david hormuth october 20, 2018 · • biomaterials: drugs, coatings/materials for implants,...

David HormuthOctober 20, 2018

What is biomedical engineering?• Biomechanics: Heart valves, knee replacements, prosthetics• Biomaterials: Drugs, coatings/materials for implants, anything that goes in or

on the body• Instrumentation: basically everything else….• Imaging, pacemakers, ECG machines, anything with computers, neuro-

stimulators, cochlear implants

Terre Haute, IndianaUndergraduate degree

Nashville, Tennessee Graduate degrees

Biomedical Engineering

Medical Imaging Mathematical Biology

Numerical Methods

High Performance Computing

Radiation Therapy

CancerResearch

Brain cancer

Computational Oncology

X-ray 1895

CT (Computed Tomography)The math ~1917The method 1963

Ultrasound1950-60s

Magnetic Resonance Imaging1973

Large magnet1 T (Tesla) = 10,000 Gauss

Fridge magnet ~ 100 Gauss. The Earth ~ .5 Gauss

3T Human (Clinical) Scanner 7T Small Animal (Pre-clinical) Scanner

Screening: Mammogram in breast cancer, dental x-rays (for cavities)

Diagnosis: Contrast-enhanced brain scan

Treatment planning: Biopsy placement, surgery, radiation therapyPredict how a patient will respond to therapy, change therapy if standard of care is not good enough.

Assessing Response: Is the tumor shrinking?Predict if a patient will respond before end of therapy.

What are some ways we can describethe tumor in this image?

Let’s assume it grows at a constant rate per day!

Volume Day 10 = 54.4 mm3

Volume Day 12 = 68.5 mm3

Difference: 14.1 mm3

(7.05 mm3/day)Predicted at Day 14 = 82.6 mm3

Measured at Day 14 = 114.4 mm3

Percent error = 27.8 % error

Error = 100% Model −MeasuredMeasured

⎛⎝⎜

⎞⎠⎟ 10 12 14 16

Time (Days)

0

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Volu

me

(mm

3 )

Model Measure

10 12 14 16Time (Days)

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me

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Model Predicted Measure

Volume(t2) =Volume(t1)+ X t2 − t1( )

ΔVChange in volume!

Δtchange in time!

=Volume(t2)−Volume(t1)

t2 − t1( ) = XGrowth Rate

(Volume Increase per day)

!

Volume(t2)−Volume(t1) = X ⋅ t2 − t1( )10 12 14 16

Time (Days)

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me

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Volume Day 10 = 54.4 mm3

Volume Day 12 = 68.5 mm3

Difference: 14.1 mm3

X = (7.05 mm3/day)

What is the change in volume over time?

What is a model?

Generic equation to find volume at any time point

Volume(ti ) =Volume(t1)+ (t − t1) ⋅X

What X works best to describe volume at both t2 and t3?

X = 7.05 X = 14 X = 21 X = 13Error at t1 0 -13.88 -27.87 -11.87Error at t2 31.86 4.06 -23.94 8.06

Total Squared Error 1015 209.02 1350 206

10 12 14 16Time (Days)

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10 12 14 16Time (Days)

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19% error

Volume(ti )Future Volume! "# $#

=Volume(t1)Initial Volume! "# $#

+Volume(ti−1)Previous Volume! "## $##

⋅X ⋅ ti − t1( )

ΔVChange in volume!

Δtchange in time!

=Volume(ti )−Volume(t1)

ti − t1( ) =Volume(ti−1) ⋅X

Constant growth didn’t work well…what else can we try?

Volume(t2) =Volume(t1)+Volume(t1) ⋅ 12( ) ⋅ 2( )Volume(t2) = 2 ⋅Volume(t1)

If X = ½ the tumor would double in volume in two days

What is the change in volume over time?

10 12 14 16Time (Days)

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me

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3 )

Model Predicted Measure

Generic equation to find volume at any time point

What X works best to describe volume at both t2 and t3?

X = 1 X = .5 X = .25 X = .125 X = .18Error at t1 -333 -79.3 -21.15 -1.32 -9.266Error at t2 -2854.5 -

287.35-33.37 24.8 3.218

Total Squared Error

8,259,300

88,858 1561 616.2 96.214 11% error

Volume(ti )Future Volume! "# $#

=Volume(ti−1)Previous Volume! "## $##

+Volume(ti−1)Previous Volume! "## $##

⋅X ⋅ ti − ti−1( )

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Prediction Error = 815 Model Error = 202

Prediction Error = 348 Model Error = 564

Prediction Error = 198 Model Error = 758

Prediction Error = 102 Model Error = 869

Prediction Error = 291 Model Error = 96

Prediction Error = 9 Model Error = 191

Prediction Error = 1925 Model Error = 195

Prediction Error = 5053 Model Error = 799

Line

arEx

pone

tial

Which model works best?

What can you tell us about how the model fits the curve?

ΔVChange in volume!

Δtchange in time!

=Volume(t2)−Volume(t1)

t2 − t1( ) =Volume(t1) ⋅X ⋅ 1−Volume(t1)Volumemax

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with volume! "###### $######

10 12 14 16Time (Days)

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me

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10 15 20Time (Days)

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Shor

t Ter

m p

redi

ctio

nsLo

ng T

erm

pre

dict

ions

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What does this plot remind you of?

What can we use this prediction for?

Volume is one thing….

What else can we model?

Hallmarks of Cancer

Modeling Immune system response

Modeling tumor genetic instability

Modeling tumor metabolism

Modeling Angiogenesis

Anderson et al Cai et al

Perez-Garcia et alEnderling et al

What can we measure non-invasively? How do these measures change with cancer?• Tumor size and shape

• Tumor cellularity

• Blood vessels, blood flow

• Oxygenation, or hypoxia

• Glucose usage

Diffusion-Weighted MRI• An imaging measurement sensitive to how freely water moves in tissue• How is that related to cells?

0 0.2 0.4 0.6 0.8 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Volume fraction

ADC

(x10

-3m

m2 /s

)

Diffusion-Weighted MRIb = 150 b = 300 b = 500

b = 700 b = 900 b = 1100

0( ) exp( )S b S b ADC= - ×ADC = 0.72 µm2/ms

Measure

Model

ΔVChange in volume!

Δtchange in time!

=Volume(t2)−Volume(t1)

t2 − t1( ) =Volume(t1) ⋅X ⋅ 1−Volume(t1)Volumemax

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with volume! "###### $######

ΔNT

Change in cell number!

Δtchange in time!

=NT (t2)−NT (t1)

t2 − t1( ) = NT (t1) ⋅X ⋅ 1−NT (t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number! "#### $####

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = NT (x,y,t1) ⋅X ⋅ 1−NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &##### '#####

Replace Volume with NTNT = the number of tumor cells

Add the location of each imaging point (x,y)

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = NT (x,y,t1) ⋅X ⋅ 1−NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &##### '#####

BUT! Cells don’t stay put over time

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = Entering − Leaving +Staying( )Amount of cells moving around

% &###### '######+NT (x,y,t1) ⋅X ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &##### '#####

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

= D ∇2NT (x,y,t)( )Amount of cells moving around

Diffusion! "###### $######

! "## $##+NT (x,y,t1) ⋅k ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell numberProliferation

! "######## $########

! "##### $#####

We model cell movement using a diffusion term

Generic equation to find volume at any time point

What X works best to describe volume at both t2 and t3?

X = 1 X = .5 X = .25 X = .125 X = .18Error at t1 -333 -79.3 -21.15 -1.32 -9.266Error at t2 -2854.5 -

287.35-33.37 24.8 3.218

Total Squared Error

8,259,300

88,858 1561 616.2 96.214

Volume(ti )Future Volume! "# $#

=Volume(t1)Initial Volume! "# $#

+Volume(ti−1)Previous Volume! "## $##

⋅X ⋅ ti − t1( )

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

= D ∇2NT (x,y,t)( )Amount of cells moving around

Diffusion! "###### $######

! "## $##+NT (x,y,t1) ⋅k ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell numberProliferation

! "######## $########

! "##### $#####Instead of finding X, we are nowgoing to find K, D, and NT,max

The model doesn’t have any death terms…what can we add?

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = Entering − Leaving +Staying( )Amount of cells moving around

% &###### '######+NT (x,y,t1) ⋅X (x,y,t) ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &####### '#######

X (x,y,t) =X > 0growing! ,High NV

X ≤ 0Dieing! ,Low NV

⎧

⎨⎪⎪

⎩⎪⎪

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = Entering − Leaving +Staying( )Amount of cells moving around

% &###### '######+NT (x,y,t1) ⋅X ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &##### '#####

How can we change this model to add vasculature?What do blood vessels do?

We could connect it to the proliferation rate!

Dynamic Contrast Enhanced MRI (DCE-MRI)

ΔCTissueΔt

= KTransCBloodAmount entering

the tissue

! "# $#− K

Trans

veCTissue

Amount leavingthe tissue

! "# $#

ΔNT x,y( )Change in cell number! "# $#

Δtchange in time!

=NT (x,y,t2)−NT (x,y,t1)

t2 − t1( ) = Entering − Leaving +Staying( )Amount of cells moving around

% &###### '######+NT (x,y,t1) ⋅X (x,y,t) ⋅ 1−

NT (x,y,t1)NT ,max

⎛

⎝⎜⎞

⎠⎟

Growth rate changes with cell number% &####### '#######

ΔNV x,y( )Change in blood vessels! "# $#

Δtchange in time!

=NV (x,y,t2)−NV (x,y,t1)

t2 − t1( ) = Entering − Leaving +Staying( )Amount of cells moving around

% &###### '######+NV (x,y,t1) ⋅XV (x,y,t) ⋅ 1−

NV (x,y,t1)NV ,max

⎛

⎝⎜⎞

⎠⎟

Vasculature Growth or Death Rate% &####### '#######

We can use a very similar model to describe how the vasculature grows, dies, and moves over time

We now have 2 equations we need to at each time point

Measured tumor cells Predicted tumor cells

t1 t2 t3

t4 t5 t6

t7

t1 t2 t3

t4 t5 t6

t7

Measured vasculature Predicted vasculature

t1 t2 t3

t4 t5 t6

t7

t1 t2 t3

t4 t5 t6

t7

What is missing in this model that nearly every patient gets?

Treatment!

• Chemotherapy

• Radiation Therapy

• Surgery

• Immunotherapy

When and what type of drugs should the patient receive?

When and how much radiation should the patient receive?

Will this person respond to the treatment?

Tumors are unique, they grow uniquely, the respond uniquely.

Website: cco.ices.utexas.eduEmail: David.Hormuth@utexas.edu