Understanding Pharmacokinetic Models

Pharmacokinetic (PK) models describe how contrast agent distributes through tissue in DCE-MRI. The choice of model affects both the parameters you can estimate and the accuracy of results.

The Central Problem

When we inject a gadolinium contrast agent, it flows through blood vessels and leaks into tissue. The concentration-time curve in tissue depends on:

1. How fast contrast arrives (arterial input function)
2. How permeable the vessels are (transfer rate)
3. How much space exists for contrast (volume fractions)

PK models describe these relationships mathematically.

Tissue Compartments

The Two-Compartment View

Most models divide tissue into two compartments:

Parameters:

• Cp(t): Plasma concentration (from AIF)
• Ce(t): EES concentration (what we estimate)
• Ktrans: Volume transfer constant (plasma → EES)
• kep: Rate constant (EES → plasma), where kep = Ktrans/ve
• vp: Plasma volume fraction
• ve: EES volume fraction

Model Equations

Standard Tofts model

The simplest model assumes negligible plasma volume (vp ≈ 0):

\[ C_t(t) = K^{trans} \int_0^t C_p(\tau) \cdot e^{-k_{ep}(t-\tau)} d\tau \]

Parameters: Ktrans, ve (2 parameters)

Assumptions:

• Blood plasma volume is negligible
• Fast exchange between plasma and EES
• First-pass effects are small

Best for: Tissues with low vascularity, simple analysis

Extended Tofts model

Adds the plasma contribution:

\[ C_t(t) = v_p \cdot C_p(t) + K^{trans} \int_0^t C_p(\tau) \cdot e^{-k_{ep}(t-\tau)} d\tau \]

Parameters: Ktrans, ve, vp (3 parameters)

Improvements over Standard Tofts:

• Accounts for contrast in blood vessels
• More accurate for highly vascular tissues
• Better first-pass characterization

Best for: Tumors, brain tissue, general use

Patlak model

Assumes unidirectional transfer (no backflux):

\[ C_t(t) = v_p \cdot C_p(t) + K^{trans} \int_0^t C_p(\tau) d\tau \]

Parameters: Ktrans, vp (2 parameters)

Assumptions:

• Contrast doesn't return from EES to plasma
• Valid when kep is very small
• Graphical analysis is possible

Best for: Early enhancement, highly permeable tissues, screening

Two-compartment exchange model (2CXM)

The most complete model separates flow and permeability:

\[ C_t(t) = F_p \cdot \left[ (1-E) \cdot R(t) + E \cdot e^{-k_{ep}t} \right] \otimes C_p(t) \]

Where E (extraction fraction) relates to PS (permeability-surface area):

\[ E = 1 - e^{-PS/F_p} \]

Parameters: Fp, PS, ve, vp (4 parameters)

Advantages:

• Separates flow (Fp) from permeability (PS)
• Most physiologically accurate
• Can identify flow-limited vs permeability-limited regimes

Requirements:

• High temporal resolution (< 2 seconds)
• Good SNR
• Sufficient timepoints

Two-compartment uptake model (2CUM)

A simplification of 2CXM that assumes unidirectional uptake (no backflux from EES):

\[ C_t(t) = v_p \cdot C_p(t) + F_p \int_0^t e^{-(F_p + PS)/v_p \cdot \tau} \cdot C_p(t - \tau) \, d\tau \]

Parameters: Fp, PS, vp (3 parameters)

Assumptions:

• No backflux from EES to plasma (like Patlak, but with explicit flow)
• Separates flow from permeability (like 2CXM, but simpler)

Best for: Tissues where contrast accumulates but does not wash out significantly during the acquisition window, or when the 4-parameter 2CXM is poorly identifiable.

Relationship to other models:

• Simplification of 2CXM (drops ve, assumes unidirectional transfer)
• Like Patlak with explicit flow and permeability separation
• 3 parameters make it more identifiable than 2CXM with fewer timepoints

Model Comparison

Model	Parameters	Assumptions	Complexity
Standard Tofts	2	vp ≈ 0	Low
Extended Tofts	3	Fast exchange	Medium
Patlak	2	No backflux	Low
2CUM	3	Unidirectional uptake	Medium
2CXM	4	None	High

Mathematical Details

Convolution Operation

The integral in these models is a convolution of the AIF with an impulse response:

\[ C_t(t) = C_p(t) \otimes h(t) \]

For Extended Tofts, the impulse response is:

\[ h(t) = K^{trans} \cdot e^{-k_{ep}t} + v_p \cdot \delta(t) \]

osipy supports several convolution methods for this (piecewise-linear, exponential, FFT).

Relationship Between Parameters

The rate constant kep connects Ktrans and ve:

\[ k_{ep} = \frac{K^{trans}}{v_e} \]

This means:

• High Ktrans + low ve → fast washout
• Low Ktrans + high ve → slow washout
• Ktrans/kep gives ve directly

Units and Scaling

osipy uses:

• Time: seconds (input), minutes (internal model)
• Ktrans: min⁻¹
• ve, vp: volume fractions in mL/100mL
• Concentration: mM (millimolar)

The time conversion happens automatically:

Automatic time unit conversion in osipy

# Public API uses seconds
result = osipy.fit_model("extended_tofts", concentration, aif, time_in_seconds)

# Internally, models use minutes for Ktrans units
time_minutes = time_seconds / 60

Physiological Interpretation

What Ktrans Tells You

Ktrans reflects the combined effect of:

1. Blood flow (F): Delivery of contrast
2. Permeability (PS): Vessel leakiness

In different regimes:

• Flow-limited (high permeability): Ktrans ≈ F
• Permeability-limited (low flow): Ktrans ≈ PS

For tumors, high Ktrans often indicates:

• Angiogenesis (new vessel formation)
• Increased vessel permeability
• Treatment response (may decrease)

What ve Tells You

ve represents the extravascular, extracellular space:

• High ve: More space for contrast accumulation
• Low ve: Dense tissue, high cellularity (like tumors)

What vp Tells You

vp represents blood plasma volume:

• High vp: Highly vascular tissue
• Low vp: Poorly perfused tissue

Model Selection Guidelines

Start with Extended Tofts

For most applications, Extended Tofts is a good default:

• 3 parameters capture most variance
• Robust to different tissue types
• Widely validated

Use Standard Tofts When

• Low vascularity expected
• Simpler interpretation needed
• Limited temporal resolution

Use Patlak When

• Only early enhancement phase
• Screening large datasets
• Graphical analysis desired

Use 2CUM When

• Need to separate F from PS but 2CXM is over-parameterized
• Tissue shows unidirectional uptake (no significant washout)
• Moderate temporal resolution (fewer timepoints than 2CXM requires)

Use 2CXM When

• High temporal resolution available (TR < 2s)
• Need to separate F from PS with full exchange
• Research applications

Arterial Delay

In practice, the AIF may arrive at different times in different tissue regions. osipy supports automatic arterial delay estimation during fitting:

Fit with automatic delay estimation

result = osipy.fit_model(
    "extended_tofts", concentration, aif, time,
    fit_delay=True  # Estimate arterial delay per voxel
)

# Access delay map (in seconds)
delay = result.parameter_maps["delay"].values

This works with any model (Tofts, Extended Tofts, Patlak, 2CXM, 2CUM). The delay parameter shifts the AIF in time before convolution with the impulse response function.

Fitting Considerations

Identifiability

Not all parameters are always identifiable:

• Low SNR: May only reliably fit 2 parameters
• Short acquisition: Limited ability to estimate ve
• No early data: Can't distinguish vp

Initial Guesses

osipy uses physiologically reasonable defaults:

Default initial guesses for fitting

# Default initial guesses
{
    'Ktrans': 0.1,   # min⁻¹
    've': 0.2,       # fraction
    'vp': 0.02,      # fraction
}

Parameter Bounds

Constraints ensure physiological plausibility:

Default parameter bounds

# Default bounds (Extended Tofts)
{
    'Ktrans': (0.0, 5.0),   # min⁻¹
    've': (0.001, 1.0),     # fraction
    'vp': (0.0, 1.0),      # fraction
}

References

1. Tofts PS, Kermode AG. "Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging." Magn Reson Med 1991.

2. Tofts PS et al. "Estimating kinetic parameters from dynamic contrast-enhanced T1-weighted MRI of a diffusable tracer: Standardized quantities and symbols." J Magn Reson Imaging 1999.

3. Sourbron SP, Buckley DL. "Classic models for dynamic contrast-enhanced MRI." NMR Biomed 2013.

4. Sourbron SP, Buckley DL. "Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability." Phys Med Biol 2012;57(2):R1-R33.

See Also

• DCE-MRI Tutorial
• How to Compare Multiple Models
• How to Choose Population AIF