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Intermediate Physics for Medicine and Biology Russell K. Hobbie • Bradley J. Roth Intermediate Physics for Medicine and Biology Fifth Edition RussellK.Hobbie BradleyJ.Roth UniversityofMinnesota OaklandUniversity Minneapolis,Minnesota Rochester,Michigan USA USA ISBN978-3-319-12681-4 ISBN978-3-319-12682-1(eBook) DOI10.1007/978-3-319-12682-1 LibraryofCongressControlNumber:2014954086 SpringerChamHeidelbergNewYorkDordrechtLondon (cid:2)c SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproduction onmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface FromthePrefacetotheThirdEdition,byRussellK.Hobbie: analysisorcomplexexponentialnotation,oftenhidesphys- icalrealityfromthestudent.Ihaveseenelectricalengineer- Between1971and1973Iauditedallthecoursesmedicalstu- ing students who could not tell me what is happening in dentstakeintheirfirst2yearsattheUniversityofMinnesota. an RC circuit but could solve the equations with Laplace IwasamazedattheamountofphysicsIfoundinthesecourses transforms. andhowlittleofitisdiscussedinthegeneralphysicscourse. Ifoundagreatdiscrepancybetweenthephysicsinsomepa- TheFourthEditionfollowedthetraditionofearliereditions. persinthebiologicalresearchliteratureandwhatIknewtobe The book added a second author: Bradley J. Roth of Oak- the level of understanding of most biology majors or premed studentswhohavetakenayearofphysics.Itwasclearthatan land University. Both of us have enjoyed this collaboration intermediatelevelphysicscoursewouldhelpthesestudents.It immensely. We added a chapter on sound and ultrasound, wouldprovidethephysicstheyneedandwouldrelateitdirectly deletingorshorteningtopicselsewhere,inordertokeepthe tothebiologicalproblemswhereitisuseful. bookonlyslightlylongerthantheThirdEdition. This book is the result of my having taught such a course since1973.Itisintendedtoserveasatextforanintermediate The Fifth Edition does not add any new chapters, course taught in a physics department and taken by a variety but almost every page has been improved and up- of majors. Since its primary content is physics, I hope that dated. Again, we fought the temptation to expand the physicsfacultywhomightshyawayfromteachingaconven- book and deleted material when possible. Some of tionalbiophysicscoursewillconsiderteachingit.Ialsohope thatresearchworkersinbiologyandmedicinewillfinditause- the deleted material is available at the book’s website: ful reference to brush up on the physics they need or to find http://www.oakland.edu/~roth/hobbie.htm.TheFifthEdition afewpointerstothecurrentliteratureinanumberofareasof has 12% more end-of-chapter problems than the Fourth biophysics.(Thebibliographyineachchapterisbynomeans Edition; most highlight biological applications of the phys- exhaustive;however,thereferencesshouldleadyouquicklyinto a field.) The course offered at the University of Minnesota is ical principles. Many of the problems extend the material takenbyundergraduatesinanumberofmajorswhowanttosee in the text. A solutions manual is available to those teach- morephysicswithbiologicalapplicationsandbygraduatestu- ing the course. Instructors can use it as a reference or dentsinphysics,biophysicalsciences,biomedicalengineering, provide selected solutions to their students. The solutions physiology,andcellbiology. Becausethebookisintendedprimarilyforstudentswhohave manual makes it much easier for an instructor to guide an taken only one year of physics, I have tried to adhere to the independent-study student. Information about the solutions followingprinciplesinwritingit: manualisavailableatthebook’swebsite. 1. Calculusisusedwithoutapology.Whenanimportantidea Chapter 1 reviews mechanics. Translational and rota- incalculusisusedforthefirsttime,itisreviewedindetail. Thesereviewsarefoundintheappendices. tionalequilibriumareintroduced,withtheforcesintheheel 2. Thereaderisassumedtohavetakenphysicsandknowthe and hip joint as clinical examples. Stress and strain, hy- basic vocabulary. However, I have tried to present a logi- drostatics, incompressible viscous flow, and the Poiseuille– caldevelopmentfromfirstprinciples,butshorterthanwhat Bernoulli equation are discussed, with examples from the wouldbefoundinanintroductorycourse.Anexceptionis foundinChaps.14–18,wheresomeresultsfromquantum circulatorysystem.Thechapterconcludeswithadiscussion mechanicsareusedwithoutderivingthemfromfirstprin- ofReynoldsnumber. ciples. (My students have often expressed surprise at this Chapter 2 is essential to nearly every other chapter in changeofpace.) the book. It discusses exponential growth and decay and 3. I have not intentionally left out steps in most derivations. Some readers may feel that the pace could be faster, par- givesexamplesfrompharmacologyandphysiology(includ- ticularly after a few chapters. My students have objected ing clearance). The logistic equation is discussed. Students strongly when I have suggested stepping up the pace in arealsoshownhowtousesemilogandlog–logplotsandto class. determine power-law coefficients using a spreadsheet. The 4. Eachsubjectisapproachedinassimpleafashionaspos- sible.Ifeelthatsophisticatedmathematics,suchasvector chapterconcludeswithabriefdiscussionofscaling. v vi Preface Chapter3isacondensedtreatmentofstatisticalphysics: nerveandmusclecells,anddefibrillation.Finally,themodel averagequantities,probability,thermalequilibrium,entropy, isextendedtotheelectroencephalogram. and the first and second laws of thermodynamics. Topics Chapter 8showshowthecurrentsinaconducting nerve treated include the following: the Boltzmann factor and its ormusclecellgenerateamagneticfield,leadingtothemag- corollary,theNernstequation;theprincipleofequipartition netocardiogramandthemagnetoencephalogram.Somebac- of energy; the chemical potential; the general thermody- teria and higher organisms contain magnetic particles used namicrelationship;theGibbsfreeenergy;andthechemical for determining spatial orientation in the earth’s magnetic potentialofasolution.Youcanplowthroughthischapterif field. The mechanism by which these bacteria are oriented you are a slave to thoroughness, touch on the highlights, or is described. The detection of weak magnetic fields and the useitasareferenceasthetopicsareneededinlaterchapters. useofchangingmagneticfieldstostimulatenerveormuscle Chapter4treatsdiffusionandtransportofsoluteinanin- cellsarealsodiscussed. finite medium. Fick’s first and second laws of diffusion are Chapter 9 covers a number of topics at the cellular and developed. Steady-state solutions in one, two, and three di- membrane level. It begins with Donnan equilibrium, where mensions are described. An important model is a spherical the presence of an impermeant ion on only one side of cell with pores providing transport through the cell mem- a membrane leads to the buildup of a potential difference brane. It is shown that only a small number of pores are across the membrane, and the Gouy–Chapman model for requiredtokeepupwiththerateofdiffusiontowardoraway how ions redistribute near the membrane to generate this from the cell, so there is plenty of room on the cell surface potential difference. The Debye–Hückel model is a sim- for many different kinds of pores and receptor sites. The ple description of the neutralization of ions by surrounding combination of diffusion and drift (or solvent drag) is also counterions.TheNernst–Planckequationprovidesthebasic discussed.Finally,asimplerandom-walkmodelofdiffusion modelfordescribingcombineddiffusionanddriftinanap- isintroduced. pliedelectricfield.ItalsoformsthebasisfortheGoldman– Chapter 5 discusses transport of fluid and neutral so- Hodgkin–Katz model for zero total current in a membrane lutes through a membrane. This might be a cell membrane, withaconstantelectricfield.Gatedmembranechannelsare the basement membrane in the glomerulus of the kidney, then discussed. Noise is inescapable in all signalling situa- or a capillary wall. The phenomenological transport equations.Afterdevelopingthebasicpropertiesofshotnoiseand tions including osmotic pressure are introduced as the first Johnson noise, we show how a properly adapted shark can (linear) approximation to describe these flows. Countercur- detect very weak electric fields with a reasonable signal-to- rent transport is described. Finally, a hydrodynamic model noise ratio. The chapter concludes with a discussion of the isdevelopedforright-cylindricalpores.Thismodelprovides basicphysicalprinciplesthatmustbekeptinmindwhenas- expressions for the phenomenological coefficients in terms sessing the possibility of biological effects of weak electric oftheporeradiusandlength.Itisalsousedtocalculatethe andmagneticfields. netforceonthemembranewhenthereisflow. Chapter 10 describes feedback systems in the body. It After reviewing the electric field, electric potential, and starts with the regulation of breathing rate to stabilize the circuits,Chap.6describestheelectrochemicalchangesthat carbon dioxide level in the blood, moves to linear feedback cause an impulse to travel along a nerve axon or along a systems with one and two time constants, and then to non- musclefiberbeforecontraction.Twomodelsareconsidered: linear models. We show how nonlinear systems described electrotonus (when the membrane obeys Ohm’s law) and bysimpledifferenceequationscanexhibitchaoticbehavior, theHodgkin–Huxleymodel(whenthemembraneisnonlin- andhowchaoticbehaviorcanariseincontinuoussystemsas ear).Saltatoryconduction inmyelinated fibers isdescribed. well.ExamplesoffeedbacksystemsincludeCheyne–Stokes The dielectric properties of the membrane are modeled in respiration, heat stroke, pupil size, oscillating white-blood- termsofitsmolecularstructure.Somesimplechangestothe cell counts, waves in excitable media, and period doubling membrane conductivity give riseto a periodically repeating andchaosintheheart. actionpotential.Finally,ageneralrelationshipisdeveloped Chapter11showshowthemethodofleastsquaresunder- betweendiffusivetransport,resistance,andcapacitancefora lies several important techniques for analyzing data. These givengeometry. range from simple curve fitting to discrete and continuous Chapter7showshowanelectricpotentialisgeneratedin Fourier series, power spectra, correlation functions, and the the medium surrounding a nerve or muscle cell. This leads Fouriertransform.Wethendescribethefrequencyresponse to the current dipole model for the electrocardiogram. The of alinear systemand thefrequency spectrumof noise.We model is refined to account for the anisotropy of the elec- conclude with a brief discussion of testing data for chaotic trical conductivity of the heart. We then discuss electrical behaviorandtheimportantconceptofstochasticresonance. stimulation,whichisimportantforpacemakers,stimulating Preface vii Armed with the tools of the previous chapter, we turn to Chapter 17 introduces nuclear physics and nuclear images in Chap. 12. Images are analyzed from the stand- medicine. The different kinds of radioactive decay are de- point of linear systems and convolution. This leads to the scribed. Dose calculations are made using the fractional useofFourieranalysistodescribethespatialfrequenciesin absorbed dose method recommended by the Medical Inter- animageandthereconstructionofanimagefromitsprojec- nal Radiation Dose Committee of the Society of Nuclear tions. Both Fourier techniques and filtered back projection MedicineandMolecularImaging.Augerelectronscanmag- arediscussed. nifythedosedeliveredtoacellortoDNA.Thiscanpoten- Chapter 13 analyzes sound, hearing, and medical ultra- tiallyprovidenewmethodsoftreatment.Diagnosticimaging sound.Thewaveequationisderived,andthewavespeedand includes single photon emission tomography and positron acoustic impedance are related to the tissue properties. The emission tomography. Therapies include brachytherapy and structureandfunctionoftheearisdescribed.Finally,meth- internal radiotherapy. A section on the nuclear physics of ods for ultrasonic imaging are discussed, including pulse radonclosesthechapter. echotechniquesandDopplerimaging. Chapter 18 develops the physics of magnetic resonance Chapter14discussesthevisible,infrared,andultraviolet imaging(MRI).Weshowhowthebasicpulsesequencesare regionsoftheelectromagneticspectrum.Thescatteringand formed and used for slice selection, readout, image recon- absorption cross sections are introduced and are used here struction, and to manipulate image contrast. We close with andinthenextthreechapters.Wethendescribethediffusion chemical shift imaging, flow effects, functional MRI, and model for photon transport in turbid media. Thermal radi- diffusionanddiffusiontensorMRI. ation emitted by the body can be detected; the emission of Biophysics is a very broad subject. Nearly every branch thermalradiationbythesunincludesultravioletlight,which of physics has something to contribute, and the boundaries injuresskin.Protectionfromultravioletlightisbothpossible between physics and engineering are blurred. Each chapter andprudent.Thedefinitionsofvariousradiometricquantities couldbemuchlonger;wehaveattemptedtoprovidethees- havevariedfromonefieldofresearchtoanother.Wepresent sentialphysicaltools.Molecularbiophysicshasbeenalmost acoherentdescriptionofradiometric,photometric,andacti- completelyignored:excellenttextsalreadyexist,andthisis nometric definitions. We then turn to the eye, showing how notourareaofexpertise.Thisbookhasbecomelongenough. spectaclelensesareusedtocorrecterrorsofrefraction.The Wewouldappreciatereceivinganycorrectionsorsugges- chaptercloseswithadescriptionofthequantumlimitations tionsforimprovingthebook. todark-adaptedvision. Finally, thanks to our long-suffering families. We never Chapter15,likeChap.3,hasfewbiologicalexamplesbut understood what these common words really mean, nor the sets the stage for later work. It describes how photons and depthofourindebtedness,untilwewrotethebook. ionizing charged particles such as electrons lose energy in traversingmatter.Theseinteractionmechanisms,bothinthe RussellK.Hobbie body and in the detector, are fundamental to the formation ProfessorofPhysicsEmeritus,UniversityofMinnesota of a radiographic image and to the use of radiation to treat ([email protected]) cancer. BradleyJ.Roth Chapter 16 describes the use of x rays for medical diag- ProfessorofPhysics,OaklandUniversity nosis and treatment. It moves from production to detection, ([email protected]) to the diagnostic radiograph. We discuss image quality and noise,followedbyangiography,mammography,fluoroscopy, and computed tomography. After briefly reviewing radio- biology, we discuss therapy and dose measurement. The chaptercloseswithasectionontherisksfromradiation. Contents 1 Mechanics ...................................................................................... 1 1.1 DistancesandSizes ........................................................................ 1 1.2 Models................................................................................... 3 1.3 ForcesandTranslationalEquilibrium.......................................................... 3 1.4 RotationalEquilibrium...................................................................... 4 1.5 VectorProduct ............................................................................ 6 1.6 ForceintheAchillesTendon................................................................. 6 1.7 ForcesontheHip .......................................................................... 7 1.8 TheUseofaCane ......................................................................... 10 1.9 Work .................................................................................... 11 1.10 StressandStrain........................................................................... 12 1.11 Shear .................................................................................... 13 1.12 Hydrostatics .............................................................................. 13 1.13 Buoyancy................................................................................. 15 1.14 Compressibility............................................................................ 15 1.15 Diving ................................................................................... 15 1.16 Viscosity ................................................................................. 15 1.17 ViscousFlowinaTube ..................................................................... 16 1.18 Pressure–VolumeWork ..................................................................... 19 1.19 TheHumanCirculatorySystem .............................................................. 20 1.20 TurbulentFlowandtheReynoldsNumber ..................................................... 22 SymbolsUsed ................................................................................... 24 Problems ....................................................................................... 25 References ...................................................................................... 30 2 ExponentialGrowthandDecay ................................................................... 33 2.1 ExponentialGrowth ........................................................................ 33 2.2 ExponentialDecay ......................................................................... 35 2.3 SemilogPaper............................................................................. 36 2.4 VariableRates............................................................................. 38 2.5 Clearance................................................................................. 40 2.6 TheChemostat ............................................................................ 40 2.7 MultipleDecayPaths....................................................................... 41 2.8 DecayPlusInputataConstantRate........................................................... 41 2.9 DecayWithMultipleHalf-LivesandFittingExponentials ........................................ 41 2.10 TheLogisticEquation ...................................................................... 42 ix x Contents 2.11 Log–logPlots,PowerLaws,andScaling....................................................... 43 2.11.1 Log–logPlotsandPowerLaws....................................................... 43 2.11.2 FoodConsumption,BasalMetabolicRate,andScaling................................... 44 SymbolsUsed ................................................................................... 45 Problems ....................................................................................... 45 References ...................................................................................... 51 3 SystemsofManyParticles ........................................................................ 53 3.1 GasMoleculesinaBox..................................................................... 54 3.2 MicrostatesandMacrostates................................................................. 56 3.3 TheEnergyofaSystem:TheFirstLawofThermodynamics ...................................... 57 3.4 EnsemblesandtheBasicPostulates........................................................... 59 3.5 ThermalEquilibrium ....................................................................... 60 3.6 Entropy .................................................................................. 62 3.7 TheBoltzmannFactor ...................................................................... 62 3.8 TheNernstEquation ....................................................................... 63 3.9 ThePressureVariationintheAtmosphere...................................................... 64 3.10 EquipartitionofEnergyandBrownianMotion.................................................. 64 3.11 HeatCapacity ............................................................................. 65 3.12 EquilibriumWhenParticlesCanBeExchanged:theChemicalPotential ............................ 66 3.13 ConcentrationDependenceoftheChemicalPotential ............................................ 67 3.14 SystemsThatCanExchangeVolume.......................................................... 68 3.15 Extensive Variables and GeneralizedForces ......................................................................... 68 3.16 TheGeneralThermodynamicRelationship..................................................... 69 3.17 TheGibbsFreeEnergy ..................................................................... 70 3.17.1 GibbsFreeEnergy ................................................................. 70 3.17.2 AnExample:ChemicalReactions .................................................... 71 3.18 TheChemicalPotentialofaSolution.......................................................... 72 3.19 TransformationofRandomnesstoOrder....................................................... 74 SymbolsUsed ................................................................................... 75 Problems ....................................................................................... 76 References ...................................................................................... 83 4 TransportinanInfiniteMedium .................................................................. 85 4.1 Flux,Fluence,andContinuity................................................................ 85 4.1.1 TheContinuityEquationinOneDimension ............................................ 86 4.1.2 TheContinuityEquationinThreeDimensions.......................................... 86 4.1.3 TheIntegralFormoftheContinuityEquation .......................................... 87 4.1.4 TheDifferentialFormoftheContinuityEquation ....................................... 88 4.1.5 TheContinuityEquationwithaChemicalReaction...................................... 89 4.2 DriftorSolventDrag ....................................................................... 89 4.3 BrownianMotion .......................................................................... 89 4.4 MotioninaGas:MeanFreePathandCollisionTime ............................................ 90 4.5 MotioninaLiquid ......................................................................... 91 4.6 Diffusion:Fick’sFirstLaw .................................................................. 92 4.7 TheEinsteinRelationshipBetweenDiffusionandViscosity....................................... 93 4.8 Fick’sSecondLawofDiffusion .............................................................. 95 4.9 Time-IndependentSolutions ................................................................. 97 4.10 Example:Steady-StateDiffusiontoaSphericalCellandEndEffects ............................... 98 4.10.1 DiffusionThroughaCollectionofPores,Corrected ..................................... 100 4.10.2 DiffusionfromaSphere,Corrected ................................................... 100 4.10.3 HowManyPoresAreNeeded?....................................................... 100 4.10.4 OtherApplicationsoftheModel ..................................................... 101 Contents xi 4.11 Example:ASphericalCellProducingaSubstance............................................... 101 4.12 DriftandDiffusioninOneDimension......................................................... 102 4.13 AGeneralSolutionfortheParticleConcentrationasaFunctionofTime ............................ 104 4.14 DiffusionasaRandomWalk................................................................. 105 SymbolsUsed ................................................................................... 107 Problems ....................................................................................... 108 References ...................................................................................... 114 5 TransportThroughNeutralMembranes ........................................................... 117 5.1 Membranes ............................................................................... 117 5.2 OsmoticPressureinanIdealGas ............................................................. 118 5.3 OsmoticPressureinaLiquid ................................................................ 120 5.4 SomeClinicalExamples .................................................................... 121 5.4.1 EdemaDuetoHeartFailure ......................................................... 122 5.4.2 Nephrotic Syndrome, Liver Disease, andAscites ....................................................................... 122 5.4.3 EdemaofInflammatoryReaction..................................................... 122 5.4.4 HeadachesinRenalDialysis......................................................... 123 5.4.5 OsmoticDiuresis .................................................................. 123 5.4.6 OsmoticFragilityofRedCells ....................................................... 123 5.5 VolumeTransportThroughaMembrane....................................................... 123 5.6 SoluteTransportThroughaMembrane ........................................................ 125 5.7 Example:TheArtificialKidney .............................................................. 126 5.8 CountercurrentTransport.................................................................... 127 5.9 AContinuumModelforVolumeandSoluteTransportinaPore ................................... 128 5.9.1 VolumeTransport.................................................................. 128 5.9.2 SoluteTransport ................................................................... 130 5.9.3 Summary......................................................................... 133 5.9.4 ReflectionCoefficient .............................................................. 133 5.9.5 TheEffectofPoreWallsonDiffusion................................................. 134 5.9.6 NetForceontheMembrane ......................................................... 134 SymbolsUsed ................................................................................... 135 Problems ....................................................................................... 135 References ...................................................................................... 139 6 ImpulsesinNerveandMuscleCells................................................................ 141 6.1 PhysiologyofNerveandMuscleCells ........................................................ 141 6.2 Coulomb’sLaw,Superposition,andtheElectricField............................................ 143 6.3 Gauss’sLaw .............................................................................. 144 6.4 PotentialDifference ........................................................................ 147 6.5 Conductors ............................................................................... 148 6.6 Capacitance............................................................................... 149 6.7 Dielectrics ................................................................................ 149 6.8 CurrentandOhm’sLaw..................................................................... 151 6.9 TheApplicationofOhm’sLawtoSimpleCircuits............................................... 153 6.10 Charge Distribution in the Resting NerveCell ................................................................................ 154 6.11 TheCableModelforanAxon................................................................ 155 6.12 ElectrotonusorPassiveSpread ............................................................... 159 6.13 TheHodgkin–HuxleyModelforMembraneCurrent............................................. 160 6.13.1 VoltageClampExperiments ......................................................... 161 6.13.2 PotassiumConductance............................................................. 163 6.13.3 SodiumConductance ............................................................... 164 6.13.4 LeakageCurrent ................................................................... 165