Linear, Power, and Exponential Growth

• Lecture 9 focuses on three types of growth: linear, power, and exponential.

Reminders

• Lab 2 Check-In 2 is due on Tuesday 2/11/2025 at midnight.
• The Midterm is next week (Week 6) on Wednesday 2/12 in class at 5:20pm

Three Kinds of Growth

• The lecture will focus on three models of growth (and decay)
• Linear
• Power Function
• Exponential Function

Linear Growth

• Formula: P(t) = mt + b
• Parameters: slope m, b = P(0)
• Change equation: ∆Ρ/ Δt = m
• The absolute rate of change is constant.
• Represented graphically as a line with slope m
• Growth: m>0
• Decay: m < 0
• Constant: m = 0
• Example: adding 3 new fish to a tank each day

Power Function Growth

• Formula: P(t) = C tp
• There are two parameters: p is the power or exponent and C controls vertical scaling
• P(0) = 0 and P(1) = C
• Quadratic growth: P(t) = 5 t²
• Cubic growth: P(t) = 7 t³
• Square root growth: P(t) = 10 †1/2

Power Function (p > 1)

• Formula: P(t) = C tp exhibits accelerating growth

Power Function (0 < p < 1)

• Formula: P(t) = C tº exhibits decelerating but unbounded growth.

Origins of Power Function Growth - Geometry

• Solid tumors start as malfunctioning cell masses.
• Early stages: exponential growth via cell division where ΔΡ/Δ† = k P
• Later stages: tumor growth is restricted to cell division on its surface, increasing the tumor radius at a constant rate
• R = m t (linear growth of the radius) with R being the radius of the tumor
• Volume is proportional to R3: V = R3 = t3

Tumor Spheroid Growth Phases

• Greenspan's mathematical model describes three phases of avascular tumor spheroid growth.
• Phase i: cells throughout the spheroid can proliferate
• Phase ii: cells near the periphery proliferate while a central region of living cells cannot proliferate, and is referred to as the inhibited region
• Phase iii: an outer region of proliferative cells, an intermediate region of living inhibited cells, and a central necrotic region composed of dead cells and cellular material in various stages of disintegration and dissolution

Mathematical Modelling

• Mathematical modelling is a powerful tool for interpretations through model calibration and mechanism deduction.
• Gompertzian growth models are used for decades to predict the growth of tumors.
• Complex mathematical models can describe internal spheroid structure over time
• Greenspan's mathematical model offers great advantages because the parameters are biologically relevant

Exponential Functions

• General forms: P(t) = C bt, P(t) = C ekt, P(t) = C •2(t/d)

Exponential Functions - P(t) = C b⁺

• Population begins at C
• If P(t) = C bt, and with a unit of time fixed (t=1)
• P(0) = C, so the population starts with C individuals
• P(1) = C b, so the population gets multiplied by b in one unit of time
• P(2) = C b², so the population gets multiplied again by b in one unit of time
• P(3) = C b³, so the population gets multiplied again by b in one unit of time
• The equation indicates the population is repeatedly multiplied by b

Exponential Functions - P(t) = C ekt

• This equation will be understood later

Exponential Functions - P(t) = C •2(t/d)

• Population begins at C
• If P(t) = C •2(t/d), and with a unit of time fixed (t=1)
• P(0) = C, so the population starts with C individuals
• P(d) = 2 C, so the population gets doubled in d units of time.
• P(2d) = 4 C, so the population gets doubled again!
• P(3d) = 8 C, so the population gets doubled again!
• The equation shows the population doubles at time intervals of d.

Rates of Change

• Linear growth exhibits a constant rate of change
• If P(t) = mt + b, then ∆P/Δt = m

Changing Rates of Change

• For P(t) = t², the rate of change depends on the time interval
• The rates of change are increasing and shows what is called acceleration

Computing Rates of Change

• For P(t) = t², the rate of change depends on the time interval
• Start at t and end at t + Δt
• Start: Time = t and P = (t)²
• End: Time = t + ∆t and P = (t + Δt)²
• Algebra yields: AP/At = 2t + At
• In the example where the rate of change is on the time interval from t=3 to t=3.1, plug in t=3 and At = 0.1
• ΔΡ/ΔΙ = 2(3) + 0.1 = 6.1

Very Small Time Interval

• If P(t) = t², then AP/At = 2t + At.
• The first equation describes the population at time t
• The second equation describes the rate of population growth, with (1) start time = t, and (2) time interval At
• As an example:
  • The rate of change on the time interval from t=3 to t=3.1 is 6.1
  • The rate of change on the time interval from t=3 to t=3.01 is 6.01
  • The rate of change on the time interval from t=3 to t=3.001 is 6.001

The Derivative

• The derivative is the rate of change at an instant.
• dP/dt is the derivative of P
• dP/dt is what AP/At becomes, if the start-time is fixed at t, and the time interval shrinks to zero
• If P(t) = 12, then AP/At = 2t + At, and dP/dt = 2t

Derivative as Slope

• If P(t) is a function, then ΔΡ/ΔΙ describes the slope of a line joining two points on the graph
• The derivative dP/dt describes the slope of the tangent line through a single point on the graph

Two Derivatives to Know: Linear and Power Rule

• If P(t) = mt + b (linear growth), then dP/dt = m (constant rate of change = slope)
• If P(t) = tⁿ (power law growth, exponent n), then dP/dt = n tⁿ⁻¹ (rate of change obeys power law, with exponent n-1)
  • Example: If P(t) = t³, then dP/dt = 3t²

Midterm on February 12th

• The study guide should be available
• The exam is 30 multiple choice questions, conceptual questions, a bit of science, and a bit of numeracy
• Fill in bubble A, B, C, D, E (bring dark pencil or pen)
• Location: in this room
• DRC Accommodations through the PBSci Testing Center, in Jack Baskin Engineering