U ovom članku razmatra se metoda računanja udaljenosti točke od eksplicitno, parametarski, te polarno zadane krivulje. U literaturi za ovaj problem postoji eksplicitno rješenje za slučaj afine funkcije, te za još neke specijalne slučajeve.This article considers the method for calculating the distance of a point to the curve given explicitly, in parameter and polar form. In literature, there exists an explicit solution to this problem
for the case of affine fuctions as well as for some other special cases

Bowers, Margaret T.

Briggs, Andrew H.

Gardner, Matthew

Graham, Felicia L.

Granger, Bradi B.

Levy, Wayne C.

Li, Yanhong

Neilson, Matthew P.

Paul, Sara C.

Polsky, Daniel E.

Reed, Shelby D.

Riegel, Barbara

Schulman, Kevin A.

Whellan, David J.

Croatian

PubMed

BACKGROUND: Heart failure disease management programs can influence medical resource use and quality-adjusted survival. Because projecting long-term costs and survival is challenging, a consistent and valid approach to extrapolating short-term outcomes would be valuable.
METHODS: We developed the Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model, a Web-based simulation tool designed to integrate data on demographic, clinical, and laboratory characteristics; use of evidence-based medications; and costs to generate predicted outcomes. Survival projections are based on a modified Seattle Heart Failure Model. Projections of resource use and quality of life are modeled using relationships with time-varying Seattle Heart Failure Model scores. The model can be used to evaluate parallel-group and single-cohort study designs and hypothetical programs. Simulations consist of 10,000 pairs of virtual cohorts used to generate estimates of resource use, costs, survival, and incremental cost-effectiveness ratios from user inputs.
RESULTS: The model demonstrated acceptable internal and external validity in replicating resource use, costs, and survival estimates from 3 clinical trials. Simulations to evaluate the cost-effectiveness of heart failure disease management programs across 3 scenarios demonstrate how the model can be used to design a program in which short-term improvements in functioning and use of evidence-based treatments are sufficient to demonstrate good long-term value to the health care system.
CONCLUSION: The Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model provides researchers and providers with a tool for conducting long-term cost-effectiveness analyses of disease management programs in heart failure

Polsky, Daniel E E.

Name not available

Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model: A Web-based program designed to evaluate the cost-effectiveness of disease management programs in heart failure.

Background:



Heart failure disease management programs can influence medical resource use and quality-adjusted survival. Because projecting long-term costs and survival is challenging, a consistent and valid approach to extrapolating short-term outcomes would be valuable.



Methods:



We developed the Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model, a Web-based simulation tool designed to integrate data on demographic, clinical, and laboratory characteristics; use of evidence-based medications; and costs to generate predicted outcomes. Survival projections are based on a modified Seattle Heart Failure Model. Projections of resource use and quality of life are modeled using relationships with time-varying Seattle Heart Failure Model scores. The model can be used to evaluate parallel-group and single-cohort study designs and hypothetical programs. Simulations consist of 10,000 pairs of virtual cohorts used to generate estimates of resource use, costs, survival, and incremental cost-effectiveness ratios from user inputs.



Results:



The model demonstrated acceptable internal and external validity in replicating resource use, costs, and survival estimates from 3 clinical trials. Simulations to evaluate the cost-effectiveness of heart failure disease management programs across 3 scenarios demonstrate how the model can be used to design a program in which short-term improvements in functioning and use of evidence-based treatments are sufficient to demonstrate good long-term value to the health care system.



Conclusion:



The Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model provides researchers and providers with a tool for conducting long-term cost-effectiveness analyses of disease management programs in heart failure.



Although economic evaluations of heart failure disease management programs are plentiful, a recent review identified only 2 formal cost-effectiveness analyses that extrapolated beyond a trial's follow-up period.1 Without extrapolation, the value of a disease management program may be underestimated. For example, an analysis of the South Texas Congestive Heart Failure Disease Management Project reported an incremental cost-effectiveness ratio &#62;$100,000 per quality-adjusted life-year (QALY) within the trial's 18-month follow-up period.2 However, extension of the time horizon with a Markov model structured using New York Heart Association (NYHA) classification reduced the incremental cost-effectiveness ratio to &#62;$50,000 per QALY.3 This example demonstrates the importance of accounting for all downstream costs and health benefits attributable to an intervention to provide a fair assessment of its cost-effectiveness.



With support from the National Institute of Nursing Research, we developed user-friendly tools to facilitate high-quality economic evaluations of patient-focused interventions. In our project, Tools for Economic Analysis of Patient Management Interventions in Heart Failure (TEAM-HF), we developed a costing tool4 and a cost-effectiveness model. In this article, we describe the TEAM-HF Cost-Effectiveness Model, a generalizable, Web-based tool designed to assist researchers, administrators, and providers in estimating short- or long-term estimates of resource use, costs, and cost-effectiveness of disease management programs or other care strategies in heart failure. We then compare predicted estimates of resource use and costs from the model to estimates from 3 studies to evaluate the internal and external validity of the model. We also evaluate the potential cost-effectiveness of 3 disease management scenarios to demonstrate how the model can be used to design more cost-effective interventions

Enlighten

English

Tools for economic analysis of patient management interventions in heart failure cost-effectiveness model: a web-based program designed to evaluate the cost-effectiveness of disease management programs in heart failure

Reed, Shelby D

Neilson, Matthew P

Briggs, Andrew H

Polsky, Daniel E

Graham, Felicia L

Bowers, Margaret T

Paul, Sara C

Granger, Bradi B

Schulman, Kevin A

Whellan, David J

Levy, Wayne C

ScholarlyCommons@Penn

Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model: A Web-Based Program Designed to Evaluate the Cost-Effectiveness of Disease Management Programs in Heart Failure

Jefferson Digital Commons

Enlighten: Publications

Shelby D. Reed

Matthew P. Neilson

Matthew Gardner

Yanhong Li

Andrew H. Briggs

Daniel E. Polsky

Felicia L. Graham

Margaret T. Bowers

Sara C. Paul

Bradi B. Granger

Kevin A. Schulman

David J. Whellan

Barbara Riegel

Wayne C. Levy

Crossref

American Heart Journal

Tools for Economic Analysis of Patient Management Interventions in Heart Failure Cost-Effectiveness Model: A Web-based program designed to evaluate the cost-effectiveness of disease management programs in heart failure

Kosmopolis

I. Kuzmanović

Hrčak - Portal of scientific journals of Croatia

Osječki matematički list 5(2005), 85–90 85Udaljenost točke do krivuljeIvana Kuzmanović∗STUDENTSKA RUBRIKASažetak.U ovom članku razmatra se metoda računanja udaljenostitočke do eksplicitno, parametarski, te polarno zadane krivulje. U litera-turi za ovaj problem postoji eksplicitno rješenje za slučaj afine funkcije,te za još neke specijalne slučajeve.Ključne riječi: udaljenost, krivulja, minimizacijaAbstract. This article considers the method for calculating thedistance of a point to the curve given explicitly, in parameter and polarform. In literature, there exists an explicit solution to this problem forthe case of affine fuctions as well as for some other special cases.Key words: distance, curve, minimization1. Udaljenost točke do grafa eksplicitno zadane funkcijeNeka su dane točke ravnine T1 = (x1, y1) i T2 = (x2, y2). L1 udaljenost točaka T1 iT2 definiramo kaod1(T1, T2) = |x2 − x1| + |y2 − y1|.L2 (euklidska) udaljenost dana je sd2(T1, T2) =√(x2 − x1)2 + (y2 − y1)2.Za udaljenost točke T0 = (x0, y0) do krivulje uzima se najkraća udaljenostizmed̄u točke T0 i točaka na krivulji. Preciznije:Neka je dana točka T0 = (x0, y0) ∈ R2 i neprekidna funkcija f : D → R, D ⊆ R.Želimo odrediti L1 i L2 udaljenost točke T0 do grafa funkcije f . Taj problem sesvodi na odred̄ivanje točke T ∗p = (x∗p, f(x∗p)) ∈ Γf takve da jedp(T0, T ∗p ) = minT∈Γfdp(T0, T ),odnosno na rješavanje problema globalnog minimuma funkcije jedne varijable, tj.treba pronaći točku x∗p ∈ D, tako da budeminx∈Dφp(x) = φp(x∗p), (1)∗Odjel za matematiku, Sveučilǐste u Osijeku, Trg Ljudevita Gaja 6, HR-31000 Osijek, e-mail:ikuzmano@mathos.hr86 Ivana Kuzmanovićgdje jeφp(x) ={ |x− x0| + |f(x) − y0| , p = 1√(x− x0)2 + (f(x) − y0)2 , p = 2.Tada je dp(T0,Γf ) = d(T0, T ∗p ). Ovaj problem je već razmatran u [3].Lako se može vidjeti da ako neprekidna funkcija ψ : D → R+0 postiže svoj minimumu točki x∗, da tada funkcija φ(x) =√ψ(x) takod̄er postiže svoj minimum u točkix∗. Zato ćemo umjesto minimizacije funkcije φp minimizirati jednostavniju funkcijuψp(x) ={|x− x0| + |f(x) − y0| , p = 1(x− x0)2 + (f(x) − y0)2 , p = 2. (2)Primjer 1. Udaljenost točke do grafa logističke funkcijeNeka je f(x) =51 + 5e−2x. Odredimo L1 i L2 udaljenost točke T0 = (2, 3)do grafa funkcije f .Treba pronaći točku x∗p ∈ R koja je rješenje problemaminx∈Rψp(x) = ψp(x∗p), p = 1, 2gdje jeψp(x) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩|x− 2| + ∣∣ 51 + 5e−2x− 3∣∣ , p = 1(x− 2)2 +( 51 + 5e−2x− 3)2, p = 2.Na Slici 1. prikazani su grafovi funkcija ψ1 i ψ2.x∗1 x∗2ψ1 ψ2Slika 1. Grafovi funkcija ψ1 i ψ2Udaljenost točke T0 do grafa Γf pri tome iznosidp(T0,Γf ) ={ψ1(x∗1) , p = 1√ψ2(x∗2) , p = 2.Udaljenost točke do krivulje 87Na Slici 2. prikazan je graf funkcije f , točka T0 i njezine L1 i L2 projekcijeT ∗p = (x∗p, f(x∗p)), p = 1, 2.1T0 T0T ∗1T ∗2Slika 2. L1 i L2 projekcije točke T0Napomena: Točka T ∗p je diralǐste krivulje Γf i kugle u Lp metrici sa sredǐstemu T0 polumjera dp(T0,Γf ). (Slika 2.)2. Udaljenost točke do parametarski zadane krivuljeNeka je dana točka T0 = (x0, y0) i krivulja Γ zadana u parametarskom oblikuΓ = {(fx(t), fy(t)) : t ∈ [t1, t2]}.Da bi odredili L1 i L2 udaljenost točke T0 do krivulje Γ, slično kao u prethodnomslučaju, treba pronaći t∗p ∈ [t1, t2] tako da budemint∈[t1,t2]ψp(t) = ψp(t∗p), (1)gdje jeψp(t) ={|fx(t) − x0| + |fy(t) − y0| , p = 1(fx(t) − x0)2 + (fy(t) − y0)2 , p = 2. (2)Tada jedp(T0,Γ) ={ψ1(t∗1) , p = 1√ψ2(t∗2) , p = 2.Pri tome je točka projekcije dana s T ∗p = (fx(t∗p), fy(t∗p)).1Svi grafovi su izrad̄eni u programskom paketu Mathematica. Za minimizaciju funkcijakorǐstena je Mathematica funkcija FindMinimum (vidjeti [4]).88 Ivana KuzmanovićPrimjer 2. Zadana je točka T0 = (1.5, 0.5) i cikloida Γ = {(t− sin t, 1− cos t) :t ∈ [0, 2π]}. Da bismo odredili L1 i L2 udaljenosti točke T0 do krivulje Γ, najprijetrebamo odrediti točke t∗1 i t∗2 u kojima funkcijeψ1(t) = |t− sin t− 1.5| + |1 − cos t− 0.5|ψ2(t) = (t− sin t− 1.5)2 + (1 − cos t− 0.5)2postižu minimum. Na Slici 3. prikazani su grafovi funkcija ψ1(t) i ψ2(t).t∗1 t∗2ψ1 ψ2Slika 3. Grafovi funkcija ψ1(t) i ψ2(t).Na Slici 4. prikazane su L1 i L2 projekcije T ∗1 = (t∗1 − sin t∗1, 1 − cos t∗1) i T ∗2 =(t∗2 − sin t∗2, 1 − cos t∗2) točke T0 na krivulju ΓT ∗1T0T0T ∗2a) L1 udaljenost b) L2 udaljenostSlika 4. L1 i L2 projekcije točke T0 na krivulju Γ3. Udaljenost točke do polarno zadane krivuljePolarno zadana krivulja je zadana jednadžbom oblikar = f(ϕ), ϕ ∈ [ϕ1, ϕ2].Kako je veza izmed̄u polarnih i Kartezijevih koordinata dana s (vidjeti [1])x = r cosϕ,y = r sinϕ,onda se problem udaljenosti točke do polarno zadane krivulje može svesti na računanjeudaljenosti do parametarski zadane krivulje.Udaljenost točke do krivulje 89Primjer 3. Neka je r = 3ϕ , ϕ ∈ [0, 3π] polarno zadana krivulja i T0 = (1, 1).Tada je u parametarskom obliku ta krivulja dana s Γϕ = {( 3ϕ cosϕ, 3ϕ sinϕ) : ϕ ∈[0, 3π]}. Da bi odredili L1 i L2 udaljenosti točke T0 do krivulje Γϕ, najprije trebamoodrediti točke ϕ∗1 i ϕ∗2 u kojima funkcijeψ1(ϕ) = | 3ϕcosϕ− 1| + | 3ϕsinϕ− 1|,ψ2(ϕ) = (3ϕcosϕ− 1)2 + ( 3ϕsinϕ− 1)2postižu globalni minimum. L1 i L2 udaljenosti točke T0 do krivulje Γϕ tada iznosed1(T0,Γf ) = ψ1(ϕ∗1),odnosnod2(T0,Γf ) =√ψ2(ϕ∗2).Na Slici 5. prikazani su grafovi funkcija ψ1(ϕ) i ψ2(ϕ).ϕ∗1 ϕ∗2ψ1 ψ2Slika 5. Grafovi funkcija ψ1(ϕ) i ψ2(ϕ).Na Slici 6. prikazane su L1 i L2 projekcije T ∗1 = (3ϕ∗1cosϕ∗1,3ϕ∗1sinϕ∗1) i T ∗2 =( 3ϕ∗2 cosϕ∗2,3ϕ∗2sinϕ∗2) točke T0 na krivulju Γf .T ∗1T0T0T ∗2a) L1 udaljenost b) L2 udaljenostSlika 6. L1 i L2 projekcije točke T0 na krivulju Γf90 Ivana KuzmanovićZadatak 1. Odredite L1 i L2 udaljenosti točke T0 = (2, 1) do pravca danogjednadžbom y = x+ 1. (Rješenje: d1 = 2, d2 =√2 )Zadatak 2. Odredite L1 i L2 udaljenosti točke T0 = (1, 4) do grafa funkcijef(x) = x2 + 1. (Rješenje: d1 = 0.732, d2 = 0.702 )Zadatak 3. L∞ udaljenost točaka T1 = (x1, y1) i T2 = (x2, y2) definira se kaod∞(T1, T2) = max{|x2 − x1|, |y2 − y1|}.Odredite L∞ udaljenosti točaka T0 do krivulja iz zadataka 1 i 2.Literatura[1] S. Kurepa, Matematička analiza, Tehnička knjiga, Zagreb, 1987.[2] K. Sabo, Minimizacija realne funkcije jedne varijable, Osječka matematičkaškola 2(2001)[3] I. Soldo, K. Sabo, Računanje udaljenosti točke do krivulje, PrimMath[2003],2003.[4] S. Wolfram, The Mathematica Book, Wolfram Media, Champaign, 1999.