March  2019, 14(1): 43-52. doi: 10.3934/nhm.2019003

Optimal stopping for response-guided dosing

Department of Mathematics, University of Portland, 5000 N Willamette Blvd, Portland, OR 97203, USA

Received  April 2018 Published  January 2019

Fund Project: The first author was funded in part by the National Science Foundation through grant #CMMI 1536717 when the author was a doctoral student at the University of Washington.

In response-guided dosing (RGD), the goal is to make optimal dosing decisions based on the stochastic evolution of a patient's disease condition. Typically, RGD is formulated as a finite-horizon problem with decision-making occurring over a predetermined time frame. In this paper we relax the latter assumption to allow for the possibility of ending treatment early. This could occur due to remission of the disease or a finding of futility in treatment of the disease. Our framework is formulated as a stochastic dynamic program (DP) where a stop/do-not-stop decision is made in discrete sessions, and if stopping is not chosen, an optimal dose is determined for that session. Numerical simulations for rheumatoid arthritis are presented, and monotonicity of the stop/do-not-stop threshold with respect to time is proven.

Citation: Jakob Kotas. Optimal stopping for response-guided dosing. Networks and Heterogeneous Media, 2019, 14 (1) : 43-52. doi: 10.3934/nhm.2019003
References:
[1]

D. A. BerryP. MüllerA. P. GrieveM. SmithT. ParkeR. BlazekN. Mitchard and M. Krams, Adaptive Bayesian designs for dose-ranging drug trials, Case Studies in Bayesian Statistics, 5 (2002), 99-181.  doi: 10.1007/978-1-4613-0035-9_2.

[2]

D. P. Bertsekas, Dynamic Programming and Optimal Control, 4th edition, Athena Scientific, Cambridge, MA, 2012.

[3]

G. L. DavisJ. B. WongJ. G. McHutchisonM. P. MannsJ. Harvey and J. Albrecht, Early virologic response to treatment with peginterferon alfa-2b plus ribavirin in patients with chronic hepatitis C, Hepatology, 38 (2003), 645-652.  doi: 10.1053/jhep.2003.50364.

[4]

M. FlendrieM. C. W. Creemers and P. L. C. M. van Riel, Titration of infliximab treatment in rheumatoid arthritis patients based on response patterns, Rheumatology, 46 (2007), 146-149.  doi: 10.1093/rheumatology/kel173.

[5]

B. I. Grigelionis and A. N. Shiryaev, On Stefan's problem and optimal stopping rules for Markov processes, Theory of Probability and its Applications, 4 (1966), 541-558. 

[6]

I. M. JacobsonP. MarcellinS. ZeuzemM. S. SulkowskiR. EstebanF. PoordadS. BrunoM. H. BurroughsL. D. PediconeN. BoparaiW. DengM. J. DiNubileK. M. GottesdienerC. A. BrassJ. K. Albrecht and J.-P. Bronowicki, Refinement of stopping rules during treatment of hepatitis C genotype 1 infection with boceprevir and peginterferon/ribavirin, Hepatology, 56 (2012), 567-575.  doi: 10.1002/hep.25865.

[7]

J. Kotas and A. Ghate, Response-guided dosing for rheumatoid arthritis, IIE Transactions on Healthcare Systems Engineering, 6 (2016), 1-21. 

[8]

J. Kotas and A. Ghate, Bayesian learning of dose-response parameters from a cohort under response-guided dosing, European Journal of Operational Research, 265 (2018), 328-343.  doi: 10.1016/j.ejor.2017.07.034.

[9]

T. M. KuijperF. B. G. Lamers-KarnebeekJ. W. G. JacobsJ. M. W. Hazes and J.J. Luime, Flare rate in patients with rheumatoid arthritis in low disease activity or remission when tapering or stopping synthetic or biologic DMARD: a systematic review, Journal of Rheumatology, 42 (2015), 2012-2022.  doi: 10.3899/jrheum.141520.

[10]

L. LojoG. BonillaD. PeiteadoA. VillalbaC. PlasenciaL. NunoA. Balsa and E. Martin-Mola, Down-titration and discontinuation of infliximab, adalimumab and etanercept in established rheumatoid arthritis, Annals of the Rheumatic Diseases, 72 (2013), 237.  doi: 10.1136/annrheumdis-2013-eular.744.

[11]

E. LouisG. Vernier-MassouilleJ.-C. GrimaudY. BouhnikD. LaharieJ.-L. DupasH. PillantL. PiconM. VeyracM. FlamantG. SavoyeR. JianM. De VosG. PaintaudE. PiverJ.-F. ColombelJ.-Y. Mary and M. Lemann, Infliximab discontinuation in Crohn's disease patients in stable remission on combined therapy with immunosuppressors: a prospective ongoing cohort study, Gastroenterology, 136 (2009), A146.  doi: 10.1016/S0016-5085(09)60659-4.

[12]

M. MarinoE. ZucchiM. FabbroI. LodoloR. MaieronS. Vadalà and M. Zilli, Outcome of infliximab discontinuation in IBD patients and therapy rechallenging in relapsers: Single centre preliminary data, Journal of Crohn's and Colitis, 8 (2014), S232-S233.  doi: 10.1016/S1873-9946(14)60521-3.

[13]

P. MüllerD. A. BerryA. P. Grieve and M. Krams, A Bayesian decision-theoretic dose-finding trial, Decision Analysis, 3 (2006), 197-207. 

[14] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley & Sons, Inc., New York, 1994. 
[15]

K. E. ShermanS. L. FlammN. H. AfdhalD. R. NelsonM. S. SulkowskiG. T. EversonM. W. FriedM. AdlerH. W. ReesinkM. MartinA. J. SankohN. AddaR. S. KauffmanS. GeorgeC. I. Wright and F. Poordad for the ILLUMINATE study team, Response-guided telaprevir combination treatment for hepatitis C virus infection, The New England Journal of Medicine, 365 (2011), 1014-1024.  doi: 10.1056/NEJMoa1014463.

[16] A. N. Shiryaev, Optimal Stopping Rules, $1^{st}$ edition, Springer-Verlag, Berlin Heidelberg, 2008. 
[17]

W. Slob, Dose-response modeling of continuous endpoints, Toxicological Sciences, 66 (2002), 298-312.  doi: 10.1093/toxsci/66.2.298.

[18]

J. SmolenA. BeaulieuA. Rubbert-RothC. Ramos-RemusJ. RovenskyE. AlecockT. WoodworthR. Alten and OP TION trial investigators, Effect of interleukin-6 receptor inhibition with tocilizumab in patients with rheumatoid arthritis (OPTION study): a double-blind, placebo-controlled, randomised trial, The Lancet, 371 (2008), 987-997.  doi: 10.1016/S0140-6736(08)60453-5.

[19]

A. SofiA. AliS. A. Khuder and A. Nawras, Meta-analysis- maintenance of remission following discontinuation of infliximab in patients with Crohn's disease, Gastroenterology, 144 (2013), S637.  doi: 10.1016/S0016-5085(13)62356-2.

[20]

M. J. SonneveldB. E. HansenT. PiratvisuthJ.-D. JiaS. ZeuzemE. GaneY.-F. LawQ. XieE. J. HeathcoteH.L.-Y. Chan and H.L.A. Janssen, Response-guided peginterferon therapy in hepatitis B e antigen-positive chronic hepatitis B using serum hepatitis B surface antigen levels, Hepatology, 58 (2013), 872-880.  doi: 10.1002/hep.26436.

[21]

M. J. SonneveldV. RijckborstC. A. BoucherB. E. Hansen and H.L. Janssen, Prediction of sustained response to peginterferon alfa-2b for hepatitis B e antigen-positive chronic hepatitis B using on-treatment hepatitis B surface antigen decline, Hepatology, 52 (2012), 1251-1257.  doi: 10.1002/hep.23844.

[22]

S. ten WoldeF. C. BreedveldB. A. C. DijkmansJ. HermansJ. P. VandenbrouckeM. A. F. J. van de LaarH. M. MarkusseM. Janssen and H. R. van den Brink, Randomised placebo-controlled study of stopping second-line drugs in rheumatoid arthritis, The Lancet, 347 (1996), 347-352. 

[23]

A. van der MaasW. KievitB. J. F. van den BemtF. H. J. van den HoogenP. L. C. M. van Riel and A. A. den Broeder, Down-titration and discontinuation of infliximab in rheumatoid arthritis patients with stable low disease activity and stable treatment: an observational cohort study, Annals of Rheumatoid Disease, 71 (2012), 1849-1854. 

[24]

N. van HerwaardenA. A. den BroederW. JacobsA. van der MaasJ. W. BijlsmaR. F. van Vollenhoven and B. J. van den Bemt, Down-titration and discontinuation strategies of tumor necrosis factor-blocking agents for rheumatoid arthritis patients with low disease activity, Cochrane Database of Systematic Reviews, 9 (2013).  doi: 10.1002/14651858.CD010455.

show all references

References:
[1]

D. A. BerryP. MüllerA. P. GrieveM. SmithT. ParkeR. BlazekN. Mitchard and M. Krams, Adaptive Bayesian designs for dose-ranging drug trials, Case Studies in Bayesian Statistics, 5 (2002), 99-181.  doi: 10.1007/978-1-4613-0035-9_2.

[2]

D. P. Bertsekas, Dynamic Programming and Optimal Control, 4th edition, Athena Scientific, Cambridge, MA, 2012.

[3]

G. L. DavisJ. B. WongJ. G. McHutchisonM. P. MannsJ. Harvey and J. Albrecht, Early virologic response to treatment with peginterferon alfa-2b plus ribavirin in patients with chronic hepatitis C, Hepatology, 38 (2003), 645-652.  doi: 10.1053/jhep.2003.50364.

[4]

M. FlendrieM. C. W. Creemers and P. L. C. M. van Riel, Titration of infliximab treatment in rheumatoid arthritis patients based on response patterns, Rheumatology, 46 (2007), 146-149.  doi: 10.1093/rheumatology/kel173.

[5]

B. I. Grigelionis and A. N. Shiryaev, On Stefan's problem and optimal stopping rules for Markov processes, Theory of Probability and its Applications, 4 (1966), 541-558. 

[6]

I. M. JacobsonP. MarcellinS. ZeuzemM. S. SulkowskiR. EstebanF. PoordadS. BrunoM. H. BurroughsL. D. PediconeN. BoparaiW. DengM. J. DiNubileK. M. GottesdienerC. A. BrassJ. K. Albrecht and J.-P. Bronowicki, Refinement of stopping rules during treatment of hepatitis C genotype 1 infection with boceprevir and peginterferon/ribavirin, Hepatology, 56 (2012), 567-575.  doi: 10.1002/hep.25865.

[7]

J. Kotas and A. Ghate, Response-guided dosing for rheumatoid arthritis, IIE Transactions on Healthcare Systems Engineering, 6 (2016), 1-21. 

[8]

J. Kotas and A. Ghate, Bayesian learning of dose-response parameters from a cohort under response-guided dosing, European Journal of Operational Research, 265 (2018), 328-343.  doi: 10.1016/j.ejor.2017.07.034.

[9]

T. M. KuijperF. B. G. Lamers-KarnebeekJ. W. G. JacobsJ. M. W. Hazes and J.J. Luime, Flare rate in patients with rheumatoid arthritis in low disease activity or remission when tapering or stopping synthetic or biologic DMARD: a systematic review, Journal of Rheumatology, 42 (2015), 2012-2022.  doi: 10.3899/jrheum.141520.

[10]

L. LojoG. BonillaD. PeiteadoA. VillalbaC. PlasenciaL. NunoA. Balsa and E. Martin-Mola, Down-titration and discontinuation of infliximab, adalimumab and etanercept in established rheumatoid arthritis, Annals of the Rheumatic Diseases, 72 (2013), 237.  doi: 10.1136/annrheumdis-2013-eular.744.

[11]

E. LouisG. Vernier-MassouilleJ.-C. GrimaudY. BouhnikD. LaharieJ.-L. DupasH. PillantL. PiconM. VeyracM. FlamantG. SavoyeR. JianM. De VosG. PaintaudE. PiverJ.-F. ColombelJ.-Y. Mary and M. Lemann, Infliximab discontinuation in Crohn's disease patients in stable remission on combined therapy with immunosuppressors: a prospective ongoing cohort study, Gastroenterology, 136 (2009), A146.  doi: 10.1016/S0016-5085(09)60659-4.

[12]

M. MarinoE. ZucchiM. FabbroI. LodoloR. MaieronS. Vadalà and M. Zilli, Outcome of infliximab discontinuation in IBD patients and therapy rechallenging in relapsers: Single centre preliminary data, Journal of Crohn's and Colitis, 8 (2014), S232-S233.  doi: 10.1016/S1873-9946(14)60521-3.

[13]

P. MüllerD. A. BerryA. P. Grieve and M. Krams, A Bayesian decision-theoretic dose-finding trial, Decision Analysis, 3 (2006), 197-207. 

[14] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley & Sons, Inc., New York, 1994. 
[15]

K. E. ShermanS. L. FlammN. H. AfdhalD. R. NelsonM. S. SulkowskiG. T. EversonM. W. FriedM. AdlerH. W. ReesinkM. MartinA. J. SankohN. AddaR. S. KauffmanS. GeorgeC. I. Wright and F. Poordad for the ILLUMINATE study team, Response-guided telaprevir combination treatment for hepatitis C virus infection, The New England Journal of Medicine, 365 (2011), 1014-1024.  doi: 10.1056/NEJMoa1014463.

[16] A. N. Shiryaev, Optimal Stopping Rules, $1^{st}$ edition, Springer-Verlag, Berlin Heidelberg, 2008. 
[17]

W. Slob, Dose-response modeling of continuous endpoints, Toxicological Sciences, 66 (2002), 298-312.  doi: 10.1093/toxsci/66.2.298.

[18]

J. SmolenA. BeaulieuA. Rubbert-RothC. Ramos-RemusJ. RovenskyE. AlecockT. WoodworthR. Alten and OP TION trial investigators, Effect of interleukin-6 receptor inhibition with tocilizumab in patients with rheumatoid arthritis (OPTION study): a double-blind, placebo-controlled, randomised trial, The Lancet, 371 (2008), 987-997.  doi: 10.1016/S0140-6736(08)60453-5.

[19]

A. SofiA. AliS. A. Khuder and A. Nawras, Meta-analysis- maintenance of remission following discontinuation of infliximab in patients with Crohn's disease, Gastroenterology, 144 (2013), S637.  doi: 10.1016/S0016-5085(13)62356-2.

[20]

M. J. SonneveldB. E. HansenT. PiratvisuthJ.-D. JiaS. ZeuzemE. GaneY.-F. LawQ. XieE. J. HeathcoteH.L.-Y. Chan and H.L.A. Janssen, Response-guided peginterferon therapy in hepatitis B e antigen-positive chronic hepatitis B using serum hepatitis B surface antigen levels, Hepatology, 58 (2013), 872-880.  doi: 10.1002/hep.26436.

[21]

M. J. SonneveldV. RijckborstC. A. BoucherB. E. Hansen and H.L. Janssen, Prediction of sustained response to peginterferon alfa-2b for hepatitis B e antigen-positive chronic hepatitis B using on-treatment hepatitis B surface antigen decline, Hepatology, 52 (2012), 1251-1257.  doi: 10.1002/hep.23844.

[22]

S. ten WoldeF. C. BreedveldB. A. C. DijkmansJ. HermansJ. P. VandenbrouckeM. A. F. J. van de LaarH. M. MarkusseM. Janssen and H. R. van den Brink, Randomised placebo-controlled study of stopping second-line drugs in rheumatoid arthritis, The Lancet, 347 (1996), 347-352. 

[23]

A. van der MaasW. KievitB. J. F. van den BemtF. H. J. van den HoogenP. L. C. M. van Riel and A. A. den Broeder, Down-titration and discontinuation of infliximab in rheumatoid arthritis patients with stable low disease activity and stable treatment: an observational cohort study, Annals of Rheumatoid Disease, 71 (2012), 1849-1854. 

[24]

N. van HerwaardenA. A. den BroederW. JacobsA. van der MaasJ. W. BijlsmaR. F. van Vollenhoven and B. J. van den Bemt, Down-titration and discontinuation strategies of tumor necrosis factor-blocking agents for rheumatoid arthritis patients with low disease activity, Cochrane Database of Systematic Reviews, 9 (2013).  doi: 10.1002/14651858.CD010455.

Figure 1.  Optimal policy of tocilizumab dosing for rheumatoid arthritis. The cost functions are $ c(d) = 0.028557d + b $, with fixed cost parameter $ b $ varying among subfigures. A dose of $ -1 $ indicates a decision to stop treatment
Figure 2.  Optimal policy with the cost function $ c(d) = 0.028557d + 0.1 $. The plot is a zoomed-in version of Figure Figure 1c around the threshold area between stopping and not stopping, with refined discretization. A dose of -1 indicates a decision to stop
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