Chemotherapy-induced cachexia and model-informed dosing to preserve lean mass in cancer treatment

Although chemotherapy is a standard treatment for cancer, it comes with significant side effects. In particular, certain agents can induce severe muscle loss, known as cachexia, worsening patient quality of life and treatment outcomes. 5-fluorouracil, an anti-cancer agent used to treat several cancers, has been shown to cause muscle loss. Experimental data indicates a non-linear dose-dependence for muscle loss in mice treated with daily or week-day schedules. We present a mathematical model of chemotherapy-induced muscle wasting that captures this non-linear dose-dependence. Area-under-the-curve metrics are proposed to quantify the treatment’s effects on lean mass and tumour control. Model simulations are used to explore alternate dosing schedules, aging effects, and morphine use in chemotherapy treatment with the aim of better protecting lean mass while actively targeting the tumour, ultimately leading to improved personalization of treatment planning and improved patient quality of life.


Introduction
2-off schedule. Mortality was reported for the daily schedule in 4/5 mice receiving 35 mg/kg. 123 For the 5-on, 2-off schedule, mortality was reported in 5/5 mice receiving 60 mg/kg and 3/5 124 mice receiving 50 mg/kg. 125 The experimental data contains some unique features of the dose response. First consider 126 the daily schedule shown in Figure 1(a). Note that there is a striking lag in mass lost over the 127 first 10 days. This lag in effect is not a true delay of response, however, because there is no mg/kg seem to all approximately track with the control, indicating no significant mass loss.
Of interest is that after 28 days of treatment, the 50 mg/kg group, which lost considerable 138 mass, recovers quite quickly and immediately with 2/5 mice surviving to the end of the 139 experiment, whereas the 60 mg/kg group did not last the full experiment.   This models the kinetic behaviour of the chemotherapy drug concentration in the plasma, 152 C 1 (t), and tissue, C 2 (t), compartments [39,40]. The pharmacokinetic equations are: Drug concentrations C 1 and C 2 are measured in µg/ml for each compartment. V 1 and V 2 are 154 the corresponding distributed volumes of the drug after one dose in each compartment (in 155 ml/kg). The drug dosage d is measured in µg/kg and administered instantaneously over the 156 set of treatment times t i ∈ T by the dirac-delta train δ(t − t i ). The rate constants k 12 and 157 k 21 (in days −1 ) describe the transfer of drug between the plasma and tissue compartments. 158 The drug clearance rate is described by k 10 in days −1 .

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To simulate the dosing schedule, we interrupt the computational simulation at each treat-  partments as volumes measured in mm 3 . The stem cells are assumed to divide symmetrically to produce either two stem cells with probability p, or two differentiated muscle cells with 170 probability 1 − p. The model equations for a healthy muscle tissue are thus: Here p(t) is the probability of satellite cell self-renewing division, ν(t) is the proliferation 172 rate, and d 0 is the natural muscle cell death rate.

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Feedback from the muscle compartment to the satellite cell compartment is required to 174 regulate tissue growth and regeneration by decreasing the probability of self-renewal for large 175 muscle sizes. We describe this feedback mathematically as: where p 0 is the base probability and p 1 is a perturbation probability activated in early 177 growth or after muscle damage. Parameter m is a half-saturation coefficient for the feedback 178 dynamic of muscle mass M (t).

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The satellite cell proliferation rate, ν(t), also has a feedback mechanism that reduces 180 proliferation as the muscle nears its equilibrium size: where ν 0 is the base proliferation rate and ν 1 is a perturbation proliferation rate activated in 182 early growth and after tissue damage. Parameter m is again a half-saturation coefficient for 183 the feedback dynamic of muscle mass M (t), assumed to be the same as in the probability 184 p(t) for simplicity.

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In order to compare our model to lean mass measurements in grams, we convert these 186 volumetric predictions to mass via the formula: lean mass = 0.002(S +M ) g. thermore, mitochondrial dysfunction can lead to senescence and stem cell aging [26]. Thus, 206 we can explore potential dysfunction of our stem-cell lineage model of muscle tissue due to 207 the applied chemotherapy's effects on mitochondria and thus stem cell regulation.

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As a simple first assumption, we assume that chemotherapy can slow the successful pro-209 liferation rate of stem cells. Biologically this mimics the effects of dysregulated mitochondria 210 (and thus the dysregulated stem cell actions) of cells exposed to 5-FU. Further, if the dif-211 ferentiation program is disrupted by the drug, then progenitor cells will die out rather than 212 form new muscle cells, impairing the regeneration cascade. We model these mechanisms 213 of mass loss as a reduction in successful proliferation and differentiation, and propose the 214 following modification to the muscle tissue model: The chemotherapy effect is included via a cubic function of a moving time-averaged expo-216 sure function y τ (t). Parameter R d (in µg/ml) is a thresholding constant to scale the cubic 217 function. The average expose y τ is defined as This τ -average exposure of the tissue to drug concentration C 2 (t) is assumed to cause the 219 disruption to muscle homeostasis. The average over τ -days is used to slow the drug's response 220 dynamics and to capture the initial delay in mass loss observed under the daily schedule.

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The cubic functions in equations (7) and (8) Here η = 20 controls the transition from the exponential growth phase to the linear growth 231 phase, and is sufficiently large enough to make the transition near-instantaneous. Parameter  Table 1.    and R d is fit using a grid search method. Then the value of τ is increased and the process is 265 repeated. Integer values of τ were tested over [1,15] days.

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The best fit for each dose schedule is shown in Figure 3. The daily schedule has a minimum 267 SSE when τ = 11 and R d = 6.18. The long value of τ here helps the model capture the lag in

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To explore the sensitivity of the two new model parameters R d and τ , we computed dynamic 283 sensitivity analyses. First, we found the resulting lean mass response following a 10% increase 284 or decrease to parameters R d = 6.8 or τ = 8. Then we computed the sensitivity coefficients 285 and relative sensitivity coefficients. For parameter ρ ∈ {R d , τ } and at time t i , the sensitivity 286 coefficient of the lean mass (LM ) computation is: and the relative sensitivity coefficient is  The parameter sensitivity for R d is shown in Figure 5. A 10% change to R d causes a 296 larger perturbation to the solution for the daily schedule than the 5-day schedule, because 297 the daily schedule applies 28 doses whereas the 5-day schedule only applies 20 doses. The    First, we consider weekly-periodic regimes delivering equal-sized doses. To do this, we 319 test the schedules listed in Table 2. To determine the optimal schedule in terms of muscle 320 mass preservation, we simulate equations (1), (2), (7), and (8) for each dosing schedule. As 321 seen in Figure 7(a), the tested schedules are all approximately equal in maintaining lean 322 mass, with about a 10% drop that is recovered upon treatment termination with overshoot. However, simply preserving lean mass is never the objective in cancer treatment -we must 324 consider the treatment effect on a tumour as well. To do this, we simulate the full model 325 Figure 7: Lean mass (a) and tumour (b) response to treatment schedules listed in Table 2. Treatment effect is computed as the fraction of treated lean mass / control (c), and treated tumour volume / initial tumour volume T 0 (d). Instantaneous therapeutic efficacy is computed by the ratio of the treatment effect fractions (e).

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(equations (1), (2), (7), (8), and (10)). Note that no cancer-induced cachexia is considered by 326 this model. The effects of the weekly standard treatments (Table 2)  Using the AUC ratios and total therapeutic efficacy one can further explore schedules deliv-398 ering a weekly standard dose. Consider the weekly schedules listed in Table 3 which deliver 399 between 168-175 mg/kg/week. These schedules further break down the dose delivery into 400 weekly patterns that alternate dosing days with drug holidays.

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The model predicted response of the lean mass, tumour volume, and total therapeutic 402 efficacy are shown in Figure 10. Spreading the doses out across the week preserves lean 403 mass, Figure 10(a), but also decreases the tumour's response to treatment, Figure 10 We now consider schedules that deliver multiple doses per day to explore metronomic schedul-  The model predicted results of metronomic therapy are shown in Figure 11. Breaking 420 Table 3: Dosing schedules that deliver a weekly dose of between 168-175 mg/kg/week. Notation: a schedule described as (4, 1, 1, 1) is a shortening of (4-on, 1-off, 1-on, 1-off), and means doses are given on days 1, 2, 3, 4, and 6 of every week. Schedule Dose (mg/kg) Weekly Total (mg/kg) Daily 24 168 5 times a week: (5, 2), (4, 1, 1, 1), (3, 1, 2, 1), (2, 1, 3, 1), (1, 1, 4, 1 Table 3. (a) The AUC ratio for lean mass compares the lean mass response from the tested schedule to that of the 24 mg/kg daily schedule. (b) The AUC ratio for tumour volume compares tumour volume response from the tested schedule to that of the 24 mg/kg daily schedule. (c) The total therapeutic efficacy is the ratio of the lean mass AUC ratio to the tumour volume AUC ratio. the daily dose up over multiple doses per day results in a small improvement in lean mass 421 preservation, Figure 11(a). It also results in less tumour control, Figure 11(b). Consequently, 422 the multiple-doses-per-day metronomic schedules are predicted to be less therapeutically 423 effective than the standard daily regime, Figure 11(c).
424 Figure 11: Treatment efficacy for metronomic chemotherapy schedules administering 168 mg/kg/week over 1, 2, 3, 4, 5, or 6 equal, and equally spaced, daily doses. Treatment efficacy is measured against the 24 mg/kg daily regime and the 35 mg/kg (5-on, 2-off) regime is included for reference. (a) The AUC ratio for lean mass compares the lean mass response from the tested schedule to that of the 24 mg/kg daily schedule. (b) The AUC ratio for tumour volume compares tumour volume response from the tested schedule to that of the 24 mg/kg daily schedule. (c) The total therapeutic efficacy is the ratio of the lean mass AUC ratio to the tumour volume AUC ratio.  Figure 12: Comparison of chemotherapy effect on lean mass in a young (6 weeks old, solid lines) and old (2 years old, dashed lines) simulated host. Predicted lean mass response to treatment (a) and the resulting stem cell ratio (b). Treatment effect on lean mass (c) defined as the ratio of treated lean mass to initial lean mass. Treatment effect on tumour volume (d) defined as the ratio of treated tumour volume to initial tumour volume. Therapeutic efficacy (e) defined as the ratio of the treatment effect on lean mass to the treatment effect on tumour volume.

Cachexia in Aged Mice
In Figure 12(a) the lean mass is simulated for the young (6 week) and old (2 year) controls.

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Whereas the young control is actively growing, the aged control experiences a small decline 442 in lean mass. The effect of chemotherapy-induced cachexia transiently reduces the lean mass. start at about 11 g and lose mass, due to the chemotherapy, until about 10% mass is lost. recovery compared to the young. The stem cell ratio is transiently perturbed by treatment, Figure 12(b), with the aged hosts experiencing an downward drift in stem ratio due to the 450 assumed decay in viability (p 1 ).

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The treatment effect on lean mass, Figure 12(c), again shows about a 10% reduction 452 in mass by the end of treatment compared to the young or old time-matched controls.

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Treatment effect on the tumour, Figure 12 Figure 13: Treatment efficacy for young (6 weeks, green-purple bars) and old (2 years, red-orange bars) simulated hosts exposed to chemotherapy schedules administering 168 mg/kg/week in 1, 2, 3, or 4 doses per week, as well as a 3-times a day schedule and an every-other-day schedule. Treatment efficacy is measured against the 24 mg/kg daily regime and the 35 mg/kg (5-on, 2-off) regime is included for reference. (a) The AUC ratio for lean mass compares the lean mass response from the tested schedule to that of the 24 mg/kg daily schedule. (b) The AUC ratio for tumour volume compares tumour volume response from the tested schedule to that of the 24 mg/kg daily schedule. (c) The total therapeutic efficacy is the ratio of the lean mass AUC ratio to the tumour volume AUC ratio. Figure 13 shows the AUC measures for lean mass and tumour volume response to treat-461 ment for the young and aged hosts, as measured against the 24 mg/kg daily treatment 462 regime. The aged host is predicted to experience slightly less mass loss than the young host, 463 Figure 13(a). The tumour response is independent of the host age, and the best treatment  Figure 15. Compared with the control (no morphine), 492 the morphine addition is predicted to cause significant and dangerous lean mass loss for 493 all tested schedules, Figure 15[a]. It also improves tumour control, Figure 15[b]. And as 494 expected, the schedule that preserves the most lean mass is the metronomic (3 times a day) 495 schedule, Figure 15[c], while the best tumour control and thus highest total therapeutic 496 efficacy is achieved with the maximum-tolerated-dose (168 mg/kg delivered 1 day a week) 497 schedule, Figure 15[d,e]. 498 Figure 15: Model simulation of various 5-FU dosing schedules with and without adjuvant morphine administration. Lean mass (a) and tumour volume (b) response to various 5-FU schedules with (dashed lines) and without (solid lines) morphine. Treatment efficacy without (green-purple bars) and with (red-orange bars) morphine for chemotherapy schedules administering 168 mg/kg/week in 1, 2, 3, or 4 doses per week, as well as a 3-times a day schedule, and an every-other-day schedule. Treatment efficacy is measured against the 24 mg/kg daily regime, and the 35 mg/kg (5-on, 2-off) regime is included for reference. (a) The AUC ratio for lean mass compares the lean mass response from the tested schedule to that of the 24 mg/kg daily schedule. (b) The AUC ratio for tumour volume compares tumour volume response from the tested schedule to that of the 24 mg/kg daily schedule. (c) The total therapeutic efficacy is the ratio of the lean mass AUC ratio to the tumour volume AUC ratio.
These model simulations represent a worst-case scenario, where morphine usage is con-499 tinuous and increasing to account for any acclimatization effects. The model predictions, 500 however, demonstrate that inadvertent interference between pain-management drugs and 501 chemotherapy regimes may exacerbate both on-target and off-target effects of the drug, and 502 thus warrant further investigation.

Discussion
Since muscle loss is correlated with poor survival rates, understanding the mechanisms of 505 cachexia, including chemotherapy-induced cachexia, and designing treatments that aim to 506 preserve muscle mass in addition to tumour control, are of great importance. This work 507 presents a first attempt to mathematically model chemotherapy-induced muscle loss in a 508 manner that captures the nonlinear dose-dependence observed in experimental data [37]. The  The parameters that couple our model together, τ and R d , were estimated by fitting 517 simulations of all doses to the data for either the daily or 5-day dosing schedules. The to treatment, and for the total therapeutic efficacy. In general, lean mass is best preserved 525 by following a metronomic schedule (doses once or more a day) and tumour control is best 526 achieved by following a maximum tolerated dose schedule (doses once a week).

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Lastly, we used our model to explore potential confounding effects of aging and morphine 528 usage. By the end of our simulated treatment, the old and young mice lost approximately 529 the same amount of lean mass, and experience similar disruptions to their stem cell ratios.

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Simulated morphine usage drastically increased the effects of 5-FU, causing a much more 531 significant lean mass loss along with a much improved tumour control.

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Together, this work highlights the difficulty in defining an optimal treatment schedule in 533 terms of reducing tumour volume while also maintaining lean muscle mass. Across all simu-534 lations, the maximum tolerated dose schedule performed best for tumour control and worst 535 for lean mass preservation. The metronomic schedules (daily or more frequent) performed 536 best for lean mass preservation but worst for tumour control. Thus, defining an optimal 537 schedule consists of a weighted balance between improved tumour control and preserved 538 lean mass.

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In mathematical modelling of chemotherapy it is standard to assume, as we do here,