Continuum approach Motion descriptionBalance lawsTraffic flow behaviorGreenshields modelLinear traffic waves Method of characteristics

If we consider an artery with obstacles thesituation can be compared to a highwaypopulated by travelling vehicles where inthis case those cars are cells of the blood.

𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁𝐁 𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐯𝐁𝐁

µmµm

𝟏𝟏𝟏𝟏−𝟒𝟒 𝟏𝟏𝟏𝟏−𝟔𝟔𝟏𝟏𝟏𝟏−𝟑𝟑𝟏𝟏𝟏𝟏𝟏𝟏𝒎𝒎

Displacement

=0

Control Volume

Trajectory

Flow direction

1-D conservation equation

Traffic flow behavior

The flowMacroscopic)

Individual cars(Microscopic)

The observed behavior of cars involvesdriving speeds

driving lane

FactorsTraffic conditionsCars characteristics

Road conditionsExternal agents

𝒒𝒒

𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎

𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎

𝒒𝒒𝒎𝒎𝒎𝒎𝒎𝒎

𝒗𝒗

𝝆𝝆 𝝆𝝆

Constitutive law 𝒗𝒗 = 𝒇𝒇 𝝆𝝆

𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎

𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆

Empirical dataGreenshields model

𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏

+ 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎

𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎

= 𝟏𝟏

𝝏𝝏𝝆𝝆𝝏𝝏𝝏𝝏

+ 𝒒𝒒𝒒(𝝆𝝆)𝝏𝝏𝝆𝝆𝝏𝝏𝒎𝒎

= 𝟏𝟏

𝒒𝒒 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝝆𝝆 − 𝝆𝝆𝟐𝟐

𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒗𝒗 𝝆𝝆 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝝆𝝆

𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎𝒒𝒒 𝝆𝝆 = 𝝆𝝆 𝒗𝒗 𝝆𝝆

𝒒𝒒′ 𝝆𝝆 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 −𝟐𝟐𝝆𝝆𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎

Quasi linearPDE

𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝜹𝜹𝝆𝝆 = 𝒒𝒒′ 𝝆𝝆𝟏𝟏 + 𝒒𝒒′′ 𝝆𝝆𝟏𝟏 𝜹𝜹𝝆𝝆 + …

Linearized form

Solution form𝒒𝒒′ 𝝆𝝆𝟏𝟏 = 𝒗𝒗𝒎𝒎𝒎𝒎𝒎𝒎 𝟏𝟏 − 𝟐𝟐𝝆𝝆𝟏𝟏

𝝆𝝆𝒎𝒎𝒎𝒎𝒎𝒎= 𝒄𝒄

𝒅𝒅𝒅𝒅𝒅𝒅𝝏𝝏𝒎𝒎𝒅𝒅𝒄𝒄𝒅𝒅 𝒎𝒎

𝟏𝟏𝒄𝒄

𝟏𝟏

𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏 = 𝒄𝒄

𝝏𝝏𝒅𝒅𝒎𝒎𝒅𝒅𝝏𝝏

Thank you for your attention