD 50 +0.3/Γ 50 for the tumour or TD 50 -0.3/Γ 50 for the normal tissue, respectively.
The fraction dose for changing the course from D-Gy m-fractions to i-fractions with the same ending survival fraction of is given by Figure 2. The modified logit function with listed parameter of A has transformed individual CSCs into straight lines and almost perfect fitting (R 2 >0.99) as shown in Figure 2a. Both UMA and LQ models have explained over 99% variability of cell survivals by the dose in the experimental dose range, but the LQ model (dashed lines) and the UMA model (solid lines) differ systematically at the doses higher than the experimental range in Figure 2b. The fitting of other 33 CSCs of human cancer cell lines for typical sites of hypofractionated radiotherapy results γ from 0.1 to 2 Gy −1 and n from 0.2 to 60, respectively. 19 The dose-independent γ and n as well as δ have confirmed the correct assumption in the derivation of Eq. (2).

The UMA model fitting of CSCs for human kidney cells (HKC), Chinese Hamster cells (CHC), human Hela cells (HC) and mouse bone marrow cells (MBM) to X-rays with marked data points redrawn from Figs. 3.3 and 3.12 in Hall's text book 23 is presented in
To anticipate the skin reactions to low doses from diagnostic X-ray imaging, 24,25 the UMA model is applied to CSCs of human skin fibroblast cells by Weichselbaum et al 26 Figure 3a and b, respectively. Skin intrinsic RS at zero dose defined by γ/n of the skin cell lines are from 0.34 to 1.14 Gy −1 that are even higher than the initial RS of 0.075/Gy to the squamous cell carcinoma. 19 More importantly, skin RS increases with dose in diagnostic imaging (<0.1 Gy) as predicted by RS

and stem cells by Schröder et al 27 as plots in
] for any skin cell lines with n > 1 to X-rays. Different γ values of 0.67, 1.08 and 2.27 Gy −1 to the fibroblasts from a normal person, a patient with D-deletion retinoblastoma and a patient with ataxia telangiectasia, with the same of n = 2, indicate that the first-order reaction activation constant not the reaction ratio varies significantly among patients with genetic alterations. Recent imaging of human bulge cell morphology with molecular markers has greatly improved our understanding of the mechanism of hair loss. 28 The temporary alopecia is likely come from the destruction of stem cells in calculations) are selected based on a recent multiple institutions' data and models' study of Liu et al. 21 The UAM model results clearly demonstrate followings: (1) the higher γ is, the lower S(10 Gy) gets since the first order reaction constant γ determines the cell survival at the high dose as S ( D ) ≈ ne γD ; (2) if fractional dose is within the straight portion of these CSCs, changing fraction number from 5 to 3 results the same dose of ~15 Gy per fraction for all of cell lines; (3) squamous cell carcinoma (SW1573) has the lowest RS(0) and BED 50 but the highest GRS, EQD2, TCP and Γ 50 mainly due to its large n or more repairing through the second-order reactions as δ = −γ ; and (4) the large-cell carcinoma (HX147) has the lowest GRS (not the initial RS), EQD2, TCP and Γ 50 but the highest S(10 Gy), D 50 and BED 50 due to its small γ and n. The large cells are ~2 times larger than the sizes of other cells and the same detectable gross tumour volume may contain ~10 times less number of the tumour cells. Having N o = 10 4 instead of 10 5 , the TCP is increased from 26.5 to 87.6% and D 50 is reduced from 10.5 to 8.9 Gy, respectively. TCP and BED 50 for the large cell carcinoma (HX147) are still worse than that for SBRT of other types of the lung cancer. Thus, SBRT of early large cell carcinoma may require more dose than other types of the lung cancer. The impact of N o on TCP, D 50 , BED 50 and Γ 50 are shown in Table 1. TCP exponentially decreases with N o as expressed in the formula (7). D 50 (or TD 50 ), BED 50 and Γ 50 are all increasing with N o as expressed in formulas (8) and (9). N o increase from 10 4 to 10 6 results significant changes of D 50 and BED 50 but less change on Γ 50 for all cell lines shown in Table 1 for the 5-fractional SBRT of lung cancer cell lines. An important conclusion is that if the BED, specified to the cancer cell line and the selected course of treatment, is greater than the tumour BED 50 by roughly 1/Γ50, complete tumour control is expected regardless to the tumour size (or number of cancer cells). Otherwise, the TCP would vary with the total number of cancer cells.
All of values in the table are reasonable predictions based on the UMA modelling of preclinical CSCs. Thus, the new UMA model are useful in the design of a new treatment scheme for radiotherapy of cancer.

DISCUSSIONS
A simple unified formula of CSC using only two doseindependent parameters of γ and n is obtained based on an assumption that radiation-activated first-and second-order reactions through multiple cell dying processes (or CD pathways) apparently satisfy a logistic equation in the dose domain. The γ Ḋ is the first-order reaction rate constant while the n is the ratio of the first-order reaction rate to the total reaction rate. The UMA model and its two parameters are conceptually different from that of the traditional MT and LQ models. More importantly, the UMA model has almost perfectly fit typical CSCs over the entire dose range and completely resolved the catastrophes of the zero slope at zero dose from MT models and the bending curve at high dose from the LQ model. The parameters γ and n could be precisely determined from a modified logit linear regression of in-vivo or close-to in-vivo CSC and the dose-independent parameters allows analytical predictions of dose-response quantities such as RS, GRS, EQD2, BED, 29 TCP, NTCP, BED 50 , D 50 and Γ 50 at any dose levels. Thus, it could hold broad implications for clinical practice, particularly for alternative fractionation schemes or new therapeutic indications such as SBRT of multiple oligometastases 30 by using the same biochemical mechanism with no modification of the parameters with low and high dose levels that differ from current practices using LQ or LQL models. [10][11][12][13]31 The UMA model with a common mechanism for all dose levels is also useful for the risk assessments of low dose radiological procedures and radioprotection that still uses contradicted models. 15 It is true that the UMA model have several facets to be explored such as the synergistic effects when combined with chemotherapy, hyperthermia, immunotherapy, radiosensitisers for the tumour cells, and/or radioprotectors for normal tissue. The dose-rate effects such as in ultra-high dose rate in FLASH radiation and blood supply changes during a course of the treatment may also influence the cell responses, biochemical reactions, or repairing processes. The UMA biochemical modeling of CD pathways is principally applicable to describe cell dying processes activated by other agents. But the continuous or pulsed activation by an agent in chemotherapy and long recovery time from hyperthermia may affect the dynamics of the cells of interest. Both theoretical and experimental investigations are required for the model extension.

CONCLUSIONS
The successful UMA model of a CSC over the entire dose range reveals possibly a common mechanism -the first-and secondorder reaction kinetics in cellular scale has integrally represented multiple compounding CD pathways activated by ionising radiation. Such a common mechanism allows us to analytically and reasonably quantify the clinical-interested doses and responses in radiotherapy and radiological imaging procedures by the UMA modeling of preclinical in-vivo CSCs.